Jehle Solutions

7/24/2019 Jehle Solutions

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ECON 5113 Advanced Microeconomics

Winter 2015

Answers to Selected Exercises Instructor: Kam Yu

The following questions are taken from Geoffrey A. Jehleand Philip J. Reny (2011)Advanced Microeconomic The-ory, Third Edition, Harlow: Pearson Education Limited.The updated version is available at the course web page:

http://flash.lakeheadu.ca/kyu/E5113/Main.html

Ex. 1.14 Let U be a continuous utility function thatrepresents. Then for allx, y Rn+, x y if and onlyifU(x) U(y).

First, suppose x, y Rn+. Then U(x) U(y) orU(y) U(x), which means that x yor y x. There-fore is complete.

Second, supposex yand y z. ThenU(x) U(y)and U(y) U(z). This implies that U(x) U(z) andso x z, which shows that is transitive.

Finally, letx Rn+ andU(x) = u. Then

U1([u, )) = {z Rn+

: U(z) u}

= {z Rn+ : z x}

= (x).

Since [u, ) is closed andUis continuous, (x) is closed.Similarly (I suggest you to try this), (x) is also closed.This shows that is continuous.

Ex. 1.17 Suppose that a and b are two distinct bundlesuch thata b. Let

A= {x Rn+ : a + (1 )b, 0 1}.

and suppose that for all x A, x a. Then is convex

but not strictly convex. Theorem 1.1 does not require to be convex or strictly convex, therefore the utilityfunction exists. Moreover, since (a) = (b) is convex,there exists a supporting hyperplane H = {x Rn+ :pTx= y} such that a, b H. Since H is an affine set,A H. This means that every bundle in A is a solutionto the utility maximization problem.

Ex. 1.341 Suppose on the contrary that E is bounded

1It may be helpful to review the proof of Theorem 1.8.

above in u, that is, for some p 0, there exists M >0such thatM E(p, u) for all u in the domain ofE.

Letu =V(p, M). Then

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

where x

is the optimal bundle. SinceU is continuous,there exists a bundlex in the neighbourhood ofx suchthat U(x) = u > u. Since U strictly increasing, E isstrictly increasing in u, so that E(p, u) > E(p, u) =M. This contradicts the assumption thatMis an upperbound.

Ex. 1.37 (a) Since x0 is the solution of the expenditureminimization problem when the price is p0 and utilitylevel u0, it must satisfy the constraint U(x0) u0. Nowby definitionE(p, u0) is the minimized expenditure whenprice is p, it must be less than or equal to pTx0 sincex0 is in the feasible set, and by definition equal when

p= p0.(b) Since f(p) 0 for all p 0 and f(p0) = 0, it

must attain its maximum value at p = p0.(c)f(p0) = 0.(d) We have

f(p0) = pE(p0, u0) x0 =0,

which gives Shephards lemma.

Ex. 1.46 Since di is homogeneous of degree zero in pandy , for any >0 and fori = 1, . . . , n,

di(p, y) = di(p, y).Differentiate both sides with respect to , we have

pdi(p, y)Tp +

di(p, y)

y y= 0.

Put = 1 and rewrite the dot product in summationform, the above equation becomes

nj=1

di(p, y)

pjpj +

di(p, y)

y y= 0. (1)
http://flash.lakeheadu.ca/~kyu/http://flash.lakeheadu.ca/~kyu/E5113/Main.htmlhttp://flash.lakeheadu.ca/~kyu/E5113/Main.htmlhttp://flash.lakeheadu.ca/~kyu/E5113/Main.htmlhttp://flash.lakeheadu.ca/~kyu/E5113/Main.htmlhttp://flash.lakeheadu.ca/~kyu/


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Dividing each term by di(p, y) yields the result.

Ex. 1.47 Suppose that U(x) is a linearly homogeneousutility function.

(a) Then

E(p, u) = minx

{pTx: U(x) u}

= minx

{upTx/u: U(x/u) 1}

= u minx

{pTx/u: U(x/u) 1}

= u minx/u

{pTx/u: U(x/u) 1} (2)

= u minz

{pTz: U(z) 1} (3)

= uE(p, 1)

= ue(p)

In (2) above it does not matter if we choose x or x/udirectly as long as the objective function and the con-

straint remain the same. We can do this because of theobjective function is linear in x. In (3) we simply rewritex/uas z.

(b) Using the duality relation between V and E andthe result from Part (a) we have

y= E(p, V(p, y)) = V(p, y)e(p)

so thatV(p, y) =

y

e(p)=v(p)y,

where we have let v(p) = 1/e(p). The marginal utilityof income is

V(p, y)

y =v(p),

which depends onp but not on y .

Ex. 1.66 (b) By definition y0 =E(p0, u0), Therefore

y1

y0 >

E(p1, u0)

E(p0, u0)

means that y1 > E(p1, u0). Since the indirect utilityfunctionVis increasing in incomey , it follows that

u1 =V(p1, y1)> V(p1, E(p1, u0)) = u0.

Ex. 1.67It is straight forward to derive the expenditurefunction, which is

E(p, u) = p2u p224p1

. (4)

(a) For p0 = (1, 2) and y0 = 10, we can use (4) toobtainu0 = 11/2. Therefore, with p1 = (2, 1),

I=u0 1/8

2u0 1 =

43

80.

(b) It is clear from part (a) that Idepends on u0.(c) Using the technique similar to Exercise 1.47, it can

be shown that if U is homothetic, E(p, u) = e(p)g(u),whereg is an increasing function. Then

I=e(p1)g(u0)

e(p0

)g(u0

)

=e(p1)

e(p0

)

,

which means thatIis independent of the reference utilitylevel.

Ex. 2.2 For i = 1, . . . , n, the i-th row of the matrixmultiplicationS(p, y)pis

nj=i

di(p, y)

pjpj +

di(p, y)

y pjdj(p, y)

=nj=i

di(p, y)

pjpj+

di(p, y)

y

nj=i

pjdj(p, y)

=

nj=i

di(p, y)pj

pj+ di(p, y)y

y (5)

= 0 (6)

where in (5) we have used the budget balancedness and(6) holds because of homogeneity and (1) in Ex. 1.46.

Ex. 2.3 By (T.1) on p. 82,

U(x) = minpRn

++

{V(p, 1) : p x= 1} .

The Lagrangian is

L = p1p2 (1 p1x1 p2x2),

with the first-order conditions

p11 p2 + x1 = 0

andp1p

12 +x2 = 0.

Eliminating from the first-order conditions gives

p2 =

x1x2

p1.

Substitute this p2 into the constraint equation, we get

p1 =

+

1

x1,

and

p2 =

+

1

x2.

The utility function is therefore

U(x) =

(+)+

x1 x

2 ,

2


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which is a Cobb-Douglas function.

Ex. 2.6 We want to maximize utility u subject to theconstraintpTx E(p, u) for all p Rn++. That is,

p1x1+p2x2 up1p2p1+p2

.

Rearranging gives

u p1+p2

p2x1+

p1+p2p1

x2

for all p Rn++. This implies that

u minp1,p2

p1+p2

p2x1+

p1+p2p1

x2

. (7)

Therefore u attains its maximum value when equalityholds in (7). To find the minimum value on the right-hand side of (7), write = p2/(p1+p2) so that 1 =

p1/(p1+ p2) and 0< 0 and x2 > 0,

lim0

x1

+ x21

=

and

lim1

x1

+ x21

=

so that the minimum value exists when 0 < 0.

From (9) we have

x2y/x2 = (x1y/x1 y)< 0,

which means that the marginal product y/x2 is nega-tive.

Ex. 4.5 Let w be the vector of factor prices and pbe the output price. Then the cost function of a typ-ical firm with constant returns-to-scale technology isC(w, y) = c(w)y where c is the unit cost function. Theprofit maximization problem can be written as

maxy

py c(w)y= maxy

y[p c(w)].

For a competitive firm, as long as p > c(w), the firm willincrease output level y indefinitely. If p < c(w), profitis negative at any level of output except when y = 0.If p = c(w), profit is zero at any level of output. Infact, market price, average cost, and marginal cost are

all equal so that the inverse supply function is a constantfunction ofy. Therefore the supply function of the firmdoes not exist and the number of firm is indeterminate.

Ex. 4.14The profit maximization problem for a typicalfirm is

maxq

[10 15q (J 1)q]q (q2 + 1),

with necessary condition

10 15q (J 1)q 15q 2q= 0.

(a) Since all firms are identical, by symmetry q = q.

This gives the Cournot equilibrium of each firm q =10/(J+ 31), with market pricep = 170/(J+ 31).

(b) Short-run profit of each firm is = [40/(J+31)]21. In the long-run = 0 so that J= 9.

Ex. 5.11 (a) The necessary condition for a Pareto-efficient allocation is that the consumers MRS are equal.Therefore

U1(x11, x12)/x

11

U1(x11, x12)/x

12

=U2(x21, x

22)/x

21

U2(x21, x22)/x

22

,

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Figure 1: Contract Curve and the Core

orx12x11 =

x222x21 . (10)

The feasibility conditions for the two goods are

x11+x21 = e

11+e

21 = 18 + 3 = 21, (11)

x12+x22 = e

12+e

22 = 4 + 6 = 10. (12)

Expressx21 in (11) and x22 in (12) in terms ofx

11 and x

12

respectively, (10) becomes

x12x11

= 10 x122(21 x11)

,

or

x12 = 10x1142 x11

. (13)

Eq. (13) with domain 0 x11 21, (11), and (12) com-pletely characterize the set of Pareto-efficient allocationsA (contract curve). That is,

A =

(x11, x

12, x

21, x

22) : x

12 =

10x1142 x11

, 0 x11 21,

x11+x21 = 21, x

12+x

22 = 10.

(b) The core is the section of the curve in (13) be-

tween the points of intersections with the consumers in-

difference curves passing through the endowment point.For example, in Figure 1, ifG is the endowment point,the core is the portion of the contract curve betweenpointsWandZ. Consumer 1s indifference curve passingthrough the endowment is

(x11x12)2 = (18 4)2,

orx12 = 72/x11. Substituting this into (13) and rearrang-

ing give5(x11)

2 + 36x11 1512 = 0.

Solving the quadratic equation gives one positive valueof 14.16. Consumer 2s utility function can be writtenas x21(x

22)2. This can be expressed in terms of x11 and

x12 using (11) and (12). The indifference curve passingthrough endowment becomes

(21 x

1

1)(10 x

1

2)

2

= (21 18)(10 4)

2

= 108.Putting x12 in (13) into the above equation and solvingforx11givex

11 = 15.21. Therefore the core of the economy

is given by

C(e) =

(x11, x

12, x

21, x

22) : x

12 =

10x1142 x11

,

14.16 x11 15.21, x11+x

21 = 21,

x12+x22 = 10.

(c) Normalize the price of good 2 to p2 = 1. The

demand functions of the two consumers are:

x11 = y1

2p1=

p1e11+p2e12

2p1=

18p1+ 4

2p1

x12 = y1

2p2=

p1e11+p2e12

2p2=

18p1+ 4

2

x21 = y2

3p1=

p1e21+p2e

22

3p1=

3p1+ 6

3p1

x22 =2y2

3p2=

2(p1e21+p2e

22)

3p2=

2(3p1+ 6)

3

In equilibrium, excess demand z1(p) for good 1 is zero.Therefore

18p1+ 42p1

+3p1+ 63p1

18 3 = 0,

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

x= (x11, x12, x

21, x

22) = (14.5, 5.27, 5.6, 4.73).

(d) It is easy to verify that x C(e).

Ex. 5.23 Let Y Rn be a strongly convex productionset. For any p Rn++, let y

1 Y and y2 Y be twodistinct profit-maximizing production plans. Therefore

p y1 =p y2 p y for all y Y. Since Y is stronglyconvex, there exists ay Ysuch that for all t (0, 1),

y> ty1 + (1 t)y2.

Thus

p y> tp y1 + (1 t)p y2

=tp y1 + (1 t)p y1

=p y1,

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which contradicts the assumption that y1 is profit-maximizing. Thereforey1 =y2.

Ex. 5.31 Let E = {(Ui, ei, ij , Yj )|i I, j J } bethe production economy and p Rn++ be the Walrasianequilibrium.

(a) For any consumer i I, the utility maximizationproblem is

maxx

Ui(x) s. t. p x= p ei +jJ

ijj(p),

with necessary condition

Ui(x) = p.

The MRS between two goods l and m is therefore

Ui(x)/xlUi(x)/xm

= plpm

.

Since all consumers observe the same prices, the MRS isthe same for each consumer.

(b) Similar to part (a) by considering the profit maxi-mization problem of any firm.

(c) This shows that the Walrasian equilibrium pricesplay the key role in the functioning of a production econ-omy. Exchanges are impersonal. Each consumer onlyneed to know her preferences and each firm its produc-tion set. All agents in the economy observe the commonprice signal and make their own decisions. This mini-mal information requirement leads to the lowest possibletransaction costs of the economy.

c2015 The Pigman Inc. All Rights Reserved.

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