The Basic New Keynesian Model
The Basic New Keynesian Model
January 11th 2012
Lecture notes by Drago Bergholt, Bergholt, Norwegian Business Business School School
[email protected] Drago.Bergho
[email protected]
I
The Basic New Keynesian Model
Contents
1.
2.
3.
4.
5.
6.
7.
Introduction Introduction .......................................................................................................................................... 1 1.1
Prologue ..................................................................................................................................................................... 1
1.2
The New Keynesian Keynesian model – Key Key features ......................................................................................................... 1
Households ........................................................................................................................................... 3 2.1
Setup ........................................................................................................................................................................... 3
2.2
Optimal consumption vector and the aggregate price index ............ ..................... ................... ................... ................... ................... ................... ................. ....... 4
2.3
Optimal allocation of consumption and labor ......................... .................................. ................... ................... .................. ................... .................... ................... .................. ......... 6
Firms .................................................................................................................................................... 11 3.1
Aggregate inflation .................. ............................ ................... .................. ................... ................... ................... ................... ................... .................... ................... .................. ................... ................... ............. .... 11
3.2
Optimal price setting ............................................................................................................................................. 12
3.3
Log-linearization ..................................................................................................................................................... 13
Equilibrium ......................................................................................................................................... 18 4.1
Market clearing clear ing................................ ............... ................................. ................................ ................................. ................................. ................................ ................................. ................................. ...................... ...... 18
4.2
The New Keynesian Keynesian Phillips curve and the Dynamic IS IS equation ................... ............................ ................... .................... ................... ................ ....... 20
Equilibrium determinacy ................................................................................................................... 26 5.1
System representation represent ation ................................. ................. ................................. ................................. ................................. ................................ ................................ ................................. ......................... ......... 26
5.2
Blanchard and Kahn conditions .......................................................................................................................... 27
Shocks .................................................................................................................................................. 30 6.1
Effects of a monetary policy shock ..................................................................................................................... 30
6.2
Effects of a technology shock .............................................................................................................................. 33
Distortions to the efficient allocation.............................................................................................. 36 7.1
The efficient steady state state ................... ............................. ................... ................... ................... .................. ................... ................... ................... .................... ................... .................. ................... .............. 36
7.2
Distortions caused by market power .................................................................................................................. 37
7.3
Distortions caused by sticky prices ..................................................................................................................... 38
7.4
Monetary policy solutions to equilibrium distortions ........................ .................................. ................... .................. ................... ................... ................... ............... ..... 39
8. The welfare loss function function .................................................................................................................. 44 8.1
Introduction ............................................................................................................................................................ 44
8.2
The simplest case – A welfare loss function when real rigidities are absent................... ............................. ................... .................. ........... 44
8.3
Introduction of cost push shocks ........................................................................................................................ 51
8.4
A welfare loss function when when real rigidities are present .................. ............................ ................... ................... ................... ................... ................... ................ ....... 53
9. Welfare based evaluation evaluation of monetary monetary policy ................................................................................ 58 9.1
Introduction ............................................................................................................................................................ 58
9.2
An efficient steady state under under discretion ................... ............................ .................. ................... ................... ................... .................... ................... .................. ................... .............. 58
9.3
An efficient steady state under under commitment.................. ............................ ................... ................... ................... ................... ................... ................... ................... ................ ....... 62
9.4
A distorted steady state under discretion discretion .................. ........................... ................... ................... ................... .................... ................... .................. ................... .................... ............. ... 66
II
The Basic New Keynesian Model
9.5
A distorted steady state under commitment commitment ................... ............................. ................... ................... ................... ................... ................... ................... ................... ................ ....... 68
10. Wage rigidities ..................................................................................................................................... 69 10.1
Introduction ............................................................................................................................................................ 69
10.2
Firms ......................................................................................................................................................................... 69
10.3
Households .............................................................................................................................................................. 70
10.4
Inflation equations and the Dynamic IS equation ................. ........................... ................... ................... ................... ................... ................... ................... .................. ........ 79
10.5
System representation and equilibrium determinacy................ determinacy.......................... .................... ................... .................. ................... ................... ................... ............... ..... 82
10.6
Shocks....................................................................................................................................................................... Shocks....................................................................................................................................................................... 83
10.7
Monetary policy design with sticky wages.......................................................................................................... 85
11. A small, open economy economy model ......................................................................................................... 93 11.1
Introduction ............................................................................................................................................................ 93
11.2
Households .............................................................................................................................................................. 93
11.3
Terms of trade, domestic domestic inflation and CPI inflation ................... ............................ ................... .................... ................... ................... ................... .................. ........... 98
11.4
The real exchange rate .................. ............................ ................... ................... ................... ................... ................... ................... ................... ................... ................... ................... ................... ................ ....... 99
11.5
International risk sharing .................................................................................................................................... 100
11.6
Uncovered interest rate parity ............................................................................................................................ 101
11.7
Firms and technologies ....................................................................................................................................... 102
11.8
Equilibrium – Aggregate Aggregate demand and output ................................................................................................. 103
11.9
Equilibrium – The The trade balance ....................................................................................................................... 109
11.10
Equilibrium – The supply side: Marginal cost and inflation dynamics ........... .................... ................... ................... ................... ................ ......109
11.11
The New Keynesian Keynesian Phillips curve and the Dynamic IS IS equation ................... ............................ ................... .................... ................... .............. .....111
11.12
Equilibrium Equili brium determinacy determi nacy ............................... ............... ................................ ................................. ................................. ................................ ................................. ................................. .................... .... 113
11.13
Equilibrium Equili brium dynamics ................................. ................. ................................. ................................. ................................. ................................ ................................ ................................. ....................... ....... 116
11.14
Optimal monetary policy in the small open economy ............ ..................... ................... ................... ................... ................... ................... ................... .............. .....118
11.15
Welfare losses ................... ............................. ................... ................... ................... ................... ................... .................. ................... .................... ................... ................... ................... .................. ................... ..........123
References ................................................................................................................................................. 127 Appendix ................................................................................................................................................... 128 A.
Dynare codes – A A monetary policy shock with sticky prices ....................................................................... 128
B.
Dynare codes – A A technology shock with sticky prices ................................................................................ 129
C.
Dynare codes – A monetary policy shock with sticky prices and wages ................... ............................ ................... ................... .............. .....131
III
The Basic New Keynesian Model
1.
Introduction
1.1 Prologue
These lecture notes take the reader through a basic New Keynesian model with utility maximizing households, profit maximizing firms and a welfare maximizing central bank. I follow Gali’s (2008) book as closely as possible. The notes were born during my participation at a couple of PhD courses in monetary policy, taught by Antti Ripatti (Bank of Finland) and Krisztina Molnar (Bank of Norway), respectively. Both courses built on the excellent book by Gali. The aim of the notes is to provide the reader with all relevant calculations which are left out of the book. In addition, the notes also go through equilibrium determinacy conditions in more detail, following benchmark articles such as Blanchard and Kahn () and Bullard and Mitra (2002). Chapters 2, 3 and 4 characterize the basic New Keynesian model. I first analyze households, then firms. Results are combined to establish general equilibrium. I derive a dynamic IS equation and a New Keynesian Phillips curve. Determinacy and shocks are discussed in chapters 5 and 6. I perform some welfare analysis of monetary policy in chapters 7, 8 and 9. Chapter 10 augments the basic model with sticky wages in addition to sticky prices, following Erceg et al. (2000). Finally, the small open economy model established by Gali and Monacelli (2005) is derived in chapter 11. Dynare codes are provided in the appendix. A few words about notation: Variables in levels are denoted with capital letters, logged variables with small letters. Percentage deviations are denoted with small letters with a hat. Let us illustrate by an example: The percentage deviation in
is presented by a first-order Taylor expansion:
1.2
from
The New Keynesian model – Key features
So, what kind of features do the New Keynesian models possess? The most important are:
Dynamic, stochastic, general equilibrium (DSGE) modeling: Agents’ behavior today affects
future environments. Agents know this and behave accordingly. Still, uncertainty arises because at least some processes in the economy are exposed to exogenous shocks. General equilibrium, in the sense that it incorporates all markets in the economy, is provided.
Monopolistic competition: Prices are set by private economic agents in order to maximize their objectives, as opposed to being determined by an anonymous Walrasian auctioneer seeking to clear all competitive markets at once.
Nominal rigidities: At least some firms are subject to constraints on the frequency with which they can adjust prices of the goods and services they sell. A lternatively, firms may face some
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The Basic New Keynesian Model
costs of adjusting those prices. The same kind of friction applies to workers in the presence of sticky wages.
Short run non-neutrality of monetary policy: As a consequence of nominal rigidities, changes in short term nominal interest rates are not matched by one-for-one changes in expected inflation, thus leading to variations in real interest rates. The latter brings about changes in real quantities. In the long run, however, all prices and wages adjust, and the economy reverts back to its natural equilibrium.
While the first bullet point is a common feature in most modern macroeconomic models, including those in the RBC literature, the last three are special ingredients in New Keynesian models. Now it is time to present the basic model.
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The Basic New Keynesian Model
2.
Households
2.1 Setup
We will study households and the implications of market power first. Consider an economy consisting of many identically, infinitely-lived households, with measure normalized to one. The representative household has an instantaneous (and time separable) money-in-utility function of the form:
∑ ∫
(2.1)
The consumption level is denoted of
,
is labor, and
is real money holdings. One can think
as a composite of many goods. We make the following assumptions about preferences: 1 ,
,
,
,
,
To simplify the analysis, we also assume that the marginal utility of one specific element in the utility function is independent of the level of other elements, i.e. that
.
A representative household maximizes lifetime utility, and discounts the future proportionally by a factor :
(2.2)
The consumption index
is the sum of consumption of all goods , and there exists a
continuum of goods represented by the interval
:
(2.3)
Note that utility is a nested function of increasing in
, where
. Thus, utility is increasing in
is increasing in
and
is
. The CES-aggregator given in (2.3) is an
assumption about preferences. Given this assumption, goods become imperfect substitutes, a
feature which equips firms with market power. 2 Households’ maximization problem is subject to a one-period budget constraint:
∫ In this setup, and
1 The
(2.4)
is the number of bonds purchased last period, each yielding a payoff of one,
is the price per bond bought today.
∫
expression
2 Equation
represents
throughout the text.
(2.3) also nests free competition as a special case. In particular, taking the limit as
(2.3) becomes
.
3
approaches infinity,
The Basic New Keynesian Model
2.2
Optimal consumption vector and the aggregate price index
The household’s decision problem can be dealt with in two stages. First, for any given level of
consumption expenditures, it will be optimal to purchase the consumption vector that maximizes total consumption
.3 Second, given this optimal bundle of consumption goods, the household
must choose the utility maximizing combination of consumption, labor and money. Let us find the optimal consumption vector first. For a given level of consumption expenditures, say
∫ ∫ ∫
, the consumption maximization problem is given by:
s.t.
(2.5)
This problem can be used to derive an aggregate price index in addition to the optimal consumption vector. Let us solve the problem:
∫ ∫ ∫ ⇒ ∫ ∫ ⇒ ⇒ ∫ ∫ ∫ FOC: :
The equality must hold for all goods, so the relationship between two different goods must be:
Insert (2.6) into the constraint and solve for
(2.6)
:
3 Alternatively,
one can find the consumption vector that minimizes t otal consumption expenditures for a given level of consumption. The two problems are equivalent and give identical results.
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The Basic New Keynesian Model
⇒ ∫
∫ [ ∫ ] | ∫ ⇒ ∫ ⇒ ⇒⇒ ∫ Insert the result above into
Define
and evaluate the result for
as the expenditure needed to purchase a unit-level of
this definition we can solve the above equation for
:
, that is
. Using
:
(2.7)
Thus, equation (2.7) can conveniently be defined as an aggregate price index. We will use it
throughout the notes. To find the optimal consumption vector, insert (2.6) into the expenditures level equation. Then, insert (2.7) and solve for consumption of good :
(2.8)
Insert (2.8) into (2.3) and rearrange:
(2.9)
Finally, we get the demand function for good by inserting (2.9) into (2.8):
(2.10)
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The Basic New Keynesian Model
Equation (2.10) is the solution to (2.5), the first stage of a representativ e household’s decision problem. Once the household knows prices and has decided on consume of each good. The next step is to decide 2.3
.
, it also knows how much to
Optimal allocation of consumption and labor
The problem in the second stage is established by using (2.2), (2.4) and (2.9):
∑
s.t.
(2.11)
Problems such as the one above are most often solved by using either Kuhn-Tucker conditions or by dynamic programming. The results should be the same, of course. I will now show both of these methods. First, the Kuhn-Tucker approach starts by setting up the Lagrangian. Let us go through the steps:
∑ ⇒ ⇒
(2.12)
FOC: :
(2.13)
:
(2.14)
:
(2.15)
:
(2.16)
From (2.16):
(2.17)
From (2.13):
(2.18)
From (2.14) and (2.13):
(2.19)
From (2.15) and (2.13):
6
(2.20)
The Basic New Keynesian Model
Equations (2.18), (2.19) and (2.20) determine the intertemporal consumption allocation (the Euler equation), the labor-leisure choice and the money demand, respectively. Together, those equations determine the rational, forward- looking household’s allocation decisions. An alternative approach to derive (2.18)-(2.20) from (2.11) is to use dynamic programming. Point of departure is
the observation that the structure of the household’s optimization problem in period is identical
⇒ to the one in period
,
beginning of period
as:
, etc. To see this, we first define total financial wealth at the
Second, rewrite the budget constraint to:
Third, assume that the budget constraint holds with equality and solve for
:
(2.21)
Fourth, recast (2.11) into a Bellman equation where
is treated as the state variable and
as the control variable:
(2.22)
Equation (2.22) captures the core idea of dynamic programming, as it already defines a necessary condition any solution to (2.11) has to fulfill. The Bellman equation basically states that the highest obtainable value of the decision problem in period , control
, is given by the
which maximizies the sum of current period utility and the discounted value of the
decision problem next period. The Euler equation for this problem states that the marginal cost of allocating more wealth today is equal to the marginal benefit of allocating more wealth tomorrow. It is written as:
When we plug (2.21) into (2.1), this optimality condition becomes: (2.23)
The envelope theorem for the problem states that the marginal change in the value function today from a change in total wealth must be equal to the marginal change in today’s utility. This
optimality condition is written as:
When we plug (2.21) into (2.1), the envelope theorem yields:
7
The Basic New Keynesian Model
(2.24)
Iterate (2.24) one period forward: (2.25)
Insert (2.25) into (2.23) and we get the following consumption Euler equation: (2.26)
Further, we characterize the remaining optimality conditions using (2.21) and (2.1): :
:
(2.27)
(2.28)
From (2.26):
(2.29)
From (2.27):
(2.30)
From (2.28):
(2.31)
Equations (2.29)-(2.31) determine the intertemporal consumption allocation (the Euler equation), the labor-leisure choice and the money demand, respectively. Notice that they are identical to (2.18)-(2.20), highlighting the fact that the household’s optimization problem should have the same solutions regardless of solution method. To proceed we need to specify utility. As an example, consider the following per-period utility function: 4
(2.32)
The marginal utilities of consumption, labor and money become:
The Euler equation given by (2.18) or (2.29) writes:
4
(2.33)
Gali (2008) excludes real money balances from the utility function, but instead imposes an ad-hoc log-linearized money demand given by , where is the interest rate elasticity in the money demand equation. We will see soon that this is equivalent to setting in (2.32).
8
The Basic New Keynesian Model
The labor-leisure choice given by (2.19) or (2.30) writes:
⇒ ⇒ () () ⇒⇒ ⇒ ⇒ ⇒
(2.34)
The money demand equation given by (2.20) or (2.31) becomes:
(2.35)
Finally, it is convenient to log-linearize (2.33)-(2.35). We denote small letter variables as the log of large letter variables. With respect to the Euler equation, define the following:
Using this, (2.33) can be rewritten to:
It is clear from the equation above that
in steady state where
. Thus,
a first-order Taylor expansion of the Euler equation around steady state yields:
(2.36)
The linearized version of the labor supply equation (2.34) is:
(2.37)
Finally, let us linearize the money demand equation given by (2.35):
9
The Basic New Keynesian Model
⇒ ⇒ If we discard the constant term and assume an income elasticity of one, where this assumption implies that
, the money demand equation can be written as (2.38), where
:
(2.38)
This ends the analysis of households in the New Keynesian model. We now turn to firms.
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The Basic New Keynesian Model
3.
Firms
3.1
Aggregate inflation
Assume Cobb-Douglas technology:5
Here, and
is the output produced by firm in period ,
(3.1) is the economy-wide technology level
is the labor force used by the firm. One key ingredient in the New Keynesian model is
price rigidity. When firms set their prices, they can do so freely. However, they do not know a priori when the next opportunity to price change emerges. The probability of being unable to
change the price in any given period is . Thus, this is the fraction of all firms that is stuck with the price they had last period while the remaining
firms reset their prices. The aggregate
∫ ⇒ ⇒ ⇒⇒ price dynamics (inflation) in period can be calculated as follows, where level,
is the aggregate price
is the optimal price set by firms who are able to reoptimize in that period, and
represent the set of firms not reoptimizing their posted price:
(3.2)
The aggregate gross inflation is defined as
. Steady state is defined by zero inflation,
implying that
. Linearizing (3.2) around steady state yields:6
and
(3.3)
Equation (3.3) makes it clear that inflation results from the fact that firms reoptimizing in any
given period choose a price that differs from the economy’s average price in the previous period.
5
The capital stock is treated as fixed and investment is set to zero in the short run. These two specifications follow McCallum and Nelson (1999), who argued that capital do not play a major role in most monetary policy and business cycle analyses. 6 Remember the first-order Taylor expansion: where is the vector of variables one wants to linearize around. Using this as a point of departure, it is often convenient to define a new variable as the log deviation in from : . This implies that , and the Taylor expansion can be
∑ ∑
rewritten to a formula for log-linearization via Taylor series expansion:
11
.
The Basic New Keynesian Model
Hence, in order to understand inflation over time one needs to analyze the factors underlying firms’ price setting decisions.
3.2
Optimal price setting
Basically, when firms are faced with the problem of setting optimal price today, they must take into consideration that this price often determine profit in the future as well, as the probability of
being stuck with today’s price periods ahead is
will choose the price
. Thus, a firm who reoptimizes in period
that maximizes current market value of the profits generated while that
price remains effective. The stochastic discount factor for nominal payoffs in period
∑ | |(|) | | |(|) | |(|) | ∑ | ∑ | , which is given by: 7
is
(3.4)
The representative firm’s maximization problem is thus given by:
s.t.
Let us spend a couple of seconds on the problem given in (3.5). for a firm that last set its price in period .
(3.5)
is the output in period
is the total cost in period
as a function of this output. The nominal, undiscounted profit in period
is thus
. The firm’s problem is subject to a sequence of demand
constraints as given by (2.10), and market clearing in period
implies that the firm produces
. The problem can be rewritten to an unconstrained one by inserting the
constraint into the profit function. We also insert for the discount factor. This gives us:
Let us find the optimal price
:
FOC:
7 Go
back to the Euler equation of the consumers to get the intuition.
12
(3.6)
The Basic New Keynesian Model
| | || | | ⇒ ∑ | | ⇒ ∑ (|) ∑ (|| ) | | ⇒ ∑() ∑ (| | ) ⇒ ∑ ∑ ∑ ⇒ ∑| ∑ | ∑ Next, we insert for
and
and solve for the optimal price
:
(3.7)
Divide both sides by
to get the optimal real price as a weighted average of future real marginal
costs:8
(3.8)
| | | Notice that in the case with flexible prices, i.e. when
, (3.6) collapses to a one period
problem and (3.7) becomes:
(3.9)
Thus, (3.9) gives the desired or frictionless markup. 3.3 Log-linearization
8
Note that the real marginal cost in period
is denoted
13
| |
.
The Basic New Keynesian Model
The next step is to log-linearize (3.7) around the steady state. In a zero inflation steady state, we must have that:
| | | | |
The last three identities follow from the zero inflation definition and from market clearing. Before log-linearizing it is convenient to divide both sides of (3.7) by
:
∑ ∑| ⇒ ∑ ∑ |
(3.10)
A first-order Taylor expansion of the LHS of (3.10): 9
9
This is just a simple first-order Taylor expansion. The first term is the LHS of (3.10) in steady state. The four last terms contain the first derivatives with respect to , , and respectively, all evaluated in steady state.
14
The Basic New Keynesian Model
A first-order Taylor expansion of the RHS of (3.10): 10
| 10
The first term is the RHS of (3.10) in steady state. The four last terms contain the first derivatives with respect to , , and respectively, all evaluated in steady state.
15
The Basic New Keynesian Model
(| ) (| ) (| ) (| ) (| ) Finally we equate LHS with RHS and solve for
:
16
The Basic New Keynesian Model
⇒ ∑ ∑ (| ) ⇒ ∑ ( | ) ∑∑ (| ) ⇒⇒ |
(3.11)
We see from (3.11) that firms will set a price that corresponds to the desired markup, 11 given by , over a weighted average of their current and expected nominal marginal costs, with
the weights being proportional to the probability of the price remaining effective at each horizon .
11 Because
, we have that
17
.
The Basic New Keynesian Model
4.
Equilibrium
4.1
Market clearing
Market clearing in the goods market implies:
∫ ⇒ ∫ ∫ ∫ ∫ ∫ ∫
(4.1)
Let aggregate output be defined as:
(4.2)
Insert (4.1) into (4.2), and then (2.10) into (4.2), to get the aggregate market clearing condition:
Finally, taking logs on both sides yields:
(4.3)
Equation (4.3) is the aggregate market clearing condition. Insert (4.3) into (2.36) and get:
(4.4)
Market clearing in the labor market:
From (3.1) we see that
(4.5)
. Insert this into (4.5) as well as the goods market clearing
condition (4.3) and the consumption demand (2.10):
Next we take the log of (4.6):
18
(4.6)
The Basic New Keynesian Model
⇒ ∫ ∫ ∫ ∫ ∫ ∫ ⇒ ∫ ⇒ ∫ ∫ ⇒ ∫ ∫ ∫ ∫ I will now show that around
because
(4.7)
up to a first-order approximation
, but first I must show that
. Recall the consumer price index
. Rearranging gives:
(4.8)
A second order approximation of (4.8) gives us:
From equation (4.9) it is also clear that
(4.9)
up to a first-order approximation. Next, let
us do a second order approximation of
:
Now, insert (4.9) and get:
19
The Basic New Keynesian Model
∫ ∫ From (4.10) we conclude that up to a first-order approximation,
(4.10)
. This
implies that:
Thus, (4.7) can be rewritten to:
4.2
(4.11)
The New Keynesian Phillips curve and the Dynamic IS equation
Next, an expression for individual firms’ marginal cost as a function of the economy’s average real marginal cost is derived. The latter is derived in (4.12), where we insert
from
(4.11):12
12 The
nominal marginal cost by using labor is the wage . The nominal marginal gain is the income increase, that is the price times the marginal increase in production by adding a little more labor. Thus, the real marginal cost is the nominal cost relative to the nominal gain, i.e. . Linearizing gives . It follows
from the average production function .
that marginal productivity is
20
. Thus,
The Basic New Keynesian Model
⇒ | | | | | | (| ) | | (| ) ⇒ | |
(4.12)
Similarly, a firm’s real marginal cost in period
is:
(4.13)
Now, the market clearing condition and the demand schedule (2.10) imply that firm output is , which in linearized terms gives
.13
Use this as well as (4.13) and (4.12) to get:
(4.14)
Notice that the last term in (4.14) disappears if there is constant returns to scale, i.e. if Then
.
, which implies that the marginal real cost is independent of the
production level; it is common across all firms. We shall now derive an expression for inflation. The point of departure is (3.11), which we rewrite to:
∑ | ⇒ ∑ ⇒ ∑ Insert (4.14):
13 Note
that
|
.
21
The Basic New Keynesian Model
Define
and subtract
on both sides.:
If we take out the
terms of each summation operator, the equation can be written more
compactly as a difference equation:
Next we insert (3.3) and solve for inflation
:
22
The Basic New Keynesian Model
⇒⇒ ⇒⇒ – –
(4.15)
Equation (4.15) expresses inflation as the sum of (discounted) expected inflation and real marginal costs, and we have defined
to ease the
notation. It is clear from (4.15) that inflation is strictly decreasing in price stickiness , in the
measure of decreasing returns , and in the demand elasticity . An alternative presentation of inflation is found by solving (4.15) forward:
Equivalently, and defining the average markup in the economy as
, we see that inflation will
be high when firms expect average markups to be below their steady state or desired level
.
In that case firms that have the opportunity to reset prices will choose a price above the
economy’s average price level in orde r to realign their markup closer to its desired level. Thus, in
the present model, inflation results from the aggregate consequences of purposeful price-setting decisions by firms, which adjust their prices in light of current and anticipated cost conditions.
⇒
Next, a relation is derived between the economy’s real marginal cost and a measure of aggregate
economic activity. We have derived earlier that
. Insert this into
, and use that
Insert for
:
and get to:
In the case with flexible prices we know from before that
(4.16)
. Define natural output level
as the equilibrium level under full price flexibility. In this case (4.16) can be rewritten to:
Solve (4.17) for natural output:
23
(4.17)
The Basic New Keynesian Model
⇒ ̃ ⇒ ̃ ⇒ ̃ ̃ ̃ ⇒ ̃ ̃
(4.18)
If we subtract (4.17) from (4.16) we get a measure of the real marginal cost gap as a function of the output gap from natural output, denoted
:
(4.19)
Finally, the New Keynesian Phillips curve is established by inserting (4.19) into (4.15):
(4.20)
The New Keynesian Phillips curve (NKPC) is one of the key building blocks of the New Keynesian model, and the parameter is defined by
. The second key equation
is the dynamic IS equation. If we use the definition of the real interest rate, equation (4.4) becomes
,
. In a similar vein, the natural output is given as
a function of the natural interest rate:
(4.21)
Subtracting (4.21) from (4.4) gives the output gap from the natural output, i.e. the dynamic IS equation (DIS):
(4.22)
Equations (4.20) and (4.22) together with an equilibrium process for the natural rate
, which in
general will depend on all exogenous forces in the model, constitute the non-policy block of the basic New Keynesian model. That block has a simple recursive structure: The NKPC determines inflation given a path for the output gap, whereas the DIS equation determines the output gap
̃ ∑
given a path for the exogenous natural rate and the actual real rate. To see the latter, assume the transversality condition
. Then one can solve (4.22) forward to yield: (4.23)
24
The Basic New Keynesian Model
Equation (4.23) emphasizes the fact that the output gap is proportional to the sum of current and anticipated deviations between the real interest rate and its natural counterpart. To gain further
̃̃ ⇒ insight into the natural interest rate, note first that (4.4) implies Second, note that the first difference of (4.18) gives for
.
. Now, solve (4.22)
and use these two observations to yield an expression for the natural real rate. From
(4.22):
(4.24)
Thus, the natural real rate is a function of households’ discount rate and expected technological
progress. In some cases it is convenient to work with deviations in the natural real rate from the
̂
discount rate, which we define as: (4.25)
Note that if one turns off technology shocks, the real rate becomes the discount rate. Once a process for the technological progress is specified, one can identify the real interest rate path in
(4.24). In order to close the model, we supplement (4.20) and (4.22) with one or more equations determining how the nominal interest rate
evolves over time, i.e. with a description of how
monetary policy is conducted. Observe from (4.23) that the equilibrium path of real variables cannot be determined independently of monetary policy when prices are sticky. The output gap is directly determined by the real interest rate gap, which is directly determined by the nominal interest rate set by central banks. This important feature of the New Keynesian Model is in contrast to classical models where monetary policy is neutral.
25
The Basic New Keynesian Model
5.
Equilibrium determinacy
5.1
System representation
Throughout this and the next sections I will look at a specific monetary policy rule, more
̃
specifically an interest rate rule. I assume the central bank follows a rule of the form:
Standard reasoning implies that
and
(5.1) are non-negative, which we assume from now on.
The first task when analyzing monetary rules is to check whether the specified policy yields a unique and stable equilibrium. While doing this, it is convenient to work with a reduced form representation of (4.20) and (4.22) who takes into account the policy rule under consideration.
̃ ̃ ̃ ̃ ( ̃ ) ̃ ̃ ̂ ̃ ̂ ̃ ̃ ⇒ ̃̃ ̃̃ ̂ ̂ ⇒
We first derive a forward looking version of the dynamic IS equation. Insert (5.1) into (4.22), and then (4.20) into (4.22). Solve the resulting equation for
:
(5.2)
Equation (5.2) shows the current output gap as a function of expected output gap, expected
inflation, and shocks. We next achieve a similar representation of current inflation. Insert (5.2)
̃ ̃ ̂ ) ( ̃ ̂ ) ̂ ̃ ( ⇒ ̃ () ̂ into (4.20) and get:
(5.3)
26
The Basic New Keynesian Model
Finally, the two equations (5.2) and (5.3) can be written as a system of forward looking difference
)̃ ̂ ̃ ( ̃ ()̂ ̃̂ ) ( ̂ ̃ | | ( ) equations:
where
(5.4)
The system given in (5.4) is a reduced form representation of the dynamic IS curve and the New Keynesian Phillips curve, which takes into account effects from the policy defined in equation (5.1). The coefficient matrix
represents effects from expectations on current output gap and
inflation while the coefficient vector monetary policy shocks in 5.2
represents effects from technology shocks in
. We have defined
and
to ease the notation.
Blanchard and Kahn conditions
In the case we consider here we have two non-predetermined variables,
and
.
Following Blanchard and Kahn (1980), the system (5.4) has a locally unique equilibrium if and only if both eigenvalues of the 2x2-matrix
are inside the unit circle. 14 Let us characterize
necessary and sufficient conditions for this property to hold. The two eigenvalues, denoted and
, are generally solutions to the following system written in matrix form, where is an
identity matrix:
In our case the system becomes:
When this is written out: 14 Consider
a recursive system of the form , where is a vector of predetermined and nonpredetermined variables and is a vector of exogenous variables. Blanchard and Kahn (1980) proved that there exists a locally unique equilibrium if and only if the number of eigenvalues of inside the unit circle is equal to the number of non-predetermined variables. If the number of eigenvalues inside the unit circle is less than the number of non-predetermined variables, then an infinite number of equilibria exist. If the number of eigenvalues inside the unit circle exceeds the number of non-predetermined variables, then no equi librium exists.
27
The Basic New Keynesian Model
[ ( ) ] ( ) ) ( ) ( ) ( ) () ( ( ) ) ( ) ( ⇒⇒ ) ) ( ( ⇒⇒ ( ) ⇒ Following LaSalle (1986)15 we know that the two eigenvalues of the
matrix are inside the unit
circle if and only if the following two inequalities are met:
and
Let us derive conditions the policy parameters
and
must meet for these two inequalities to
hold. From the first inequality:
(5.5)
It is clear that condition (5.5), and consequently the first inequality, are fulfilled as long as
,
which we assume. Thus, the only relevant inequality is the second one, which we rewrite to:
(5.6)
We see from condition (5.6) that the equilibrium is unique as long as the policy parameters and
have sufficiently high values, i.e. as long as monetary authorities respond to deviations of
15 LaSalle
||
(1986) showed that both solutions to the same time as .
are smaller than one if and only if
28
||
at
The Basic New Keynesian Model
̃ ̃ ̃
inflation and output with adequate strength. Note also that our assumptions about the other parameters imply that
is sufficient for (5.6) to hold. This is referred to as the Taylor
principle. Let us give (5.6) some intuition. Suppose the economy is exposed to a permanent change in inflation;
. From (4.20) we see that without any policy this leads to a permanent
change in the output gap equal to
. However, with the policy rule described here,
we can find the nominal interest rate response by inserting
into the differentiated
version of (5.1):
Furthermore, by rearranging (5.6) we get that
. This implies that the change in
inflation should be met by a larger change in the nominal interest rate. Eventually this will drive
the real rate upwards and act as a stabilizing force. Thus, from (5.6) we see that when the central bank responds aggressively enough to changes in output gap and inflation, i.e. when are large enough, output is forced back to natural output and inflation back to zero.
29
and
The Basic New Keynesian Model
6.
Shocks
6.1
Effects of a monetary policy shock
Assume that the exogenous component of (5.1) follows an AR(1) process, where
Notice that a positive (negative) realization of
: (6.1)
is interpreted as a contractionary (expansionary)
monetary policy shock, leading to a rise (decline) in the nominal interest rate for given levels of
̃ ̃ ⇒ ̃ ⇒⇒ ⇒ ̃ ̃ ̃ ( ̃ ) ̃ ( ̃ ) ( ) ( )
inflation and output gap. We want to find the contemporaneous effects of a monetary policy shock
on the output gap
and inflation
. One way to identify these effects is by using the
method of undetermined coefficients. Let us start by making the following guess:
The coefficients
and
(6.2) (6.3)
are yet to be determined. First, insert (6.1)-(6.3) into the New
Keynesian Phillips curve given by (4.20). Find an expression for
:
(6.4)
Then, insert the monetary policy rule (5.1) into the dynamic IS equation (4.22): 16
Insert (6.1)-(6.4) and solve for the coefficient
16 To
make the analysis as transparent as possible we set technology shocks.
:
̂
. That is, we turn off the
30
The Basic New Keynesian Model
⇒⇒ ( ) ⇒ ⇒ ⇒ ̃ ̃ ̃
(6.5)
Insert (6.5) into (6.4) to obtain:
(6.6)
Finally, this means that the solutions to (6.2) and (6.3) are:
To ease the notation,
(6.7) (6.8)
. It is also straight forward to show that
as long as (5.6) is satisfied. Note that if we insert (6.1) into (6.7) and (6.8), we get:
Hence, an exogenous increase in the interest rate leads to a persistent decline in both output gap and inflation. Because the natural level of output is unaffected by the monetary policy shock, the response of output matches that of the output gap. Furthermore, (4.22) and (6.2) can be used to obtain an expression for the real interest rate deviation from its steady state counterpart, the
̂ ̃ ̃ ̂ ̂ ̂ natural real rate:
The response on nominal interest rate combines both the direct effect of
(6.9)
and the indirect
effect induced by reduced output gap and inflation. From (6.8) and (6.9):
Note that if the persistence of the monetary policy shock, interest rate will decline in a response to a rise in
(6.10)
, is sufficiently high, the nominal
. In that case, and despite the lower nominal
rate, the policy shock still has a contractionary effect on output, because the latter is inversely related to the real rate, which goes up unambiguously. Finally, one can use (2.38) and (4.3) to determine the change in the money supply required to bring about the desired change in the interest rate. From
, where we insert (6.8), (6.7) and (6.10):
31
The Basic New Keynesian Model
⇒
(6.11)
The sign of the change in money supply that supports the exogenous policy intervention is, in principle, ambiguous. Note however, that
is a sufficient condition for a contraction in
the money supply. Let us simulate the effects of a monetary policy shock. Parameters are calibrated as follows:
Table 1
0.99
1
1
1/3
6
4
2/3
1.5
0.5/4
0.5
0.0625
Setting = 0.99 implies a steady state real return on financial assets of about =
Log utility is implied by = 1 and = 1. = 2/3 implies an average price duration of quarters.
= 1.5 and
≈ 4%.
= 3
= 0.5/4 is roughly consistent with observed variations in the federal
funds rate over the Greenspan era. Finally,
= 0.0625 corresponds to a monetary policy shock
of 25 basis points. Simulated impulse responses are shown in the figure 1. 17 Consistent with the analytical results, it is seen that the policy shock leads to an increase i n the real interest rate, and a decrease in inflation and output. The latter two effects correspond to that of the output gap because the natural level of output is not affected by the monetary policy shock. Under the calibration given in table 1 the nominal interest rate goes up, though by less than its exogenous component, as a result of the downward adjustment induced by the decline in inflation and output gap. In order to bring about the observed interest rate response, the central bank must engineer a reduction in the money supply. The calibrated model thus displays a liquidity effect. Note also that the response of the real rate is higher than that of the nominal rate as a result of the decrease in expected inflation. Overall, figure 1 shows dynamic responses which are qualitatively similar to those estimated using structural vector auto regressive methods. Figure 1: A monetary policy shock
17 Simulations
are done with Dynare and Matlab. See Appendix A for the Dynare codes.
32
The Basic New Keynesian Model
6.2
Effects of a technology shock
Assume that the technology parameter
follows an AR(1) process, where
̂ ̃ ̃ ⇒⇒ ⇒ ⇒ Here we want to find contemporaneous effects of the technology shock and inflation
:
̃
(6.12)
on the output gap
. Given (4.25), the implied natural rate expressed in terms of deviations from
steady state, is given by:
(6.13)
Again we use the method of undetermined coefficients. Guess the following:
(6.14) (6.15)
Insert (6.12) and (6.14)-(6.15) into the New Keynesian Phillips curve (4.20). Find an expression for
:
33
(6.16)
The Basic New Keynesian Model
̃ ̃ ̃ ( ̃ ) ̃ ( ̃ ̂) ( ) ( ) ⇒⇒ ( ) ⇒ ⇒ ⇒ ̃ ̃ ̃ ̃ ̃ Then, insert the monetary policy rule (5.1) into the dynamic IS equation (4.22): 18
Insert (6.12)-(6.15) and solve for the coefficient
:
(6.17)
Insert (6.17) into (6.16) to obtain:
(6.18)
Finally, this means that the solutions to (6.14) and (6.15) are:
(6.19) (6.20)
To ease the notation,
. Note that if we insert (6.12) into
(6.19) and (6.20), we get:
Hence, a positive technology shock leads to a persistent decline in both output gap and inflation.
To find the implied equilibrium response of output, decompose output into . Then insert for
from (4.18) and
18 To
from (6.19):
make the analysis as parsimonious as possible we now set monetary policy shocks.
34
. That is, we turn off the
The Basic New Keynesian Model
( )
(6.21)
To find the equilibrium response of employment, insert for (6.21) into (4.11):
(6.22)
Hence, the sign of the response of output and employment to a positive technology shock is in general ambiguous, depending on the configuration of parameter values, including the policy parameters
and
. Parameters are calibrated as follows in the simulation exercise:
Table 2
0.99
1
We assume that
1
1/3
6
4
2/3
1.5
0.5/4
0.9
1
= 0.9, i.e. that it takes some time for a technology shock to die out. Simulated
impulse responses are shown in the figure 2. 19 Notice that the improvement in technology is partly accommodated by the central bank, which lowers nominal and real rates while increasing the money in circulation. That however is not enough to close the negative output gap, which is responsible for a decline in inflation. Output increases, but less than its natural counterpart. Figure 2: A technology shock
19 Simulations
are done with Dynare and Matlab. See Appendix B for the Dynare codes.
35
The Basic New Keynesian Model
7.
Distortions to the efficient allocation
7.1
The efficient steady state
I will now consider monetary policy design in the New Keynesian framework. This section investigates distortions to the efficient allocation and how monetary authorities can cope with these distortions. First we need to determine the efficient allocation. A natural benchmark is the problem faced by a benevolent social planner seeking to maximize the representative households’
social welfare, given preferences and technology. Using (2.3) and (4.5), this problem reads as: 20
∑ ∑ ∫ ∫ ⇒ ∑ ⇒ ⇒
s.t.
(7.1)
The constraint is the resource constraint coming from all the firms. Notice how all goods enter the utility function symmetrically, at the same time as utility is concave in each good. Also, all goods are produced with identical technology. Thus, by symmetry,
can never be
optimal. This gives the following efficiency conditions:
(7.2) (7.3)
The problem therefore simplifies to:
(7.4)
Let us solve the social planner’s problem. FOC: :
(7.5)
From the firm’s problem in free competition we also have the following:
(7.6)
FOC: :
(7.7)
From (7.5) and (7.7):
20 Here
(7.8)
we depart from the MIU-specification used previously in order to keep the analysis as simple as possible.
36
The Basic New Keynesian Model
Thus, (7.8) is the relevant efficient benchmark monetary authorities should opt for. However, there are two sources of inefficiencies built into the New Keynesian model setup. The first one is firm’s market power, which allows firms to set prices individually instead of being price takers.
The second is staggered price setting, which prevent firms from adjusting optimally to shocks in the economy in the short run. I will now study these two inefficiencies in turn. 7.2
Distortions caused by market power
Market power, which yields monopolistic competition, stems from the construction that each firm perceives an imperfectly elastic demand for its differentiated product. This gives firms the opportunity to set prices above marginal costs. Market power is unrelated to the presence of sticky prices. To illustrate this, suppose for the moment that prices are fully flexible so that
⇒ . Firm ’s problem then becomes:
FOC:
⇒ ⇒ ⇒ ⇒ Because all firms behave in the same way,
:
37
(7.9)
The Basic New Keynesian Model
⇒
As before,
is the gross optimal markup chosen by firms and
(7.10) is the marginal cost. If
we insert (7.10) into the efficient allocation (7.8), where the first equality follows from the optimality conditions of the household, we immediately see that: (7.11)
Thus, the presence of market power not only leads to higher prices than optimal, but also an inefficiently low level of employment, and therefore also of output. This kind of distortion to the efficient equilibrium can be dealt with in a simple way by means of an employment subsidy. Let
denote the rate at which the cost of employment is subsidized, and let outlays associated with the subsidy be financed by a lump-sum tax. If the subsidy is set to
, then, by construction, the
equilibrium under flexible prices yields efficiency. 21 Equation (7.10) becomes:
7.3
(7.12)
Distortions caused by sticky prices
The assumed constraints on the frequency of price adjustment constitute a source of inefficiency on two grounds. First, the fact that firms do not adjust their prices continuously implies that the economy’s average markup will vary over time in response to shocks, and will generally differ
from the constant frictionless markup
. Denote the economy’s average markup, i.e. the ratio
of average price to average marginal cost, as
. Then, from (7.12): (7.13)
The last equality follows from the assumption that the subsidy in place exactly offsets the monopolistic competition distortion, which allows us to isolate the role of sticky prices. Insert (7.13) into the efficient benchmark allocation (7.8):
Thus, (7.8) is violated whenever
(7.14)
. Efficiency can only be restored if policy manages to
stabilize the economy’s average markup to its frictionless level. In addition to the inefficiency
described above, staggered price setting is a source of a second type of inefficiency. The latter has
21 In
much of the analysis below it is assumed that such an optimal subsidy is in place.
38
The Basic New Keynesian Model
to do with the fact that relative prices of different goods will vary in a way unwarranted by
changes in preferences or technologies, as a result of the lack of synchronization in price adjustments. Whenever
we also get
, and consequently
, which
violates (7.2) and (7.3). To cope with distortions caused by staggered price setting, one should therefore opt for markups that are equal across all firms at all times. I will now analyze how this goal can be achieved by monetary authorities. 7.4
Monetary policy solutions to equilibrium distortions
To keep the analysis simple, assume that in the last period, had an efficient allocation at
, implying that we
. Then this efficient allocation can be attained by a policy that
stabilizes marginal costs at a level consistent with firms desired markup, i.e.
, given the prices in place. If that policy is expected to be in place indefinitely, no firm
has an incentive to adjust its price because it is currently charging its optimal markup and expects to keep doing so in the future. As a result,
and, hence,
. In other words,
the aggregate price level is fully stabilized and no relative price distortions emerge. In addition,
̃
, and output and employment matches their counterparts in the flexible price
equilibrium allocation with a subsidy in place. From (4.19) and (4.20) we immediately see that implies the following, where inflation is given by (3.3): (7.15) (7.16)
From the dynamic IS equation (4.22) we see that once (7.15) and (7.16) are expected to take place indefinitely, the nominal interest rate becomes the natural real rate:
(7.17)
Two features of the optimal policy are worth emphasizing. First, stabilizing output is not desirable in and of itself. Instead, output should vary one for one with the natural level of output. Whenever real shocks cause natural output to fluctuate a lot, so should also output. Second, price stability emerges as a feature of the optimal policy even though, a priori, the policy maker does not attach any weight on such objective. The next step is to analyze how to implement (7.15) and
̃
(7.16) in practice. Because (7.15) and (7.16) imply (7.17), one could think of (7.17) as a natural candidate for monetary policy. Although one obvious equilibrium is
, we need to
whether this equilibrium is unique. Treat (7.17) as an exogenous interest rate rule and insert it into (4.22). Combine with (4.20) to yield a system of difference equations:
̃ ̃ ̃ 39
The Basic New Keynesian Model
⇒ ̃ ̃ ̃ ̃ ⇒ ̃ ̃ | | where
(7.18)
Let us calculate the characteristic equation:
With respect to the two conditions necessary for smaller-than-unity eigenvalues derived by
||
LaSalle (1986), we get:
and
Clearly, the last condition does not hold, so both eigenvalues of
cannot lie inside the unit
circle. Thus, by the Blanchard and Kahn (1980) conditions, there exists a multiplicity of equilibria because the number of eigenvalues inside the unit circle is smaller than the number of nonpredetermined variables. The zero output gap and zero inflation target is only one of them, and there is nothing in the policy (7.17) that drives the economy back to the desired equilibrium given by (7.15) and (7.16). The second policy rule we consider is an interest rate rule with an
̃ ̃ ̃ ̃ ( ̃ ) ̃ ̃ ̃ ̃ ̃ ⇒ ̃ ̃ endogenous component:
(7.19)
We first derive a forward looking version of the dynamic IS equation. Insert (7.19) into (4.22) and solve for
:
40
The Basic New Keynesian Model
⇒ ̃ ̃ ̃ ̃ ) ( ̃ ) ̃ ( ⇒ ̃ () )̃ ( )̃ ̃ ( ̃ ) ( ̃ ̃ ̃ ̃ ( ̃ ) ̃ ̃ ̃ ̃ ̃ ⇒ ̃
(7.20)
Next, insert (7.20) into (4.20) and solve for
:
(7.21)
The two equations (7.20) and (7.21) can be written as a system of difference equations:
where
(7.22)
Note that the transition matrix
in (7.22) is identical to the one in (6.4). Thus, whenever
condition (6.6) holds, which it does as long as one follows the Taylor principle policy rule given by (7.19) yields a unique and stable equilibrium with
, the
. A last
monetary policy rule worth considering is a forward-looking interest rate rule:
(7.23)
Now, the monetary authorities adjust the nominal interest rate to variations in expected inflation and output gap, as opposed to their current values. Insert (7.23) into (4.22):
(7.24)
Insert (7.24) into (4.20):
(7.25)
41
The Basic New Keynesian Model
̃ ̃ ̃ | | ( ) ()() ()
Write as a system of forward looking difference equations:
where
(7.26)
The characteristic equation:
In our case:
Written out:
The inequalities from LaSalle (1986) which should be met by the two eigenvalues of
and
42
:
The Basic New Keynesian Model
From the first inequality:
⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ If
:
Condition (7.27) cannot be binding when
(7.27)
is non-negative because
. If
:
(7.28)
From the second equality:
If
:
If
(7.29)
:
(7.30)
Condition (7.30) turns out to be identical to (6.6). However, in this case the two conditions (7.28) and (7.29) must hold in addition. We see from (7.28)-(7.30) that the policy responses should be
neither too weak nor too strong. In particular, from (7.28) and (7.30) we see that a very high value on
leads to indeterminacy, quite independently of
modest, then rules with too large a value of be set to zero.
. If the response to output is
can lead to determinate equilibria. However, from (7.29) we see that
again leads to indeterminacy. Finally, note that if
43
, then
could
The Basic New Keynesian Model
8.
The welfare loss function
8.1 Introduction
I will now derive measures of the society’s welfare losses caused by deviations in output and inflation from their steady state targets. The result will be a quadratic loss function that represents a quadratic second-order Taylor series approximation to the level of expected utility of the representative household in equilibrium with a given monetary policy. First I look at the simplest case where the only distortions in the economy are the presence of monopolistic competition and sticky prices. Then I look at an empirically more appealing case where what one refers to as cost push shocks exist. 8.2
The simplest case – A welfare loss function when real rigidities are absent
The first case we consider is the one analyzed previously, where the government implements an employment subsidy that removes the distortions caused by monopolistic competition. Thus, we assume that the subsidy given by (7.12) is in place. This case will also serve as a methodological
framework for the welfare analysis conducted later. In order to lighten the notation, denote the period utility as
and the steady state utility as
. We will use the
following second order approximation of relative deviation in consumption from its steady state counterpart, where logged consumption is approximated around logged steady state consumption:
̂ ̂ ̂
The same kind of second order approximation is performed on labor
, so that:
We need some more results as well. From (2.32) we have that
and
. From the market clearing condition we have that
these results, a second-order Taylor approximation of
around steady state
. Using all leads us to
the following criterion for welfare losses: 22
22 From
(19) we see that utility is separable in consumption and labor, i.e. simple and assume away money in the utility function.
44
. Also, we make the analysis
The Basic New Keynesian Model
̂ ̂ ̂ ̂ ̂ ⇒ ̂ ̂ ⇒
(8.1)
Our goal is to find a way to express (8.1) in terms of steady state deviations only, that is with the gap in output from natural output and the gap in inflation from zero inflation. The way to such a representation contains several steps. First, note from (4.7) that:
∫ ⇒ ∫ ̂ ̂ ̂ ̂ ̂ ∫ ∫ ̂ ̂ ⇒ ̂ ̂
(8.2)
As before:
The next step is to get an alternative expression for
(8.3)
. In the welfare analysis we do a second-
order approximation. Thus, while we earlier found that
up to a first-order, this result can
no longer be used. The following second-order approximation of
will be useful, where
is approximated around zero:
From
, we have that
. Thus, when taking
expectations on both sides of the above, where
denotes the expectations operator with respect
to good , we get:
45
The Basic New Keynesian Model
⇒ ̂ ̂ ̂ ̂ ̂ ⇒ ̂ ̂ ̂ ̂ ̂ ̂ ̂ ̂ ̂ ̂ ̂ ̂ ̂ ̂ ̂ ⇒
(8.4)
The price dispersion is denoted in
. Next, let us do a second-order approximation of
:
(8.5)
Finally, insert (8.4) and (8.5) into (8.3) to get the following second-order approximation of
:
(8.6)
As before, we have defined
. The next step is to insert (8.2) and (8.6) into (8.1),
rearrange and also get rid of non-policy terms whenever possible:
46
The Basic New Keynesian Model
⇒
The notation
(8.7)
is shorthand for terms independent of policy. To proceed we must find a
way to get rid of
on the RHS of (8.7). From (7.5) we see that the undistorted steady state
equilibrium implies:
(8.8)
Using the specified production function,
, and
the above can be rewritten to:
(8.9)
Insert (8.9) into (8.8):
47
The Basic New Keynesian Model
⇒ ⇒ ̃
(8.10)
From (4.18) we have:
(8.11)
Furthermore, by the definition of
and
:
(8.12)
Insert (8.11) and (8.12) into (8.10), and write up a discounted sum of lifetime welfare losses as a function of output gap from natural output and inflation gap from zero inflation:
48
The Basic New Keynesian Model
( ) ̃ ̃ ⇒ ∑ ∑ ̃ ⇒
(8.13)
The final step consists in rewriting the terms involving price dispersion in (8.13) as a function of inflation. Note that because a fraction
of firms are able to reset their price in period
while the remainding firms are stuck with last periods price, we can rewrite the expected price for good to:
Rewrite this:
(8.14)
Before proceeding, we refresh a simple, useful result from basic statistics. Consider a random variable with expected value or mean equal to
. Then the variance of is given by
. This expression can by expanded as follows:
Thus, the variance of is the same as the mean of the square minus the square of the mean. Now, using this, the price dispersion measure writes as:
Furthermore, because only an exogenous draw of
(8.15)
firms are able to reset their price:
Insert (8.16), and then (8.14), into (8.15). Then simplify:
49
(8.16)
The Basic New Keynesian Model
∑ ∑ ∑ ̃ ̃ ̃ ̃ ̃ ⇒ ∑ ∑ ∑ ̃ Iterating backward, and collecting terms for every period , yields:
Thus, if one takes the discounted value of these terms over all periods:
(8.17)
Now we can insert (8.17) into (8.13):
The parameter is defined as
(8.18)
, just as it was in equation
(4.15). Finally, we are ready to establish a quadratic welfare loss function
which consists only
of those terms in (8.18) that are relevant to monetary policy:
(8.19)
50
The Basic New Keynesian Model
Welfare losses are expressed in terms of the equivalent permanent consumption decline, measured as a fraction of steady state consumption. The average welfare loss per period is thus given by a linear combination of the variances of output gap and inflation:
̃
(8.20)
From (8.19) and (8.20) we see that the relative weight of output gap fluctuations in the loss function is increasing in , and . The reason is that larger values of those curvature parameters amplify the effect of any given deviation of output from its natural level on the size of the gap between the marginal rate of substitution and the marginal product of labor, which is a measure of the economy’s aggregate inefficiency. On the other hand, the weight of inflation
fluctuations is increasing in the elasticity of substitution among goods, , and the degree of price
stickiness, . The former amplifies the welfare losses caused by any given price dispersion, the latter amplifies the degree of price dispersion resulting from any given deviation from zero inflation. The optimal monetary policy in the case considered here achieves, as we saw earlier, the flexible price equilibrium with zero output gap and inflation. This is referred to in the literature as a divine coincident. However, such an allocation is rarely seen in practice. Most of the time, monetary authorities face a real tradeoff between stabilizing inflation and stabilizing the output gap. Typically, reducing inflation comes at the cost of a negative output gap, given an initial equilibrium allocation. This observation gives us a motivation for the introduction of cost push shocks. 8.3
Introduction of cost push shocks
I will now consider a world in which the central bank has to consider policy tradeoffs between minimizing output gap and minimizing inflation. When nominal rigidities coexist with real imperfections, the flexible price equilibrium is generally inefficient. In that case, there is no longer optimal for the central bank to seek that allocation. On the other hand, any deviation of economic activity from its natural, flexible price level generates variations in inflation, with consequent relative price distortions. Now we assume existence of some real imperfections that generate a time-varying gap between output and its efficient counterpart, even in the absence of price rigidities. The resulting monetary policy under this environment is referred to as flexible inflation targeting. We will first look at two alternative ways to include real imperfections in the New Keynesian model. One is through variations in desired price markups. Assume that the elasticity of substitution among goods varies over time according to some stationary stochastic process
. Let the associated desired markup for firms be
firm’s optimal price setting, previously given by ( 3.11), becomes:
51
. Then, the representative
The Basic New Keynesian Model
⇒ ∑∑(|( ) ) | | | | | | ⇒ ⇒ ⇒
(8.21)
The only difference between (3.11) and (8.21) is that optimal prices in the former was determined by
instead of
. The
resulting inflation equation previously given by (4.15) rewrites to:
(8.22)
Let us denote
as the efficient equilibrium level of output under flexible prices and a constant
markup , whereas
denotes the equilibrium under flexible prices and a time-varying markup
.23 Then, whereas (4.17) describes the real marginal cost in equilibrium with price flexibility,
derived directly from (4.16), the efficient marginal cost version of (4.16) now becomes:
(8.23)
Thus, whereas the real marginal cost gap previously was given by (4.19), it is now given by:
We define
(8.24)
as the output gap from efficient output. Furthermore:
(8.25)
Insert (8.24) and (8.25) into (8.22) to yield the following structural equation for inflation:
23 We
still denote as the equilibrium output with flexible prices, but this output level is now associated with a time-varying price markup instead of as in (3.9). If the labor market subsidy previously discussed is in place, then
.
52
The Basic New Keynesian Model
⇒
As long as
(8.26) , then it is impossible to attain
simultaneously zero inflation and an efficient level of activity. Thus, the disturbance
, which is
exogenous to monetary policy, generates a tradeoff for monetary authorities. Refer to push shock and assume it follows an AR(1) process:
as a cost
(8.27)
The efficient level of output, which previously was given by (4.21), now becomes:
Solve for the efficient real interest rate
(8.28)
, the rate that supports the efficient allocation and is
invariant to monetary policy:
(8.29)
Subtract (8.28) from (4.4):
(8.30)
Equations (8.26) and (8.30) constitute the New Keynesian Phillips curve and the dynamic IS equation in this setting. The next step is to obtain a welfare loss function for this case such as the one in (8.19). 8.4
A welfare loss function when real rigidities are present
At this point it is useful to distinguish between an environment with an efficient steady state and an environment with a distorted steady state. The former is the special case in which the inefficiencies associated with the flexible price equilibrium do not affect the steady state, which remains efficient. Then
by definition, and the flexible price steady state does not involve
any markup for firms. One way to obtain such a steady state is to implement the employment subsidy introduced in (7.12). The latter environment is the arguably less restrictive. Here, a steady
state distortion generates a permanent gap between the actual and efficient levels of output. Let us first look at the case where the steady state is distortive, i.e. where
. The situation with
an efficient steady state follows immediately afterwards because the distortive steady state outcome nests the efficient steady state. In the case of a steady state distortion, we introduce a
parameter
which represents the wedge between the marginal product of labor and the marginal
rate of substitution between consumption and hours, both evaluated at the steady state: (8.31)
53
The Basic New Keynesian Model
One example of such a steady state distortion is firms’ market power when the fiscal policy given
⇒ by (7.12) is absent. Comparing (7.11) to (8.31), this distortion would be given by:
Using that the production function implies
when markets clear, (8.31) can be rewritten to:
(8.32)
Even though the economy now is subject to real rigidities, the derivation given in (8.1)-(8.27)
remains unchanged because it does not involve natural or efficient output, only actual output.
⇒
The point of departure is therefore (8.7). Insert (8.32) into (8.7):
We proceed by assuming that the steady state distortion
is of the same order of magnitude as
fluctuations in the output gap and inflation. This implies that the product of order term can be ignored as negligible. Thus, we can rewrite (8.33) to:
54
(8.33)
and a second-
The Basic New Keynesian Model
⇒
(8.34)
The efficient level of of output as a function of the technology level is derived in the the same way as we did with natural natural output. Instead Instead of (8.11), (8.16)-(8.18) (8.16)-(8.18) now leads leads to:
Furthermore, Furthermore, by the definition of
and
(8.35)
:
Insert (8.35) into (8.34). Then use (8.36). Solve out:
55
(8.36)
The Basic New Keynesian Model
( ) ∑ ∑ ⇒ ∑ ∑
We then take the sum sum over all discounted discounted time periods: periods:
(8.37)
Finally we split the terms and make use of the result in (8.17), which is valid even though real
rigidities are present:
As before,
(8.38)
. The last step is to normalize (8.38) (by multiplying both sides by
) ) and to establish a quadratic welfare loss function
are relevant to monetary policy:
56
which consists only of those terms that
The Basic New Keynesian Model
⇒ ∑ ⇒ ∑ ⇒ ∑
Here,
and
as earlier and
(8.39) . The period losses write as:
Let us finally look at the case where the steady state is efficient, i.e. where
(8.40)
and
. In
this case (8.8) still holds, and therefore also (8.9). Thus, also (8.10) holds. However, the
∑ ̃
definitions of
and
now implies that:
(8.41)
Thus, (8.39) collapses collapses to:
(8.42)
The period losses losses follows directly from from (8.42):
(8.43)
Equations (8.39) and (8.40) also nest the case without real rigidities. That is, when the steady state
is efficient (
and
) ) and cost push shocks are absent (
), ), then:
It follows immediately that (8.39) and (8.40) then collapse to (8.19) and (8.20), respectively. Now
that we have derived the relevant welfare loss functions under different environment assumptions, it is time to look for the optimal monetary policy.
57
The Basic New Keynesian Model
9.
Welfare based evaluation of monetary policy
9.1 Introduction
In this section I characterize the optimal monetary policy given the second-order approximation to welfare losses derived above. In the first case. i.e. the c ase with an efficient steady state without real rigidities, it is straightforward to show that the optimal policy is to opt for zero inflation and zero output gap in all periods. This is done by responding sufficiently aggressively to any price change in order to keep zero inflation. The result should come as no surprise given the analysis conducted earlier, where it was shown that a policy that seeks to replicate the flexible price allocation is both feasible and optimal. With zero inflation output equals its natural level, which in turn, under the assumptions made, is also the efficient level. Thus, under that environment, the central bank does not face a meaningful policy tradeoff and strict inflation targeting emerges as the optimal policy. In cases with cost push shocks however, we will see that there is a short run tradeoff between zero inflation and zero output gap. In this environment the central bank should allow for only partial accommodation of inflationary pressures in order to avoid too large instability of output and employment. This kind of policy is often referred to as flexible inflation targeting. One critical point is whether the agents in the economy believe that the central bank will commit to its stated policy or whether they think the central bank will deviate in order to achieve short run gains. If agents trust the policy makers’ statements, we typically refer to the
policy as policy under commitment. If not, we typically refer to the policy as discretionary policy. This section studies the cases with an efficient steady state and a distorted steady state, both under full commitment and under full discretion. 9.2
An efficient steady state under discretion
The approximated welfare loss function in the case of an efficient steady state is derived in (8.42). Monetary authorities want to maximize (8.42) subject to (8.26) and (8.30). The problem reads as:
s.t.
(9.1)
The forward-looking nature of the constraints implies that one must specify to which extent the central bank can credibly commit in advance to future policies. With full commitment, the central bank can credibly manipulate private sector’s beliefs, and therefore commit to a policy that
58
The Basic New Keynesian Model
influences pri vate sector’s expectations about the future. Full discretion on the other hand, implies that the central bank cannot credibly manipulate private sector’s beliefs, and therefore
⇒
takes expectations as given. In the latter case, the problem becomes a period by period problem in time variables only. First I show the optimality condition under full discretion. The Lagrangian is given by:
FOC: :
(9.2)
:
(9.3)
:
(9.4)
From (9.2) we get
. From (9.4) we get
. Insert these two into (9.3):
(9.5)
Equation (9.5) states the optimal combination of output gap and inflation in a discretionary setting. In the face of inflationary pressures resulting from a cost push shock the central bank responds by driving output below its efficient level, thus creating a negative output gap, with the objective of dampening the rise in inflation. This policy goes on up to the point where (9.5) is satisfied. To derive an expression for the equilibrium inflation under discretionary policy, first insert (9.5) into (8.26):
⇒ ⇒ Iterate forward using (8.27):
59
The Basic New Keynesian Model
⇒
(9.6)
Insert (9.6) into (9.5) to get an analogous expression for the output gap:
(9.7)
Thus, under the optimal discretionary policy, the central bank lets the output gap and inflation
deviate from their targets in proportion to the current value of the cost push shock. One might think that (9.6) and (9.7) could be inserted into (8.30) directly to derive a monetary policy rule. Let us do that:
⇒⇒
(9.8)
Combine the policy rule (9.8) with (8.26) and (8.30) to get a forward looking system. Equation (9.8) into (8.30):
This expression into (8.26):
The system representation becomes:
where
60
(9.10)
The Basic New Keynesian Model
Note that
in (9.10) is identical to
in (7.18). Thus, a policy as the one in (9.8) yields a
multiplicity of solutions, only one of which corresponds to the desired outcome given by (9.6) and (9.7). However, one can always derive a rule that guarantees equilibrium uniqueness. The rule can be derived by adding a term proportional to any deviation in (9.6) into (9.8). Let us see how
⇒ ⇒ ⇒ this works:
Insert (9.11) and (8.26) into (8.30):
This expression into (8.30):
The system representation becomes:
61
(9.11)
The Basic New Keynesian Model
where
(9.12)
Note that the transition matrix
is a special case of
in (5.4) where
analysis of (5.4) we know that (9.12) has a stable, unique solution as long as
. Thus, from the . However,
following an interest rate rule like (9.11) is difficult because we do not observe the efficient output level 9.3
, at least not in real time.
An efficient steady state under commitment
Next we consider a central bank that is able to commit, with full credibility, to a policy plan. In the context of the model, such a plan consists of a specification of the desired levels of inflation
and the output gap at all possible dates and states of nature, current and future. More specifically, the monetary authority is assumed to choose a state-contingent sequence
that solves
(9.1). Let us solve the problem:24
FOC: :
(9.13)
:
(9.14)
:
(9.15)
:
(9.16)
From (9.13) we get
. From (9.15),
. From (9.16),
. From
(9.14):
24 The
law of iterated expectations is used to eliminate the conditional expectation that appeared in each constraint.
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The Basic New Keynesian Model
⇒ Combining
and
:
(9.17)
(9.18)
Equations (9.17) and (9.18) constitute the optimal combinations of output gap and inflation with commitment. It is convenient to combine those two equations into a single expression. First,
̂
define the log deviation between the price level and an implicit target given by the price prevailing one period before the central bank chooses its optimal plan, as
. Then, note that
̂ ̂ ⇒ ̂ inflation can be rewritten to:
(9.19)
This observation together with (9.17) and (9.18) yield:
(9.20)
Equation (9.20) can be viewed as a targeting rule that the central bank must follow period by period in order to implement the optimal policy under commitment. To find a solution under commitment, note that relation (9.19) also holds for forward-looking variables. Thus, the New Keynesian Phillips curve (8.26) can be rewritten, using (9.19) and (9.20), to:
̂ ̂ ̂ ̂ ̂ ̂ ̂ ⇒ ̂ ̂ ̂̂ ̂ ̂ ⇒ ̂ ̂ ̂ ⇒
We have defined
(9.21)
. Equation (9.21) is a second order difference equation.
Since it involves a forward-looking variable, the stability requirement is that one of the roots is above unity, and the other is below unity. To find the solution, divide both sides of (9.21) with and evaluate the result one period backward:
63
The Basic New Keynesian Model
̂ ̂ ̂ ̂ ̂ ̂ ⇒ √ √ ̂ ̂ ̂ ̃̃ ̂ ̃ ⇒ ̃ ̃ ⇒ ̃ ̃ ̃ ̃ ̃ ̂ ̃ ̂ ̂ ̂ ̂ ̂
(9.22)
A solution should satisfy
,
and
, where is the
root. Thus, the characteristic equation of (9.22) is:
The solutions are:
(9.23) (9.24)
One can show that
and
. To proceed we make use of the lag operator
. First, express the solution to (9.22) with lag operators:
(9.25)
Second, define
. Then shift (9.25) one period forward and solve for
:
Iterate forward:
Then insert for
(9.26)
and solve for
:
Using (9.23)-(9.24) we get:
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The Basic New Keynesian Model
̂ ̂ ̂ √ √ ̂ (√)(√ ) ̂ ⇒ ̂ ̂ ̂ ̂ ̂ ⇒
(9.27)
Finally, insert (9.27) into (9.20) to get the output gap solution:
The response of a shock at
(9.28)
is:
The key difference between cost push shock responses under discretion and under commitment, that is between the solutions (9.6)-(9.7) and (9.27)-(9.28), is that output gap and inflation are only determined by current shocks in the former, while lagged variables are relevant in the latter in addition to the shocks. Thus, under discretionary monetary policy the effect of a shock dies out once the shock is gone, while the effect will persist even in succeeding periods in the case with commitment. To see how this occurs, one can iterate (8.26) forward:
∑
(9.29)
We see from (9.29) that the central bank can offset the inflationary impact of a cost push shock by lowering the current output gap, but also by committing to lower future output gaps. If
credible, such promises will bring about a downward adjustment in the sequence of expectations for
. As a result, and in response to a positive cost push shock, the central
bank may achieve any given level of current inflation output gap
with a smaller decline in the current
. That is in the sense in which the output gap and inflation tradeoff is improved by
the possibility of commitment. Although this strategy comes at the cost of worse tradeoff in succeeding periods, it is still better from a welfare perspective because of the convexity in the loss function with respect to output gap and inflation. A feature of the economy’s response under discretionary policy is the attempt to stabilize the output gap in the medium term more than the optimal policy under commitment calls for, without internalizing the benefits in terms of short
65
The Basic New Keynesian Model
term stability that result from allowing larger deviations of the output gap at future horizons. This characteristic is often referred to as the stabilization bias associated with the discretionary policy. 9.4
A distorted steady state under discretion
Next we characterize the optimal policy when the steady state is distorted, i.e. when (8.31) is in place. In that case, (8.26) and (8.30) become:
⇒
The cost push shock term is now given by
(9.30) (9.31)
. The problem is then to optimize
(8.39) subject to (9.30) and (9.31):
s.t.
(9.32)
Under full discretion, the Lagrangian becomes:
FOC: :
(9.33)
:
(9.34)
:
(9.35)
From (9.33) we get
. From (9.35) we get
. Insert these two into (9.34):
(9.36)
Equation (9.36) states the optimal combination of output gap and inflation in this setting. In the face of inflationary pressures resulting from a cost push shock the central bank responds by driving output below its efficient level, thus creating a negative output gap, with the objective of dampening the rise in inflation. This policy goes on up to the point where (9.36) is satisfied. Note that (9.36) is similar to (9.5) except for a positive constant term. Thus, for any given level of inflation, the policy is more expansionary than that given in the absence of a steady state
66
The Basic New Keynesian Model
distortion. To derive an expression for the equilibrium inflation under discretionary policy, first insert (9.36) into (9.31):
⇒ ⇒ ⇒ ⇒ Iterate forward:
(9.37)
Insert (9.37) into (9.36) to get an analogous expression for the output gap:
(9.38)
67
The Basic New Keynesian Model
Thus, under the optimal discretionary policy, the presence of a steady state distortion does not affect the response of the output gap and inflation to shocks. However, the steady state distortion has an effect on the average levels of inflation and the output gap in which the economy fluctuates. In particular, when the natural level of output and employment are inefficiently low, i.e. when
, the optimal discretionary policy leads to positive average
inflation as a consequence of the central bank’s incentive to push output above its natural steady
state level. That incentive increases with the inefficiency of the natural steady state, which explains the fact that the average inflation is increasing in phenomenon. 9.5
A distorted steady state under commitment
TBA
68
, giving rise to the inflation bias
The Basic New Keynesian Model
10.
Wage rigidities
10.1 Introduction
This section expands the baseline model with monopoly supply power and sticky wages in the labor market. A continuum of differentiated labor services is assumed, all of which are used by each firm. Each household specialize in one type of labor, which it supplies monopolistically. Wages are sticky in an analogous way to goods prices. Each period only a constant fraction of households can adjust their posted nominal wage. As a result, the aggregate nominal wage responds sluggishly to shocks, generating inefficient variations in the wage markup. 10.2 Firms
We saw earlier that households face two maximization problems. First, for any given level of consumption expenditures, they have to find the consumption vector that maximizes total consumption. Second, and given this optimal consumption vector, they have to find the optimal combination of consumption and labor. Now that households have monopoly power in the labor market, firms face a two-dimensional problem as well. First, for any given level of labor costs, they have to find the output maximizing combination of labor. Second, and given this optimal
labor vector, firms have to set prices such that profit is maximized. Firms’ output is still given by (3.1). However, total labor
used by firm , where
employed, is now defined by:
∫
is the quantity of type -labor
The elasticity of substitution among labor types is denoted labor is an imperfect substitute as long as worker types implies an indexation
∫ ∫
(10.1) . Specification (10.1) implies that
. Note that the assumption of individual
. Let
denote the nominal wage for type -worker
in period . In an analogous way to the optimal consumption vector derived in (2.10), firm ’s demand for labor is given by:
This holds for all
(10.2) , where: (10.3)
In exactly same manner as we derived the aggregation result for consumption expenditures, one can derive the following aggregation re sult for firms’ labor expenditures: (10.4)
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The Basic New Keynesian Model
∑ | Denote
as the price stickiness parameter in the goods market and
as the elasticity of
substitution between consumption goods. Then, the profit maximization problem given in (3.6) writes as:
(10.5)
As shown in (4.15), the aggregation of the resulting price setting rules yields, to a first order approximation and in a neighborhood of the zero inflation steady state, the following equation
̂ ̂ ()() for price inflation
:
The notations are
and
(10.6) . From (10.6) we see
that when the average price markup is below its steady state value, firms that are adjusting prices set them higher, thus generating positive inflation. 10.3 Households
Assume a continuum of households indexed by
∑ ( ) ∫
. As before, given a sequence of budget
constraints household seeks to maximize lifetime utility:25
(10.7)
The consumption index is now given by: (10.8)
We now assume that households specialize in different types of labor, and therefore face some monopoly power in the labor market given by
. They post the nominal wage at which
they are willing to supply labor services to firms who demand them, and, because of monopoly
power, this wage contains a markup. However, households also face a constraint with respect to the frequency in which they can change wages. A constant fraction with the wage they had last period, while the remaining
of the households is stuck
households can reoptimize the
price of their labor services. Thus, a household who reset its wage in period will choose order to maximize:
∑(| |) 25 We
in
(10.9)
now depart from the money in utility specification we had in (2.1) to keep the analysis as simple as possible.
70
The Basic New Keynesian Model
| | and
denote consumption and labor at time
for a household that last set the
wage at time . Because of the wage rigidity, (10.9) can be interpreted as the expected discounted sum of utilities generated over the uncertain period during which the wage remains unchanged at the level
set the current period. Note that the utility generated under any other wage set in
the future is irrelevant from the point of view of the optimal setting of the current wage, and thus can be ignored in (10.9). Maximization of (10.9) is subject to a sequence of labor demand
| | || ∫ | | ∫⇒ ||| | ∫ | | | | ( ) schedules and flow budget constraints that are effective while
remains in place:
(10.10)
and
(10.11)
Equation (10.10) follows directly from (10.2), and
is the amount of labor done in time
for a household that last set its price in period .
employment at time
denotes the aggregate
. The integrals in (10.11) are removed by (2.9) and (10.4), respectively.
is the market value in period
of the portfolio of securities held at the beginning of
that period by households that last reoptimized in period . Thus, corresponding market value as of period yields a random payoff
is the
of the portfolio purchased in that period, which
. Given the concavity of the utility function (10.11) holds with
equality. Furthermore, it can be solved for
:
| | | |
Insert (10.12), and then (10.10) into (10.9) to get an unconstrained problem:
71
(10.12)
The Basic New Keynesian Model
(| |) (| | |) | | | |
Thus, the household’s maximization problem when it comes to wage setting is described as:
Let us solve the problem. FOC:
(| |) (| |) (| |) (| |) (| |) | (| |) | Multiply both sides by
:
72
(10.13)
The Basic New Keynesian Model
∑ (|( | )) | (| |)| | (||||) ∑ |(| |) | | (| |) (| |)| (| |) (| |)| ∑ ⇒ ∑ ((||||))| ((||||)) | | | Define
as the marginal rate of substitution between consumption
and hours in period
for a household resetting the wage in period . Then, the equation
above can be rewritten to:
(10.14)
To solve for
, first move the terms with
over to the other side and insert for
(10.10):
Then take out the optimal wage of the expectation terms on both sides and solve for
:
Notice that in the case with flexible wages, i.e. when
(10.15)
, (10.13) becomes a one period
problem and (10.15) becomes:
Thus,
(10.16)
is the wedge between the real wage and the marginal rate of substitution that prevails
in the absence of wage rigidities, i.e. the desired gross wage markup. Note also that in a perfect foresight zero inflation steady state we have:
(10.17)
In a similar manner as we log-linearized the optimal price equation (3.7) around steady state, we now log-linearize the optimal wage equation (10.15). First, it is convenient to rewrite it to:
73
The Basic New Keynesian Model
(|| ||) (|| ||)|| A first-order Taylor expansion of the LHS of (10.18):
(|| ) (|| ) Rewritten in terms of log deviations:
74
(10.18)
The Basic New Keynesian Model
(|| ) (|| ) (|| ) (|| ) (|| ) (|| )
The last equality follows follows from (10.17). (10.17). A first-order first-order Taylor expansion of the RHS of of (10.18):
75
The Basic New Keynesian Model
( (|| ) (|| ) ( (|| ) (|| ) (|| )(|| ) (|| ) (|| ) (|| ) (|| )(|| ) ⇒⇒ ∑∑ ∑ ∑ (||( |) | ) ⇒ ∑ ( || ) ⇒ ∑ (|| ) Finally we equate LHS with RHS and solve for
:
76
The Basic New Keynesian Model
( | ) ⇒ ∑
We have defined
straight forward. First,
(10.19)
. The intuition behind the wage setting rule is
is increasing in expected future prices because households care about
the purchasing power of their nominal wage. Second,
is increasing in the expected average
marginal disutilities of labor in terms of goods over the life of the wage, because households want to adjust their expected average real wage accordingly, given expected future prices. Without money in the utility equation (2.32) becomes:
|(| |) | | (||) ⇒ | | | ⇒⇒ || | (| ) ⇒⇒ ∑ ∑ ⇒ ∑
(10.20)
The assumed separability between consumption and hours, combined with the assumption of complete asset markets, implies that consumption is independent of the wage history of a household, i.e.
. Thus, from (10.20) we get:
Also, from (10.10) we get:
Let
def ine the economy’s average marginal rate of substitution, so
that:
(10.21)
Insert (10.21) into (10.19) and solve for
, using that
average wage markup:
77
is the economy’s
The Basic New Keynesian Model
⇒ ∑ ̂ ⇒ ∑ ̂ ⇒ ̂ ⇒ ⇒⇒ ̂ ̂ ⇒ ̂ ⇒ ̂ ⇒ ̂ ̂ ⇒
(10.22)
We have denoted
as the deviation of the economy’s log average markup from its
steady state level. The next step is to derive the wage inflation equation. Given the wage setting structure, the evolution of the aggregate wage index is given by:
(10.23)
A first order Taylor approximation around zero wage inflation steady state, followed by logging the resulting equation, yields:
(10.24)
From (10.24) we see that wage inflation
is given by:
(10.25)
Finally, insert (10.22) into (10.24) and use (10.25):
(10.26)
78
The Basic New Keynesian Model
We have defined
. Notice that the wage inflation equation (10.26) has a form
analogous to the gods price inflation equation (10.6). The intuition is also the same. When the average wage in the economy is below the level consistent with maintaining (on average) the desired markup, households readjusting their nominal wage will tend to increase the latter, thus generating positive wage inflation. Here, (10.26) replaces condition
, one of the
optimality conditions associated with the household’s problem used earlier. The imperfect
adjustment of nominal wages will generally drive a wedge between the real wage and the marginal rate of substitution for each household, and as a result, between the average real wage and the average marginal rate of substitution. This leads to variation in the average wage markup, and
|
given (10.26), also to variation in wage inflation. The last dimension worth considering with respect to households is the Euler equation. It becomes the similar as before because
(|) (|) ( )
, but because hours worked now depends on when the wage last was set, ( 2.18) writes as:
However, given the specification (10.20), log-linearization yields: (10.27)
This expression is the exact same as before, i.e. identical to (2.36). Thus, the Euler equation does not depend on wage rigidities in this setting. 10.4 Inflation equations and the Dynamic IS equation
̃
I will now derive the price inflation equation, the wage inflation equation and the output gap equation. Start with wage inflation. Let
be the output gap from the level of natural
output, with the latter now being defined as the equilibrium level of output in absence of both price and wage rigidities. Define also the real wage as
wage gap is:
The natural real wage, denoted
rigidities. To get an expression for
, which implies that the real
, is the real wage that would prevail in the absence of nominal , note first that the production function (3.1) implies that
⇒
output, employment and marginal productivity in their linearized versions are given by:
(10.28)
(10.29) (10.30)
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The Basic New Keynesian Model
– – ) ( ⇒ – ⇒ – – ̂ ⇒ ̂ ̃ ⇒ ̃ ̃ ̃ Use the fact that the natural real marginal cost is given by the identity insert (10.28)-(10.30). Solve for
and
:
(10.31)
We have defined
and
. Note also that the change in the
natural real wage is given by:
(10.32)
Equations (10.28)-(10.30) will also be used to derive an expression for the real marginal cost gap from its steady state counterpart, where
. Insert for
(10.28)-(10.30):
(10.33)
Finally, insert (10.33) into (10.6) to get the New Keynesian Phillips curve:
(10.34)
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The Basic New Keynesian Model
Thus, increased output and increased real wage lead to higher price inflation. Next, the wage inflation equation is derived. Given the utility function (10.20), marginal rate of substitution in its
̂ ⇒ ̂ ̃ ⇒ ̃ ̃ ̃ ⇒ linearized form is given by:
(10.35)
Thus, given market clearing in the goods market and in the labor market, the log deviation in the economy’s average wage markup from its steady state counterpart, where the wage markup writes
as
, is given by:
(10.36)
Insert (10.36) into (10.26) to get the New Keynesian Wage Phillips curve:
(10.37)
With wage rigidity in addition to price rigidity, there is an identity relating changes in the real wage gap to wage inflation, price inflation and changes in the natural real wage:
(10.38)
The last term in (10.38) is given by (10.32). In order to complete the non-policy block of the model, equilibrium conditions (10.34), (10.37) and (10.38) must be supplemented with a dynamic IS equation. Given (10.27) and market clearing in the goods market, aggregate output is given by:
( ) ̃ ( ) ⇒ ̃ ̃ ( )
(10.39)
As before we subtract (4.21) and get:
The natural real rate
(10.40)
and its deviation from the steady state are given as in (4.24) and (4.25):
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The Basic New Keynesian Model
̂
(10.41) (10.42)
10.5 System representation and equilibrium determinacy
In order to close the model, we must specify how the nominal interest rate is determined. Let us
̃ ̃ ̃ ( ̃ ) ̃ ̃ ( ) ̂ ̃ ̃ ( ) ̂ ⇒⇒ ( )̃ ̃ ̂ ̃ ̃ ̃ ̃ ̂ [ ] [ ] postulate an interest rate rule of the form:
(10.43)
The five equations (10.34), (10.37), (10.38), (10.40) and (10.43) constitute the New Keynesian model with sticky prices and sticky wages. To gather them all into a forward looking system with period variables on the LHS and period
variables on the RHS,26 we first insert (10.43)
into (10.40):
(10.44)
Equation (10.44) is the first row in the system. Second, we rewrite (10.34):
(10.45)
Equation (10.45) is the second row in the system. Third, we rewrite (10.37):
(10.46)
Equation (10.46) is the third row in the system. Fourth, we rewrite (10.38):
(10.47)
Equation (10.47) is the fourth and last row in the system. Thus, (10.44)-(10.47) can be written as:
or
where
26 Note
(10.48)
that the structure of (10.38) is backward looking, so that the element associated with (10.38) in the LHS vector of the system will consist of and the RHS element of .
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The Basic New Keynesian Model
̃ ̂ ̃ ̂ ,
,
,
and
In general, (10.48) does not have a solution satisfying
, not even under
the assumption that the intercept of the interest rate rule adjusts one-for-one to variations in the natural real interest rate, i.e. that
. An implication of that is that the allocation associated
with the equilibrium with flexible prices and wages cannot be attained in the presence of nominal rigidities in both goods and labor markets. The intuition for the previous result rests on the idea that in order for the constraints on price and wage setting not to be binding all firms and workers should view their current prices and wages as the desired ones. This makes adjustment unnecessary and leads to constant aggregate price and wage levels, i.e. zero inflation in both markets. Note, however, that such an outcome implies a constant real wage, which will generally be inconsistent with the flexible price and wage level allocation. Only when the natural real wage is constant, i.e. when
, which according to (10.32) requires complete absence of
̂
̃
technology shocks, and at the same time the central bank adjusts the nominal rate one-for-one with changes in the natural rate, i.e.
, the outcome
is a solution
to (10.48). Another question of interest relates to the conditions that the rule (10.43) must satisfy to guarantee a unique stationary equilibrium or, equivalently, a unique stationary solution to the system of difference equations (10.48). Given that vector
contains three non-predetermined
variables and one predetermined variable, local uniqueness requires that three eigenvalues of lie inside, and one outside, the unit circle. If
uniqueness can be shown, using numerical analysis, to be:
, the condition for
(10.49)
Thus, the central bank must adjust the nominal rate more than one-for-one in response to variations in any arbitrary weighted average of price and wage inflation. This can be seen as an extension of the Taylor principle to the case with sticky wages. Furthermore, the region consistent with a determinate equilibrium in the increases from zero.
( )
parameter space becomes larger as
10.6 Shocks
I will now consider a monetary policy shock within the framework of both sticky prices and sticky wages. Parameters are calibrated as follows in the simulation exercise:
83
The Basic New Keynesian Model
Table 3
0.99
1
1
1/3
6
6
2/3
3/4
= 3/4 implies an average duration of wage spells of
empirical evidence. Policy parameters
and
1.5
0
0
0.5
0.0625
= 4 quarters, consistent with
are set to zero as the Taylor rule (10.49) is
satisfied anyway. Simulated impulse responses are shown in the figure 2. 27 Notice that the presence of sticky wages and prices generates a much smaller inflation decline than with only sticky prices (figure 1). The reason is that when wages are flexible, a monetary policy shock leads to a large decline in wage inflation. Here, instead, since also wages are sticky, the inflationary contraction is divided between prices and wages. As a result, real wage does not change much. This in turn reduces the impact of decline in activity on the real marginal cost and, hence, the limited size on inflation response. Thus, there is only a moderate endogenous response of monetary authority to the lower inflation, implying higher interest rates, which in turn account for the larger decline in output compared to figure 1. Figure 3: A monetary policy shock
27 Simulations
are done with Dynare and Matlab. See Appendix C for the Dynare codes.
84
The Basic New Keynesian Model
Overall, the limited responses of price and wage inflation following a monetary policy shock seem more in line with data than the large responses we found in figure 1. Also, the effect on the real wage in figure 3 seems much more plausible. 10.7 Monetary policy design with sticky wages The benevolent social planner seeks to maximize households’ utility, but this maximization is
∑ ∑ ∫ ∫
subject to (10.8), (10.1) and (3.1). Thus, the maximization problem reads as:
s.t.
(10.50)
The constraint is the resource constraint coming from all the firms, as in the case with flexible wages. As before both consumption goods and labor enter the utility function symmetrically.
Thus:
(10.51) (10.52)
The problem therefore simplifies to the one we had before, and the efficient solution becomes: (10.53)
Thus, (10.51)-(10.53) are the relevant efficient benchmark equations monetary authorities should opt for. However, optimal wage and price setting (when monopolistic competition is present in both markets, but price and wage rigidities are absent) follows from (10.16) and (7.10), and implies that:
(10.54) (10.55)
As earlier one can get rid of the distortions caused by market power, and obtain the outcome in (10.53), by imposing a labor subsidy financed by lump-sum taxes. In this case, the appropriate tax
is given by
, which changes (10.55) to:
85
The Basic New Keynesian Model
⇒
(10.56)
Combining (10.56) and (10.54), we see that (10.53) is obtained, thus guaranteeing the efficiency of the flexible price and wage equilibrium. Next I derive a second-order approximation to the average welfare losses experienced by households in the economy with sticky wages and prices. Point of departure is (8.1) integrated across households:
⇒ ∫ ∫ ∫ ∫ ( ) ( ) ( ) ⇒ ∫ ∫ ⇒ ∫ ∫ ∫ ∫ ∫ ⇒ ∫ ∫
(10.57)
Next, define aggregate employment as:
First, note that in terms of log deviations from a steady state and up to a second-order approximation:
(10.58)
Insert (10.58) into (10.57):
(10.59)
In order to proceed, a couple of results are needed. First, a first-order approximation of (10.2) gives:
( ) ⇒⇒ ) ∫( ) ∫⇒ ∫( ∫ ∫ ∫ Second, and using the result above:
86
(10.60)
The Basic New Keynesian Model
∫ ∫ ∫ ∫ () ( ) ( ) ⇒ ∫ ∫ ∫ ∫ ∫ ∫ ( ) ⇒ ∫
Third, we have to take care of the term
somehow. From (10.3), that is from
, we have that
the notation
, a second-order approximation with respect to
Fourth, taking expectations on both sides of the above, where
. With
yields:
denotes the expectations
operator with respect to labor for firm , we get:
From (10.61) it is clear that
(10.61)
is of second order. Thus, to a first order,
(10.60) can be written:
(10.62)
Finally, insert (10.62) into (10.59):
(10.63)
The next step is to derive a relationship between aggregate employment and output. Using (10.2), then (3.1), and then (2.10):
87
The Basic New Keynesian Model
⇒ ∫ ∫
We have defined
and
(10.64)
. Taking the log of (10.64), and
then a second-order approximation to the relationship between log aggregate output and log aggregate employment, we get an expression similar to (8.2):
⇒⇒ ∫ ∫ () ( ) ( )
(10.65)
where
(10.66)
(10.67)
We derived in (8.3)-(8.6) that (10.67) can be written as:
Now we do a similar derivation for
. Let us do a second-order approximation of
:
Insert this result into (10.66) and use (10.61):
88
in
The Basic New Keynesian Model
⇒ ( ) ⇒
(10.68)
Finally, inserting for
and
into (10.65):
Now we are ready to update the welfare loss function (10.63). Inserting for (10.69) gives:
By introducing the parameter
we get the following:
89
(10.69)
The Basic New Keynesian Model
(10.70)
Throughout,
stands for terms independent of policy. As earlier we take a general approach
and allow for a distorted steady state. With this framework we can insert for (8.32) into (10.70):
Using the assumption about a small steady state distortion, implying that we can neglect second order moments containing
, the derivation becomes similar as the one in (8.34):
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The Basic New Keynesian Model
Note that the expression above is the same as (8.34), except for the additional term with wage dispersion. Thus, using (8.35) and (8.36), and aggregating over time using the discounting factor
⇒ ∑ ∑ ) ∑ ∑ ()( ∑ ∑ , we get an expression analogous to (8.37):
(10.71)
As before,
is the log difference between the
output gap from efficient output and its steady state counterpart. From (8.17) we know that:
The equivalent derivation to (8.14)-(8.17) for wages gives us:
(10.72)
Using (8.17) and (10.72), (10.71) can be rewritten to:
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The Basic New Keynesian Model
) ()( ⇒ ∑ ∑
(10.73)
Finally, the welfare loss function follows from (10.73) as:
∑ ∑ ̃ ̃ ∑ ̃ ̃
(10.74)
The corresponding period utility losses becomes:
Note that in the particular case of an efficient steady state,
and
(10.75)
. Then the welfare
loss function and the period losses writes as:
Moreover, if the subsidy suggested in (10.56) is in place, then
and
(10.76) (10.77) where
. In that case, we get:
92
(10.78) (10.79)
The Basic New Keynesian Model
11.
A small, open economy model
11.1 Introduction
Existence of more than one economy makes the basic New Keynesian model more complex because of a significant amount of new notation. One must take into account how foreign economies affect domestic behavior. In particular, the modeler has to decide whether the economy is large or small, the nature of international asset markets, whether they are autarkies or complete markets, the existence of discrimination between domestic and foreign markets, tradable versus non-tradable goods, trading costs, international policy coordination, and exchange rate regimes. Here we model small economies, complete asset markets, existence of discrimination between domestic and foreign goods, even though all goods can be traded internationally, and a world economy without international policy coordination. The world economy consists of a continuum of small open economies represented by the unit interval. Since each economy is of measure zero, its performance does not have any impact on the rest of the world. Different economies are subject to imperfectly correlated productivity shocks, but it is assumed that they share identical preferences, technology and market structure. Firms are identical across countries and have the simplest Cobb-Douglas production function with constant returns to scale. 11.2 Households
A typical small open economy household is inhabited by a representative household who seeks to
∑ maximize:
The variable
(11.1)
is a composite consumption index determined by both home and foreign goods.
Because of the complexity of the model it is convenient to set up a graphical representation of the goods structure: Figure 4: The consumption goods structure in an open economy Continuum of domestic goods
Home goods: Parameters: , Aggregate consumption:
Continuum of foreign goods
Continuum of foreign countries
Imported goods from country : Parameter:
Imported goods: Parameters: ,
93
Domestic good : Parameter:
Foreign good country : Parameter:
from
The Basic New Keynesian Model
Figure 4 illustrates the goods structure which we shall specify below. The composite consumption index is defined by: 28
The parameter
measures the degree of openness in the economy. Equivalently,
measures the home bias. The closer
is to one, the more open is the economy. If
(11.2)
we
get the closed economy case described earlier. Trade restrictions imposed by governments,
geographical barriers such as distance and mountainous terrain, etc, is assumed to be reflected in . The substitutability between domestic and foreign goods from the viewpoint of domestic
consumers is denoted
.
is an index of consumption of domestic goods, given by the CES-
function (constant elasticity of substitution):
∫
The parameter
(11.3)
is interpreted as before, that is as the elasticity of substitution between varieties
produced within any given country , including the home country. goods given by the CES-function:
is an index of imported
∫ ∫ ∫ ∫ ∫ Here,
denotes the elasticity of substitution between importing countries. Finally,
index of the different goods imported from country , given by the CES-function:
(11.4) is an
(11.5)
Note that the nested system of equations (11.1)-(11.5) characterizes preferences of a representative household. Maximization of (11.1) is subject to a sequence of budget constraints:
The domestic price on good is denoted is denoted period . 28 In
.
(11.6)
while the price on good imported from country
is the nominal payoff in period
from a portfolio held at the end of
is the stochastic discount factor for one-period ahead nominal payoffs of the
the particular case of
, the consumption index is given by
94
.
The Basic New Keynesian Model
domestic household. As before the optimization problem can be dealt with in several stages. First, for any given level of consumption expenditures on home goods, the household must decide how much to buy of each. The utility maximizing combination of which determines all the elements in
is the solution,
. An equivalent decision has to be made about imported
goods from each of the foreign countries. For instance, for any given level of consumption
expenditures on imported goods from country , the household must decide how much to
consume of each import good from that country. This determines the optimal combination of , i.e. all the elements in
. Second, for any given level of consumption expenditures on
imported goods, the household must decide how much to import from each foreign country. This decision determines the optimal combination of
, i.e. all the elements in
. Third,
for any given level of total consumption expenditures, the household must decide how much to consume of home goods relative to imported goods. This decision determines
and
.
Finally, the household must decide how much to consume and how much to work. This decision
∫ ∫ ∫ determines
. As before, we get an aggregate price index and optimal demand for every specific
consumption unit at each stage in the nested system. For home goods, the aggregate price index is given by:
(11.7)
The price index (11.7) follows from the CES-aggregator (11.3) in exactly the same way as (2.7)
follows from (2.3). The optimal demand for home good is:
(11.8)
In a similar vein, the aggregate price index for imported goods from country is given by:
(11.9)
The optimal consumption of good imported from country is given by: (11.10)
The aggregate price index for all imported goods is given by:
(11.11)
Optimal basket of import consumption from country is:
Finally, the aggregate consumption price index (CPI) in the home country is given by:
95
(11.12)
The Basic New Keynesian Model
(11.13)
Optimal consumption of home goods is: (11.14)
Optimal consumption of imported goods is: 29 (11.15)
Given the market equilibrium for all these aggregators, the total consumption expenditure is derived in the same way as we derived (2.9) from (2.8) and (2.3). From the optimality condition (11.8) and the domestic price index (11.7):
∫ ∫ ∫ ⇒
(11.16)
From the optimality condition (11.10) and the import price index from country (11.9):
(11.17)
From the optimality condition (11.12) and the aggregate import price index (11.11):
(11.18)
Finally, from the optimality conditions (11.14) and (11.15), and from the CPI index (11.13):
(11.19)
Thus, the left hand side of the period budget constraint (11.6) can be rewritten as:
(11.20)
Given the optimality conditions (11.8), (11.10), (11.12), (11.14) and (11.15) 30, the household must decide on the allocation of total consumption and labor. Analytically the problem is to maximize (11.1) subject to (11.20). As before we specify the utility function to be:
(11.21)
To derive the Euler equation, note first that the budget constraint can be rewritten, assuming that it holds with equality, to: 31
29 In
the particular case of , the CPI takes the form . 30 That is the optimal demand of good from domestic production and from country production, the optimal demand of goods from country , the optimal demand of home goods and imported goods, respectively. 31 This setup closely follows Cochrane (2005:4-5).
96
The Basic New Keynesian Model
Here, the left hand side represents consumption expenditures in period , the right hand side represents available gross income in period , while
the time investment in a portfolio with nominal payoff
in period
on
represents
. Thus, the
constraint above tells us that whatever income is left after the portfolio investment, is used for consumption. The intertemporal problem for the household with respect to the optimal one-
(∫ ∫ ) ∫ ⇒ ⇒ ⇒ period portfolio purchase writes as:
subject to
Here,
(11.22)
is the market price of the one-period portfolio yielding a random payoff
, where we integrate over all possible states of nature indexed by .
is the period
price of an Arrow security, i.e. a one-period security that yields one unit of domestic currency if a specific state of nature is realized in period
, and zero otherwise.
that a given state of nature is realized in period
is the probability
. Equivalently, the price can be written as
. Thus, the stochastic discount factor can be defined as
To solve (11.22), insert the constraints into the maximum. Then take the first order conditions and find the optimal intertemporal allocation:
FOC:
Taking conditional expectations on both sides, the Euler equation becomes the same as in (2.18):
(11.23)
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The Basic New Keynesian Model
The first order condition which summarizes the labor-leisure choice becomes identical to (2.19):
(11.24)
As before we can log-linearize equations (11.23) and (11.24) and get: (11.25) (11.26)
11.3 Terms of trade, domestic inflation and CPI inflation
Bilateral terms of trade between the domestic economy and country is defined as the price of
country ’s goods in terms of home goods:
∫ ( )( ) ⇒ ∫ ∫ ⇒⇒
(11.27)
The effective terms of trade are thus given by:
(11.28)
A first order approximation around a symmetric steady state satisfying
gives us:
(11.29)
Similarly, log-linearization of the CPI (11.13) around the same symmetric steady state where :
(11.30)
Domestic inflation is given by:
(11.31)
Thus, using (11.30) CPI inflation is given by:
(11.32)
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The Basic New Keynesian Model
We see from (11.32) that the gap between domestic inflation and CPI inflation is proportional to the percentage change in terms of trade, with the coefficient of proportionality given by the openness index
.
11.4 The real exchange rate
The next step is to look at exchange rates. Define
as the bilateral nominal exchange rate, i.e.
the price of country ’s currency in terms of domestic currency. Thus,
measures how many
domestic currency units one country currency unit is worth. As an example, the bilateral
( ) ⇒ ∫ nominal exchange rate between Norway and US could be
. Define
as the
price of country ’s good in terms of its own currency, for example the price of an iPhone ( ) in terms of US dollars ( ). Assume that the law of one price holds for individual goods at all times for both import and export prices. 32 Thus, for all goods
in every country
:
(11.33)
Suppose the price of an iPhone ( ) in the US ( ) in terms of US currency is
.
Then, the law of one price implies that the Norwegian price on iPhone in terms of Norwegian currency is
. Aggregation across all goods using
(11.9) gives:
(11.34)
Here,
is defined as the aggregate price level in country in terms of
country currency, i.e. country ’s domestic price index . Thus, (11.34) is the law of one price at the country level where to
represents the domestically produced goods in country , in contrast
which represents all goods in country . Insert (11.34) into (11.11):
32 This
law loosely states that the relative price on a good is equal to the nominal exchange rate, i.e. that
For example, if an Iphone costs twice as many for the law of one price to hold.
in Norway as
99
.
in US, the nominal exchange rate must be 2
The Basic New Keynesian Model
∫ ∫ () ∫( ) ∫ ∫ ∫ ∫ ∫
Then, log-linearize around a symmetric steady state:
(11.35)
The (log) domestic price index for country expressed in terms of it own currency is denoted , the effective nominal exchange rate is denoted
world price index is denoted
, and the (log)
. Notice that for the world as a whole, there is no
distinction between CPI and domestic price level, nor between their corresponding inflation rates. Insert (11.35) into (11.29):
(11.36)
Equation (11.36) expresses the terms of trade as a linear function of the effective nominal exchange rate, the world price and the price on domestically produced goods. Next, define the
bilateral exchange rate between the home country and country , i.e. the ratio of the two countries CPI’s, both expressed in terms of domestic currency, as:
(11.37)
In logs:
∫ ( ) ⇒
(11.38)
Then, let the (log) effective real exchange rate be:
(11.39)
Insert (11.38) into (11.39), using (11.36) and (11.30):
(11.40)
Notice that the last equality holds only up to a first order approximation when
.
11.5 International risk sharing
Under the assumption of complete markets for securities traded internationally, a condition analogous to (11.23) must also hold for the representative household in any other country, say
country :
100
(11.41)
The Basic New Keynesian Model
⇒ ⇒ ⇒ ∫ Divide (11.23) by (11.41) and solve for
:
(11.42)
is some constant that will generally depend on initial conditions regarding
relative net asset positions. Without loss of generality, assume symmetric initial conditions, i.e. zero net foreign asset holdings and an ex-ante identical environment. This implies . Taking logs of both sides of (11.42):
(11.43)
Equation (11.43) is at the household level. Note that world consumption is given by
Integrating (11.43) over all and using (11.39) and (11.40) yields:
(11.44)
Thus, the assumption of complete markets at the international level leads to a simple relationship linking domestic consumption with world consumption and the terms of trade, where relative home consumption to world consumption is given by
.
11.6 Uncovered interest rate parity
Allow households to invest both in domestic and foreign bonds; constraint may be written as:
101
and
. The budget
The Basic New Keynesian Model
⇒ ⇒
(11.45)
The optimality conditions with respect to these assets are:
(11.46) (11.47)
Divide (11.46) by (11.47) to obtain:
(11.48)
Log-linearizing (11.48) gives:
(11.49)
This is the familiar uncovered interest rate parity equation, which states that the nominal interest rate at home is equal to the world nominal interest rate plus expected rate of depreciation of the
⇒ ⇒ home currency. Now, from (11.36) we have that:
Thus, using (11.49) we get the following stochastic difference equation:
(11.50)
Given that the terms of trade are pinned down uniquely in the perfect foresight steady state, and given the assumptions of stationarity in the models driving forces and unit relative prices in
⇒ ∑ steady state, it follows that
Hence, (11.50) can be solved forward to obtain:
(11.51)
Equation (11.51) expresses the terms of trade as the expected sum of real interest rate differentials between the world market and the home market. 11.7 Firms and technologies
Now we turn to the supply side of the economy. Because domestic firms take the business environment as given, including state of affairs in foreign economies, the individual firm still only
102
The Basic New Keynesian Model
takes into account its own marginal cost. Assume that a typical firm in the home economy produces a differentiated good with linear technology represented by the production function:
(11.52)
Assume that an employment subsidy identical to the one in (7.12) is in place. From (4.11), using CRS, we get
and
. Thus, (7.12) now becomes:
The employment subsidy is captured in the term
(11.53)
. The firms optimal price
setting behavior is identical to the one described in the closed economy case. As in (3.11) the optimal price is: 33
̅ ∑ | ∫
(11.54)
The log of the gross markup, or equivalently, the equilibrium markup in the flexible price economy, is denoted
.
11.8 Equilibrium – Aggregate demand and output
Market clearing for good in the home economy implies:
The supply of domestically produced good is denoted
(11.55)
, the domestic demand is denoted
, and country ’s demand for good produced in the home economy is denoted
. Due to
the nested structure one can express demand in sub-markets in terms of total demand by
combining all demand functions from each level. For instance, insert (11.14) into (11.8) and get: (11.56)
Furthermore, the demand for domestically produced good in country is expressed by nesting up across different demand layers in country . First, note that the consumption of domestically produced good in country is a function of country ’s consumption of goods produced in the home economy, given as in (11.8):
Second, note that country ’s consumption of goods produced in the home economy is a function of country ’s consumption of foreign goods, given as in ( 11.12):
33 Now
̅
denotes the optimal price, instead of
, which denotes the world price.
103
The Basic New Keynesian Model
Third, note that consumption of imported goods in country is a function of total consumption in that country, given as in (11.15):
Combining all these yields the demand for domestically produced good in country as a function of total consumption in that country:
∫ ⇒ ∫
(11.57)
Thus, we can insert (11.56) and (11.57) into (11.55) and get:
(11.58)
To aggregate, start with the definition of aggregate domestic output:
∫
(11.59)
Insert (11.58) into (11.59) and solve out:
104
The Basic New Keynesian Model
} [ ] 105
The Basic New Keynesian Model
⇒ ∫ ⇒ ∫
(11.60)
Next, factorize out the elements in the integral and insert for (11.37):
(11.61)
Define the effective terms of trade for country as:
(11.62)
Use this, and also insert for the bilateral terms of trade between the domestic economy and
⇒ ∫ ( ) ∫ ( ) ⇒ country from (11.27), and for
from (11.42):
(11.63)
Log-linearization of (11.63) around a symmetric steady state yields the following:
106
(11.64)
The Basic New Keynesian Model
⇒ ∫ ∫ ∫ ∫ ∫ ∫ ⇒ ⇒ Insert for (11.40):
(11.65)
Note that
is reasonable because
. A condition
analogous to (11.65) will hold for all countries. Thus, for a generic country it can be rewritten as . By aggregating over all countries, a world market clearing condition can be
derived as:
(11.66)
This result follows from the fact that
. Inserting (11.44) and (11.66) into (11.65)
yields:
(11.67)
Note that
. Finally, insert for
from (11.65) into the Euler equation (11.26)
to get the IS equation:
(11.68)
Thus, the IS equation is similar to the one in a closed economy except that now there is an additional term linking domestic output to the international environment. An alternative
⇒
representation including domestic goods inflation is found by inserting (11.32) into (11.68):
Note that
107
(11.69)
The Basic New Keynesian Model
⇒ ⇒ ⇒ ⇒ if
and
are sufficiently high. Yet another representation is found by inserting for
from
(11.67):
Use that
and
:
(11.70)
The last term, openness
, is exogenous to domestic allocations. Note that in general, the degree of
influences the sensitivity of output to any given change in the domestic real rate . Also note from (11.69) that if
, i.e. if
and
are sufficiently high, we have that
, and output is more
sensitive to real rate changes than in the closed economy case. The reason is the direct negative
108
The Basic New Keynesian Model
effect of an increase in the real rate on aggregate demand and output is amplified by the induced real appreciation and the consequent switch of expenditure toward foreign goods. This will be partly offset by any increase in CPI inflation relative to domestic inflation induced by the expected real depreciation, which would dampen the change in the consumption based real rate,
, which is the one ultimately relevant for aggregate demand, relative to
.
11.9 Equilibrium – The trade balance
Next, we can define net exports
as the difference between total domestic production and
total domestic consumption, relative to steady state output :
(11.71)
A first-order approximation around a symmetric steady state with price level output level
and
, i.e. zero net export, yields:
The last equality follows from (11.30). To get an even simpler expression, insert for
from
(11.65):
(11.72)
In the special case with
,
, though the latter property will also hold
for any configuration satisfying
. More generally,
the sign of the relationship between the terms of trade and net export is ambiguous, depending on the relative size of ,
and
.
11.10 Equilibrium – The supply side: Marginal cost and inflation dynamics
As before market clearing in the labor market requires that:
∫ ⇒ ∫ ∫ ∫
(11.73)
Thus, using (11.52) and the market clearing condition:
109
(11.74)
The Basic New Keynesian Model
⇒ ⇒
As shown in (4.6)-(4.11), log-linearization gives, up to a first order:
(11.75)
Domestic inflation is derived as in (4.15), and is given by:
(11.76)
where
and
.
The real marginal cost is now derived from (11.53). Insert (11.25) and (11.30):
Next, insert (11.44) and (11.75) and make use of world market equilibrium:
(11.77)
Thus, we see that the real marginal cost is increasing in terms of trade and the world output. These variables end up influencing the real wage through the wealth effect on labor supply resulting from their impact on domestic consumption. In addition, changes in the terms of trade have a direct effect on the product wage for any given level of consumption wage. The influence of technology (through its direct effect on labor productivity) and of domestic output (through its effect on employment and, hence, the real wage for given output) is analogous to what we get in the closed economy setting described in (4.16). Finally we can use (11.67) to insert for
:
( ) ()
(11.78)
From (11.78) we see that domestic output affects marginal costs through its impact on employment (captured by ) and the terms of trade (captured by
, which is a function of the
degree of openness and the substitutability between domestic and foreign goods). World output
on the other hand, affects marginal costs through its effect on consumption (and hence, the real wage as captured by ) and the terms of trade (captured by marginal costs is positive if
This is because with sufficiently high
). The effect of world output on
, that is if
and
, the size of the real appreciation needed to
absorb the change in relative supplies is small, with its negative effect on marginal costs more
than offset by the positive effect from a higher real wage. What about the natural level of output
110
.
The Basic New Keynesian Model
( ) () ( ) () () ⇒ ⇒ ⇒ , i.e. the output when prices are flexible? We know from earlier that in this case,
.
Thus, the flexible price version of (11.78) is simply:
(11.79)
Solve (11.79) for natural output and use that
:
(11.80)
Here,
,
, and
. Again the effect of world output on
natural output is ambiguous, depending on the effect of world output on domestic marginal
costs, which in turn depends on the relative importance of the terms of trade effect discussed above. 11.11 The New Keynesian Phillips curve and the Dynamic IS equation
̃ (( ) () ) () ⇒ ( )̃
In this section I set up a canonical representation of the small open economy version of the basic New Keynesian model. First we denote
as the domestic output gap from flexible
price output. Second, if we subtract (11.79) from (11.78) the real marginal cost gap emerges:
(11.81)
Then insert (11.81) into (11.76) to get the New Keynesian Phillips curve for the small open
⇒ ()̃ ̃ economy:
Note that (11.82) nests the special case of a closed economy because
(11.82)
implies that
and then (11.82) becomes identical to (4.20). In general, the relation between the degree
of openness parameter
, an increase in the output gap, and domestic inflation, depends on the
sign on . This is because
. If
(i.e. if
and
are sufficiently high), an
increase in the openness will make domestic inflation less responsive to a change in the output gap. On the other hand, if
, then more openness will make domestic inflation more
111
The Basic New Keynesian Model
responsive to output gap changes. To derive the open economy dynamic IS equation (DIS) we have to do some additional steps. First, note that the real interest rate is defined as:
̃ ⇒ ̃ ̃ Using this, (11.70) can be written as:
In a similar vein, the natural output is given as a function of the natural real interest rate: (11.83)
The DIS equation emerges by subtracting (11.83) from (11.70):
(11.84)
Equations (11.82) and (11.84), together with an equilibrium process for the natural real rate
,
constitute the non-policy block of the small open economy version of the New Keynesian model. The natural real rate can be extracted from (11.84), but first one should note that (11.70) implies that:
̃ Second, (11.80) implies that:
Third, (11.84) implies:
Using these results, one can solve for
:
112
The Basic New Keynesian Model
̃ () ⇒
(11.85)
Thus, we see that the New Keynesian Phillips curve and the DIS equation in the small open economy equilibrium is similar to the counterparts in the closed economy. A couple of
differences appear however. First, the degree of openness influences the sensitivity of the output gap to interest rate changes. Second, openness generally makes the natural real interest rate depend on expected world output growth, in addition to domestic productivity. Finally, it is convenient to define the real rate gap as:
̂
(11.86)
As in the closed economy case the real rate converges to the discount rate once technology shocks and world output growth is turned off. Note however, that the real rate will typically by higher than the discount rate because the world experiences a positive growth on average. 11.12 Equilibrium determinacy
In order to close the model one must specify a monetary policy rule. Suppose the central bank
̃
follows an interest rate rule of the form:
where
(11.87)
̃
is a monetary policy shock. To set up the equilibrium system we first insert (11.87) and
(11.82) into (11.84) and solve the resulting equation for
113
:
The Basic New Keynesian Model
̃ ̃ ̃ ( ̃ ) ̃ ( ̃)̃ ̃ ( ̃)̃ ̂ ̃ ̃ ̂ ⇒ ̃ ̃ ̂ ⇒ ̃ ̃ ̂
(11.88)
Equation (11.88) is the reduced form version of the DIS equation, and shows the current output gap as a function of expected output gap, expected domestic inflation, and shocks. We next
̃ ̃ ̂ ̃ ̂ ̃ ( ) ̂ ⇒ ̃ () ̂
achieve a similar representation of current inflation. Insert (11.88) into (11.82) a nd solve the resulting equation for
:
(11.89)
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The Basic New Keynesian Model
The equilibrium dynamics above is represented as a system of difference equations, and is written
̃ ()̃ ̂ ()̃ ̂ ̃ ̂ ( ) in matrix form as:
where
We have defined
(11.90)
to ease the notation. The system (11.90) is a reduced form
representation of the dynamic IS curve and the New Keynesian Phillips curve, which takes into account effects from the monetary policy defined in (11.87). The Blanchard and Kahn (1980) conditions state that the system (11.90) has a locally unique equilibrium if and only if both eigenvalues of the 2x2-matrix characteristic equation:
are inside the unit circle. To check this we derive the
() (( )) ( ) ( ) ( ) (( )) ( )
When this is written out:
115
The Basic New Keynesian Model
Following LaSalle (1986), the two eigenvalues of
lie inside the unit circle if and only if:
( ) ( ) () and
The first condition holds because
and all elements in the denominator are positive.
The second condition can be rewritten to:
Thus, the second condition holds as long as the policy parameters high. In fact, whenever
, then
and
are sufficiently
can actually be zero. Therefore the Taylor principle
holds in this open economy setting as well. 11.13 Equilibrium dynamics
In order to get an analytical expression for the dynamic processes one must specify how the
̂ ̃ ⇒ ̃ ⇒ ⇒
monetary policy shock evolves. Suppose it follows an AR(1)-process:
For simplicity and without loss of generality we now set
(11.91)
. We use the method of
undetermined coefficients and guess that the solution takes the form: (11.92) (11.93)
To proceed we insert (11.91)-(11.93) into (11.82):
Then, insert the monetary policy rule into (11.84):
116
(11.94)
The Basic New Keynesian Model
̃ ̃ ̃ ( ̃ ) ̃ ( ̃ ) ⇒ ( ) ⇒ ⇒ ⇒ ⇒ ̃ ̃⇒ ̃ ̃ ⇒
We have defined
(11.95)
to ease the notation. Insert (11.95)
back into (11.94) to obtain:
(11.96)
Using (11.95)-(11.96) the guessed solutions (11.92)-(11.93) become:
(11.97) (11.98)
To get an AR(1) representation we insert (11.91) into (11.97) and (11.98):
(11.99)
(11.100)
Thus, a positive monetary policy shock gives a decline in both the output gap and domestic inflation. Further, it can be shown that
is increasing in the degree of openness, implying that a
given monetary policy shock will have a larger impact in the small open economy than its closed economy counterpart. The response on the real interest rate is found by inserting (11.97)-(11.98) into the policy rule (11.87): 34
34 Note
(11.101)
that we look at the effect on the nominal interest rate, not at the interest rate level. Thus, the constant not part of the expression.
117
is
The Basic New Keynesian Model
The sign of the response of the nominal interest rate is ambiguous and depends on parameter values. The response of the real interest rate:
⇒ ( ( ) )
(11.102)
Using (11.67) we find the effect on the terms of trade:
(11.103)
Thus, a monetary contraction leads to an improvement in the terms of trade, i.e. a decrease in the relative price on foreign goods. Using (11.36) we find the effect on the change in the nominal
⇒ exchange rate:
(11.104)
Thus, a monetary contraction leads to a nominal exchange rate appreciation. Using (11.40) we find the effect on the effective real exchange rate:
(11.105)
Thus, the effective real exchange rate appreciates as well. Using (291) we find the effect on net exports:
The sign on the response of net exports to a monetary contraction is negative whenever
(11.106)
.
11.14 Optimal monetary policy in the small open economy
In the following I will characterize the optimal monetary policy for the small open economy. In order to get analytical results I make some assumptions regarding parameter coefficients. These assumptions are as follows:
First we characterize the optimal allocation from the viewpoint of the social planner. The optimal allocation maximizes household utility (11.1) subject to the technological constraint (11.52), a consumption/output possibilities set implicit in the international risk-sharing conditions (11.42), and the market clearing condition (11.63). The latter condition is somewhat changed under the
⇒ ⇒
parameter restrictions above. First, from (11.44),
implies that:
(11.107)
118
The Basic New Keynesian Model
⇒ ⇒⇒ ∑ ∑ Furthermore, when
, the CPI given by (11.13) takes the Cobb-Douglas form:
(11.108)
When we rewrite (11.108), and then insert (11.28):
Thus, (11.63) becomes:
(11.109)
The period optimization problem of the social planner follows as:
subject to
(11.110)
It is useful the make the problem simpler by getting rid of some constraints. Insert (11.107) into (11.109) and combine with (11.66), which states that
. The result is an equilibrium
identity linking domestic consumption to domestic and world output:
(11.111)
Finally, to achieve an consumption expression useful to the social planner we insert (11.52) into (11.111) and use that the optimal allocation implies
just as in the closed economy case:
The period optimization problem of the social planner now becomes a problem in
FOC:
119
(11.112) only: (11.113)
The Basic New Keynesian Model
⇒ ⇒ ⇒ ⇒ ⇒ :
Using the specified utility with
, which implies that
(11.114) , the LHS of
(11.114) becomes:
(11.115)
Thus, the optimal optimal employment is constant. From home firms optimization optimization problem problem in flexible price, free competition, we also have the following:
(11.116)
FOC: :
(11.117)
From (11.114) and (11.117) we get the optimal allocation of domestic quantities in the economy: (11.118)
As a comparison, let us first study the the distortion in in a market equilibrium equilibrium where firms firms have monopolistic power, but where prices are flexible. This is what we refer to as the natural equilibrium (illustrated by top script ). ). Home firms’ maximization problem follows from firm production (11.52), the demand for home good , given by (11.8), the aggregated version of (11.52), and finally market clearing conditions. We know from the closed economy case that
monopolistic competition competition yields a distorted equilibrium which, in the absence of sticky prices, can be fixed by a labor subsidy. Thus, we also a lso add the labor subsidy with size yet to be determined:
120
The Basic New Keynesian Model
⇒ () ⇒ ( ) ⇒ ⇒ ⇒ ̃ ̃
(11.119)
FOC:
The LHS can be rewritten by inserting inserting for (11.24) and (11.109):
(11.120)
If we insert (11.115) to get the social planner’s solution, the optimal subsidy is found as:
(11.121)
Note that (11.121) nests the closed economy e conomy case where
to
. In addition, and because
. In this case, (11.121) collapses
, a sufficiently open economy (and
) )
implies a wage tax as the optimal fiscal policy instead of the subsidy. As in the closed economy, the optimal monetary policy requires stabilizing the output gap, i.e. Phillips curve given by (11.82) the implies that
. The New Keynesian
. Thus, in the special case under
consideration, (strict) domestic inflation targeting targeting (DIT) is indeed the optimal policy. From the dynamic IS equation (11.84) we see that
implies
in equilibrium, with all
variables matching their their natural levels levels at all times. As discussed earlier, an interest interest rate rule of of the
121
The Basic New Keynesian Model
̃
form
is associated with an indeterminate equilibrium, equilibrium, and hence, does not guarantee that
the outcome of full price stability is attained. However, the central bank can get to the desired outcome if it commits to a rule of the form:
(11.122)
where
Under strict domestic inflation targeting, the behavior of real variables in the small open
economy corresponds to the one that would be observed in the absence of nominal rigidities. Hence, from (11.80), that is
(where
,
, and
), we see that domestic domestic output always increases in response response to a positive technology technology
shock at home. The sign of the response to a rise in world output depends on the sign of
,
however. The natural level of the terms of trade is found by inserting (11.80) into the natural
⇒ level (flexible price) version of (11.67):
Because
(11.123)
, an improvement in domestic technology leads to a real depreciation through its
expansionary effect on domestic output. On the other hand, and since
, an
increase in world output generates an improvement in the terms of trade, (i.e. a real appreciation), given domestic technology. Because domestic prices are fully stabilized under DIT, it follows from (11.36) that it can be written as:
(11.124)
122
The Basic New Keynesian Model
Thus, the nominal exchange rate moves one for one with the natural terms of trade and inversely with the price level. Assuming constant world prices, the nominal exchange rate will inherit all the statistical properties of the natural terms of trade. Accordingly, the volatility of the nominal exchange rate under DIT will be proportional to the volatility of the gap between the natural level of domestic output, which in turn is related to productivity, and world output. In particular, the nominal exchange rate volatility will tend to be low when domestic natural output displays a strong positive comovement with world output. When that comovement is low or even negative, possibly because of a large idiosyncratic componenent in domestic productivity, the volatility of the terms of trade and the nominal exchange rate under DIT will be enhanced. The implied
equilibrium process for the CPI can also be derived, by substituting (11.124) into (11.30): (11.125)
Thus, it is seen that under the DIT regime, the CPI level will also vary with the natural terms of trade and will inherit its statistical properties. If the economy is very open, and if domestic productivity and hence, the natural level of domestic output, is not much synchronized with world output, CPI prices could potentially be highly volatile, even if the domestic price level is constant. One lesson from this analysis is that potentially large and persistent fluctuations in the nominal exchange rate, as well as in some inflation measures like the CPI, are not necessarily undesirable, nor do they acquire a policy response aimed at dampening such fluctuations. Instead, and especially for an economy that is very open and subject to large idiosyncratic shocks, those fluctuations may be an equilibrium consequence of the adoption of an optimal policy, as illustrated by the model above. 11.15 Welfare losses
In the following I will derive a welfare loss function for the special case with log utility and unit
̂ ̂ ̂ ̂ ̂ elasticity of substitution between goods of different origin, i.e. for:
With log consumption, a derivation similar to the one who previously lead to (8.1) now gives:
123
The Basic New Keynesian Model
̂ ̂ ⇒
(11.126)
A few results are needed to proceed. First, notice that in the special case considered here, (11.67) can be rewritten to:
( ) ⇒ ⇒ ∫
where I have used that the parameter restrictions above implies:
Thus, (11.44) becomes:
Insert (11.127) into (11.126) and use that
(11.127)
in the log consumption case:
stands for terms independent of policy as usual. The next step is to rewrite
of the output gap and price dispersion. From the production function (11.52),
(11.128)
as a function . Thus,
using (11.8), market clearing in the labor market and the goods market requires:
(11.129)
Using the same derivation as in (8.3)-(8.6):
Here, I have used that
(11.130)
in (8.6) because of the CRS assumption. By combining
(11.129) and (11.130), the employment gap from steady state employment writes as:
The next step is to insert (11.131) into (11.128):
124
(11.131)
The Basic New Keynesian Model
⇒ ( ) ⇒ ̃ Notice that the steady state version of (11.114) becomes:
When we insert this:
(11.132)
To proceed, note that with CRS and the parameter restrictions above, (4.18) becomes:
Thus, from the definition of the natural output gap from its steady state counterpart, the technology level, which previously was specified in (8.11), now satisfies:
(11.133)
Insert (11.133) into (11.132):
(11.134)
The last line follows from (8.12). When we take the sum over all discounted periods and make use of (8.17):
125
The Basic New Keynesian Model
̃ ̃ ̃ ̃ ⇒ ∑ ∑ ̃
(11.135)
The parameter
is defined as in (11.76). Thus, we can write the second-order
approximation to the utility losses of the domestic representative consumer resulting from deviations in optimal policy, expressed as a fraction of steady state consumption, as:
∑ ̃
Taking unconditional expectations on (11.136) and letting
(11.136)
, the expected welfare losses for
any policy that deviates from strict inflation targeting can be written in terms of the variances of inflation and the output gap:
̃
(11.137)
Text
126
The Basic New Keynesian Model
References
Blanchard and Kahn () Bullard and Mitra (2002) Erceg et al. (2000) Gali, Jordi (2008), Monetary Policy, Inflation and the Business Cycle . Oxford: Princeton University Press. Gali, Jordi and Monacelli Tommasso (2005), Monetary Policy and Exchange Rate Volatility in a Small Open Economy. Review of Economic Studies 72(707-734) Hamilton (1994) LaSalle (1986) Molnar, (2011) Ripatti, Antti (2011) Sydsæter (2006) Woodford ()
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The Basic New Keynesian Model
Appendix
A.
Dynare codes – A monetary policy shock with sticky prices
// The basic New Keynesian model - A monetary policy shock // Gali ch. 3, fig. 3.1 // Modified by Drago Bergholt //-----------------------------------------// Preamble //-----------------------------------------// Variables var pi y Y rn i m_r n a v; varexo eps_v eps_a; // // Parameters parameters beta epsilon theta sigma rho phi alpha phi_pi phi_y eta PSI_yan THETA lambda kappa rho_v rho_a LAMBDA_v LAMBDA_a; beta = 0.99; sigma = 1; phi = 1; alpha = 1/3; epsilon = 6; eta = 4; theta = 2/3; phi_pi = 1.5; phi_y = 0.5/4; PSI_yan = (1+phi)/(sigma*(1-alpha)+phi+alpha); THETA = (1-alpha)/(1-alpha+alpha*epsilon); lambda = (1-theta)*(1-beta*theta)*THETA/theta; kappa = lambda*(sigma+(phi+alpha)/(1-alpha)); rho = 1/beta-1; rho_v = 0.5; rho_a = 0.9; LAMBDA_v = 1/((1-beta*rho_v)*(sigma*(1-rho_v)+phi_y)+kappa*(phi_pirho_v)); LAMBDA_a = 1/((1-beta*rho_a)*(sigma*(1-rho_a)+phi_y)+kappa*(phi_pirho_a)); //-----------------------------------------// Model //-----------------------------------------model(linear); // Taylor-Rule i = rho+phi_pi*pi+phi_y*y+v; // eq'n. (25), p. 50 // IS-Equation y = y(+1)-1/sigma*(i-pi(+1)-rn); // y is output gap (22) rn=rho+sigma*PSI_yan*(a(+1)-a); // natural rate of interest (23) Y = PSI_yan*(1-sigma*(1-rho_a)*(1-beta*rho_a)*LAMBDA_a)*a; // actual output; 3rd eq'n from bottom, p. 54 // Phillips Curve pi = beta*pi(+1)+kappa*y; // (21) // Money Demand m_r = y-eta*i; // ad hoc money demand; m_r = m-p // Employment n = (((PSI_yan-1)-sigma*PSI_yan*(1-rho_a)*(1-beta*rho_a)*LAMBDA_a)/(1alpha))*a; // bottom p. 54 // Autoregressive Error a = rho_a*a(-1) + eps_a; // technology shock (28) v = rho_v*v(-1) + eps_v; // shock to i (bottom p. 50) end; //
128
The Basic New Keynesian Model
//-----------------------------------------// Steady State //-----------------------------------------check; //-----------------------------------------// Shocks //-----------------------------------------shocks; var eps_v = 0.0625; var eps_a = 0; end; tech = 0; policy = 1; //-----------------------------------------// Computation //-----------------------------------------stoch_simul(irf=12); //stoch_simul(periods=1000,irf=12); //-----------------------------------------// Plots //-----------------------------------------if policy==1; // Gali's figure 3.1 figure(2); clf; subplot(3,2,1); plot(y_eps_v, '-o'); title('Output gap'); subplot(3,2,2); plot(4*pi_eps_v, '-o'); title('Inflation'); subplot(3,2,3); plot(4*i_eps_v, '-o'); title('Nominal interest rate'); subplot(3,2,4); plot(4*i_eps_v(1:end-1)-4*[pi_eps_v(2:end)], '-o'); title('Real interest rate'); subplot(3,2,5); plot(4*(m_r_eps_v-[0;m_r_eps_v(1:end-1)]), '-o'); title('Real money growth'); subplot(3,2,6); plot(v_eps_v, '-o'); title('v'); end; if tech==1; // Gali's figure 3.2 figure(2); clf; subplot(4,2,1); plot(y_eps_a); title('Output gap'); subplot(4,2,2); plot(4*pi_eps_a); title('Inflation'); subplot(4,2,3); plot(Y_eps_a); title('Output'); subplot(4,2,4); plot(n_eps_a); title('Employment'); subplot(4,2,5); plot(4*i_eps_a); title('Nominal interest rate'); subplot(4,2,6); plot(4*rn_eps_a); title('Real interest rate'); subplot(4,2,7); plot(4*(m_r_eps_a-[0;m_r_eps_a(1:end)])); title('Real money growth'); subplot(4,2,8); plot(a_eps_a); title('a'); end;
B.
Dynare codes – A technology shock with sticky prices
// The basic New Keynesian model - A technology shock // Gali ch. 3, fig. 3.2 // Modified by Drago Bergholt //-----------------------------------------// Preamble //-----------------------------------------// Variables var pi y Y rn i m_r n a v; varexo eps_v eps_a; // // Parameters
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The Basic New Keynesian Model
parameters beta epsilon theta sigma rho phi alpha phi_pi phi_y eta PSI_yan THETA lambda kappa rho_v rho_a LAMBDA_v LAMBDA_a; beta = 0.99; sigma = 1; phi = 1; alpha = 1/3; epsilon = 6; eta = 4; theta = 2/3; phi_pi = 1.5; phi_y = 0.5/4; PSI_yan = (1+phi)/(sigma*(1-alpha)+phi+alpha); THETA = (1-alpha)/(1-alpha+alpha*epsilon); lambda = (1-theta)*(1-beta*theta)*THETA/theta; kappa = lambda*(sigma+(phi+alpha)/(1-alpha)); rho = 1/beta-1; rho_v = 0.5; rho_a = 0.9; LAMBDA_v = 1/((1-beta*rho_v)*(sigma*(1-rho_v)+phi_y)+kappa*(phi_pirho_v)); LAMBDA_a = 1/((1-beta*rho_a)*(sigma*(1-rho_a)+phi_y)+kappa*(phi_pirho_a)); // //-----------------------------------------// Model //-----------------------------------------model(linear); // Taylor-Rule i = rho+phi_pi*pi+phi_y*y+v; // eq'n. (25), p. 50 // IS-Equation y = y(+1)-1/sigma*(i-pi(+1)-rn); // y is output gap (22) rn=rho+sigma*PSI_yan*(a(+1)-a); // natural rate of interest (23) Y = PSI_yan*(1-sigma*(1-rho_a)*(1-beta*rho_a)*LAMBDA_a)*a; // actual output; 3rd eq'n from bottom, p. 54 // Phillips Curve pi = beta*pi(+1)+kappa*y; // (21) // Money Demand m_r = y-eta*i; // ad hoc money demand; m_r = m-p // Employment n = (((PSI_yan-1)-sigma*PSI_yan*(1-rho_a)*(1-beta*rho_a)*LAMBDA_a)/(1alpha))*a; // bottom p. 54 // Autoregressive Error a = rho_a*a(-1) + eps_a; // technology shock (28) v = rho_v*v(-1) + eps_v; // shock to i (bottom p. 50) end; // //-----------------------------------------// Steady State //-----------------------------------------check; // //-----------------------------------------// Shocks //-----------------------------------------shocks; var eps_v = 0; var eps_a = 1; end; // tech = 1; policy = 0;
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The Basic New Keynesian Model
//-----------------------------------------// Computation //-----------------------------------------stoch_simul(irf=12); //stoch_simul(periods=1000,irf=12); // //-----------------------------------------// Plots //-----------------------------------------if policy==1; // Gali's figure 3.1 figure(2); clf; subplot(3,2,1); plot(y_eps_v, '-o'); title('Output gap'); subplot(3,2,2); plot(4*pi_eps_v, '-o'); title('Inflation'); subplot(3,2,3); plot(4*i_eps_v, '-o'); title('Nominal interest rate'); subplot(3,2,4); plot(4*i_eps_v(1:end-1)-4*[pi_eps_v(2:end)], '-o'); title('Real interest rate'); subplot(3,2,5); plot(4*(m_r_eps_v-[0;m_r_eps_v(1:end-1)]), '-o'); title('Real money growth'); subplot(3,2,6); plot(v_eps_v, '-o'); title('v'); end; if tech==1; // Gali's figure 3.2 figure(2); clf; subplot(4,2,1); plot(y_eps_a, '-o'); title('output gap'); subplot(4,2,2); plot(4*pi_eps_a, '-o'); title('inflation'); subplot(4,2,3); plot(Y_eps_a, '-o'); title('output'); subplot(4,2,4); plot(n_eps_a, '-o'); title('employment'); subplot(4,2,5); plot(4*i_eps_a, '-o'); title('nominal interest rate'); subplot(4,2,6); plot(4*i_eps_a(1:end-1)-4*[pi_eps_a(2:end)], '-o'); title('real interest rate'); subplot(4,2,7); plot(4*(m_r_eps_a-[0;m_r_eps_a(1:end-1)]), '-o'); title('real money growth'); subplot(4,2,8); plot(a_eps_a, '-o'); title('a'); end;
C.
Dynare codes – A monetary policy shock with sticky prices and wages
// The basic New Keynesian model - A monetary policy shock // Gali ch. 6, fig. 6.3 // Modified by Drago Bergholt //-----------------------------------------// Preamble //-----------------------------------------// Variables var y Y n yn i pip piw rn w wn v a; varexo eps_v eps_a; // Parameters parameters alpha gammap gammaw sigma beta phi thetap thetaw psinya psinwa kappap kappaw lambdap lambdaw rho phip phiw phiy rhoa rhov ew ep; beta = 0.99; sigma = 1; phi = 1; alpha = 1/3; ew = 6; ep = 6; thetaw = 3/4; thetap = 2/3;
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The Basic New Keynesian Model
phip = 1.5; phiw = 0; phiy = 0; rho = 1/(beta-1); psinya = (1+phi)/(sigma*(1-alpha)+phi+alpha); psinwa = (1-alpha*psinya)/(1-alpha); lambdap = ((1-thetap)*(1-beta*thetap)/thetap)*((1-alpha)/(1alpha+alpha*ep)); lambdaw = ((1-beta*thetaw)*(1-thetaw))/(thetaw*(1+ew*phi)); kappap = lambdap*alpha/(1-alpha); kappaw = lambdaw*(sigma + phi/(1-alpha)); rhoa = 0.9; rhov = 0.5;
//-----------------------------------------// Model //-----------------------------------------model(linear);
y = y(+1) - (1/sigma)*(i - pip(+1) - rn); y = Y - yn; yn = psinya*a; Y = a + (1-alpha)*n; rn = sigma*psinya*(rhoa - 1)*a; pip = beta*pip(+1) + kappap*y + lambdap*(w-wn); piw = beta*piw(+1) + kappaw*y - lambdaw*(w-wn); w = w(-1) + piw - pip; wn = psinwa*a; i = rho + phip*pip + phiw*piw + phiy*y + v; v = rhov*v(-1)+eps_v; a = rhoa*a(-1)+eps_a; end;
//-----------------------------------------// Steady State //-----------------------------------------check; //-----------------------------------------// Shocks //-----------------------------------------shocks; var eps_v = 0.0625; var eps_a = 0; end; tech = 0; policy = 1;
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