Chapter 3 Equilibrium analysis in Economics
Chapter 3 Equilibrium Analysis in Economics --------------------------------------------------------------------- Studies or analysis of Economics 1. Static Analysis ‐ Studies focus only on a particular period of time. 2. Comparative Analysis ‐ Studies focus on the external forces that make equilibrium move to a new one. 3. Dynamic Analysis ‐ Studies focus on the change of time and how the equilibrium change with time.
P
S Static Analysis
E
P*
D Q
Q*
P P1
S E´ Comparative Analysis
E
P0
D´ D Q0 Q1
Q
3-1
Chapter 3 Equilibrium analysis in Economics
Dynamic Analysis
Price
S1
P
S2
P1
S3
P2 P3
Pt
D
Time
Q
0
1
2
3
Variable
Converge
Time
Variable
Diverge
Time
Example of dynamic analysis: Cobb-web Theorem St (Pt-1)
P P1
P
St (Pt-1)
P1
P2 Dt (Pt)
P2 Dt (Pt)
Q2
Q1
Q2
Q
3-2
Q1
Q
Chapter 3 Equilibrium analysis in Economics
3.1
Partial Market Equilibrium - A linear model Identify equilibrium price & equilibrium quantity (endogenous) Constructing the model 1. Equilibrium condition Conditional equation
Qd = Qs
2. Demand equation: a decreasing linear function of P (P ↑, Qd ↓) Behavioral equation Qd = a – bP (a, b > 0) 3. Supply equation: an increasing linear function of P (P ↑, Qs ↑) Behavioral equation Qs = -c + dP (c, d > 0)
Q
Qd= a - bP
Qs= -c + dP
Q*
P*
P
Solution Qd = a – bP QS = -c + dP The equilibrium condition Qd = Qs a – bP = -c + dP a + c = bP + dP a + c = (b + d)P a+c P* = b+d
3-3
The solution values are P* and Q*
Chapter 3 Equilibrium analysis in Economics
Substitute P* into Qd = Q*
a – bP* a+c ) = a–b( b+d a(b+d) - b(a+c) = b+d ab+ad - ba-bc = b+d ad - bc = b+d Since b+d > 0 ∴ (ad - bc) must be positive in order to have positive Q* ad – bc > 0 ad > bc (Restriction) •
What happen if (b + d) = 0, can an equilibrium solution be found by using P*,Q* why or why not ? - No, there will be division by zero. The solution is undefined
•
What can you conclude regarding the position of D & S curve? - D & S would be parallel, with no equilibrium
3.2
Partial Market Equilibrium: A nonlinear model
Quadratic function: numerical example Qd = Qs Qd = 4 – P2 Qs = 4P – 1 4 – P2 = 4P - 1 P2 + 4P - 5 = 0 The Quadratic formula
ax2 + bx + c = 0
a 0
The roots are
x1* , x*2 =
-b± b 2 -4ac 2a 3-4
Chapter 3 Equilibrium analysis in Economics
According to the above equation P2 + 4P - 5 = 0
∴ The equilibrium price = P1* ,P2* =
-4± 42 -4(1)(-5) 2(1)
-4±6 2 = -5, 1 * The equilibrium quantity = Q = Qs = 4P*- 1 = 4(1) - 1 =3 =
Qs = 4P - 1
(1, 3) 3 -5 1 (-5, -25) -25 Q Example:
Qd
=
Qs
Qd
=
8 – P2
Qs
=
P2 – 2
8 - P2 =
P2 – 2
2P2
10
=
(P- 5)(P+ 5) = P* =
5
0 Q* = 3
3-5
P Qd = 4 – P2
Chapter 3 Equilibrium analysis in Economics
3.3
General Market Equilibrium
Last 2 sections: an isolated market. ∵ consider only 1 good In the real world, there are more than one good in the economy
Consider more than 1 good market
General Equilibrium with n commodities, the equilibrium conditions are Qdi – Qsi = 0
Ei
(i = 1, 2, ….. n)
Two commodity mkt. model
Qd1
=
10 – 2P1 + P2
……. (1)
Qs1
=
-2 + 3P1
……. (2)
Qd2
=
15 + P1 – P2
……. (3)
Qs2
=
-1 + 2P2
……. (4)
d
s
1
=
Q1
Qd2
=
Qs2
Q
(1) = (2)
10 – 2P1 + P2 = -2 + 3P1
(3) = (4)
P2 15 + P1 - P2 P1
= -12 + 5P1 = -1 + 2P2 = -16 + 3P2
Substitute (6) into (5)
14P2
= = =
P2*
=
P2
-12 + 5 (-16 +3P2) -12 – 80 + 15P2 92 92 14 3-6
…….(5) ……. (6)
Chapter 3 Equilibrium analysis in Economics
Substitute P2* into (6) P1*
Q1*
= = =
Q2*
= = =
-16 + 3P2* 92 = -16 + 3 ( ) 14 52 = 14 -2 + 3P1* 52 -2 + 3( ) 14 64 7 -1 + 2P2* 92 -1 + 2( ) 14 85 7
=
Good 1 Q1
Good 2 Q2
Q s1
Q s2
85 7
64 7 Q d1
52 14
Q d2
P1
92 14
P2
Solution of a General Equilibrium System
To guarantee that the model yields a unique solution, the equations should have the following properties. 1. Consistency ‐ The satisfaction of any one equation in the model will not preclude the satisfaction of another x+y =8 x+y =9 2. Functional independence
3-7
Chapter 3 Equilibrium analysis in Economics
‐
No equation is redundant which means that one can be derived from the other. 2(2x + y) = 2(12) 4x + 2y = 24
(Try exercise 3.4 in the text book)
3.4
Equilibrium in National Income Analysis
Keynesian national income model Y=C+I+G C = a + bY I = I0 G = G0 Y (1 - b)Y
= = =
Y*
=
Then C*
= = = =
restriction
(a > 0, 0 < b < 1)
C+I+G a + bY + I0 + G0 a + I0 + G0 1 ( ) (a + I0 + G0) 1-b a + b Y* 1 a+b( ) (a + I0 + G0) 1-b a(1 - b) + ba + bI0 + bG 0 1-b a + b(I0 + G 0 ) 1-b
b 1
IS – LM Framework
• •
Equilibrium in good market – IS Equilibrium in money market - LM
a. Good market C = = Yd T =
a + bYd Y–T T0 + tY
; [a > 0, 0 < b < 1] ; [0 < t < 1] 3-8
Chapter 3 Equilibrium analysis in Economics
I G
= =
I0 – er G๐
X
=
X๐
M
=
M๐
; [e > 0]
Conditional equation (Equilibrium) Y
= = = = (1 – b + bt)Y = Y*
=
Y*
=
C + I + G + X –M a + bYd + I0 – er + G0 + X0 – M0 a + b (Y – T0 - tY) + I0 – er + G0 + X0 – M0 a + bY – bT0 – btY + I0 + er +G0 + X0 – M0 a – bT0 + I0 + G0 + X0 – M0 – er a – bT0 + I0 + G 0 + X 0 – M 0 er 1- b + bt 1 - b + bt α 0 – α1 r
According to the expression of Y*, National income has a negative relationship with interest r
IS
Y
b. Money market
Money demand 1. Transaction Demand 2. Precautionary Demand 3. Speculative Demand Md = N0 + myY – mrr
; my , mr > 0
Money Supply – Monetary policy (constant) Ms = Mo Conditional equation (Equilibrium) Md N0 + myY – mrr
= =
Ms M0
3-9
Chapter 3 Equilibrium analysis in Economics
r* rewrite
=
N0 + m y Y - M0 mr
Y
=
M0 - N0 mr +( )r my my
Y*
=
β0 + β1r
According to the expression of Y*, National income has a positive relationship with interest. r
LM
Y c.
IS – LM Framework
IS α 0 – α1 r
= =
LM β0 + β1r
(β1 + α1 ) r
=
α 0 - β0
r*
=
α0 - β0 β1 + α1
Y*
=
α 0 – α1 r*
=
α 0 – α1
=
α 0β1 + α1β 0 β1 + α1
From IS
Substitute r*
=
α 0 , α1 , β 0 , β1
α0 - β0 β1 + α1
by their expressions
m y [a - bT0 + I0 + G 0 + X 0 - M 0 ] - [1 - b + bt](M 0 - N 0 ) (1 - b + bt)m r + em y
3-10
Chapter 3 Equilibrium analysis in Economics
Y*
m r [a - bT0 + I0 + G 0 + X 0 - M 0 ] + e(M 0 - N 0 ) (1 - b + bt)m r + em y
=
r LM
r* =
α0 - β0 β1 + α1 IS
α β + α1β 0 Y= 0 1 β1 + α1
Y
*
Exercise Chapter 3
1. Let the demand and supply functions be as follow: (a) Qd = 51 – 3P (b) Qd = 30 – 2P find P*and Q*
Qs = -10 + 6P Qs = -6 + 5P
2. Find the zeros of the following functions graphically: (a) f(x) = x2 – 8x + 15 (b) g(x) = 2x2 – 4x – 16 3. Find the equilibrium solution for each of the models: Qd = Qs (a) Qd = 3 – P2 (b) Qd = 8 – P2
Qs = 6P – 4 Qs = P2 – 2
4. The demand and supply functions of a two-commodity market model are as follows: Qd1 = 18 – 3P1 + P2 Qd2 = 12 + P1 – P2 Qs1 = -2 + 4P1 Qs2 = -2 + 3P2 Find and Pi* and Q*i 5. Given the following model: Y = C + I0 + G0 C = a + b(Y – T) T = d + tY
(a > 0, 0 < b < 1) (d > 0, 0 < t < 1) 3-11
Chapter 3 Equilibrium analysis in Economics
(a) How many endogenous variables are there? (b) Find Y*, T* and C* 6. Let the national-income model be: Y = C + I0 + G C = a + b(Y – T0) G = gY (a) (b) (c) (d)
(a > 0, 0 < b < 1) (0 < g < 1)
Identify the endogenous variables. Give the economic meaning of the parameter g Find the equilibrium national income. What restriction on the parameters is needed for a solution to exist?
3-12
Chapter 3 Equilibrium analysis in Economics
7. Using the following money market information to derive an equation of LM: M d = 1375 + 0.25Y − 25r M s = 2500 Md = Ms
8. Find national-income and aggregate consumption at the equilibrium of the following model Y = C + I 0 + G0 C = 25 + 6Y I 0 = 16
1
2
G0 = 14
3-13
