ECON 5113 Advanced Microeconomics Winter 2011 Answers to Selected Exercises
Instructor: Kam Yu
The following questions are taken from Geoffrey A. Jehle Ex. 1.45 Since di is homogeneous of degree zero in p and Philip J. Reny (2001) Advanced Microeconomic The- and y , for any α > 0 and for i = 1, . . . , n, n, Boston: Addison Addison Wesley Wesley.. The upory , Second Edition, Boston: di (αp, αy) αy) = di (p, y). dated version is available at the course web page: http://flash.lakeheadu.ca/∼kyu/E5113/Main.html
Differentiate both sides with respect to α, we have
∇p di (αp, αy) αy)T p +
∂d i (αp, αy) αy) y = 0. 0. ∂y
Ex. 1.14 Let U be a continuous utility function that n represents . Then for all x, y ∈ R+ , x y if and only Put α = 1 and rewrite the dot product in summation if U ( U (x) ≥ U ( U (y). form, the above equation becomes n First, First, suppose x, y ∈ R+ . Then U ( U (x) ≥ U ( U (y) or n U ( U (y) ≥ U ( U (x), which means that x y or y x. There∂d i (p, y) ∂d i (p, y) pj + y = 0. 0. (1) fore is complete. ∂p ∂y j j =1 Second, suppose x y and y z. Then U ( U (x) ≥ U ( U (y) and U ( U (y) ≥ U ( U (z). This implie impliess that U ( U (x) ≥ U ( U (z) and Dividing each term by di (p, y) yields the result. so x z, which shows that is transitive. n Finally, let x ∈ R+ and U ( U (x) = u. Then Ex. 1.46 Suppose that U ( U (x) is a linearly homogeneous n utility function. U −1 ([u, ([u, ∞)) = {z ∈ R+ : U ( U (z) ≥ u} (a) Then n = {z ∈ R+ : z x} = (x). E (p, u) = min{pT x : U ( U (x) ≥ u}
x
Since [u, [u, ∞) is closed and U is continuous, continuous, (x) is closed closed.. Simila Similarly rly (I sugges suggestt you you to try this), this), (x) is also closed. This shows that is continuous. 1
Ex. 1.33 Suppose on the contrary that E is bounded above in u, that is, for some p 0, there exists M > 0 such that M ≥ E (p, u) for all u in the domain of E . Let u∗ = V ( V (p, M ). ). Then
= min{upT x/u : U ( U (x/u) /u) ≥ 1} x
= u min{pT x/u : U ( U (x/u) /u) ≥ 1} x
= u min{pT x/u : U ( U (x/u) /u) ≥ 1}
(2)
= u min{pT z : U ( U (z) ≥ 1}
(3)
x/u
z
= uE (p, 1) = ue( ue(p)
E (p, u∗ ) = E (p, V ( V (p, M )) )) = M = pT x∗ ,
In (2) above it does not matter if we choose x or x/u where x∗ is the optimal optimal bundle. Since U is continuous, directly as long as the objective function and the conthere exists a bundle x in the neighbourhood neighbourhood of x of x∗ such straint remain the same. We can do this because of the (3) we simply rewrite that U ( U (x ) = u > u∗ . Since Since U strictly increasing, E is objective function is linear in x. In (3 x /u as z . strictly increasing in u, so that E (p, u ) > E (p, u∗ ) = (b) Using the duality relation between V and E and M . This contradicts the assumption that M is an upper the result from Part (a) we have bound. 1
It may be helpful to review the proof of Theorem 1.8.
y = E (p, V ( V (p, y)) = V ( V (p, y)e(p)
so that V ( V (p, y) =
y = v (p)y, e(p)
Ex. 2.3 By (T.1’) (T.1’) on p. 78
U ( U (x) = min {V ( V (p, 1) : p · x = 1} . where we have let v (p) = 1/e( /e(p). The marginal marginal utility utility p∈R ++ of income is ∂V ( ∂V (p, y) The Lagrangian Lagrangian is = v (p), ∂y L = − p1α p2β − λ(1 − p1 x1 − p2 x2 ), which depends on p but not on y . n
with the first-order conditions
Ex. 1.65 (b) By definition y0 = E (p0 , u0 ), Therefore
−αp1α−1 p2β + λx1 = 0
y1 E (p1 , u0 ) > y0 E (p0 , u0 )
and
0. −βp 1α p2β −1 + λx2 = 0. means that y > E (p , u ). Since Since the indirec indirectt utility utility function V is increasing in income y, it follows that Eliminating λ from the first-order conditions gives 1
1
0
u1 = V ( V (p1 , y 1 ) > V ( V (p1 , E (p1 , u0 )) = u0 .
β x1 p1 . α x2
p2 = Ex. 1.65 It is straight forward to derive the expenditure function, which is p2 E (p, u) = p2 u − 2 . 4 p1
Substitute this p2 into the constraint equation, we get
(4)
α 1 , α + β x1
p2 =
β 1 . α + β x2
and
(a) For p0 = (1, (1, 2) and y 0 = 10, we can use (4 ( 4) to 0 1 obtain u = 11/ 11 /2. Therefore, with p = (2, (2 , 1), I =
p1 =
The utility function is therefore
u0 − 1/8 43 = . 0 2u − 1 80
U ( U (x) =
0
αα β β (α + β )α+β
x1−α x2−β ,
(b) It is clear from part (a) that I depends on u . (c) Using the technique similar to Exercise 1.46, it can be shown that if U is homothetic, E (p, u) = e(p)g (u), which is a Cobb-Douglas function. where g is an increasing function. Then Ex. 2.6 We want to maximize utility u subject to the 1 0 1 e(p )g (u ) e(p ) constraint pT x ≥ E (p, u) for all p ∈ Rn++. That is, I = = , e(p0 )g (u0 ) e(p0 ) up1 p2 p1 x1 + p2 x2 ≥ . which means that I is I is independent of the reference utility p1 + p2 level. Rearranging gives Ex. Ex. 2.2 For i = 1, . . . , n, n, the i-th row of the matrix multiplication S (p, y)p is n
u≤
∂d (p, y) p + ∂d (p, y) p d (p, y) ∂p ∂y ∂d (p, y) p + ∂d (p, y) p d (p, y) = ∂p ∂y ∂d (p, y) p + ∂d (p, y) y = (5) i
j
j =i
n
j =i n j =i
=0
i
j
i
j
i
∂p j
j
i
∂y
u ≤ min
n
p1 ,p2
j j
j =i
j
for all p ∈ Rn++ . This implies that
j j
i
p1 + p2 p1 + p2 x1 + x2 p2 p1
p + p 1
p2
2
p1 + p2 x1 + x2 . p1
(7)
Therefore u attains its maximum value when equality holds in (7 (7). To find the minimum value value on the righthand side of (7 ( 7), write α = p2 /( p1 + p2 ) so that 1 − α = p1 /( p1 + p2 ) and 0 < α < 1. The minimization problem (6) becomes
where in (5 (5) we have used the budget balancedness and (6) holds because of homogeneity and (1 ( 1) in Ex. 1.45.
min α
2
x
x2 + : 0 < α< 1 . α 1−α 1
(8)
Notice that for any x1 > 0 and x2 > 0, lim
α→0
x
x2 + α 1−α 1
C (w, y) = c(w)y where c is the unit cost function. function. The profit maximization problem can be written as
=∞
max py − c(w)y = max y[ p p − c(w)]. )]. y
y
For a competitive firm, as long as p > c( c (w), the firm will x2 increase output level y indefinitely indefinitely.. If p < c(w), profit 1 lim + =∞ is negative at any level of output except when y = 0. α→1 α 1−α If p = c(w), profi profitt is zero zero at any any leve levell of outpu output. t. In so that the minimum value exists when 0 < α < 1. The fact, market price, average cost, and marginal cost are first-order condition for minimization is all equal so that the inverse supply function is a constant x1 x2 function of y . Therefore the supply function of the firm − 2+ = 0, 2 α (1 − α) does not exist and the number of firm is indeterminate.
and
x
which can be written as
Ex. 4.14 The profit maximization problem for a typical firm is
α2 x2 = (1 − α)2 x1 .
max [10 − 15 15q q − (J − 1)¯ q q ]q − (q 2 + 1), 1), q
Taking the square root on both sides gives 1/2 2
αx
1/2 1
= (1 − α)x
with necessary condition .
10 − 15 15q q − (J − 1)¯ q q − 15 15q q − 2q = 0.
Rearranging gives 1/2
α=
1/2
x1 1/2
x1
(a) Since all firms are identical, by symmetry q = q q¯ . This gives the Cournot equilibrium of each firm q ∗ = 10 10//(J + 31), with market price p∗ = 170/ 170/(J + 31). (b) Short-run profit of each firm is π = [40/ [40/(J +31)] J +31)]2 − 1. In the long-run π = 0 so that J = 9.
1/2
+ x2
and 1 − α =
x2 1/2
x1
1/2
+ x2
.
It is clear that α is indeed indeed between between 0 and 1. Puttin Puttingg α and 1 − α into the objective function in (8 ( 8) give the Ex. Ex. 5.11 5.11 (a) The necess necessary ary condit condition ion for a Paret Paretoodirect utility function efficient allocation is that the consumers’ MRS are equal. Therefore 2 1/2 1/2 U ( U (x1 , x2 ) = x1 + x2 , ∂U 1 (x11 , x12 )/∂x 11 ∂U 2 (x21 , x22 )/∂x 21 = , ∂U 1 (x11 , x12 )/∂x 12 ∂U 2 (x21 , x22 )/∂x 22 which is the CES function with ρ = 1/2. You shoul should d or verify with Example Example 1.3 on p. 38–39 that the expenditure expenditure x12 x22 = . (10) function is indeed as given. x11 2x21 The feasibility conditions for the two goods are Ex. 3.2 Constant returns-to-scale means that f is linearly homogeneous. So by Euler’s theorem x11 + x21 = e11 + e21 = 18 + 3 = 21, 21, (11)
x1 ∂y/∂x1 + x2 ∂y/∂x2 = y.
x12 + x22 = e12 + e22 = 4 + 6 = 10. 10.
(9)
(12)
Express x21 in (11 11)) and x22 in (12 12)) in terms of x11 and x12 respectively, (10 (10)) becomes
Since average product y/x1 is rising, its derivative respect to x1 is positive, that is,
x12 10 − x12 = , x11 2(21 − x11 )
(x1 ∂y/∂x1 − y )/x21 > 0. or
From (9) (9) we have
x12 =
x2 ∂y/∂x2 = −(x1 ∂y/∂x1 − y ) < 0,
10 10x x11 . 42 − x11
(13)
Eq. (13 (13)) with domain 0 ≤ x11 ≤ 21, (11 (11), ), and (12 (12)) completely characterize the set of Pareto-efficient allocations which means that the marginal product ∂y/∂x2 is negA (contract curve). That is, ative. 10 10x x11 , 0 ≤ x11 ≤ 21 21,, A = (x , x , x , x ) : x = 1 42 − x1
Ex. Ex. 4.5 Let w be the vector vector of factor factor prices prices and p be the output output price. price. Then Then the cost functi function on of a typtypical firm with constant constant returns-to-sc returns-to-scale ale technolo technology gy is
1 1
1 2
2 1
2 2
1 2
x11 + x21 = 21 21,, x12 + x22 = 10. 10 . 3
demand functions of the two consumers are: y1 2 p1 y1 x12 = 2 p2 y2 x21 = 3 p1 2y 2 x22 = 3 p2 x11 =
p1 e11 + p2 e12 18 p1 + 4 = 2 p1 2 p1 1 1 p1 e1 + p2 e2 18 p1 + 4 = = 2 p2 2 2 2 p1 e1 + p2 e2 3 p1 + 6 = = 3 p1 3 p1 2( p1 e21 + p2 e22 ) 2(3 p1 + 6) = = 3 p2 3 =
In equilibrium, equilibrium, excess excess demand demand z1 (p) for good 1 is zero. Therefore 18 p1 + 4 3 p1 + 6 + − 18 − 3 = 0, 2 p1 3 p1
Figure 1: Contract Curve and the Core
which gives p1 = 4/ 4 /11 (check that market 2 also clears). The Walrasian equilibrium is p = ( p1 , p2 ) = (4/ (4/11 11,, 1). From the demand functions above, the WEA is (b) The core is the section of the curve in (13 ( 13)) between the points of intersections with the consumers’ inx = (x ( x11 , x12 , x21 , x22 ) = (14. (14.5, 5.27 27,, 5.6, 4.73). 73). difference curves passing through the endowment point. For example, in Figure 1, if G is the endowment point, (d) It is easy to verify that x ∈ C (e). the core is the portion of the contract curve between points W and Z . Consumer 1’s indifference curve pass- Ex. 5.21 Let Y ⊆ Rn be a strongly convex production ing through the endowment is set. For any p ∈ Rn++ , let y1 ∈ Y and y2 ∈ Y be two distinct distinct profit-max profit-maximizin imizingg production production plans. plans. Therefore Therefore 1 2 p · y = p · y ≥ p · y for all y ∈ Y . Y . Since Y is strongly (x11 x12 )2 = (18 · 4)2 , ¯ ∈ Y such that for all t ∈ (0, convex, there exists a y (0, 1), ¯ > ty1 + (1 − t)y2 . y
or x12 = 72/x 72 /x11 . Substituting this into (13 (13)) and rearranging give 1 2 1
Thus
1 1
5(x 5(x ) + 36x 36x − 1512 = 0. 0.
¯ > tp · y1 + (1 − t)p · y2 p·y
= tp · y1 + (1 − t)p · y1 Solving the quadratic equation gives one positive value = p · y1 , of 14. 14.16. Consumer Consumer 2’s utility utility function function can be written written as x21 (x22 )2 . This can can be expres expressed sed in terms terms of x11 and which which contra contradic dicts ts the assump assumptio tion n that that y1 is profitprofitx12 using (11 (11)) and (12 (12). ). The indifference indifference curve curve passing passing 1 2 maximizing. Therefore y = y . through endowment becomes Ex. 5.29 Let E = {(U i , ei , θij , Y j )|i ∈ I , j ∈ J } be (21 − x11 )(10 − x12 )2 = (21 − 18)(10 − 4)2 = 108. 108. the production economy and p ∈ Rn++ be the Walrasian equilibrium. (a) For any consumer i ∈ I , the utility utility maximizati maximization on Putting x12 in (13) (13) into the above equation and solving problem is for x11 give x11 = 15 15..21. Therefore Therefore the core of the econ- problem omy is given by max U i (x) s. t. t. p · x = p · ei + θ ij πj (p),
x
C (e) = (x11 , x12 , x21 , x22 ) : x12 =
1 1
10 10x x , 42 − x11
j ∈J
with necessary condition
∇U i (x) = λp.
14 14..16 ≤ x11 ≤ 15 15..21 21,, x11 + x21 = 21, 21 , x12 + x22 = 10. 10 .
The MRS between two goods l and m is therefore ∂U i (x)/∂x l pl = . i ∂U (x)/∂x m pm
(c) Normalize the price of good 2 to p2 = 1. 1. The 4
Since all consumers observe the same prices, the MRS is the same for each consumer. (b) Similar to part (a) by considering the profit maximization imization problem of any firm. (c) This shows that the Walrasian equilibrium prices play the key role in the functioning of a production economy.. Exchange omy Exchangess are impersonal. impersonal. Each Each consumer consumer only need to know her preferences and each firm its production set. All agents in the economy observe the common price price signal and mak makee their their own own decisio decisions. ns. This This miniminimal information requirement leads to the lowest possible transaction costs of the economy.
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