ECON 5113 Advanced Microeconomics

Winter 2018

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. 1.11 Suppose that p ≥ 0 is a limit point of A. Thenfor every ε > 0, there exists a point q 6= p in (p− ε, p+ ε)such that q ∈ A. This means that for every neighbour-hood Bε(pe) ∈ Rn+, the bundle qe is in % (x). Hencepe is a limit point of % (x). Since % is continuous sothat %(x) is a closed set, pe ∈ %(x), which implies thatp ∈ A. Therefore A is closed.

Ex. 1.14 Let U be a continuous utility function thatrepresents %. Then for all x,y ∈ Rn+, x % y if and onlyif U(x) ≥ U(y).

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

Second, suppose x % y and y % z. Then U(x) ≥ U(y)and U(y) ≥ U(z). This implies that U(x) ≥ U(z) andso x % z, which shows that % is transitive.

Finally, let x ∈ Rn+ and U(x) = u. Then

U−1([u,∞)) = {z ∈ Rn+ : U(z) ≥ u}= {z ∈ Rn+ : z % x}= % (x).

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.

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

A = {x ∈ Rn+ : αa + (1− α)b, 0 ≤ α ≤ 1}.

and suppose that for all x ∈ A, x ∼ a. Then % is convexbut not strictly convex. Theorem 1.1 does not require% to be convex or strictly convex, therefore the utility

function 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 boundedabove in u, that is, for some p � 0, there exists M > 0such that M ≥ E(p, u) for all u in the domain of E.

Let u∗ = V (p,M). Then

E(p, u∗) = E(p, V (p,M)) = M = pTx∗,

where x∗ is the optimal bundle. Since U is continuous,there exists a bundle x′ in the neighbourhood of x∗ 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 that M is 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 definition E(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 whenp = p0.

(b) Since f(p) ≤ 0 for all p � 0 and f(p0) = 0, itmust attain its maximum value at p = p0.

(c) ∇f(p0) = 0.

(d) We have

∇f(p0) = ∇pE(p0, u0)− x0 = 0,

which gives Shephard’s lemma.

Ex. 1.46 Since di is homogeneous of degree zero in pand y, for any α > 0 and for i = 1, . . . , n,

di(αp, αy) = di(p, y).

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

Differentiate both sides with respect to α, we have

∇pdi(αp, αy)Tp +∂di(αp, αy)

∂yy = 0.

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

n∑j=1

∂di(p, y)

∂pjpj +

∂di(p, y)

∂yy = 0. (1)

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}

= uminx{pTx/u : U(x/u) ≥ 1}

= uminx/u{pTx/u : U(x/u) ≥ 1} (2)

= uminz{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/u as 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 that

V (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 on p but not on y.

Ex. 1.54 The utility maximization problem is

maxx

A

n∏i=1

xαii

subject to pTx = y,

where A > 0 and∑ni=1 αi = 1.

(a) The Lagrangian is

L = A

n∏i=1

xαii − λ(pTx− y).

The necessary conditions for maximization are the bud-get constraint and

∂L∂xj

= αjA

∏ni=1 x

αii

xj− λpj = 0,

for j = 1, . . . , n. Consider two goods i and j, the abovenecessary condition implies that

αi/xiαj/xj

=pipj,

which can be rearranged to

xj =

(αjαi

)(pipj

)xi.

Substitute this relation for j = 1, . . . , n in the budgetconstraint, we have

p1

(α1

αi

)(pip1

)xi+· · ·+pixi+· · ·+pn

(αnαi

)(pipn

)xi = y,

or (piαi

)( n∑k=1

αk

)xi = y.

Since∑nk=1 αk = 1, the above equation gives the Mar-

shallian demand function of good i as

di(p, y) = xi =αiy

pi,

for i = 1, . . . , n.

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 utilityfunction V is increasing in income y, it follows that

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

Ex. 1.67 It 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) toobtain u0 = 11/2. Therefore, with p1 = (2, 1),

I =u0 − 1/8

2u0 − 1=

43

80.

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(b) It is clear from part (a) that I depends 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),where g is an increasing function. Then

I =e(p1)g(u0)

e(p0)g(u0)=e(p1)

e(p0),

which means that I is independent of the reference utilitylevel.

Ex. 2.1 Consider the case of one good. Let the demandfunction be

d(p, y) =

√y√p.

It is homogenous of degree zero but it does not satisfybudget balancedness.

Conversely, consider the two-good case that the de-mand function is given by

d(p, y) =

(y log p2

p1 log p2 + p2 log p1,

y log p1p1 log p2 + p2 log p1

).

Then

p · d(p, y) =yp1 log p2

p1 log p2 + p2 log p1+

yp2 log p1p1 log p2 + p2 log p1

= y

and therefore satisfies budget balancedness. It is straightforward to verify that d(p, y) is not a homogenous func-tion.

Ex. 2.2 For i = 1, . . . , n, the i-th row of the matrixmultiplication S(p, y)p is

n∑j=i

[∂di(p, y)

∂pjpj +

∂di(p, y)

∂ypjdj(p, y)

]

=

n∑j=i

∂di(p, y)

∂pjpj +

∂di(p, y)

∂y

n∑j=i

pjdj(p, y)

=

n∑j=i

∂di(p, y)

∂pjpj +

∂di(p, y)

∂yy (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) = minp∈Rn

++

{V (p, 1) : p · x = 1} .

The Lagrangian is

L = −pα1 pβ2 − λ(1− p1x1 − p2x2),

with the first-order conditions

−αpα−11 pβ2 + λx1 = 0

and−βpα1 p

β−12 + λx2 = 0.

Eliminating λ from the first-order conditions gives

p2 =β

α

x1x2p1.

Substitute this p2 into the constraint equation, we get

p1 =α

α+ β

1

x1,

and

p2 =β

α+ β

1

x2.

The utility function is therefore

U(x) =

(ααββ

(α+ β)α+β

)x−α1 x−β2 ,

which is a Cobb-Douglas function.

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

p1x1 + p2x2 ≥up1p2p1 + p2

.

Rearranging gives

u ≤ p1 + p2p2

x1 +p1 + p2p1

x2

for all p ∈ Rn++. This implies that

u ≤ minp1,p2

{p1 + p2p2

x1 +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 α = p2/(p1 + p2) so that 1− α =p1/(p1 + p2) and 0 < α < 1. The minimization problembecomes

minα

{x1α

+x2

1− α: 0 < α < 1

}. (8)

Notice that for any x1 > 0 and x2 > 0,

limα→0

(x1α

+x2

1− α

)=∞

and

limα→1

(x1α

+x2

1− α

)=∞

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so that the minimum value exists when 0 < α < 1. Thefirst-order condition for minimization is

−x1α2

+x2

(1− α)2= 0,

which can be written as

α2x2 = (1− α)2x1.

Taking the square root on both sides gives

αx1/22 = (1− α)x

1/21 .

Rearranging gives

α =x1/21

x1/21 + x

1/22

and 1− α =x1/22

x1/21 + x

1/22

.

It is clear that α is indeed between 0 and 1. Putting αand 1−α into the objective function in (8) give the directutility function

U(x1, x2) =(x1/21 + x

1/22

)2,

which is the CES function with ρ = 1/2. You shouldverify with Example 1.3 on p. 39–41 that the expenditurefunction is indeed as given.

Ex. 3.2 Constant returns-to-scale means that f is lin-early homogeneous. So by Euler’s theorem

x1∂y/∂x1 + x2∂y/∂x2 = y. (9)

Since average product y/x1 is rising, its derivative respectto x1 is positive, that is,

(x1∂y/∂x1 − y)/x21 > 0.

From (9) we have

x2∂y/∂x2 = −(x1∂y/∂x1 − y) < 0,

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

Ex. 4.1 If preferences are identical and homothetic, eachconsumer’s preferences can be represented by a linearlyhomogeneous utility function. Using an argument sim-ilar to exercise 1.47, the ordinary demand function ofconsumer i is separable in p and yi, that is,

di(p, yi) = f(p)yi.

Market demand is therefore∑i

di(p, yi) = f(p)∑i

yi = f(p)y,

where y =∑i yi is aggregate income. It is clear that

market demand depends on aggregate income y but noton income distribution. The market level income elastic-ity of demand for good j is

ηj =∂(fj(p)y)

∂y

y

fj(p)y= 1.

Ex. 4.2 If preferences are homothetic but not identical,the demand function of consumer i is

di(p, yi) = f i(p)yi.

Market demand is therefore∑i

di(p, yi) =∑i

f i(p)yi.

In this case market demand depends on income distribu-tion.

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

Ex. 4.7 Suppose that all firms have the same technologyand therefore the same cost function. Given market pricep, profit of a typical firm j is pqj − c(qj).

(a) Suppose that a > 0, b < 0, and there are J firms inthe industry. The short-run profit maximization for firmj is

maxqj

(α− β

J∑i=1

qi

)qj − (aqj + bq2j ).

The necessary condition for profit maximization is

α− βqj − βJ∑i=1

qi − a− 2bqj = 0. (10)

By symmetry, q1 = q2 = · · · = qJ . Equation (10) be-comes

α− β(J + 1)qj − a− 2bqj = 0,

4

which gives

qj =α− a

β(J + 1) + 2b.

The market output is

q =J(α− a)

β(J + 1) + 2b,

and the market price is

p = α− βJ(α− a)

β(J + 1) + 2b,

(b) The average cost of production is

c(q)

q= a+ bq,

which is a decreasing function if a > 0 and b < 0. Sincethere is no fixed cost, a firm can potentially increase out-put until the average (or total) cost of production is zero,which is at the output level q = −a/b. Notice that at zeromarket price, consumer demand is α/β. Therefore thelong-run equilibrium market price and number of firmsdepend on the relative values of −a/b and α/β.

(c) If a > 0 and b > 0, the minimum efficiency scaleis at q = 0. Therefore the long-run equilibrium marketprice and number of firms are indeterminate.

Ex. 4.12 In the Bertrand duopoly model of section 4.2.1,the firms have no fixed costs and equal marginal cost c.In equilibrium both firms charge p = c and share themarket equally.

(a) Now suppose that firm 1 has fixed costs F > 0. Ifit stays in production, in equilibrium the price it chargesmust be the same as firm 2. This implies that p1 = p2 =p. But the zero profit condition for firm 1 is

pq1 − cq1 − F = 0,

or

p = c+F

q1> c.

Assume that the firm share the market equally, q1 = q2 =Q/2 so that

p = c+2F

Q= c+

2F

α− βp. (11)

Since firm 2 has no fixed cost, it can charge a priceslightly lower than this and capture the whole market.In equilibrium firm 2 charges a price that satisfies equa-tion (11), which can be rearranged to the quadratic equa-tion

βp2 − (α+ βc)p+ (αc+ 2F ) = 0. (12)

Therefore p2 = p, the solution in equation (12), q1 = 0,and q2 = α− βp.

(b) Suppose that both firms have fixed costs F > 0.Then both firms stay in production in equilibrium andcharge a price equal to the solution in equation (12).They share the market with q1 = q2 = (α−βp)/2. Noticethat in equation (12), p = c is the solution if F = 0.

(c) If firm 1 has a lower marginal cost so that c2 >c1 > 0, then firm 1 will charge a price p1 = c2 just tokeep firm 2 out of the market. Therefore q2 = 0 andq1 = α− βc2.

Ex. 4.14 The 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 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.

Ex. 4.19 The Marshallian demands are x∗ = 1/p andm∗ = y − 1. The indirect utility function is V (p, y) =y − log p − 1. If the price of x rises from p0 to p1, thecompensating variation is implicitly defined as

V (p1, y + CV) = V (p0, y),

ory + CV− log p1 − 1 = y − log p0 − 1.

This gives CV = log(p1/p0), which is equal to the changein consumer surplus.

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

∂U1(x11, x12)/∂x11

∂U1(x11, x12)/∂x12

=∂U2(x21, x

22)/∂x21

∂U2(x21, x22)/∂x22

,

orx12x11

=x222x21

. (13)

The feasibility conditions for the two goods are

x11 + x21 = e11 + e21 = 18 + 3 = 21, (14)

x12 + x22 = e12 + e22 = 4 + 6 = 10. (15)

Express x21 in (14) and x22 in (15) in terms of x11 and x12respectively, (13) becomes

x12x11

=10− x12

2(21− x11),

5

Figure 1: Contract Curve and the Core

or

x12 =10x11

42− x11. (16)

Eq. (16) with domain 0 ≤ x11 ≤ 21, (14), and (15) com-pletely characterize the set of Pareto-efficient allocationsA (contract curve). That is,

A =

{(x11, x

12, x

21, x

22) : x12 =

10x1142− x11

, 0 ≤ x11 ≤ 21,

x11 + x21 = 21, x12 + x22 = 10.}

(b) The core is the section of the curve in (16) be-tween the points of intersections with the consumers’ in-difference curves passing through the endowment point.For example, in Figure 1, if G is the endowment point,the core is the portion of the contract curve betweenpoints W and Z. Consumer 1’s indifference curve passingthrough the endowment is

(x11x12)2 = (18 · 4)2,

or x12 = 72/x11. Substituting this into (16) and rearrang-ing give

5(x11)2 + 36x11 − 1512 = 0.

Solving the quadratic equation gives one positive valueof 14.16. Consumer 2’s utility function can be writtenas x21(x22)2. This can be expressed in terms of x11 andx12 using (14) and (15). The indifference curve passingthrough endowment becomes

(21− x11)(10− x12)2 = (21− 18)(10− 4)2 = 108.

Putting x12 in (16) into the above equation and solvingfor x11 give x11 = 15.21. Therefore the core of the economy

is given by

C(e) =

{(x11, x

12, x

21, x

22) : x12 =

10x1142− x11

,

14.16 ≤ x11 ≤ 15.21, x11 + x21 = 21,

x12 + x22 = 10.}

(c) Normalize the price of good 2 to p2 = 1. Thedemand functions of the two consumers are:

x11 =y1

2p1=p1e

11 + p2e

12

2p1=

18p1 + 4

2p1

x12 =y1

2p2=p1e

11 + p2e

12

2p2=

18p1 + 4

2

x21 =y2

3p1=p1e

21 + 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 + 4

2p1+

3p1 + 6

3p1− 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 y1 ∈ Y and y2 ∈ Y be twodistinct profit-maximizing production plans. Thereforep · y1 = p · y2 ≥ p · y for all y ∈ Y . Since Y is stronglyconvex, there exists a y ∈ Y such 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,

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

Ex. 5.31 Let E = {(U i, ei, θij , Y j)|i ∈ I, j ∈ J } bethe production economy and p ∈ Rn++ be the Walrasianequilibrium.

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

maxx

U i(x) s. t. p · x = p · ei +∑j∈J

θijπj(p),

6

with necessary condition

∇U i(x) = λp.

The MRS between two goods l and m is therefore

∂U i(x)/∂xl∂U i(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.

Other Exercises

Theorem 2.1 Here is a suggested proof that the utilityfunction generated by the expenditure function is un-bounded:

On the contrary suppose that U is bounded. That is,there exists a utility level u such that for all x ∈ Rn+,U(x) ≤ u. Let u0 > u and p0 � 0. Then by theconcavity of E in p, for any p� 0,

E(p, u0) ≤ E(p0, u0) +∇pE(p0, u0)T(p− p0)

= E(p0, u0) +∇pE(p0, u0)Tp

−∇pE(p0, u0)Tp0

= E(p0, u0) +∇pE(p0, u0)Tp− E(p0, u0)

= ∇pE(p0, u0)Tp,

where in the second last equality we apply Euler’s theo-rem to E since it is linearly homogeneous in p. Definex0 = ∇pE(p0, u0) so that we have E(p, u0) ≤ pTx0 forall p � 0. In other words, u0 is feasibility set of themaximization problem

U(x0) = max{u : pTx0 ≥ E(p, u) ∀ p� 0}.

Therefore U(x0) ≥ u0, which contradicts the assumptionthat u is an upper bound of U .

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