• Author / Uploaded
• patipet2753

Citation preview

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, ory, Second Edition, Boston: Addison Wesley. The updi (αp, 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)T p +

∂di (αp, αy) y = 0. ∂y

Ex. 1.14 Let U be a continuous utility function that represents %. Then for all x, y ∈ Rn+ , x % y if and only Put α = 1 and rewrite the dot product in summation if U (x) ≥ U (y). form, the above equation becomes n First, suppose x, y ∈ R+ . Then U (x) ≥ U (y) or n U (y) ≥ U (x), which means that x % y or y % x. ThereX ∂di (p, y) ∂di (p, y) y = 0. (1) pj + fore % is complete. ∂p ∂y j j=1 Second, suppose x % y and y % z. Then U (x) ≥ U (y) and U (y) ≥ U (z). This implies that U (x) ≥ U (z) and Dividing each term by di (p, y) yields the result. so x % z, which shows that % is transitive. Finally, let x ∈ Rn+ and U (x) = u. Then Ex. 1.46 Suppose that U (x) is a linearly homogeneous −1 n utility function. U ([u, ∞)) = {z ∈ R+ : U (z) ≥ u} (a) Then = {z ∈ Rn+ : z % x} = % (x).

E(p, u)

Since [u, ∞) is closed and U is continuous, % (x) is closed. Similarly (I suggest you to try this), - (x) is also closed. This shows that % is continuous.

=

min{pT x : U (x) ≥ u}

=

min{upT x/u : U (x/u) ≥ 1}

=

u min{pT x/u : U (x/u) ≥ 1}

=

u min{pT x/u : U (x/u) ≥ 1}

∗

x

x

(2)

x/u

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 (p, M ). Then

x

=

u min{pT z : U (z) ≥ 1}

=

uE(p, 1)

=

ue(p)

z

(3)

T ∗

E(p, u ) = E(p, V (p, M )) = M = p x ,

In (2) above it does not matter if we choose x or x/u directly as long as the objective function and the conwhere x is the optimal bundle. Since U is continuous, straint remain the same. We can do this because of the 0 ∗ there exists a bundle x in the neighbourhood of x such objective function is linear in x. In (3) we simply rewrite 0 0 ∗ that U (x ) = u > u . Since U strictly increasing, E is x/u as z. 0 ∗ 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 (p, y)) = V (p, y)e(p)

so that V (p, y) =

y = v(p)y, e(p)

Ex. 2.3 By (T.1’) on p. 78 U (x) = min {V (p, 1) : p · x = 1} . n

where we have let v(p) = 1/e(p). The marginal utility of income is ∂V (p, y) = v(p), ∂y

p∈R++

The Lagrangian is β L = −pα 1 p2 − λ(1 − p1 x1 − p2 x2 ),

which depends on p but not on y.

with the first-order conditions

Ex. 1.65 (b) By definition y 0 = E(p0 , u0 ), Therefore

−αpα−1 pβ2 + λx1 = 0 1

E(p1 , u0 ) y1 > y0 E(p0 , u0 )

and β−1 −βpα + λx2 = 0. 1 p2

means that y 1 > E(p1 , u0 ). Since the indirect utility function V is increasing in income y, it follows that 1

1

1

1

1

0

Eliminating λ from the first-order conditions gives

0

u = V (p , y ) > V (p , E(p , u )) = u . p2 =

β x1 p1 . α x2

Ex. 1.65 It is straight forward to derive the expenditure Substitute this p into the constraint equation, we get 2 function, which is α 1 p1 = , p22 α + β x1 E(p, u) = p2 u − . (4) 4p1 and β 1 (a) For p0 = (1, 2) and y 0 = 10, we can use (4) to p2 = . obtain u0 = 11/2. Therefore, with p1 = (2, 1), α + β x2 I=

The utility function is therefore   αα β β −β U (x) = x−α 1 x2 , (α + β)α+β

43 u0 − 1/8 = . 2u0 − 1 80

(b) It is clear from part (a) that I depends on u0 . (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 e(p1 )g(u0 ) e(p1 ) constraint pT x ≥ E(p, u) for all p ∈ Rn++ . That is, = , I= 0 0 0 e(p )g(u ) e(p ) up1 p2 p1 x1 + p2 x2 ≥ . p1 + p2 which means that I is independent of the reference utility level. Rearranging gives Ex. 2.2 For i = 1, . . . , n, the i-th row of the matrix multiplication S(p, y)p is  n  X ∂di (p, y) ∂di (p, y) pj + pj dj (p, y) ∂pj ∂y j=i =

n X ∂di (p, y) j=i n X

∂pj

u≤

p1 + p2 p1 + p2 x1 + x2 p2 p1

for all p ∈ Rn++ . This implies that   p1 + p2 p1 + p2 u ≤ min x1 + x2 . p1 ,p2 p2 p1

n

∂di (p, y) X pj + pj dj (p, y) ∂y j=i

(7)

Therefore u attains its maximum value when equality holds in (7). To find the minimum value on the right= (5) hand side of (7), write α = p2 /(p1 + p2 ) so that 1 − α = j=i p1 /(p1 + p2 ) and 0 < α < 1. The minimization problem =0 (6) becomes   where in (5) we have used the budget balancedness and x2 x1 min + :0 0,   x1 x2 lim + =∞ α→0 α 1−α

max py − c(w)y = max y[p − c(w)]. y

y

For a competitive firm, as long as p > c(w), the firm will increase output level y indefinitely. If p < c(w), profit lim =∞ is negative at any level of output except when y = 0. α→1 If p = c(w), profit is zero at any level of output. 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 x2 x1 function of y. Therefore the supply function of the firm = 0, − 2+ α (1 − α)2 does not exist and the number of firm is indeterminate. and



x1 x2 + α 1−α



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 − 15q − (J − 1)¯ q ]q − (q 2 + 1), q

Taking the square root on both sides gives 1/2

αx2

with necessary condition

1/2

= (1 − α)x1 .

10 − 15q − (J − 1)¯ q − 15q − 2q = 0.

Rearranging gives 1/2

1/2

α=

x1 1/2

x1

(a) Since all firms are identical, by symmetry q = q¯. This gives the Cournot equilibrium of each firm q ∗ = 10/(J + 31), with market price p∗ = 170/(J + 31). (b) Short-run profit of each firm is π = [40/(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 between 0 and 1. Putting α and 1 − α into the objective function in (8) give the Ex. 5.11 (a) The necessary condition for a Paretodirect utility function efficient allocation is that the consumers’ MRS are equal. Therefore  2 1/2 1/2 U (x1 , x2 ) = x1 + x2 , ∂U 2 (x21 , x22 )/∂x21 ∂U 1 (x11 , x12 )/∂x11 = , ∂U 1 (x11 , x12 )/∂x12 ∂U 2 (x21 , x22 )/∂x22 which is the CES function with ρ = 1/2. You should or verify with Example 1.3 on p. 38–39 that the 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, (11) x1 ∂y/∂x1 + x2 ∂y/∂x2 = y.

x12 + x22 = e12 + e22 = 4 + 6 = 10.

(9)

(12)

Express x21 in (11) and x22 in (12) in terms of x11 and x12 Since average product y/x1 is rising, its derivative re- respectively, (10) becomes spect to x1 is positive, that is, x12 10 − x12 = , 2 1 (x1 ∂y/∂x1 − y)/x1 > 0. x1 2(21 − x11 ) or From (9) we have 10x11 x12 = . (13) 42 − x11 x2 ∂y/∂x2 = −(x1 ∂y/∂x1 − y) < 0, Eq. (13) with domain 0 ≤ x11 ≤ 21, (11), and (12) comwhich means that the marginal product ∂y/∂x2 is neg- pletely characterize the set of Pareto-efficient allocations A (contract curve). That is, ative.  10x11 A = (x11 , x12 , x21 , x22 ) : x12 = , 0 ≤ x11 ≤ 21, Ex. 4.5 Let w be the vector of factor prices and p 42 − x11 be the output price. Then the cost function of a typ x11 + x21 = 21, x12 + x22 = 10. ical firm with constant returns-to-scale technology is 3

demand functions of the two consumers are: y1 2p1 y1 x12 = 2p2 y2 x21 = 3p1 2y 2 x22 = 3p2

x11 =

p1 e11 + p2 e12 18p1 + 4 = 2p1 2p1 p1 e11 + p2 e12 18p1 + 4 = = 2p2 2 p1 e21 + p2 e22 3p1 + 6 = = 3p1 3p1 2(p1 e21 + p2 e22 ) 2(3p1 + 6) = = 3p2 3 =

In equilibrium, excess demand z1 (p) for good 1 is zero. Therefore 18p1 + 4 3p1 + 6 + − 18 − 3 = 0, 2p1 3p1

Figure 1: Contract Curve and the Core

which gives p1 = 4/11 (check that market 2 also clears). The Walrasian equilibrium is p = (p1 , p2 ) = (4/11, 1). From the demand functions above, the WEA is

(b) The core is the section of the curve in (13) between the points of intersections with the consumers’ inx = (x11 , x12 , x21 , x22 ) = (14.5, 5.27, 5.6, 4.73). 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 profit-maximizing production plans. Therefore 1 1 2 2 p · y1 = p · y2 ≥ p · y for all y ∈ Y . Since Y is strongly (x1 x2 ) = (18 · 4) , ¯ ∈ Y such that for all t ∈ (0, 1), convex, there exists a y ¯ > ty1 + (1 − t)y2 . y

or x12 = 72/x11 . Substituting this into (13) and rearranging give 5(x11 )2

+

36x11

Thus

− 1512 = 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 14.16. Consumer 2’s utility function can be written as x21 (x22 )2 . This can be expressed in terms of x11 and which contradicts the assumption that y1 is profitx12 using (11) and (12). The indifference curve passing maximizing. Therefore y1 = y2 . through endowment becomes (21 −

x11 )(10

−

x12 )2

Ex. 5.29 Let E = {(U i , ei , θij , Y j )|i ∈ I, j ∈ J } be the production economy and p ∈ Rn++ be the Walrasian equilibrium. (a) For any consumer i ∈ I, the utility maximization problem is X max U i (x) s. t. p · x = p · ei + θij π j (p),

2

= (21 − 18)(10 − 4) = 108.

Putting x12 in (13) into the above equation and solving for x11 give x11 = 15.21. Therefore the core of the economy is given by

x

 C(e) =

(x11 , x12 , x21 , x22 )

:

x12

10x11 , = 42 − x11

j∈J

with necessary condition ∇U i (x) = λp.

14.16 ≤ x11 ≤ 15.21, x11 + x21 = 21, x12 + x22 = 10.

The MRS between two goods l and m is therefore pl ∂U i (x)/∂xl = . i ∂U (x)/∂xm pm

(c) Normalize the price of good 2 to p2 = 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 problem of any firm. (c) This shows that the Walrasian equilibrium prices play the key role in the functioning of a production economy. Exchanges are impersonal. Each consumer only need to know her preferences and each firm its production set. All agents in the economy observe the common price signal and make their own decisions. This minimal information requirement leads to the lowest possible transaction costs of the economy.

c

2011 The Pigman Inc. All Rights Reserved.

5