Advanced Microeconomics: Problems

Atsushi Kajii�

Institute of Economic Research, Kyoto University

September 1, 2013

Abstract

This is a master copy - do not think my students do all of them! I do cut and paste from this

master copy to create assignments for my advanced microeconomics course. Some of the problems

are original, but many are copied from various textbooks and slightly modi�ed to my taste.

There is a solution manual I wrote. It contains not only solutions but also discussions on

related topics. It is available upon request.

1 Mathematics

1. Let f and g be functions on R2 given by f (x1; x2) = �x1+�x2 and g (x1; x2) = (x1 � )2+(x2)2,where �; �; and are constants.

(a) Let � = 1, � = 2, and draw�(x1; x2) 2 R2 : f (x1; x2) = 3

:

(b) Let = 1, and draw�(x1; x2) 2 R2 : g (x1; x2) � 1

: Also write the vector Dg (x1; x2) +

(x1; x2) for (x1; x2) = (1; 1) ; whereDg (x1; x2) is the gradient vector�

@@x1g (x1; x2) ;

@@x2g (x1; x2)

�:

(c) Solve minx1;x2 f (x1; x2) subject to g (x1; x2) � 1, using the Kuhn-Tucker method. Explainthe �rst order condition graphically for � = 1; � = 2; = 1.

(d) What about minx1;x2 f (x1; x2) subject to g (x1; x2) � 1? What is the �rst order conditionfor this problem? Is it necessary and/or su¢ cient for minimization?

2. A set X in RL is convex if for any x; y 2 X, and for any t 2 [0; 1]; tx+ (1� t) y 2 X.

(a) Show thatn(x1; x2) 2 R2 : (x1)2 + (x2)2 � 1

ois a convex set, by verifying the property

above (i.e., not just drawing a picture).

(b) Let p 2 RL. Show that�x 2 RL : p � x = 0

is a convex set, by verifying the property above

(i.e., not just drawing a picture).

(c) Prove that if X and Y are convex sets in RL, then X \ Y is a convex set in RL.

3. Let C be a convex set in RL. A function f on C is convex if for any x; y 2 C, and for anyt 2 [0; 1]; tf (x) + (1� t) f (y) � f (tx+ (1� t) y).

(a) Let p 2 RL. Verify that the function x 7! p � x is convex.

(b) Let f be a convex function on C. Prove that the set f(x; �) 2 C � R : f (x) � �g is a convexset (remark. this set is called the epigraph of f).

�Most problems are written by Atsushi Kajii, but some of them are taken form variaous textbooks or academicpapers without acknowledgement (in fact he does not remember where they are taken from).

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(c) Let f be a function on C. Suppose that f(x; �) 2 C � R : f (x) � �g is a convex set. Verifythat f is a convex function.

(d) Let f and g be convex functions on C. Show that max ff; gg (that is, the function x 7!max ff (x) ; g (x)g) is a convex function. [Hint. look at the epigraph.]

4. Show that:

(a) g (x1; :::; xK) �PK

k=1 gk (xk) is a concave function if each gk, k = 1; :::;K is a concave

function.

(b) if f (x1; :::; xK) is quasi concave and homogeneous of degree one, f is concave.

5. [Homogeneity and convexity] A function f : RL+ ! R is homogeneous of degree � if f (tx) =

t�f (x) for any x and t > 0. A set C in RL is said to be convex if for any x; y 2 C and t 2 [0; 1] ;tx + (1� t) y 2 C. A function f : C ! R is said to be convex if for any x; y 2 C, and for anyt 2 [0; 1]; tf (x) + (1� t) f (y) � f (tx+ (1� t) y). A function f : C ! R is said to be concaveif �f is convex. A function f : C ! R is said to be quasi-convex if fx : f (x) � �g is a convexset for any �. A function f : C ! R is said to be quasi-concave if fx : f (x) � �g is a convexset for any �.

(a) Show that f (x1; :::; xK) = (x1)�1 (x2)

�2 � � � (xK)�K with �k > 0, k = 1; :::;K and �1 +

�2 + � � ��K = 1 is homogeneous with degree 1.

(b) Show that if a function f : R! R is non-decreasing, it is quasi-convex and quasi-concave.

(c) Give a simple example of a quasi-concave function which is not concave.

(d) Show that g (x1; :::; xL) �PK

k=1 gk (x1; :::; xL) is a concave function if each gk, k = 1; :::;K

is a concave function.

(e) Show that if h : R! R is increasing and f : C ! R is concave, then h (f (x)) is a quasi-concave function of x.

(f) Show that if f (x1; :::; xK) is quasi concave and homogeneous of degree one, f is concave.

(g) Show that f (x1; :::; xK) = (x1)�1 (x2)

�2 � � � (xK)�K is concave if �k > 0, k = 1; :::;K and

�1 + �2 + � � ��K � 1.

6. Duality. Let u be a function on RL+ and p 2 RL++ is a vector. Fix a number � and assume thatboth �m := inf fp � x : u (x) � �g and �v := supfu (x) : p � x � �mg exist (note that since u maynot be continuous, inf and sup are di¤erent from min and max in general).

(a) Prove that if fx : u (x) � �g is closed, �v � � holds Give an example which shows v < �

can occur if fx : u (x) � �g is not closed.

(b) Prove that if fx : u (x) > �g is open, �v � � holds. Give an example which shows v > �

can occur if fx : u (x) > �g is not open.

(c) Show that �v = � if u is continuous.

7. A simpler version of the Gale Nikaido theorem. Let 4 be the L� 1 dimensional simplexin RL, i.e., 4 :=

n�p1; :::; pL

�� 0 :

PLl=1 p

l = 1o, and let f : 4! RL be a continuous function

such that p � f (p) � 0 for any p 2 4. (Note that f is de�ned even for the case where some�prices�are zero.) Let �f (p) := maxl f l (p), and � (p) := fq 2 4 : ql = 0 if f l (p) < �f (p)g: So,q 2 � (p) if and only if q � f (p) = �f (p).

(a) Show that � satis�es the conditions for Kakutani�s �xed point theorem.

Microeconomic Theory problems by A. Kajii 3

(b) So by Kakutani�s theorem, there is p� 2 � (p�). Show that f (p�) � 0 (i.e., f l (p�) � 0 forevery l) must hold.

(c) Assume in addition that p � f (p) = 0 for any p 2 4, and if pl = 0 for some l, there is l0

(may be the same as l) such that pl0= 0 and f l

0(p) > 0. Show that a �xed point p� found

above in fact must satisfy f (p�) = 0.

8. A �xed point theorem. Let 4 be the L� 1 dimensional simplex in RL, and let f : 4! RL

be a continuous function such thatPL

l=1 fl (p) = 0 for any p 2 4. The Fan-Brouwer theorem

says that f has a �xed point if f is inward pointing, that is, if pl = 0 implies f l0(p) > 0 for

some l0 with pl0= 0 (l0 may be same as l). This question asks you to establish this result from

the Gale-Nikaido theorem:

(a) Show that if f is inward pointing, then pl = 0 implies f l (p) � 0. [Hint: consider a sequencepn convergent to p, such that elements of pn are positive except for l]

Assume on the contrary that f does not have a �xed point from now on.

(b) De�ne � (p) :=�q 2 4 : ql = 0 if f l (p) < pl

. Show that � is non-empty convex valued

and its graph is closed (i.e., upper hemicontinuous).

(c) So by Kakutani�s theorem there is �p 2 4 such that �p 2 � (�p). Show that this contradictsthe assumption that f is inward - pointing.

2 Producer Theory

1. For a Cobb-Douglas production function f (z1; z2) = (z1)�(z2)

� where z1 and z2 are inputs,

�+ � � 1; �; � > 0,

(a) Find the pro�t function, supply correspondence, cost function, factor demand function.

(b) Suppose z2 is �xed, but not sunk. That is, the producer can use either z2 units of factor

2 or choose zero input (and then f = 0 by construction). Find the average cost curve, the

marginal cost curve, and the supply curve. Draw graphs.

(c) As z2 changes, how do they change?

(d) What happens if � > 1? And if �+ � > 1? Interpret.

2. Show that:

(a) g (x1; :::; xK) �PK

k=1 gk (xk) is a concave function if each gk, k = 1; :::;K is a concave

function.

(b) if f (x1; :::; xK) is quasi concave and homogeneous of degree one, f is concave.

3. For a CES production function withK factors of production f (z) � f�z1; z2; :::; zK

�=hPK

k=1 ak�zk��i 1�

;

where ak > 0; k = 1; :::;K; are constants withP

k ak = 1, and � � 1; � 6= 0;

(a) Show that this technology exhibits constant returns to scale.

(b) Show that f is a concave function.

(c) Find the cost function and the conditional factor demand function

4. Let c (w; y) be the cost function and z (w; y) be the conditional factor demand function. Prove:

(a) c (tw; y) = tc (w; y) for any t > 0;

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(b) Dwc (w; y) = z (w; y)

(c) if the corresponding technology exhibits constant returns to scale, c (w; ty) = tc (w; y) for

any t > 0.

(d) c (w; y) is concave in w:

(e) c (w; y) is convex in y, if the corresponding production function f is concave.

5. Consider a general class of linear activity production set Y =nPK

k=1 tkak : tk � 0o, where ak,

k = 1; :::;K, is a �xed technology vector in RL.

(a) For an example, let L = 2, K = 3, and a1 = (�3; 1) ; a2 = (�2; 2) ; a3 = (1;�5). Drawproduction set Y . If price vector p is given by p = (1; 1), what is the supply (set)? What

if p = (5; 1) ; or p = (2; 1)?

(b) Show that production set Y exhibits CRS.

(c) Show that for a given price vector p: if p � ak > 0 for some k, then the maximum pro�t is

+1. If p �ak � 0 for all k, then the maximum pro�t is zero, and the supply correspondenceis given by y (p) =

nPKk=1 tkak : tk � 0, and tk = 0 if p � ak < 0

o.

6. Consider a competitive �rm with a concave production function f�z1; :::; zK

�; where zk is the

amount of good k used as an input. The price of output is one, and the price of good k is wk, so

the total cost of inputs isPK

k=1 wkzk. The �rm is under a �nancial constraint so that there is

a maximum amount it can spend for purchase of factors of production. Let be this maximum

amount. So the total cost of �rm�s inputs must be no larger than . Answer the following

questions.

(a) Write this �rm�s pro�t maximization problem.

(b) Show that if is large enough, the total cost of pro�t maximizing combination of inputs

will be less than . (That is, the �nancial constraint does not matter.)

(c) Show that the �rm�s pro�t is non decreasing in ; that is, as the �nancial constraint gets

less severe, the pro�t tends to increase.

3 Consumer Theory

1. For a Cobb-Douglas utility function u�x1; x2

�=�x1�� �

x2��with � + � > 0; �; � > 0, �nd

the demand function, the minimum expenditure function, Hicksian demand function, and the

indirect utility function.

2. Consider utility function u�x1; x2

�= min

��1x1; �2x2

where �1 and �2 are positive constants.

Draw the indi¤erence curve corresponding to u�x1; x2

�= 1. Find the demand function, the

minimum expenditure function, Hicksian demand function, and the indirect utility function.

3. Consider utility function u�x1; x2

�= �1x1+�2x2 where �1 and �2 are positive constants. Draw

the indi¤erence curve corresponding to u�x1; x2

�= 1. Find the demand function, the minimum

expenditure function, Hicksian demand function, and the indirect utility function.

4. For a quasi-linear utility function u�x1; x2

�= v

�x1�+ x2, where v0 > 0; v00 < 0; show that

the demand for good 1 does not depend on income. Can you say anything about the form of

the minimum expenditure function, Hicksian demand function, and the indirect utility function?

[Do not worry about the boundary: do as if negative consumption is allowed.]

Microeconomic Theory problems by A. Kajii 5

5. Let x (p;m) =�xk (p;m)

�Lk=1

be a demand function.

(a) Show thatPL

k=1 pk @@mx

k = 1.

(b) � @@plxl (p;m) =

�xl

pl

�is called the price elasticity of demand for good l (with respect to its

own price). Show that the consumer will spend less on good l as pl (marginally) goes up if

and only if the price elasticity is more than one.

(c) If @@mx

l (p;m) � 0; good l is said to be normal at (p;m). A good is called a normal goodif it is normal at any (p;m), otherwise it is called an inferior good. If @

@plxl (p;m) > 0

occurs, good l is called a Gi¤en good. Show that a Gi¤en good cannot be normal. Show

graphically that an inferior good is not necessarily a Gi¤en good.

6. Consider an additively separable utility function u (x) = u1�x1�+ � � �+ uL

�xL�:

(a) Show that the preference relation represented by a Cobb-Douglas utility function in (1)

can be represented by an additively separable utility function.

(b) Show that if each ul is concave, so is u.

(c) Show that goods are normal.

7. Prove that the derivatives of the minimum expenditure function give the Hicksian demand.

8. Revealed Preference. Let there be two goods and consider a consumer with income m facing

prices p =�p1; p2

�. Suppose that this consumer exhibits the consumption pattern as in the

following table:

prices pk income mk chosen bundle

(a) (1; 2) 5 (3; 1)

(b) (2; 1) 5 (1; 3)

(c) (1; 1) 4�y1; y2

�(d)

�12 ;

13

�3 (4; 3)

(e) (1; 3) 6 (1; 2)

For instance, this consumer chooses to consume (3; 1) when prices are (1; 2) and m = 5: The

chosen bundle y =�y1; y2

�for case (c) is missing in the data.

(a) By drawing a picture, show that choices (a) and (b) above are consistent with utility

maximization.

(b) By drawing a picture, show that choices (a) and (e) above are inconsistent with utility

maximization.

(c) By drawing a picture, �nd the area for x that choices (a), (b) and (c) above are consistent

with utility maximization.

(d) Now assume that�y1; y2

�= (2; 2). De�ne utility function u on R2 by the rule u (x) =

mink2fa;b;c;dg f�k (pk � x) + �kg ; where k denote each case (i.e., pa = (1; 2)), each �k is

some positive number, and �k is a constant. Show that by appropriately choosing �k�s and

�k�s, u (x) is a concave function which justi�es cases (a) to (d). [Hint. Let �a = �b = 1 and

�c =54 , and �a = �b = �c = 0: Draw the indi¤erence curve for utility level 5, pretending

that k = d is never a minimizer, i.e., as if u (x) = mink2fa;b;cg f�k (pk � x) + �kg : Play with�d to accommodate (d).]

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4 Risk and Uncertainty

1. Consider an agent with concave vNM utility function u. He can spend W for a gamble (x1; x2)

- he receives x1 with probability p and x2 with probability 1 � p. Show that he accepts moregambles as the ratio r (W ) � �u00(W )

u0(W ) [Arrow-Pratt measure of absolute risk aversion] gets

smaller. Do the same exercise for relative gambles (i.e., x1 and x2 represent the fractions of the

initial wealth and so he receives (x1W;x2W )) and�u00(W )Wu0(W ) [Arrow-Pratt measure of relative

risk aversion.]

2. Prove that the EU representation is unique up to positive a¢ ne transformation. [You may

assume that the set of outcomes is [x; �x]. ]

3. Let u be a vNM utility function. The certainty equivalent for a random income X is a number

c that is de�ned by the rule E [u (X)] = u (c). Assume that X is a discrete random variable.

(a) Interpret c.

(b) Let X be the lottery which yields 4 with probability p and 1 with probability 1� p. Findthe certainty equivalent of X for the following utility functions:

i. u (x) =px

ii. u (x) = ax where a > 0 is a constant.

iii. u (x) = x2

(c) Show that if v is a positive a¢ ne transformation of u, the certainty equivalent of X for v

is the same as that for u. That is, the certainty equivalent depends on the underlying risk

preferences, not a particular choice of vNM function.

(d) Show that c � E [X] if u is concave (thus risk averse).What about the converse?

Suppose that there is another random variable S that may be correlated with X. De�ne the

conditional certainty equivalent C (S) by the rule E [u (X)� u (C (S)) jS] = 0, where E [�jS]denotes the conditional expectation given S. That is, C (S) is the certainty equivalent after

observing the realization of signal S. So S can be considered as a piece of information about

X that is revealed before the level of income gets known.

(e) Show that E [u (C (S))] = u (c).

(f) Show that E [C (S)] � c.

(g) Is information S valuable?

4. By a lottery X we mean that a random variable whose cumulative distribution function is FX .

We assume that the lotteries are bounded, so there is numbers � and �, � < �; such that every

outcome of each lottery we consider is included in [�; �]. A lottery X is said to (weakly) �rst

order stochastically dominates (FOSD) a lottery Y if FX (z) � FY (z) � 0 for any z 2 [�; �].A lottery X is said to (weakly) second order stochastically dominates (SOSD) a lottery Y ifR z�[FX (s)� FY (s)] ds � 0 for any z 2 [�; �], and

R ��[FX (s)� FY (s)] ds = 0.

(a) Let X be the lottery which yields 1 with probability one. Let Y be a lottery which yields

2 with probability q and 0 with probability 1� q, where 0 < q < 1:

i. Does X FOSD Y ? Does Y FOSD X?

ii. Does X SOSD Y ? Does Y SOSD X?

(b) Show that:

Microeconomic Theory problems by A. Kajii 7

i. If X FOSD Y , then E (X) � E (Y ), i.e., X yields a higher outcome on average.

ii. If X SOSD Y , then V ar (Y ) � V ar (X) :[Hint. You may assume that X and Y are continuous random variable. Use integration

by parts.]

5. Consider the following situation. Your income today is zero, and that for tomorrow is given by

W , which may be a random variable. If you invest in a risky project, it will yield an additional

random income X. Let r be the (compensation) value of this investment for you, in the sense

that r is a constant that solves E [u (X +W )] = E [u (r +W )] :

(a) Show that if W is not random and u is risk averse, we have r � E [X].

(b) Show that if W and X are independent, we have r � E [X]. [Hint. Use the conditional

Jensen�s inequality]

(c) Give an example to show that if W and X are not independent, r > E [X] is possible.

6. Simple investment problem. Suppose your income today is m0 and tomorrow is m1, and

you can invest part of your income today. The gross return from a unit of investment is given by

a random variable R. You are risk averse and your vNM utility function for income is denoted

by u; and you discount the future utility level by factor �. So, to sum up, you would solve

maxxu (m0 � x) + �E [u (xR+m1)]

where E is the expectation with respect to R.

(a) Write down the �rst order condition for this problem.

(b) Suppose your income today m0 increases. Does it imply that the demand for the risky

asset x increases? If not, give a su¢ cient condition for this to be true.

(c) Suppose the return R gets more risky in the sense of the second order stochastic dominance.

Does it mean that the demand for the risky asset decrease? If not, give a su¢ cient condition

for this to be true. (Hint: consider the property of the function xu0 (x)

7. Prudence and precautionary saving. Suppose your income is m today and Y tomorrow.

You can save part of your income m today, and the rate of interest is r, r > �1. Your incomeY tomorrow may be random but you may not insure against the income risk generated by

this randomness. Your vNM utility function for today�s income is denoted by u and that for

tomorrow is denoted by v. Assume that both are concave. To sum up, you would solve

maxxu (m� x) +E [v ((1 + r)x+ Y )]

where E is the expectation with respect to random variable Y .

(a) Write down the �rst order condition for this problem.

(b) Show that when your future income is not random, i.e., Y = y0 for sure, your saving

increases if y0 decreases.

(c) If today�s income m increases, how does your saving change?

(d) How does your saving change when interest r increases? Discuss.

From now on, we are only concerned with change in income so set r = 0 to simplify notation.

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(e) Suppose that Y0 and Y1 are random variables of the same mean and Y1 is riskier than

Y0. (i.e., Y0 second order stochastically dominates Y1) Show that you save more under the

income risk Y1 than you do under Y0 if v000 > 0 (i.e., v0 is convex). Interpret.

(f) The coe¢ cient of absolute prudence of v is de�ned as the coe¢ cient of absolute risk aversion

for�v0. Assume that u = v. Show that if the coe¢ cient of absolute prudence for v increases,you save more.

8. Variance and risk. Let X; Y , Z be random variables with Y = X + Z. We say Y is at least

as risky as X if E [ZjX] = 0.

(a) Show that if Y is at least as risky as X, E [Z] = 0 and Cov [X;Z] = 0, and V ar [Y ] �V ar [X].

(b) Show by an example that E [Z] = 0 and Cov [X;Z] = 0 do not imply that Y is at least as

risky as X.

9. Simple insurance problem. Suppose your income is yH when the state of nature is H and

yL when the state of nature is L. You believe that state H will occur with probability �H and

the state L will occur with probability �L (= 1 � �H). There is an insurance which pays onedollar per unit if state is s, s = H;L. You may buy or sell these insurance contracts, and the

price of insurance that pays out in state s is given by ps. Writing xs for the amount of insurance

which pays out in state s (so negative xs means you �sell�), you are interested in maximizing

expected utility by choosing appropriate amounts of xH and xL, i.e.,

�Hu (yH + xH) + �Lu (yL + xL)

subject to pHxH + pLxL = 0, where u is a concave vNM utility function.

(a) Setting cs = ys + xs for s = H;L, and write some indi¤erence curves exhibiting your

preferences among (cH ; cL) pairs.

(b) What is the marginal rate of substitution when cH = cL?

(c) Show that if pHpL =�H�L, you will choose xH and xL in such a way yH + xH = yL + xL.

(d) If pHpL >�H�L, in which state will you consume more?

10. Risk and demand. Suppose there are S equally likely states, and write x 2 RS for a contingentconsumption plan: that is, xs, s = 1; :::; S, is consumption in state s.

(a) Assuming S = 2, write the set of contingent consumption plans with average consumption

equal to c in a graph measuring x1 horizontally and x2 vertically.

(b) Assuming S = 2, and �x �x 2 R2 with �x1 > �x2 > 0. Write the graph of cumulative

distribution function induced by �x. (Hint: Pr(z � �) = 12 if �x

1 > � > �x2.)

(c) Assuming S = 2, and �x �x 2 R2 with �x1 > �x2 > 0. Write the set of contingent consumptionplans which �rst order stochastically dominates �x. Also write the set of contingent plans

which second order stochastically dominates �x.

(d) Imagine an investor with an increasing and concave vNM utility u, u0 > 0 and u00 < 0, with

positive income w. There are S states, all equally likely. Denote the price of consumption

good in state s by ps > 0, and denote by p the vector of prices, i.e., p = (� � � ; ps; � � � ) :Thus the investor wants to choose a contingent plan x =

�x1; ::; xS

�; where xs is the

amount of consumption in state s, which maximizes the expected utilityPS

s=1 �su (xs)

given the budgetPS

s=1 psxs � w, where �s = 1

S for s = 1; :::; S. Denote by x (p; w) :=�x1 (p; w) ; :::; xS (p; w)

�the contingent plan which maximizes the expected utility.

Microeconomic Theory problems by A. Kajii 9

i. Write the �rst order condition of the maximization problem above.

ii. Show that when w increases, the investor will increase demand xs for all s; that is,

xs (p; w) is increasing in w for all s.

iii. Show that if w0 > w, the random consumption resulting from x (p; w0) �rst order

stochastically dominates that from x (p; w).

iv. Show that if ps = ps0, then xs (p; w) = xs

0(p; w) : (i.e., if the price in state s and that

in state s0 are the same, the investor consumes the same amount in these states.)

v. Show that the price and the consumption must be inversely related; that is, ps > ps0

if and only if xs (p; w) < xs0(p; w).

vi. [Harder] Show that if the price p and the consumption x are inversely related, then

there is a vNM utility function u such that x is utility maximizing.

11. The common ratio paradox. Denote by A the lottery that yields $3000 for sure, by B the

lottery that yields $4000 with probability 0:8, and $0 otherwise. Also denote by C the lottery

that yields $3000 with probability 0:25, and $0 otherwise, and by D the lottery that yields

$4000 with probability 0:2, and $0 otherwise. Many studies have shown a systematic tendency

for subjects to express a preference for A over B and for D over C. Show that this choice pattern

violates the expected utility hypothesis.

12. The Ellsberg paradox. There are two urns identical balls. Urn 1 contains 49 white and 51red balls. Urn 2 has 100 balls, but with unknown proportion of white and red balls. Consider

the following two bets: Bet 1: if red, $1000, otherwise 0; Bet 2: if white $1000, otherwise 0.

Write these problems in the Savage framework, and show that if an agent prefer urn 1 for both

bets, his preferences violate the sure-thing principle.

5 Classical Partial Equilibrium Analysis

1. Consider a quasi-linear utility function u�x1; x2

�= v

�x1�+ x2, where v0 > 0; v00 < 0: Suppose

the price of good one changes from p to �p. Verify that the change in consumer surplus measures

the exact change of utility level.

2. Consider a quasi-linear utility function u�x1; x2

�= v

�x1�+ x2, where v0 > 0; v00 < 0: Suppose

the price of good one changes from p to �p. Verify that the change in consumer surplus measures

the exact change of utility level. Also show that the equivalent variation and compensated

variation coincide, and relate these to the change in consumer surplus.

3. Aggregation of producers. Consider J producers, producing a homogeneous good from an

homogenous input. Price of output is p and that of input is 1. Producer j using a technology

represented by a cost function cj (qj). Assume for each j, cj (0) = 0 and c0j > 0 and c00j > 0,

and hence the supply is characterized by c0j (qj) = p. Write sj (p) =hc0

j

i�1(p), i.e., si (p) is the

quantity supplied by the �rm j at price p. By the assumptions, each sj is increasing function,

and so is the aggregate supply S (p) :=PJ

j=1 sj (p). So we can de�ne the inverse aggregate

supply function P (q) := S�1 (q). Notice that by construction S (P (q)) = q.

Now consider a �ctitious, �representative �rm�, whose cost function is given by C (q) =PJ

j=1 cj (sj (P (q))).

Notice that this function has the following interpretation. For the target output level q, calculate

the price level P (q), and ask each �rm j to produce taking P (q) as given. Then C (q) is the

total cost incurred by the �rms.

Show that the representative �rm with cost function C as given above will supply S (p) when

price is p (thus, S constitutes the supply curve for this �ctitious �rm.)

10

4. Producer surplus. Consider a �rm with cost function c (q) = f (q)+ v (q), where (1) v (0) = 0

and v0 > 0 and v00 < 0; (2) f (q) = K > 0 if q > 0, and f (0) = 0: So the constant K represents

a �xed (but not sunk) cost of production.

(a) Find the quantity q this �rm will supply when the price of output is p > 0.

(b) We shall write q (p) for this quantity supplied found above, and p (q) for its inverse. Write

the graph of p (q) taking q horizontally.

(c) Suppose the �rm produces a positive amount at price �p > 0. The area to the left of the

marginal cost curve is called producer surplus: That is, the producer surplus is �pq (�p) �R q(�p)0

v0 (q) dq. Is the producer surplus the same as pro�ts?

5. Ine¢ ciency of taxation. Let qD (p) be a downward sloping demand curve and qS (p) be

upward sloping supply curve, and denote by p� the equilibrium price. Consider tax t per unit

of trade, and write p (t) for the price for the buyers after the taxation (so p (0) = p�), and let

q (t) be the quantity traded. Assume these functions are di¤erentiable, and q0 (0) < 0:

(a) Show graphically that taxation leads to a loss in social welfare (consumer surplus + pro-

ducer surplus + tax revenue).

(b) Write the amount of social welfare explicitly, and di¤erentiate it with respect to t. Show

that the derivative is zero at t = 0.

(c) Di¤erentiate the social welfare twice in t. Can you sign the derivative at t = 0?

(d) Interpret your results above.

6. Monopoly. Consider a monopoly �rm with a convex cost function c (q). Given decreasing

inverse demand function p (q), write q� for the amount that the �rm produces.

(a) Assume that the revenue function p (q) q is concave in q. Write the �rst and second order

conditions that characterize q�.

(b) Show that the price elasticity of demand, � pp0q , must be more than one at q

�.

(c) Show that the revenue function p (q) q is concave if p (q) = a � bq, where a > 0, b > 0 arepositive constants (that is, the concavity assumption in 6a is satis�ed for linear demand

models). Show that when a increases, both quantity q� and the �rm�s pro�ts increase.

(d) Find the price elasticity of demand for linear demand p (q) = a � bq, and verify that it isdecreasing in q. (So the elasticity of demand is di¤erent from the slope of demand curve.)

(e) Derive a demand function of constant elasticity ". Con�rm that the monopoly problem has

no solution if " < 1.

7. A Large �rm and small �rms. Consider an industry with one large �rm, and N small �rms.

Small �rms are price takers, but the large �rm behaves as a monopoly �rm, setting the price

taking the demand as well as the supply of small �rms into account. Each small �rm has an

identical cost function c (q) = 12q2. The large �rm can produce costlessly (i.e., zero marginal

cost and no �xed cost). The demand for the product is given by p = 1�Q, p is the price of theproduct and Q is the total demand. So if QL is the amount the large �rm produces, and QS is

the sum of quantities the small �rms produce, then the market price will be p = 1� (QL +QS).

(a) Find the total quantity produced by the small �rms when the price is p.

(b) Write the problem that the large �rm solves, and �nd the quantity which the large �rm

will produce.

Microeconomic Theory problems by A. Kajii 11

(c) If the social welfare is to be maximized, how much should the large �rm produce? What

about the small �rms?

(d) As the number of small �rms increases, does the equilibrium approach a socially desirable

state? What if the large �rm�s technology were also given by a convex cost function?

Discuss.

8. Third degree price discrimination. Consider a monopoly �rm with a cost function with

constant marginal cost c. There are two types of consumers, and the demand of type t, t = 1; 2,

consumers is given by a downward sloping demand curve dt (p).

(a) Suppose the �rm cannot price discriminate; that is, the �rm must charge the same price p

for both types. Characterize the monopoly price.

(b) Suppose that the �rm can price discriminate, thus the �rm charges pt for type t consumers.

Write down the �rm�s problem and the corresponding �rst order condition.

(c) Show that the ability of price discrimination is advantageous to the �rm (i.e., the �rm will

get at least as much pro�ts as in the case of non-discrimination).

(d) Show that if the �rm sets p1 > p2, the demand of type 1 at p1 is more elastic than that of

type 2 at p2.

9. Ramsey taxation. Consider a central authority who needs to raise tax revenue of M by

speci�c (per unit) taxes on L goods. There are L+ 1 goods, and the representative consumer�s

utility function is linear in the L+1 st good. (This L+1st good is not taxed). More speci�cally,

assume that it is given byPL

l=1 vl�xl�+ xL+1. Write the demand for good l as xl

�pl�. Given

price pl, the demand for good l when tax � l is levied will be xl�pl + � l

�, and so the tax revenue

from good l will be xl�pl + � l

�� l. Currently, the prices of goods are given by �p =

��p1; :::; �pL

�and the price of good L+ 1 is one.

(a) Write the problem to minimize the loss of consumer surplus, given that the tax revenue

must be at least as much as M .

(b) Show that price elastic goods tend to get taxed less. Interpret.

10. Policy mix. Consider a monopoly �rm with a positive constant marginal cost c. A decreasing

inverse demand function p (q) is given. Assume that the revenue function p (q) q is concave in

q. Write q (c) for the amount that the �rm produces. Also denote by q� (c) the socially optimal

amount of output.

(a) Write the �rst order condition which determines q (c) : Show that q� (c) > q (c) :

(b) Compare the following two policies. (1) quantity control polity, which requires the �rm

to produce an amount �q; (2) tax/subsidy policy, which requires the �rm to pay t to the

governement per unit produced. Show that both policies can induce the socially optimal

amount. What is the sign of t? Discuss.

From now on assume a linear inverse demand function p (q) = 1� �q, where � > 0.

(c) Suppose now that the government does not know c although the �rm does: the government

knows c is either cH or cL, where 1 > cH > cL > 1, and these are equally likely.

i. Find the quantity control policy which maximize the social welfare. Explain why the

social optimal amount is not necessarily induced?

12

ii. Also �nd the tax/subsidy policy which maximize the social welfare, and explain the

source of ine¢ ciency.

iii. Consider the following policy mix: there is a �xed �target production�level q, and the

�rm must produce at least this amount. But the �rm may produce more, and in such

a case the �rm pays t per unit of the amount for the excess production: i.e., the �rm

pays t (q � q) to the government if (q � q) > 0. Show that this policy mix is better

than the two policies above.

6 Competitive Markets: General Equilibrium

1. Characterization of Pareto e¢ ciency. Consider a pure exchange economy where L goodsare traded. The total endowments are �e 2 RL++, and assume each ui is strictly increasing andcontinuous.

(a) Give the de�nition of a Pareto e¢ cient allocation of this economy.

Now consider the following maximization problems: 1) Given �i > 0, i = 1; :::; I;

Maximizex12RL++;:::;xI2RL++

Xi

�iui (xi) subject toXi

xi = �e; (1)

2) Given �ui > 0, i = 2; :::; I,

Maximizex12RL++;:::;xI2RL++

u1 (x1) subject to ui (xi) � �ui, i = 2; :::; I andXi

xi = �e (2)

(b) Show that (xi)Ii=1 2

�RL++

�Iis Pareto e¢ cient if and only if (xi)

Ii=1 solves problem 2 above

with some �u. Is this true if ui were not strictly increasing?

(c) Suppose every ui is concave, di¤erentiable, and Dui (xi) >> 0, ui (xi) > 0 for any xi >> 0.

Show that (xi)Ii=1 2

�RL++

�Isolves problem 1 for some � if and only if (xi)

Ii=1 solves problem

2 with some �u. (Hint. Both problems are then �concave� so compare the Kuhn-Tucker

condition.)

2. Characterization of Pareto e¢ ciency in a production economy. Consider an economywith L goods. The total resource endowments are given by a vector �e 2 RL++, and there is oneproducer with an increasing and quasi-convex production transformation function F : that is, a

production plan y 2 RL can be carried out by the �rm if and only if F (y) � 0. Assume for eachconsumer i, Xi = RL+ and utility function ui is strictly increasing and continuous.

(a) Give the de�nition of a Pareto e¢ cient allocation of this economy.

Now consider the following maximization problems: 1) Given �i > 0, i = 1; :::; I;

Maximizex12RL++;:::;xI2RL++

Xi

�iui (xi) subject toXi

xi = �e; (3)

2) Given �ui > 0, i = 2; :::; I,

Maximizex12RL++;:::;xI2RL++

u1 (x1) subject to ui (xi) � �ui, i = 2; :::; I andXi

xi = �e (4)

(b) Show that (xi)Ii=1 2

�RL++

�Iis Pareto e¢ cient if and only if (xi)

Ii=1 solves problem 2 above

with some �u. Is this true if ui were not strictly increasing?

Microeconomic Theory problems by A. Kajii 13

(c) Suppose every ui is concave, di¤erentiable, and Dui (xi) >> 0, ui (xi) > 0 for any xi >> 0.

Show that (xi)Ii=1 2

�RL++

�Isolves problem 1 for some � if and only if (xi)

Ii=1 solves problem

2 with some �u. (Hint. Both problems are then �concave� so compare the Kuhn-Tucker

condition.)

3. Pareto e¢ cient allocations and competitive equilibria in Edgeworth Box. Consider�Edgeworth box economies� given as follows. For each economy, show graphically, the set

of Pareto e¢ cient allocations, the set of weakly Pareto e¢ cient allocations, and the set of

competitive equilibria.

(a) u1�x11; x

21

�= min

�x11; x

21

�, u2

�x12; x

22

�= x12 + x

22, e1 = (1; 0), e2 = (0; 1)

(b) u1�x11; x

21

�= min

�x11; x

21

�, u2

�x12; x

22

�= min

�x12; x

22

�, e1 = (1; 0), e2 = (0; 1)

(c) u1�x11; x

21

�= x11, u2

�x12; x

22

�= x12; e1 = (1; 0), e2 = (0; 1) :

(d) u1�x11; x

21

�= x21, u2

�x12; x

22

�= x12; e1 = (1; a), e2 = (0; 1� a), where a 2 [0; 1]

4. E¢ ciency charcterization in Robinson Crusoe: In the Robinson Crusoe economy, provethat a feasible consumption x =

�x1; x2

�is Pareto E¢ cient if and only if it is a solution of the

following maximization problem:

max(x1;x2)

u�x1; x2

�subject to x2 � f

�e� x1

�:

5. Pateto E¢ cient share of random income. Consider an economy with I agents. Eachagent�s preferences are represented by a vNM utility function ui. The aggregate income of this

economy is denoted by y. For simplicity, suppose that there are only �nitely many possible

income level, and denote by p (y) the probability that the aggregate income is y. Denote by

xi (y) the amount agent i receives when aggregate income is y, and we call (xi)Ii=1 an income

sharing rule if it is feasible allocation of income, i.e.,PI

i=1 xi (y) = y for any y. (Assume an

interior throughout.)

(a) Give the de�nition of Pareto e¢ cient income sharing rule.

(b) Assuming each ui satis�es u0i > 0 and u00i < 0, write the Pareto maximization of weighted

sum of utility functions and the �rst order condition for Pareto e¢ ciency.

(c) Fix a positive �i, for i = 1; :::; I, and let g (t) =PI

i=1 (u0i)�1�t�i

�; where (u0i)

�1 is the

inverse of u0i. Show that if we set xi (y) = (u0i)�1�g�1(y)�i

�, then (xi)

Ii=1 is a Pareto e¢ cient

sharing rule.

(d) Assume further ui (z) = � 1�iexp [��iz], where �i is a positive constant, for every agent i.

Find the (class of) Pareto e¢ cient income sharing rule.

(e) Do the same exercise for ui (z) = 11� i z

1� i where where i is a positive constant, for every

agent i.

6. State and prove the �rst fundamental theorem of welfare economics. Discuss why completeness

of markets is important in this result.

7. weak e¢ ciency to e¢ ciency. Say that the budget is tight at prices p and income w for uiif p � x0 < w and x0 2 Xi implies that x0 is not a utility maximizing demand vector (i.e., tomaximize utility, all the given income must be spent).

14

(a) Show that the budget is tight at prices p and income w for ui if and only if any demand

vector is cost minimizing in the following sense: if x maximizes utility ui in the budget set,

then for any x0 2 Xi, ui (x0) � ui (x) implies p � x0 � p � x.(b) Assume that Xi = RL+ and ui is non-decreasing. Given positive prices p and income w

suppose that the following holds: if x is a demand vector given p and w, then it is cost

minimizing. Show that the budget must be tight.

(c) Argue graphically that the budget might not be tight when ui is not strictly increasing

(i.e., a thick indi¤erence curve is possible)

(d) Let (x�; y�; p�) be a competitive equilibrium of a private ownership economy ((ui)Ii=1 ; (Yj)

Jj=1 ;�

!i; (�ij)Jj=1

�Ii=1). Show that (x�; y�) is Pareto E¢ cient if the budget is tight at x�i (given

prices p� and income p� � !i +P

j �ij�p� � y�j

�) for ui for all consumer i:

8. Second fundamental theorem. Let p 2 RL be a price system, p 6= 0. We say that x� is a

quasi-utility maximizer if ui (x0) > ui (x�) implies p � x0 � p � x�. (Note: this will be triviall if allthe prices are zero)

(a) Show that if x� is utility maximizing given prices p and income w, it is quasi utility

maximizing.

(b) Set L = 2 and let the consumption set to be RL+. Consider a consumer with utility functionu�x1; x2

�= x1 + x2 with initial endowments (1; 0). Show that a quasi-utility maximizer is

not necessarily a utility maximizer under some prices.

(c) In general, suppose that Xi is convex, ui is continuous, and there is x 2 Xi such thatp � x < p � !i. Show that if x� is quasi-utility maximizing, then it is utility maximizing.

[Hint. Suppose this does not hold. Then there will be x0 2 Xi such that ui (x0) > u (x�)but p � x0 = p � x�. Consider (1� t)x0 + tx with small t > 0].

(d) Comment on the second fundamental theorem of welfare economics discussed in the class.

9. the quasi-linear economy. There are two goods, x and m, and there are n traders. Thereare I traders and each trader i has a quasi-linear utility function ui (xi)+mi and endowed with

(�xi; �mi), where ui is strictly increasing and concave. The price of good x in units of good y is

denoted by p. To avoid a boundary consumption, assume that u0i (0) = +1, and also that anegative consumption of good m is allowed.

(a) Show that (xi;mi)Ii=1 is Pareto e¢ cient if and only if (a) (xi)

Ii=1 maximizes

PIi=1 ui (xi)

subject toP

i xi =P

i �xi, and (b)P

imi =P

i �mi hold.

(b) Show that the (general) equilibrium price p� depends on �x �P

i �xi only (so in particular,

it is independent of the way the total resource is initially allocated).

(c) Show that the (general) equilibrium price p� maximizes the social welfare (i.e., the sum of

consumer surplus and the pro�ts cannot be increased further.)

(d) Suppose �x1 increases by a small amount, whereas the other �xi, i = 2; :::; n, are kept �xed.

What can you say about (i) the change in the equilibrium price and (ii) who bene�ts from

this change?

10. the quasi-linear economy. There are two goods, x and m, and there are n traders. Thereare I traders and each trader i has a quasi-linear utility function ui (xi)+mi and endowed with

(�xi; �mi), where ui is strictly increasing and concave. The price of good x in units of good y is

denoted by p. To avoid a boundary consumption, assume that u0i (0) = +1, and also that anegative consumption of good m is allowed.

Microeconomic Theory problems by A. Kajii 15

(a) Show that a trader may be interpreted as a producer with a convex cost function, if we allow

a negative consumption of xi as well. [Hint: think of ui as the negative of cost function,

and check that utility maximization given budget is equivalent to pro�t maximization.]

(b) Show that (xi;mi)Ii=1 is Pareto e¢ cient if and only if (a) (xi)

Ii=1 maximizes

PIi=1 ui (xi)

subject toP

i xi =P

i �xi, and (b)P

imi =P

i �mi hold.

(c) Show that the (general) equilibrium price p� depends on �x �P

i �xi only (so in particular,

it is independent of the way the total resource is initially allocated).

(d) Show that the (general) equilibrium price p� maximizes the social welfare (i.e., the sum of

consumer surplus and the pro�ts cannot be increased further.)

(e) Suppose �x1 increases by a small amount, whereas the other �xi, i = 2; :::; n, are kept �xed.

What can you say about (i) the change in the equilibrium price and (ii) who bene�ts from

this change?

11. �Symmetric�Quasi-linear economies

There are two goods, x and y, and there are 2 consumers. The consumers characteristics are

summarized below: Consumer 1: utility function u (x1) + y1, endowed with ((1� �)!; �!),Consumer 2: utility function x2 + u (y2), endowed with (�!; (1� �)!), where u0 > 0 and

u00 < 0, and ! > 0 and � are �xed numbers. Notice that the total supply of each good is !,

independent of �. Assume that negative consumption of �linear good�is allowed (so consumers

have di¤erent consumption sets). Normalize the price of y good to be one, and write p for the

price of good x. Assume that u is increasing and concave. Write � = (u0)�1. Since u0 is a

positive and decreasing function, � is also a positive and decreasing function.

(a) Find the demand for good x for both consumers, and the market excess demand function

z (p) for good x.

(b) Show that p = 1 is an equilibrium, i.e., z (1) = 0. Find the equilibrium allocation of

consumption goods.

(c) Compute ddpz (1). Show that if the equilibrium allocation is close to the initial endowments

(1� �; �) and (�; 1� �), then ddpz (1) < 0 (i.e., the index of p = 1 is +1).

(d) Show that ddpz (1) > 0 is possible for some �. What can you conclude from this observation?

12. There is one consumer and one producer in the economy. The consumer�s preferences are

represented by utility function u (x1; x2) = 12 lnx1 + x2. Good 2 is �leisure� and initially the

consumer owns one unit of good 2, and no good 1. The producer is a price taker (competitive

�rm) and it�s production function is f (z) = kz, where k > 0 is a constant; that is, the producer

produces kz units of good 1 from z units of good 2 (i.e., with one unit of �labor�k units of good

1 is produced.) All the pro�ts are distributed to the consumer. Good 2 is the numéraire; i.e.,

p2 = 1 in the following, and write p instead of p1 for simplicity.

(a) What does the constant k represent?

(b) Argue that the �rm cannot make pro�ts (nor losses) in equilibrium.

(c) Derive the demand curve for good 1, assuming that there is no pro�t distribution (so his

income is always 1).

(d) Derive the supply curve for good 1.

(e) Find the (partial) equilibrium price and quantity in the good 1 market. Is this a general

equilibrium, too?

16

(f) As k changes, how the equilibrium quantity change? Interpret.

(g) Let k = 1. Suppose the producer has to pay $1 per unit of input (so e¤ectively, the producer

needs two units of labor to produce one unit of good 1). The tax revenue will be given to

the consumer as a lump-sum subsidy. Find the general equilibrium. Does this tax-subsidy

policy make the consumer better o¤? Do you think that your conclusion might change if

preferences and/or production technology were di¤erent form the ones above?

13. In the same setting as in question 12, but with production function f (z) = z�, � 2 (0; 1);

(a) Find a competitive equilibrium price and the consumer�s equilibrium consumption bundle.

(b) We can transform this economy to a CRS economy by adding an additional good, good 3,

to the economy. The consumer does not care good 3 per se; his utility function does not

depend on the amount of good 3 he consumes. The consumer owns one unit of good 3.

The �rm�s production function is f (z; ") = "�z"

��= "��+1z�, where " is the amount of

good 3 used for production. Notice that if " = 1, we have the same production function as

before. [This is called the McKenzie transformation]

i. Check that the technology is CRS.

ii. So in equilibrium, the pro�ts to the �rm will be zero. Interpret the equilibrium price

of good 3.

14. There are two countries, A and B. There are two consumers, 1 and 2, in country A and there is

one consumer, 3, in country B. Price of good 2 is set to be 1 throughout, and write p instead of

p1. Utility and endowments of consumers are summarized below:

utility endowments

1 2 lnx1 + 2 lnx2 (6; 4)

2 lnx1 + 3 lnx2 (8; 4)

3 lnx1 + lnx2 (4; 16)

(a) Find individual excess demand functions.

(b) First assume that there is no international trade. Find the equilibrium in each country.

(c) Suppose consumers can trade internationally. Find the equilibrium.

(d) Does free world trade make everybody better o¤? Is this consistent with the �rst funda-

mental theorem of welfare economics? Discuss.

15. Linear model of international trade and gains from trade. In the following, assume thatthere are two goods, and consumers�consumptions sets are R2+. Normalize the price of good 2to be 1, and denote by p the price of good 1. Here is the list of consumers to be considered in

the following questions (0 < � < 12 ):

utility function u�x1; x2

�endowments

�e1; e2

�consumer A (1� �)x1 + x2 (1; 1)

consumer B (1� �)x1 + x2 (1; 0)

consumer 1 x1 (1; 1)

consumer 2 x2 (1; 1)

(a) Find the demand for good 1 for consumer A and consumer 1, as a (possibly set valued)

function of p.

Microeconomic Theory problems by A. Kajii 17

(b) Find a competitive equilibrium of the pure exchange economy consisting of consumer 1 and

consumer 2. Draw an Edgeworth box to illustrate the equilibrium.

(c) Find a competitive equilibrium of the pure exchange economy consisting of consumer A

and consumer 1. Draw an Edgeworth box to illustrate the equilibrium.

(d) Is there a competitive equilibrium of the pure exchange economy consisting of consumer A

and consumer 2? If there is, �nd one and illustrate it in an Edgeworth box. If not, verify

nonexistence.

(e) Is there a competitive equilibrium of the pure exchange economy consisting of consumer B

and consumer 2? If there is, �nd one and illustrate it in an Edgeworth box. If not, verify

nonexistence.

(f) Find a competitive equilibrium of the pure exchange economy consisting of consumer A,

consumer 1 and consumer 2.

(g) Imagine that consumers A and 1 live in the same country, and consumer 2 lives in another

country. Discuss the welfare e¤ect of a free trade agreement of the two countries, i.e., before

the agreement, only consumers in the same country could trade but after the agreement

all three consumers can trade.

16. Consider two �rms, each producing a consumption good from two inputs, z1 and z2, by pro-

duction function f1�z11 ; z

21

�=�z11�� �

z21�1��

and f2�z12 ; z

22

�=�z12�� �

z22�1��

, respectively, and

0 < � < � < 1. The total amount of inputs available in the economy is �z1 and �z2, respectively.

The price of consumption good is given at p1 and p2. We can interpret that this is a small coun-

try with two outputs and two inputs, and output prices are determined in the world markets,

whereas inputs are not mobile and their prices are determined domestically.

(a) Give the condition for pro�t maximizing inputs. (recall section 1, (1))

(b) Given factor (input) prices w1 and w2, �nd the ratio of inputs used in each �rm. Interpret.

Will this property hold when production function is not Cobb-Douglas?

(c) Derive equations that characterize the factor prices that clear the input markets. (Do not

forget the pro�t maximization condition.)

(d) Show that the equations above can be solved as follows: �rst, �nd the input ratio�z1=z2

�that satis�es the pro�t maximization condition for each �rm. Then �nd a unique allocation

of inputs that has the ratio. Draw a picture like Edgeworth box to explain how this is done.

(e) Show that for given p1 and p2, the factor prices that clear the input markets do not depend

on �z1 or �z2.

(f) Stolper-Samuelson theorem. How do the equilibrium factor prices w1 and w2 change

when p1 increases? State this in terms of factor intensity � and �.

(g) Rybcszynski�s theorem. How do the equilibrium factor inputs z1 and z2 change when

�z1 increases? State this in terms of factor intensity � and �.

17. Consider an economy with three goods one consumer and two �rms as follows. The prices of

good 1 and 2 are denoted by w1 and w2 respectively, and the price of good 3 is denoted by p.

� Consumer. utility function u�x3�+ x2, and he does not care good 1. He is endowed with

L units of good 2. (Think of good 3 as a consumption good, good 2 as leisure, good 1 as

a productive intermediate good, necessary for production of good 3, which is not initially

endowed thus must be produced from labor. ) The function u satis�es the standard

conditions. The consumer receives pro�ts from the �rms.

18

� Firm 1 produces good 3 with production function f�z1; z2

�=�z1�� �

z2�1��

where z1 and

z2 are the amount of good 1 and 2 used as inputs. So the conditional factor demand given

y are

z1 (w; y) =

��w2

(1� �)w1

�1��y:

z2 (w; y) =

�(1� �)w1�w2

��y

� Firm 2 produces kz2 units of good 1 from z2 units of good 2, where k is a given constant.

(a) Find the market clearing conditions for good 1 and good 3.

(b) Find a general equilibrium price system.

(c) How will the prices change when k and/or �.

18. Consider an economy with three goods one consumer and two �rms as follows. The prices of

good 1 and 2 are denoted by w1 and w2 respectively, and the price of good 3 is denoted by p.

Assume that the prices are all positive.

� Consumer. His utility function is u�x3�+ x2, where xl is the consumption of good l.

Assume u0 > 0 and u00 < 0. For simplicity allow x2 < 0. Note that he does not care good 1.

He is endowed with 1 unit of good 1. The consumer receives pro�ts from the �rms. Denote

by � the total amount of pro�ts he receives.

� Firm 1 produces good 2 from good 1. It produces k1z1 units of good 2 from z1 units of

good 1, where k1 is a given positive constant.

� Firm 2 produces good 3 from good 2. It produces k2z2 units of good 3 from z2 units of

good 2, where k2 is a given positive constant.

(a) Write the utility maximization problem of the consumer. Find the �rst order condition

which characterize the demand for goods.

From now on, write � (y) := (u0)�1 (y).

(b) To �nd a general competitive equilibrium, we can assume � = 0. Explain why.

(c) To �nd a general competitive equilibrium, we can assume p = 1. Explain why.

From now on assume p = 1 and � = 0.

(d) Assuming that each �rm produces a positive quantity in equilibrium, �nd the conditions

which prices w1 and w2 must satisfy.

(e) From the market clearing condition (i.e., demand = supply) for good 1, �nd the quantity

of good 2 that �rm 1 produces in equilibrium.

(f) Write the market clearing conditions for good 2 and good 3.

(g) Show that if the market clearing condition for good 3 is satis�ed, the market for good 2

automatically clears. Is this a coincidence? Explain.

(h) Suppose that k2 increases. What will happen to the equilibrium prices? Is this good for

the consumer? Interpret your �nding.

(i) Suppose that k1 increases. What will happen to the equilibrium prices? Is this good for

the consumer? Interpret your �nding.

Microeconomic Theory problems by A. Kajii 19

19. In Section 6 (12), suppose the �rm behaves as a monopoly �rm, and assume that the consumer

has utility function u (x1; x2) = 2px1 + x2; that is, the �rm takes the demand for x1 as given,

and set the price. Note in particular that the dividend � is also taken as given by the �rm,

although it won�t matter in the following since the demand does not depend on �.

(a) Given �, �nd the price that the �rm sets.

(b) Compute the pro�t level at the price you found above, and �nd the dead weight loss.

20. There is one consumer and one �rm in the economy. There are two periods and there is one

perishable good in each period. The price of the good is always normalized to be one. The con-

sumer�s preferences are represented by utility function u�x0; x1

�= x0 + 1

2 lnx1. The consumer

owns one unit of the consumption good in period 0, and nothing in period 1. The �rm is a price

taker (competitive �rm) and it�s production function is x1 = kp�x0, (x0 � 0) where k > 0 is a

constant; that is, it produces kp� units of good in period 1 from � units of good in period 0.

In period 0, the consumer may save by purchasing a discount bond issued by the �rm. Denote

by b the amount of bond the consumer holds at the end of period 0, and by q the price of bond.

The �rm issues s units of bond on period 0, and so purchases qs units of good for input. The

consumer owns the �rm in the sense that he receives the entire realized pro�ts in period 1.

(a) Given q, �nd the supply of the discount bond by the �rm.

(b) Given q, �nd the pro�t distribution (dividend) �.

(c) Find the demand b for bond as a function of q and �.

(d) Find the equilibrium price of the discount bond.

(e) Study how the equilibrium interest rate changes as k changes. Interpret.

21. Consider an economy as follows. There are two consumers, i = 1; 2: There are two periods, and in

each period, consumers trade a single, perishable commodity (i.e., they cannot store the good).

There is no uncertainty. Consumer i�s preferences are represented by ui�x0i ; x

1i

�= lnx0i +�i lnx

1i

where �i 2 (0; 1) is a constant. Every consumer is endowed with one unit of the good in eachperiod. Thus consumers�preferences are identical, except for the discount factor �i. The price

of good in period 0 is p0, and that in period 1 is p1. In period 0, consumers can save: denote by

si the amount that consumer i saves in period 0. Let r be the interest rate. That is, if consumer

i saves si dollars, he receives (1 + r) si dollars at the beginning of period 1 before the trade of

the consumption good takes place. Hence consumer i chooses�x0i ; x

1i ; si

�2 R+�R+�R. Notice

that negative saving, i.e., borrowing is allowed.

(a) Consumer i faces budget constraint p0x0i + si = p0 with x0i � 0 in the �rst period. Write

the second period budget constraint.

(b) Find demand x0i , x1i , and si as functions of p

0, p1, and r.

(c) Show that when the saving market clears, i.e., s1�p0; p1; r

�+ s2

�p0; p1; r

�= 0, the good

market clears in both periods 0 and 1.

(d) Find all the equilibrium prices and the corresponding interest rate. Show that any interest

rate r, r > �1 can arise in equilibrium.(e) What about the real interest rate - the relative price of period 2 good - in equilibrium?

22. First Fundamental Theorem of Welfare Economics. Let x maximize utility ui given

prices p and income w. Say that the budget is tight at prices p and income w for ui if p � x0 < wimplies that x0 is not a utility maximizing demand vector (i.e., to maximize utility, all the given

income must be spent).

20

(a) Show that the budget is tight at prices p and income w for ui if and only if any demand

vector is cost minimizing in the following sense: if x maximizes utility ui in the budget set,

then ui (x0) � ui (x) implies p � x0 � p � x.

(b) Argue graphically that the budget might not be tight when ui is not strictly increasing

(i.e., a thick indi¤erence curve is possible)

(c) Let (x�; y�; p�) be a competitive equilibrium of a private ownership economy ((ui)Ii=1 ; (Yj)

Jj=1 ;

�!i; (�ij)

Jj=1

�Ii=1).

Show that (x�; y�) is Pareto E¢ cient if the budget is tight at x�i (given prices p� and income

p� � !i +P

j �ij�p� � y�j

�) for ui for all consumer i:

23. Altruism and the welfare theorem: Consider an exchange economy with 2 consumers and Lgoods, with initial endowments ! = (!1; !2). Each consumer�s utility function can be expressed

as U1 (x1; u2 (x2)) for consumer 1 and U2 (x1; u2 (x2))

24. Uniquness of no-trade equilibrium. Consider an exchange economy with I consumers andL goods. Suppose that the initial endowments ! = (!1; :::; !I) >> 0 constitute a competitive

equilibrium; that is, there is no trade in this equilibrium. Show that ! is in fact a unique

equilibrium allocation if utility functions are strictly quasi-concave.

25. Gross substitution. Let f : RL++ ! RL be a di¤erentiable market excess demand function.The excess demand function exhibits the gross substitution property if the demand for good l

goes up if the price of good l0 goes up.

(a) Show that if the market excess demand exhibits the gross substitution property, an equi-

librium must be unique.

(b) Consider an exchange economy where each consumer is endowed with ei with ei � 0 andits preferences are represented by a utility function:

ui (x) =

"KXk=1

aki�xki��i# 1

�i

; where aki > 0; k = 1; :::;K;Xk

aki = 1, and �i � 1; �i 6= 0:

Does the market excess demand function exhibit the gross substitution property? If not in

general, �nd some su¢ cient condition.

26. The Negishi Method. Consider an exchange economy with I consumers and L goods with dif-ferentiable utility functions. Assume that for each � 2 4I�1 :=

n(�1; :::; �I) � 0 :

PIi=1 �i = 1

o,

there is a unique Pareto E¢ cient allocation x� (�) = (� � � ; x�i (�) ; � � � ) which is obtained by max-imizing the weighted utility sum

Pi �iui (xi) subject to

Pi (xi � !i) = 0, and the corresponding

supporting price vector p� (�) (that is, if ui (xi) > ui (x�i (�)), p� (�)�xi > p� (�)�x�i (�)). Assume

further that x� (�) and p� (�) are continuous functions of �. Set �i (�) := p� (�) � (x�i (�)� !i),and � (�) := (�i (�))

Ii=1.

(a) Show thatP

i �i (�) = 0 for any �.

(b) Show that if �����= 0 (i.e., �i

����= 0 for all i),

�x�����; p�

�����constitutes a competitive

equilibrium.

(c) Assume that I = 2. Argue using a graph, that under the standard assumptions, there will

be �� such that �����= 0.

27. The full insurance theorem. Consider an exchange economy where each household h receivesa random endowment vector esh 2 RL if state s occurs. Household h, h = 1; ::;H, has utility

Microeconomic Theory problems by A. Kajii 21

functionPS

s=1 �suh (x

sh) where uh is strictly concave, and x

sh is the consumption vector in state

s. Note that probability weights �s, s = 1; ::; S, do not depend on h. Assume that there is

no aggregate risk; that is, there is a vector �e such that �e =PH

h=1 eh (s) for all s. Show that

consumption does not depend on states in any Pareto e¢ cient allocation.

28. Insurance Markets. Consider an exchange economy withH households, where each household

h receives a random endowment vector esh 2 RL if state s occurs. Household h, h = 1; ::;H, hasutility function

PSs=1 �

suh (xsh) where uh is strictly concave, and x

sh is the consumption vector

in state s. Note that probability weights �s, s = 1; ::; S, do not depend on h. Assume that there

is no aggregate risk; that is, there is a vector �e such that �e =PH

h=1 esh for all s.

(a) Show that the consumption does not depend on states in any Pareto e¢ cient allocation;

that is, if x is a Pareto e¢ cient allocation, xsh = xs0

h for any s; s0 for every h.

(b) From now on, assume that S = H,and consider the sequential trade model with H assets,

where the asset h pays $1 if state s = h occurs. (i.e., these are the Arrow securities).

Assume moreover that esh = � if s = h, esh = � otherwise, where � > � > 0. Notice

that asset h can be interpreted as insurance for household h�s income risk. Write zsh for

the amount of asset s household h holds at the beginning of period 1, and there is no

consumption in period 0 (so only assets are traded).

i. Show that household h must buy the insurance for h (i.e., zhh > 0) in any equilibrium.

ii. Does any household other than h buy insurance h in some equilibrium?

(c) Let H = S = 2, and L = 1, u1 (z) = u2 (z) = ln z, e1 = (1; 0), e2 = (0; 1). Write��1; �2

�= (�; 1� �). Consider the sequential trade model, and �nd the equilibrium price

of the Arrow security which pays $1 in state s, as a function of �, with normalization that

the sum of Arrow security prices is one. Also �nd the price of a discount bond which pays

$1 in every state.

29. Sequential trade and asset structure. Consider an I -consumers exchange economy underuncertainty, with one good in each state s = 1; :::; S. Let A be a linear subspace of RS . Considerthe following optimization problem:

(*) maxx ui (x) subject toPS

s=0 ps (xsi � !si ) = 0 and

�x1i � !1i ; :::; xSi � !Si

�2 A:

We say that (p; x) 2 RL ��RL�Iconstitute an A-equilibrium if every consumer i solves the

problem above at xi andPI

i=1 (xsi � !si ) = 0 for s = 1; :::; S. So an Arrow - Debreu equilibrium

of this economy is an A-equilibrium where A = RS .

(a) Now consider a two period sequential trade model: in the �rst period before the realization

of states, the consumers trade J assets,�r1; :::; rJ

�, which pay in the good (that is, these

are vectors of real payo¤s). Show that a rational expectation equilibrium of this economy

is an A-equilibrium for some linear space A.

(b) Establish the following converse implication as well: Given an A-equilibrium, for any lin-

early independent vectors r1; :::; rJ of real payo¤s which span A, in the economy with these

assets there is a rational expectations equilibrium with equilibrium consumption identical

to the A-equilibrium consumption.

(c) Show that if two collections of linearly independent assets,�r1; :::; rJ

�and

hr1; :::; rJ

iof

real payo¤s, generate the same linear subspace A, then the corresponding sets of rational

expectations equilibrium consumption allocations must coincide.

22

30. No arbitrage pricing for long and short bond. Consider an exchange economy which lastsfor three periods, 1,2, and 3. There is one perishable good in each period. There are I consumers

and consumer i has preferences represented by a discounted sum of payo¤s: Ui�x1i ; x

2i ; x

3i

�:=

ui�x1i�+ �ui

�x2i�+ �2ui

�x3i�, where � > 0. Each consumer i is endowed with ei =

�e1i ; e

2i ; e

3i

�.

In period 1, two assets, called S1 and L, are traded in addition to the good. Asset S1 yields one

unit of the consumption good in period 2, and asset L yields one unit of the consumption good

in period 3 (but nothing in period 2 - so this is a �long term bond�). In period 2, one asset,

asset S2 is traded in addition to the good. Asset S2 yields one unit of the consumption good in

period 3. The net supply of any of these assets is zero. Denote by zS1i , zS2i , and z

Li the amount

of asset S1; S2, and L held by consumer i at the end of each respective trading period. Write

qS1, qS2, and qL for the price of assets in units of the good (so qS1, qS2 corresponds short term

interest rates in periods 1 and 2, and qL corresponds to a long term interest rate in period 1).

Then the budget constraint for consumer i in period 1 is x1i + qS1zS1i + qLzLi � e1i .

For simplicity, you may assume that all the budget inequalities hold with equality at an optimum.

and you may ignore the non negativity constraint for consumption.

(a) Write the budget constraints for consumer i in period 2 and 3.

(b) Using the law of one price, derive a condition which relates qS1, qS2, and qL.

(c) If in addition, an additional asset which yields one unit of the good in both periods 2 and

3 is traded in period 1, how is the price of this asset related to qS1, and qS2?

(d) Write the de�nition of a one shot (Arrow Debreu) equilibrium for this economy.

(e) If qS1, qS2, and qL are dynamic equilibrium prices, what are the corresponding one shot

equilibrium prices?

31. Sequential Trade and Consumption Smoothing. There are �nitely many periods, t =0; 1; :::; T . There is a single perishable consumption good in each period t = 1; :::; T . There are

I consumers. The preferences of consumer i, i = 1; :::; I, is represented by a discounted sum of

utilities:PT

t=1 (�i)tui (x

ti), where �i 2 (0; 1] is a constant, ui is strictly increasing and strictly

concave function on [0;1), and xti is the consumption in period t. There are T �trees� in

this economy, and the consumption good is produced by these trees. Tree t, t = 1; :::; T , yields

yt > 0 units of the consumption good in period t, and nothing in the other periods. Thus the

total supply of the consumption good in period t is yt. The trees are owned by the consumers.

Initially, consumer i owns a share ��ti of tree t: thusPI

i=1��ti = 1 for every t. In period 0, the

ownership shares of the trees are traded. Write pt for the price of the tree t share. So consumer

i�s income in the period 0 markets isPT

t=1 pt��ti .

(a) Write �ti for the share of tree t which consumer i owns at the end of period 0. Thus for

instance consumer i will consume �tiyt units of the consumption good in period t. Write

the utility maximization problem for consumer i in period 0.

(b) De�ne a competitive equilibrium for the markets of the shares. Is a competitive equilibrium

e¢ cient?

(c) De�ne an anonymous (in net trade) allocation of the shares. Is a competitive equilibrium

necessarily anonymous?

(d) Consider a new asset which promises to pay rt units of the consumption good in period t;

for t = 1; :::; T . How should this asset be priced in the markets?

(e) Suppose that yt � �y for every t, and �i � �� for every i. Show that in any competitive equi-librium, the consumption of any consumer is constant over time (i.e., xti will be independent

of t).

Microeconomic Theory problems by A. Kajii 23

32. Consumption Smoothing Consider an economy with two periods, where a single consumptiongood is available in each period. Each consumer i receives eti units of the good in period t = 1; 2.

Consumer i, i = 1; ::; I, has utility function ui�x1i�+ �iui

�x2i�where ui is smooth, increasing

and strictly concave, �i 2 (0; 1] is a constant (discount factor) and xti is the consumption inperiod t. Denote by �et :=

PIi=1 e

ti the total amount of the good available in period t. The good

cannot be stored, so the feasibility of consumption will requirePI

i=1 xti = �e

t for both t = 1 and

2. There are markets where each consumer can buy/sell the good to be consumed in period t,

before these time periods. Denote by pt for the price of the good to be consumed in period t.

So the income of consumer i is p1e11+ p2e21 in the markets. You may assume an interior solution

throughout, i.e., ignore the non-negativity constraints.

(a) (5 points) Write the de�nition of a competitive equilibrium.

(b) (5 points) Saving is implicit in this model. Discuss brie�y how the rate of interest is

determined if there is a market for saving and loan.

(c) (5 points) Write the Pareto problem to �nd a Pareto e¢ cient allocation, and derive the

�rst order condition for e¢ ciency, assuming an interior solution.

(d) (5 points) Suppose that utility function ui is the same for all the consumers, and it is

given as ui (z) = 11��z

1��, where � > 0. (when � = 1, set ui (z) = ln (z)). Show that in

any competitive equilibrium, if �i > �j thenx2ix1i>

x2jx1jholds, i.e., more patient consumer

consumes relatively more in the second period.

(e) (5 points) Suppose that �i is the same for all the consumers and that �e1 = �e2, i.e., the total

amount of the good is constant over periods. Show that in any competitive equilibrium,

the consumption must be perfectly smoothed, i.e., x1i = x2i holds for every consumer i.

33. Comonotonicity of e¢ cient risk sharing. Consider an exchange economy with a single

good and S states. Each consumer i receives a esi units of the good if state s occurs. Consumer

i, i = 1; ::; I, has utility functionPS

s=1 �sui (x

si ) where ui is increasing and strictly concave, and

xsi is the consumption in state s. Note that probability weights �s, s = 1; ::; S, do not depend

on i. Denote by �es :=PI

i=1 esi the total resource available in state s.

(a) Denote by �i the welfare weight for consumer i, and write the �welfare maximization

problem�which characterizes the Pareto e¢ cient allocations.

(b) Write the �rst order condition for Pareto e¢ ciency.

(c) Show that if x is a Pareto e¢ cient allocation, xsi � xs0

i implies xsj � xs

0

j for all consumers.

That is, at e¢ cient allocation, consumption is monotonic to each other (this property is

called comonotonicity).

(d) Show that consumption must be monotonic with respect to the total resource in any Pareto

e¢ cient allocation, i.e., for every i, xsi � xs0

i holds if and only if �es � �es0i .

(e) Show that the set of Pareto e¢ cient allocations is invariant with respect to the probability

assignment �; that is, an allocation is e¢ cient for a common belief �, it is e¢ cient for any

common belief.

(f) Do the results above hold if the beliefs are not common?

34. Comonotonicity of e¢ cient risk sharing and contingent markets. Consider an economywith a single good and S uncertain states. Each consumer i receives a esi units of the good if

state s occurs. Consumer i, i = 1; ::; I, has utility functionPS

s=1 �sui (x

si ) where ui is increasing

and strictly concave, and xsi is the consumption in state s. Note that probability weights �s,

s = 1; ::; S, do not depend on i. Denote by �es :=PI

i=1 esi the total resource available in state s.

24

(a) Denote by �i the welfare weight for consumer i, and write the �welfare maximization

problem�which characterizes the Pareto e¢ cient allocations.

(b) Write the �rst order condition for Pareto e¢ ciency.

(c) Show that if x is a Pareto e¢ cient allocation, xsi � xs0

i implies xsj � xs

0

j for all consumers.

That is, at e¢ cient allocation, consumption is monotonic to each other (this property is

called comonotonicity).

(d) Show that consumption must be monotonic with respect to the total resource in any Pareto

e¢ cient allocation, i.e., for every i, xsi � xs0

i holds if and only if �es � �es0i .

(e) Show that the set of Pareto e¢ cient allocations is invariant with respect to the probability

assignment �; that is, an allocation is e¢ cient for a common belief �, it is e¢ cient for any

common belief.

(f) Do the results above hold if the beliefs were not common?

From now on, imagine that there are markets for the good before the uncertainty is re-

vealed1 . Denote by ps the price of the good available in state s. So consumer i has

incomeP

s psesi to spend in these markets, and the total expenditure of consumption plan

xi = (� � � ; xsi ; � � � ) isP

s psxsi .

(g) Write the de�nition of a competitive equilibrium in this setup.

(h) Show that a competitive equilibrium is Pareto E¢ cient.

(i) Suppose �es = �es0, i.e., in states s and s0 the total consumption good available is the same.

How �s and �s0are related with ps and ps

0?

(j) Now suppose that S = I, i.e., the number of the individuals is the same as the number of

the states. Suppose further that esi = 1 if s 6= i, and eii = 0. That is, state i is the bad statefor consumer i where his income drops to 0. Show that at any competitive equilibrium xsiis independent of s, and �s and ps are monotonically related. Interpret.

35. Gains from trade in general equilibrium. Consider a pure exchange economy with I con-sumers and L goods. Denote by �e 2 RL++ the total endowment vector. Write xi =

�x1i ; :::; x

Li

�� 0 for consumer i�s consumption, and write x = (x1; :::; xi; ::::; xI). Each consumer i is char-acterized by utility function ui. Assume that utility functions are concave, di¤erentiable, and

Dui >> 0. Answer the following questions.

(a) Write the de�nition of a competitive equilibrium of this economy.

We say that a feasible consumption allocation (x1; :::; xI) 2 RL+ � � � � � RL+ allows no

gains from bilateral trade, if for any pair of consumers i; j, there is no z 2 RL such thatui (xi + z) > ui (xi) and uj (xj � z) > uj (xj).

(b) Show that if a feasible consumption allocation (x1; :::; xI) is Pareto e¢ cient, it allows no

gains from bilateral trade.

(c) Let (x1; :::; xI) be a feasible consumption allocation such that xi >> 0 for each i. Show

that if (x1; :::; xI) allows no gains from bilateral trade, (x1; :::; xI) is Pareto e¢ cient. [Hint:

express the �rst order condition for no gains from bilateral trade by a constrained maxi-

mization problem.]

1This setup is often referred to as the contingent commodity markets model.

Microeconomic Theory problems by A. Kajii 25

36. An Equilibrium model for interest rate. Consider a two period economy with one per-ishable good in each period, 0 and 1. There is a representative consumer with a concave,

di¤erentiable vNM utility function u and his utility is additively separable with discount factor

� 2 (0; 1). The consumer is endowed with e0 units of the good in period 0, and his endowmentof the good in period 1 is random, and it is represented by a random variable Y . There is a

market for a riskless discount bond, which is a security which promises to pay one unit of good

in the second period for sure. The net supply of the bond is zero. Denote by x the amount of

the discount bond the consumer chooses to own. The price of the bond is q. To sum up, the

representative consumer solves

maxxu�e0 � qx

�+ �Eu (x+ Y )

where E is the expectation with respect to Y .

(a) Write down the �rst order condition for the consumer�s problem.

(b) Show that a competitive equilibrium occurs when the representative consumer wants to

choose x = 0; that is, when x = 0 is chosen then both of the markets for the good in period

0 and the good in period 1 are clearing.

(c) Derive the equilibrium bond price and the interest rate of this economy.

(d) When the consumer becomes more patient, i.e., the discount factor increases, what happens

to the equilibrium bond price?

(e) When the second period endowment gets riskier, that is, the random variable Y changes

to Y 0 and Y 0 is a riskier random variable than Y , what happens to the equilibrium bond

price?

37. An Equilibrium model for interest rate with background labor productivity shock.There are two periods. Your gross income is m in the �rst period, part of which may be invested.

Writing x for the amount of investment, your net income in the �rst period is m� x. The rateof return on the investment is r. You own L units of labor available in the second period. Your

utility is independent of labor, thus you will supply the whole L units in the labor market. But

the productivity of your labor is subject to some random shock. The shock is given by a positive

random variable � with E [�] = 1. Denote by w the wage per productivity unit in the labor

market, so your income from labor will be w�L. To sum up, your income in the second period

is (1 + r)x + w�L. Your vNM utility function for today�s income is denoted by u and that for

tomorrow is denoted by v, and you are interested in maximizing the sum of u and the expected

value of v. Assume that both functions are smooth, increasing, and concave. You may ignore

the non-negativity condition for income throughout.

(a) (5 points) Write down the maximization problem you need to solve to determine how much

to invest.

(b) (15 points) How does your investment change for the following cases?

i. wage w goes up.

ii. labor endowment L increases.

iii. period 1 income m increases.

iv. the shock parameter � gets riskier in the sense of the second order stochastic dominance.

v. the rate of return r increases.

26

38. Equilibrium asset pricing model. Consider an economy with a single (representative) con-sumer with concave vNM utility index u and discount factor �. There is one good in each period,

period 0 and 1. The representative consumer�s endowment in period 0 is w0 2 R, and the totalrandom endowment is W in period 1. There are J assets with zero net supply, which may be

traded before the uncertainty is resolved. Denote by Dj the (random) dividend of asset j and

by pj the price of asset j. The price of good is normalized to be one in each period, and also

the units of assets are normalized so that the expected dividend E [Dj ] = 1 for all assets. Thus

the problem of the consumer is to solve, given p,

maxx;y

u (x) + �E

24u0@W +

JXj=1

yjDj

1A35 subject to (5)

x+Xj

pjyj = w0:

(a) Assume that the �rst order condition is su¢ cient for utility maximization. Write the FOC.

(b) Since there is a single consumer and assets are in zero net supply, yj = 0 must hold in a

competitive equilibrium. Find the equilibrium price pj of asset j.

(c) Note that for any random variables X and X, we have E [XY ]�E [X]E [Y ] = COV [X;Y ],where COV indicates the covariance. Using this relation, re-write the equilibrium pricing

formula above.

(d) Now assume that utility function is quadratic, u (x) = ax � 12x

2, where a is positive. Re-

write the formula you obtained above. Among those assets j and asset k have the same

expected dividend. at type of assets tend to have high market price?

39. Equilibrium asset pricing model. Consider an economy with a single (representative) con-sumer with concave vNM utility index u and discount factor �. There is one good in each period,

period 0 and 1. The representative consumer�s endowment in period 0 is w0 2 R, and the totalrandom endowment is W in period 1. There are J assets with zero net supply, which may be

traded before the uncertainty is resolved. Denote by Dj the (random) dividend of asset j and

by pj the price of asset j. The price of good is normalized to be one in each period. Thus the

problem of the consumer is to solve, given p,

maxx;y

u (x) + �E

24u0@W +

JXj=1

yjDj

1A35 subject to (6)

x+Xj

pjyj = w0:

(a) Find the equilibrium price of asset j.

(b) From now on, assume D1 = 1 for sure; that is Asset 1 is riskless. Suppose the random

endowment W �improves�in the sense the new random endowment W 0 �rst order stochas-

tically dominates the original W . What will happen to the price of asset 1? Can you say

anything about the prices of the other assets?

(c) Suppose the random endowment W �improves� in the sense the new random endowment

W 0 second order stochastically dominates the original W , i.e., the endowment is less risky.

Assume in addition that u000 > 0. What will happen to the price of asset 1? Can you say

anything about the prices of the other assets?

40. Consider an agent with concave vNM utility function u. His total wealth is W and he can invest

(save) in a riskless or a risky asset. The riskless asset pays 1 per unit in the next period and costs

Microeconomic Theory problems by A. Kajii 27

q1 per unit in this period. The risky asset pays r with probability 1� p and �r with probabilityp, and it costs q2. Denote by y1 (resp. y2) his demand for the riskless (resp. risky) asset.

(a) Write the budget constraint, for the case where short sales is allowed (i.e., he can hold a

negative amount of asset), and for the case it is not.

(b) What is the relationship between q1 and interest rate?

(c) Assuming short sales are allowed, show that assets are normal goods.

(d) Suppose he owns one unit of each asset. So his initial wealth is q1 + q2. Suppose further

that he is the only trader in the markets (i.e., he is a representative trader (consumer).

Find the equation that characterizes the equilibrium asset prices.

(e) Assume u (z) = z � 1 z

2. Find the capital asset pricing formula in the question above.

41. Risk sharing. Consider an economy with I agents. Each agent�s preferences are representedby a vNM utility function ui. Assume that each ui satis�es u0i > 0 and u

00i < 0. The aggregate

income of this economy is denoted by y > 0, and it will be distributed among the agents. Denote

by xi (y) � 0 the income the agent i receives when the aggregate income is y, soP

i xi (y) = y

must hold. For simplicity, suppose that there are only �nitely many possible aggregate income

levels, y1; � � � ; yK and denote by p (yk) > 0, k = 1; :::;K, the probability that the aggregate

income is yk (soPK

k=1 p (yk) = 1).

(a) Assume I = K = 2. So the utility of agent i; i = 1; 2; can be written as ui (xi (y1)) p (y1)+

ui (xi (y2)) p (y2).

i. Let y1 > y2. Think of xi (yk) as consumption of good k, and draw an �Edgeworth box�

which represents feasible income distributions among the agents. Explain graphically

the conditions a Pareto e¢ cient allocation must satisfy.

ii. Show that if y1 = y2, xi (y1) = xi (y2) must hold for both i, for any Pareto e¢ cient

allocation.

(b) Now assume I > 2; K > 2: So agent i receives a vector of income xi := (xi (yk))Kk=1, which

yields the expected utility Ui (xi) :=PK

k=1 ui (xi (yk)) p (yk)

i. Give the de�nition for a feasible allocation x :=�� � � ; (xi (yk))Kk=1 ; � � �

�. Give the

de�nition of a Pareto e¢ cient allocation, which does not rely on special properties of

utility functions such as di¤erentiability.

ii. Fix �i > 0, i = 1; :::; I, and write the Pareto maximization problem of weighted sum of

expected utility functions,PI

i=1 �iUi, and derive the �rst order condition for Pareto

e¢ ciency for interior points.

iii. Assume that ui (z) = 11� i z

1� i where where i is a positive constant, for every agent

i. Show that if x =�� � � ; (xi (yk))Kk=1 ; � � �

�is Pareto e¢ cient, then xi (yk) = �iyk for

k = 1; :::;K where �i > 0 is a constant.

42. Contingent commodities and risk sharing. Consider an exchange economy with a singlegood and S states. Each consumer i receives esi units of the good if state s occurs. Consumer

i, i = 1; ::; I, has utility functionPS

s=1 �sui (x

si ) where ui is smooth, increasing and strictly

concave, and xsi is the consumption in state s. Note that probability weights �s, s = 1; ::; S, do

not depend on i. Denote by �es :=PI

i=1 esi the total resource available in state s. Assume an

interior solution throughout, i.e., ignore the non-negativity constraints.

28

(a) Suppose that there are competitive markets for the contingent consumption goods. That

is, each consumer can buy/sell the good to be consumed in state s. Denote by ps for the

price of the good to be consumed in state s, so the income for consumer i isPS

s=1 psesi .

Give the de�nition of a competitive equilibrium in these markets.

(b) Denote by �i the welfare weight for consumer i, and write the �welfare maximization

problem� which characterizes the Pareto e¢ cient allocations, and write the �rst order

condition for Pareto e¢ ciency.

(c) Show that if x is a competitive equilibrium allocation, then a consumer with a higher

income consumes more in every state: that is, if p is an equilibrium price and x is the

corresponding consumption allocation,PS

s=1 psesi >

PSs=1 p

sesj implies xsi > x

sj at every s.

(d) Show that in any Pareto e¢ cient allocation x, the consumption must be monotonic with

respect to the total resource, i.e., if x is a competitive equilibrium allocation, for every i,

xsi � xs0

i holds if and only if �es � �es0i .

43. Consider an economy as follows. There are two consumers, i = 1; 2: There are two periods, and in

each period, consumers trade a single, perishable commodity (i.e., they cannot store the good).

There is no uncertainty. Consumer i�s preferences are represented by ui�x0i ; x

1i

�= lnx0i +�i lnx

1i

where �i 2 (0; 1) is a constant. Each consumer is endowed with one unit of the good in eachperiod. Thus consumers�preferences are identical, except for the discount factor �i. The price

of good is normalized to be 1 in both periods. In period 0, consumers can save: denote by si the

amount that consumer i saves in period 0. Let r be the interest rate. That is, if consumer i saves

si units of the good, he receives (1 + r) si units of good at the beginning of period 1 before the

trade of the consumption good takes place. Hence consumer i chooses�x0i ; x

1i ; si

�2 R+�R+�R.

Notice that negative saving, i.e., borrowing is allowed.

(a) Write the consumer i �s budget constraint.

(b) For each i, derive the saving function si (r), that is, si (r) is the amount of good consumer

i saves in period 0 when the interest rate is r.

(c) Show that when the saving market clears, i.e., s1 (r) + s2 (r) = 0, the good market clears

in both periods 0 and 1.

44. no gains from bilateral trade. Consider a pure exchange economy with I consumers and Lgoods. Each consumer i is characterized by utility function ui and endowments ei =

�e1i ; :::; e

Li

�.

Write xi =�x1i ; :::; x

Li

�� 0 for consumer i�s consumption, and write x = (x1; :::; xi; ::::; xI).

Denote by p =�p1; :::; pL

�the prices of goods. Assume that utility functions are concave,

di¤erentiable, and Dui >> 0. Answer the following questions.

(a) Write the de�nition of a competitive equilibrium of this economy.

We say that a feasible consumption allocation (x1; :::; xI) 2 RL+ � � � � � RL+ allows no

gains from bilateral trade, if for any pair of consumers i; j, there is no z 2 RL such thatui (xi + z) > ui (xi) and uj (xj � z) > uj (xj).

(b) Show that if a feasible consumption allocation (x1; :::; xI) is Pareto e¢ cient, it allows no

gains from bilateral trade.

(c) Let (x1; :::; xI) be a feasible consumption allocation such that xi >> 0 for each i. Show

that if (x1; :::; xI) allows no gains from bilateral trade, (x1; :::; xI) is Pareto e¢ cient. [Hint:

express the �rst order condition for no gains from bilateral trade by a constrained maxi-

mization problem.]

Microeconomic Theory problems by A. Kajii 29

45. [core (non)-convergence] Consider an economy with two goods, two consumers. The con-sumers 1 and 2 have an identical utility function ui

�x1i ; x

2i

�= min

�x1i ; x

2i

, and their endow-

ments are (2; 1) and (1; 2), respectively.

(a) Find the core of this economy (you may answer graphically).

(b) Suppose that this economy is replicated. What will happen to the core?

46. [Replica economy] Consider an economy with two goods, two types of consumers. The type 1and 2 consumers have an identical utility function ui

�x1i ; x

2i

�= lnx1i + lnx

2i , and their endow-

ments are (8; 2) and (2; 8), respectively.

(a) Verify that the allocation x1 = (4; 4), x2 = (6; 6) belongs to the core of the two consumers

exchange economy where there is one consumer of each type.

(b) Suppose that this economy is replicated once so that there are two consumers of each

type. Show that the allocation where type one consume x1 = (4; 4), and type 2 consume

x2 = (6; 6) is not in the core of this economy.

47. [fair allocation] Consider an exchange economy with I consumers and L goods, where the totalendowments of goods is �! >> 0. A feasible allocation x = (� � � ; xi; � � � ) is called envy-free iffor any i and j, ui (xi) � ui (xj). A feasible allocation is called a fair allocation if it is Paretoe¢ cient and envy-free.

(a) Give an example where a competitive equilibrium allocation is not a fair allocation.

(b) Show that under the standard assumptions, there exists a fair allocation.

48. [Matching] There are two types of agents, type M and type F. There are three type M agents

and four type F agents, and consider make pairs of one M agent and one F agent. A matching

is a function � : f1; 2; 3g ! f1; 2; 3; 4g where � (i) is the type F agent to whom type M agent i

is paired with. Every type M agent wants to be paired with some F agent while some F agent

prefers not being paired at all to being matched with some agent. Type M agent�s preferences

over type F agents as well as Type F agents�preferences over M agents are given in the following

two tables (0 means not being paired):

M agent�s name

1 1 2 3 4

2 1 3 4 2

3 1 4 2 3

F agent�s name

1 3 2 0 1

2 1 3 2 0

3 1 2 0 3

4 1 2 3 0

For instance, M agent 1 prefers F1, F2, F3 and F4 in this order, and F agent 1 prefers M3, M2,

but she would rather not matched than paired with M1.

(a) Give the de�nition of an e¢ cient matching.

(b) Consider � (i) = i, i = 1; 2; 3: Is this e¢ cient?

(c) Consider � (1) = 2; � (2) = 3; � (3) = 1. Is this e¢ cient?

(d) Find all the e¢ cient matchings.

30

7 Strategic Market and Game theory

7.1 Nash Equilibrium of strategic form games

1. In a game in strategic form, a strategy si 2 Si is strictly dominated if there is a strategy s0i 2 Sisuch that ui (si; s�i) < ui (s0i; s�i) for all s�i 2 S�i. A strategy is said to be weakly dominatedif if there is a strategy s0i 2 Si such that ui (si; s�i) � ui (s

0i; s�i) for all s�i 2 S�i, where <

holds for some s�i 2 S�i.

(a) Show that a strictly dominated strategy cannot be part of a Nash equilibrium strategy

pro�le.

(b) Give a 2-person game example where there is a Nash equilibrium in which both players

choose weakly dominated strategies.

2. Find all (i.e., pure and mixed) Nash equilibria of the following two person games.

(a)L R

T (2; 2) (0; 0)

B (0; 0) (1; 1)

(b)L C R

T (2; 2) (�1; 3) (1; 0)

M (1;�1) (�2; 0) (�1;�2)B (0; 1) (1;�2) (�1;�1)

(use the domination argument as much as possible)

(c)L R

T (z;�1) (�1; 1)B (�1; 1) (1;�1)

;

where z is a positive number. Observe that the equilibrium strategy of ROW player is

independent of z! Interpret.

3. A $100 bill is to be sold in a simple second price auction with I participants. Formulate this as

a game in strategic form, and show that bidding $100 is a weakly dominant strategy but not a

strictly dominant strategy.

4. Show that in a �nite strategic form game, a mixed strategy �i of a player is a best response to

other players strategy pro�le ���i if and only if �i (si) = 0 for any pure strategy si 2 Si thatis not a best response to ���i. Consider a game in strategic form, represented by the following

table.player 2

a b c d

x (2; 2) (�1; 1) (1; 0) (�1; 1)player 1 y (1;�1) (�2; 4) (0; 5) (�3; 0)

z (0; 1) (�5;�2) (�5;�3) (1; 3)

Answer the following questions about this game. For questions (a) to (g), you do not haveto explain your answers.

Microeconomic Theory problems by A. Kajii 31

(a) How many strategy pro�les are there?

(b) Is Strategy x is a dominant strategy?

(c) Is Strategy a is a dominant strategy?

(d) Is Strategy d is a dominated strategy?

(e) Is Strategy b is a best response to strategy x?

(f) Is there any strategy for which a best response is not unique?

(g) Does iterative deletion of dominated strategies result in a single strategy pro�le?

(h) Find all Nash equilibria of this game.

(i) Now consider mixed strategies as well. Find all mixed strategy equilibria of this game, ifany, which are di¤erent from the Nash equilibria you found above.

5. Mixed Strategy. Consider a sporting contest between two players, row and column. So eitherrow wins or column wins, not both. If loose, the player receives 0 yen. If row wins, he receives

x yen (x > 0) and if column wins, he receives y yen (y > 0). So, this strategic situation is

characterized by the following matrix.

L R

U(1� �11; y)

(�11; x)

(1� �12; y)(�12; x)

D(1� �21; y)

(�21; x)

(1� �22; y)(�22; x)

where for instance if row plays U and column plays L, row wins x with probability �11: It is

assumed that �11 > �21 and �12 < �22. Row�s vNM utility index is u and column�s vNM utility

index is v, and we normalize u (0) = v (0) = 0. Hence for instance if row plays U and column

plays L, row receives an expected utility of �11u (x).

(a) Show that there is no pure strategy Nash equilibrium.

(b) Write p for the chance row plays U , and denote by q the probability column plays L. When

the players choosing these mixed strategies, write the expected utility of playing U for the

row player.

(c) Write all the conditions that p and q must satisfy in a mixed strategy equilibrium.

(d) Show that an equilibrium strategy pro�le of this game does not depend on x and y (thus

the equilibrium play of this sporting event is invariant of the size of prize).

6. A $100 bill is to be sold in a �rst price auction with 2 participants. Formulate this as a game

in strategic form, and show that there is no Nash equilibrium in pure strategy but there is a

(symmetric) mixed strategy equilibrium.

7. Nash Demand Game. Consider the following �demand game�of two players. Each of twoplayers announces the share si 2 [0; 1] he demands out of an amount of money that may be splitbetween them. If both of the demands can be satis�ed, i.e., if the sum of the demanded amount

does not exceed one, then the money is split as is demanded. If not, neither player receives any

money, which shall be referred to as a disagreement.

(a) Describe this game formally.

(b) Show that any pro�le (s1; s2) with s1 + s2 = 1 is a Nash equilibrium.

32

(c) Is (1; 1) a Nash equilibrium?

(d) Show that a pro�le (s1; s2) with s1 < 1 and s2 < 1 such that s1 + s2 6= 1 is not a Nash

equilibrium.

(e) Let �s be a number with 0 < �s < 12 . Show that there is a mixed strategy equilibrium where

both players randomly demand �s or 1� �s.

(f) Notice that in the mixed strategy equilibrium above, a disagreement occurs with posi-

tive probability. Prove that in any mixed strategy equilibrium (i.e., at least one player

randomizes his strategy) a disagreement must occur with a positive probability.

(g) Show that any mixed strategy equilibrium is Pareto dominated by a pure strategy equilib-

rium (i.e., there is a pure strategy equilibrium where each player receives a payo¤ higher

than his expected payo¤ in the mixed strategy equilibrium).

7.2 Simple Models of Strategic Competition

1. Cournot Competition. There are two identical �rms with constant marginal cost of produc-tion c. The total demand for the product is given by q = a � bp, where a; b > 0 are constants.Each �rm j; j = 1; 2, freely determines the quantity qj to produce.

(a) Formulate this problem as a strategic form game.

(b) Find a Nash equilibrium. Is it in dominant strategies?

(c) Find quantity qM per �rm that maximizes the sum of pro�ts of the two �rms. The formulate

a strategic form game where each �rm can choose either qM or the quantity you found in

( 1b) above. Find a Nash equilibrium of this game. Is it in dominant strategies? Discuss.

2. Cournot Competition and iterative deletion of dominated strategies. There are twoidentical �rms with constant marginal cost of production 0. The total demand for the product

is given by q = 1� p. Each �rm j; j = 1; 2, freely determines the quantity qj to produce.

(a) Formulate this problem as a strategic form game.

(b) Find a Nash equilibrium. Is it in dominant strategies?

(c) Find dominated strategies. (Hint. Will the �rm ever produce less than a monopoly �rm

produces?)

(d) Assuming that the other �rm will never choose a dominated strategy, �nd dominated

strategy. (Hint. If a �rm is sure that the other �rm never produces more than the monopoly

amount, it should produce some positive amount.)

(e) Repeat this process of iteratively eliminating dominated strategies What do you get in the

limit?

3. Bertrand Competition. There are two identical �rms with constant marginal cost of produc-tion c. The total demand for the product is given by q = a� bp, where a; b > 0 Each �rm freely

sets its price, but if they set di¤erent prices, every consumer chooses to buy the product from

the �rm with the cheaper price. The demand will be split evenly if the prices are the same.

(a) Formulate this problem as a strategic form game.

(b) Find a Nash equilibrium.

Microeconomic Theory problems by A. Kajii 33

4. Price competition with capacity constraint. In the setting in (3), we shall assume thateach �rm j, j = 1; 2; can sell only up to a preset capacity limit kj > 0. When �rm j sets a

higher price pj than that of the other �rm, �rm j does not necessarily lose all the demand if the

other �rm is selling at its capacity ki. In such a case, demand for �rm j is the residual demand

1� ki � pj . To simplify computation, set a = b = 1 for the demand function, and c = 0 for themarginal cost.

(a) Formulate this problem as a strategic form game.

(b) Show that if k1 � 1 and k2 � 1, p1 = p2 = 0 constitute a Nash equilibrium.(c) Show that p1 = p2 = 0 is not a Nash equilibrium if k1 < 1 or k2 < 1.

(d) Assume k1 = k2 = 13 . Find a Nash equilibrium.

5. Monopolistic Competition. There are two �rms, 1 and 2, with constant marginal cost ofproduction cj , 0 < cj < 1, j = 1; 2. Each �rm j sets the price pj of its own product. The

demand for �rm j�s product is given by qj = 1 � pj + jp�j , where j are positive constants,j = 1; 2, and p�j is the price of the other �rm�s product.

(a) Formulate this problem as a strategic form game.

(b) Find a Nash equilibrium.

(c) Do comparative statics on j . Discuss.

6. Free Rider Problem. Consider a community with 2 individuals. Each individual owns 1 unitof a consumption good. Individual i�s utility depends on private consumption of the good as

well as the amount of public good available in the community. Speci�cally, individual i�s utility

is xi + y, where xi is the amount of the good privately consumed, and y is the amount of the

public good. The public good can only be produced from the consumption good: from z units of

the consumption good, f (z) units of the public good can be produced. So when each individual

i decides to consume xi units of the consumption good privately, f (2� (x1 + x2)) units of thepublic good is produced. Assume that f (0) = 0; f 0 > 0; and f 00 < 0. Each individual i chooses

xi strategically, and negative consumption is not allowed.

(a) De�ne a feasible allocation of the consumption good and the public good; that is, describe

(x1; x2; y) which can be achieved in this community. Then write the de�nition of a Pareto

e¢ cient allocation of this community.

(b) Assume that f 0 (2) > 12 . Find all Pareto e¢ cient allocations.

(c) Will a Pareto e¢ cient allocation be realized when f 0 (2) > 12? Explain.

(d) Will a Pareto e¢ cient allocation be realized when f 0 (0) < 12? Explain.

7. Consider a community with 2 individuals. Each individual owns 1 unit of consumption good.

Individual i�s utility depends on private consumption of the good as well as the amount of public

good available in the community. Speci�cally, individual i�s utility is xi + y, where xi is the

amount of good privately consumed, and y is the amount of the public good. The public good

can only be produced from consumption good: from z units of consumption good, f (z) units

of public good can be produced. So when each individual i decides to consume xi units of the

consumption good privately, f (2� (x1 + x2)) units of public good is produced. Assume thatf (0) = 0; f 0 > 0; and f 00 < 0, and f 0 (0) > 1

2 > f0 (2) : Negative consumption is not allowed.

(a) De�ne a feasible allocation of consumption good and public good, and write the de�nition

of a Pareto e¢ cient allocation of this community.

34

(b) Find the �rst order condition which characterizes Pareto e¢ cient allocations.

(c) Consider a game where both individuals simultaneously choose x1 and x2. Give the de�n-

ition of a Nash equilibrium of this game.

(d) Is a Nash equilibrium Pareto e¢ cient? Why?

(e) Consider a game where the game above is repeated twice as follows: in each of the periods,

each individual is endowed with one unit of consumption good, and simultaneously choose

private consumption level. The utility is a discounted sum of utilities with discount factor

� 2 (0; 1). Is there a subgame perfect Nash equilibrium where the �rst period allocation

or the second period allocation are e¢ cient? Why?

8. E¢ ciency loss by campaigning. N players compete for an award by campaigning. The

award is worth A if acquired, and 0 if not. If xi hours are spent for campaigning by player

i, i = 1; :::; N , player i�s chance of getting the award is xi=PN

j=1 xj (by assumption it is 0 if

nobody campaigns). The cost of campaigning is c per hour. So player i�s expected bene�t from

campaigning xi hours is AxiPNj=1 xj

� cxi:

(a) Find a symmetric Nash equilibrium.

(b) What will happen when N increases for campaigning hours in the equilibrium above?

(c) Suppose that a player is selected at random, and give the award to the selected player.

Argue that this rule is better than the competitive campaign as far as the total welfare is

concerned. What about by the Pareto criterion?

(d) Suppose that the award is valued di¤erently across the players, i.e., the value of the award

to player i is Ai. How would you modify the answer to the question above?

9. An Exhaustible Resource Commons Problem. (Dutta) Suppose two players, 1 and 2,share a �xed supply of y �sh. Each player lives for exactly two periods. In the �rst period, each

player i can consume a non-negative amount of �sh, ci, provided that c1+ c2 � y. In the secondperiod, any remaining �sh, y�c1�c2, are divided equally between the players. Player i�s utilityis given by

ui(ci; c�i) = ln ci + ln

�y � c1 � c2

2

�,

where lnx is the natural logarithm of x. By convention, utility is �1 if consumption of �sh is

zero in the second period.

(a) Consider this as a simultaneous move game where each player chooses a strategy si such

that 0 � si < y (that is, choosing si = y is not possible). If s1 + s2 � y then ci = si, butif s1 + s2 > y then c1 = c2 = y=2. Find each player�s best response rule and �nd the Nash

equilibrium of the game.

(b) Suppose that a central planner can set c1 and c2 (subject to c1 + c2 � y). Assume the

planner aims to maximize social welfare given by u1 + u2. Find her choice of c1 and c2.

Now consider an analogous situation with N players. Player i�s utility is now given by

ui(ci; c�i) = ln ci + ln

�y � c1 �

Pj 6=i cj

N

�where

Pj 6=i cj is the sum of �other�players��rst period consumptions.

Microeconomic Theory problems by A. Kajii 35

(c) Consider the corresponding N -player game where 0 � si < y, as before, and

ci =

(si if

Pi si � y

y=N otherwise

Find the Nash equilibrium. What happens to total �sh consumption in the �rst period as

N goes to in�nity? Give an intuition.

(d) Consider the social planner who aims to maximize the sum of the N players utilities:P

i ui.

What ci�s will she choose? Compare your answer to part (b) and give an intuition.

(e) How do the Nash and social welfare maximization outcomes compare when N = 1? Give

an intuition.

(f) Compare all your answers above to Cournot competition. Comment brie�y.

7.3 Dynamic games and commitment

1. A dynamic oligopoly game. There are three �rms, 1, 2, and 3 producing a consumptiongood whose inverse demand is given by p = 1�Q. Each �rm j, j = 1; 2; 3; chooses quantity qjas a strategic variable. The marginal cost of production is zero.

(a) Suppose three �rms choose quantities simultaneously. Find a Nash equilibrium.

(b) Suppose that the decisions are made sequentially by �rms 1, 2, and 3 in this order. That is,

after �rm 1 chooses q1; �rm 2 chooses q2 observing q1, and so on. Find a Nash equilibrium

consistent with the backward induction principle.

(c) Suppose that after �rm 1 chooses its quantity, �rms 2 and 3 choose their quantities simul-

taneously. Thus at the time of decision making both �rms 2 and 3 know q1 but do not

know each other�s choice of quantity. Find a subgame perfect Nash equilibrium.

(d) Suppose that �rm 1 chooses its quantity after �rms 2 and 3 choose their quantities simul-

taneously. Thus �rm 1 knows q2 and q3, but �rms 2 and 3 must decide without knowing

any other �rm�s decision. Find a subgame perfect Nash equilibrium.

2. A dynamic oligopoly game with a global player: Consider two countries, 1 and 2. Thereis one domestic �rm in each country. There is one global �rm, which operates in both countries.

These �rms produce an identical good at zero marginal cost. Demand for the product is given

by an inverse demand function p = 1 � q in each county (so the demand is identical). Denoteby q1 and q2 the quantity produced by the domestic �rms in country 1 and 2, respectively, and

denote by qG1 and qG2 the quantities the global �rm produces in country 1 and 2, respectively.

The price of the good in a country is determined by the demand and the supply in the domestic

market: for instance, the price of good in country 1 will be 1 ��q1 + q

G1

�. The global �rm

maximizes the sum of pro�ts in the two countries. (You may take it for granted that a solution

to maxx (a� x)x is x = a2 )

(a) (5 points) Suppose that the �rms behave strategically a la Cournot. Find equilibrium

quantities q1,q2, and�qG1 ; q

G2

�.

(b) (5 points) Suppose that the global �rm is the leader in both countries: that is, the global

�rm chooses�qG1 ; q

G2

�and then the domestic �rms make their decisions. Find quantities

each �rm produces in equilibrium.

36

(c) (5 points) Suppose that the global �rm is a follower in both countries, and moreover it

must produce exactly the same amount due to some technical reasons. That is, after q1and q2 are chosen, qG1 = q

G2 = q

G is chosen. Find an equilibrium strategy pro�le as well as

the equilibrium quantities induced by the pro�le.

(d) (10 points) Suppose that the �rms behave strategically a la Cournot as in (a), but suppose

that the global �rm (but not the domestic �rm) is taxed in both countries as follows: �rst,

each country i �x a per unit tari¤ ti on the global �rm, and then the �rms compete a la

Cournot fashion. Thus the global �rm must pay tiqGi to country i�s tax o¢ ce.

i. Find equilibrium quantities after (t1; t2) is selected.

ii. The tax o¢ ce in country i is interested in the welfare in country i�s consumers and its

domestic �rm as well as the tax revenue. Show that there is a dominant strategy for

each country�s tax o¢ ce.

3. Entry deterrence. Consider two �rms, Incumbent and Entrant. Both �rms can produce aconsumption good with a constant marginal cost c, 0 � c < 1. The inverse demand for the

good is given by 1�Q, where Q is the total production of the good. Entrant however incurs a

�xed entry cost K � 0 if it chooses to produce. Denote by qE and qI the level of production ofEntrant and Incumbent, respectively.

(a) Suppose that once Entrant pays the entry cost, both �rms will compete in the Cournot

fashion.

i. Find a subgame perfect equilibrium in which Entrant does enter, when K is small

enough.

ii. Find a subgame perfect equilibrium where Entrant chooses not to enter if K is large.

iii. Is subsidizing entry cost K (thus Entrant�s e¤ective cost for entry is zero) a good policy

from the point of view of social welfare?

(b) Suppose that Incumbent can commit to its quantity produced qI before Entrant makes its

entry decision, and its level of production.

i. Let K = 0. Find a subgame perfect equilibrium.

ii. Let K > 0. Will Incumbent produce more than the amount for the case of K = 0? If

so, why? [in this example this is going to be a degenerate case]

iii. Is subsidizing entry cost K (thus Entrant�s e¤ective cost for entry is zero) a good policy

from the point of view of social welfare?

(c) Suppose that Incumbent can invest to reduce its marginal cost of production to 0, before

Entrant makes its entry decision. The �xed cost of the investment is � > 0. Once entry

takes place, both �rms compete a la Cournot.

i. Show that if � is small and K is large, there is a subgame perfect equilibrium where

Incumbent invests, and Entrant does not enter.

ii. Is the cost reducing investment above socially desirable?

4. Imagine a developer building houses near a lake. There are two periods, t = 1 and 2. In each

period, the developer can construct houses, and the construction cost per unit is c > 0. Denote

by xt the total unit the developer chooses to construct in period t. Also denote by pt the price of

house in period t which the developer determines. The developer is interested in the discounted

sum of pro�ts with discount factor �, 0 < � � 1. That is, the developer maximizes the sum of

period 1 pro�t and � times period 2 pro�t. Because of �nancial reasons, the developer must

Microeconomic Theory problems by A. Kajii 37

sell all units of houses in the period they are built. In other words, unsold houses at theend of period t = 1 will be con�scated without compensation and so the houses to be sold in

period 2 must be constructed in period 2.

In each period, after houses are constructed, many potential buyers come to see the houses.

Each buyer buys at most one unit, and the house cannot be sold later for other buyers. Also,

a buyer who comes in period 1 cannot choose to come back in period 2. That is, it is assumed

that a buyer who comes in period 1 cannot wait till period 2 for a bargain. If a potential buyer

buys a house, his payo¤ (in terms of money) is a � (x1 + x2), where a is a positive constantwith a > c. If he does not buy a house, his payo¤ is 0. Note that in period 1, the second period

houses are yet to be constructed, thus the buyer�s decision will depend on the expectation of x2.

Assume that if a buyer is indi¤erent between buying a house and not buying, he will buy. Also

assume that if the demand for the houses exceeds the supply of the houses then the buyers will

be rationed (say by a lottery). Assume that xt�s are real numbers to simplify the question.

(a) Note that the payo¤ from a house depends is decreasing in the sum of houses x1 + x2.

Interpret this assumption in economic terms.

(b) Suppose that �x1 units have already been sold in period 1.

i. Consider the stage where the developer has constructed x2 units of houses. So, each

buyer�s payo¤ from a house is a � (�x1 + x2), so all the x2 houses will be sold if andonly if p2 � a� (�x1 + x2). What p2 should the developer choose?

ii. Find the number of houses x2 the developer constructs.

(c) Suppose that at the beginning of period 1, the developer can somehow convince the buyers

that the number of houses sold in the next period will be x2, and assume that the developer

keeps his promise and construct x2 units of houses in the second period.

i. If the developer wishes to sell x1 units of houses in period 1, what will be the prices of

houses in period 1 and period 2?

ii. What will be the number of houses constructed in period 1?

iii. Find x2 which maximizes the total pro�t.

iv. In the solution above, will the developer has incentive to construct x2 units in period

2, after the �rst period houses are all sold?

(d) Suppose that the developer cannot commit to the number of houses to be constructed in the

second period, thus the buyers will take into account what the developer will do in period

2. How many houses will be constructed in period 1, and what will the total number of

houses?

5. Trade war. Consider two countries, 1 and 2. There is one domestic �rm in each country and

they produce an identical good at zero marginal cost. Demand for the product is given by an

inverse demand function p = 1 � q in each county (so the demand is identical). Denote by q1and q2 the quantity produced towards the domestic market in each country and denote by e1and e2 the quantity exported form country 1 and 2 to the other country. So for instance the

price of good in country 1 will be 1� (q1 + e2).

(a) Suppose the �rms behave as price takers, and there is no international trade. Find the

equilibrium production level for each country.

(b) Suppose the �rms behave strategically a la Cournot. That is, �rm 1 maximizes the sum

of pro�ts from country 1 and 2 by changing q1 and e1, given q2 and e2, for instance. Find

equilibrium quantities qi; ei, i = 1; 2:

38

(c) Consider the following game: �rst, each country i �x a per unit tari¤ ti on import, and

then the �rm compete a la Cournot fashion. Thus for instance the �rm in country 1 must

pay t2e1 to country 2�s tax o¢ ce. Solve this game by �rst �nding the second stage (i.e.,

the game where (t1; t2) is already selected) equilibrium.

(d) Suppose that each country�s tax o¢ ce is interested in the social welfare, de�ned as (con-

sumer surplus) + (pro�ts of domestic �rm) + (tari¤ revenue). Finding a pair of tari¤

(t1; t2) which arises in a subgame perfect equilibrium of the game where tari¤s are set �rst,

and then the �rms make decisions as in (5c).

6. In�ation targeting. Consider two players, the public (P) and the monetary authority (M).The payo¤ of P is given by � (�e � �)2, where �e is the expected rate of in�ation held by Pand � is the actual rate of in�ation. The payo¤ of M is given by ��2 � (y � �y)2, where y isthe realized GDP of the economy and �y is the natural (full employment) level of GDP. That

is, in�ation as well as over/under production are costly to M. Suppose that there is a trade-o¤

between the rate of in�ation and GDP (i.e., a Phillips curve) given by (��y � y) + (� � �e) = 0where 0 < � � 1 is a constant, and this relation is known to the both players. So if � < 1, thefull employment GDP can be realized only with a �surprise in�ation�, i.e., � � �e > 0.

(a) Suppose P selects �e and M controls � simultaneously. Find a Nash equilibrium of this

game.

(b) Suppose P moves �rst. That is, M can select the rate of in�ation after observing the

expected rate of in�ation. What happens?

(c) Suppose M moves �rst. That is, M can commit to its �target� rate of in�ation before P

forms its expectation. What happens?

7. Alternative for pro�t maximization. Consider two �rms, 1 and 2, producing an identicalgood at marginal cost c (su¢ ciently low, say c < 1). The demand for the product is given by

p = 1�q. The quantity produced q1 and q2 by the �rms 1 and 2 are simultaneously determined.But for each �rm, the production decision is done by a manager who is independent of the

owner of the �rm; that is, the managers hired by the �rms decide q1 and q2. The �rm needs

to pay for the manager, thus the total cost of production from the view point of the �rm is cqjplus the payment to the manager. The manager of �rm j is paid " times �j�(revenue of �rmj)+(1� �j)�(pro�ts of �rm j), where " is a given parameter which is determined exogenously

in labor market and �j 2 [0; 1] is determined by the owner of �rm j before production takes

place: that is, �1 and �2 are chosen simultaneously by the owners who want to maximize the

�rm�s pro�t, and then q1 and q2 are chosen by the managers simultaneously. For instance if

the revenue of �rm i is R and pro�t is �, the manager of �rm i is paid " (�iR+ (1� �i)�).So if �j = 0; the manager of �rm j wants to maximize the pro�t of the �rm, but otherwise

he may not be interested in pro�t maximization. If �j = 1 the manager wants to maximize

the revenue, so the �rm will look as if it adopts revenue maximization rule. The parameter "

represents the share of managerial labor, but in what follows let�s assume " is negligibly small

from the viewpoint of the �rm. Thus when you compute the �rm�s pro�t, ignore the payment

to the manager.

(a) How much will the managers choose to produce after (�1; �2) have been set (and the pair

(�1; �2) is mutually observable)?

(b) Suppose that �1 and �2 are chosen simultaneously. What will happen?.

(c) Suppose that �rm 1 choose �1 �rst and then �rm 2 choose �2. What will happen?

Microeconomic Theory problems by A. Kajii 39

8. (30 points) Consider two �rms, 1 and 2, producing an identical good at constant marginal cost

c = 1. The inverse demand for the product is given by p = 2 � q. The �rms simultaneouslydetermine their quantity produced q1 and q2. Suppose that for each �rm, the production deci-

sion is done by a manager who is independent of the owner of the �rm. The �rm needs to pay

for the manager, thus the total cost of production from the view point of the �rm is cqj (= qj)

plus the payment to the manager. The manager of �rm j is paid " times �j�(revenue of �rmj)+(1� �j)�(pro�ts of �rm j), where " > 0 is a given parameter which is determined exoge-

nously in labor market and �j 2 [0; 1] is determined by the owner of �rm j before production

takes place. For instance, if the �rm j�s revenue is Rj and pro�t (= revenue minus cost) is �j ,

the manager of �rm j receives "(�jRj + (1 � �j)�j). The manager of �rm j will chooses qj in

such a way to make this number as large as possible. So if �j = 0, the manager will be interested

in maximizing the pro�t, and if �j = 1, he will be interested in maximizing the revenue. The

parameter " represents the share of managerial labor, but in what follows let�s assume " is neg-

ligibly small from the viewpoint of the �rm. Thus when you compute the �rms�pro�ts, ignore

the payments to the managers. That is, for each �rm j, assume that Rj = (2� q1 � q2) qj and�j = Rj � qj . Firm j is interested in maximizing �j .

(a) (2 points) Show that the payo¤ for the manager of �rm 1 is "q1 ((1 + �1)� q1 � q2) :

(b) (2 points) In this economy, the total surplus is the consumer surplus plus the total pro�ts,

since payments to the managers are ignored. Show that the total surplus when q1 and q2are chosen is 1

2 (q1 + q2) (4� (q1 + q2))

(c) (5 points) How much will the managers choose q1 and q2 produce after (�1; �2) have been

set (and the pair (�1; �2) is observable to everybody)?

(d) (5 points) Suppose that �1 and �2 are chosen simultaneously by the �rms, and then the

managers behave as in 8c. Find �1 and �2 which will be chosen in a subgame perfect

equilibrium.

(e) (5 points) Suppose that �rm 1 choose �1 �rst and �rm 2 choose �2 after observing �1. And

then the managers behave as in 8c. Find �1 and �2 which will be chosen in a subgame

perfect equilibrium.

(f) (4 points) Interpret your results 8d and 8e above from the view point of pro�t maximization

of �rms.

(g) (7 points)In the sequential choice case 8e above, suppose that a government wants to

maximize the total surplus (see 8b), but it can only regulate �1 2 [0; 1]. How should thegovernment choose �1?

9. Centipede Game. Consider the two player game in the �gure below. (For instance, the gameends if player I chooses S at the �rst node, and player I�s payo¤ is 1.)

(a) Find all subgame perfect Nash equilibria.

(b) Write the strategic form of this game. Find all Nash equilibria. Are there any Nash

equilibrium that is not subgame perfect?

40

I

10

S

C II C I

02

S

C II

31

S

53

C

24

S

Centipede Game

10. A Strategic Bargaining Problem. There are N 100 Yen coins to be allocated to two players.

The game goes as follows.. Player 1 makes an o¤er y (integer) to player 2, and if player 2 accepts,

then the game ends and player 2 receives y coins and player 1 receives the rest. If player 2 rejects

the o¤er, then the game continues it becomes player 2�s turn to make an o¤er x, but the total

number of the coins will be reduced by one, thus player 2 receives N � 1� x coins if his o¤er isaccepted by player 1. If rejected, player 1 makes an o¤er to divide N � 2 coins, and so on. Thegame ends if there is no coin left. Assume that the players have common discount factor � < 1

that is very close to one.

(a) Find a subgame perfect equilibrium strategy pro�le of this game.

(b) Find a Nash equilibrium strategy pro�le that is not subgame perfect.

11. Simple model of incentive contracts. Consider 2 agents who are to invest in a joint project.Each agent can invest up to one unit of some productive resource. Denote by xi the amount

agent i invests (thus xi 2 [0; 1]) and then k (x1 + x2) will be obtained as the �nal output of thejoint project, where k is a constant with 1 < k < 2. Let yi the amount agent i receives after the

project is completed, so y1 + y2 � k (x1 + x2) must hold. Agent i�s utility is given by yi � xi.

(a) De�ne a feasible allocation of the resource and the output ((x1; y1) ; (x2; y2)) ; and describe

the set of Pareto e¢ cient allocations.

(b) Suppose that the �nal output is divided equally, i.e., yi = k2 (x1 + x2). Find a Nash

equilibrium of the game where x1 and x2 are chosen simultaneously. Interpret.

(c) Suppose that the �nal output is paid out proportionally to the individual investment: e.g.,y1y2= x1

x2when both xi are positive. Find a Nash equilibrium of the game where x1 and x2

are chosen simultaneously. Interpret.

(d) Suppose that investments are done sequentially: �rst, agent 1 decides x1, and agent 2

decides x2. The �nal output is divided equally, i.e., yi = k2 (x1 + x2). Write the strategy

pro�le which is obtained by the backward induction. Interpret.

(e) Suppose that investments are done sequentially as above, but this time agent 1 can promise

the amount w paid to agent 2: �rst, agent 1 decides both x1 and w � 0. Then agent 2

decides x2, and consequently y2 = w, and y1 = k (x1 + x2). Write the strategy pro�le

which is obtained by the backward induction. Interpret.

(f) Suppose that investments are done sequentially as above, but this time agent 1 can demand

the amount r to be paid to himself as well: �rst, agent 1 decides both x1 and r � 0. Thenagent 2 decides x2, and consequently y1 = r and y2 = k (x1 + x2) � r. Write the strategypro�le which is obtained by the backward induction. Interpret.

Microeconomic Theory problems by A. Kajii 41

(g) Suppose that investments are done sequentially as above, but this time agent 1 can set up

a sharing scheme: �rst, agent 1 decides both x1 and the shares (s1; s2) with s1 + s2 = 1,

s1 � 0 and s2 � 0. Then agent 2 decides x2, and consequently yi = si� k (x1 + x2). Writethe strategy pro�le which is obtained by the backward induction. Interpret.

12. Suppose Bertrand competition (question 3) is repeated in�nitely may times (with the same

demand every period). Show that if discount factor is su¢ ciently close to one, there is an

equilibrium where the monopoly price emerges on an equilibrium path.

13. Consider the game where the following one shot game is repeated twice.

C D M

C 3,3 -1,4 0,0

D 4,-1 0,0 0,0

M 0,0 0,0 �,�

;

where � is a positive constant with � < 3. The payo¤ of the game is given by the sum of payo¤s

from the results of the two one shot games.

(a) How many strategies are they for each player?

(b) Suppose one player believes that the opponent will play M regardless of the result of �rst

round in a subgame perfect equilibrium. Is it possible that a player chooses C in the �rst

round in this equilibrium?

(c) Consider the following strategy: �play C in the �rst round. In the second round, if (C,C) is

the result of the �rst round, play C. Otherwise, play D.�If both players adopt this strategy,

does it constitutes a subgame perfect Nash equilibrium?

(d) Consider the following strategy: �play C in the �rst round. In the second round, if (C,C)

is the result of the �rst round, play M. Otherwise, play D.� If both players adopt this

strategy, does it constitutes a subgame perfect Nash equilibrium?

(e) Consider the following strategy: �play M in the �rst round. In the second round, if neither

player plays D in the �rst round, play M. Otherwise, play D.� If both players adopt this

strategy, does it constitutes a subgame perfect Nash equilibrium?

14. Repeated prisoner�s dilemma. Consider two shops. Every day, each can either try maintaina monopoly price pM , or cut price to some competitive level pC . The resulting pro�ts are

summarized by the following table.

pM pC

pM 2; 2 0; 3

PC 3; 0 1; 1

:

They do not see each other�s prices directly but they learn the choice of prices in the past. They

are interested in maximizing the sum of discounted pro�ts, with discount factor � 2 (0; 1].

(a) Suppose that there are two days for sales, and the price can be set each day independently.

Describe a subgame perfect Nash equilibrium

15. Role of commitment in a supermodular game: Suppose that (S; u) is a two person supermodular game, where Si = [0;1) and payo¤s are increasing in all players�strategy. Assume thatpayo¤ function ui as well as the best response function are di¤erentiable, and there is a Nash

equilibrium s� which is in the interior (so the usual �rst order condition for a interior solution is

42

necessary). Suppose now player 1 moves �rst, and player 2 chooses their actions after observing

player 1�s action. So player 1 maximizes u1 (s1; BR2 (s1)), and assume that it is concave in s1.

Prove that in the dynamic game, player 1�s choice is larger than s�1.

8 Economics of Information

1. Auction. There is a single seller who has one unit of an indivisible good to be sold in a �rstprice auction. There are two buyers, 1 and 2. The good has no value to the seller, whereas

the value to the buyer i is Vi: Although there is no value, the seller may commit not to sell

under a speci�ed minimum price p: That is, if both bids from the buyers are less than p, no

transaction takes place and the good will be discarded. Each buyer i knows his own valuation

Vi; but not the other�s. But both knows that each Vi is uniformly distributed on [0; 1] and Vi�s

are independently distributed. This auction can be modeled as a 2 player Bayesian game: each

vi 2 [0; 1] is regarded as a type of player i, and a strategy of player i is a function bi from [0; 1]

to [0; 1] with the understanding of bi (vi) is the bid of player i when his type is vi. Since the

minimum price p is �xed exogenously for the players, it is clear that if vi < p, player i has no

incentive to bid. So for simplicity, assume that 0 � p < 1 and bi must be di¤erentiable, and

increasing on [p; 1] and bi (vi) = vi if vi < p, and that its inverse b�1i is well de�ned on a suitable

domain.

(a) Assuming that buyer 2 uses strategy b2, write the interim payo¤ to bid x1 2 [p; 1] forbuyer 1 who has learned that his value V1 is v1: Then write the �rst order condition that

characterizes the payo¤ maximizing bid.

(b) Assuming that buyer 2 uses strategy b2, write the ex ante payo¤ to strategy b1 for buyer

1.

(c) Find a symmetric, Bayesian Nash equilibrium. [Hint. Look at the �rst order condition you

derived is a di¤erential equation. Try b(v) = k(v + cv ): Since a buyer with value less than

p has no reason to bid, you may assume that bi (p) = p:]

(d) Show that the expected revenue from this auction is maximized at a positive minimum

price. Discuss the e¢ ciency of transaction.

2. Joint Venture under asymmetric information. Consider two players, i = 1; 2; who are

to invest in a joint project. Denote by xi � 0 the amount invested by player i. The payo¤ toplayer i is � (x1 + x2)� 1

2 (xi)2, where � is a positive number. Before investment, each player i

observes a private signal si about the pro�tability of the project. It is commonly known that

each si is independently, uniformly distributed on [0; 1], and � = s1 + s2:

(a) Suppose that each player can observe the other�s private signal as well, thus each player

already knows � at the time of investment decision. Suppose investment decisions are done

simultaneously. Find a Nash equilibrium.

(b) Suppose that each player cannot observe the other�s signal.

i. Denote by f2(s2) the amount player 2 invests if he observes s2. Write the interim

payo¤ maximization of player 1 who has observed s1, and �nd the amount player 1

should invest.

ii. Find a Bayesian Nash equilibrium.

(c) Find the �rst best outcome, that is, the amount of investment which maximizes the sum

of the payo¤s of the two players. Compare this with the total amount of investment (i.e.,

x1 + x2) in the two cases (a) and (b) above. Discuss.

Microeconomic Theory problems by A. Kajii 43

3. Investment with privately known pro�tability Consider two players, i = 1; 2; who are

to invest in a joint project. Denote by xi � 0 the amount invested by player i. The payo¤ toplayer i is � (x1 + x2) � 1

2 (xi)2, where � is a non negative number. So we can think of � as

the pro�tability of the joint project per person, and 12 (xi)

2 as the cost of investment. Each

player i observes a private signal si about the pro�tability of the project. Here s1; s2, and � are

random variables, and it is commonly known that each si, i = 1; 2 is independently, uniformly

distributed on [0; 1], and there is a relation � = s1 + s2: So for instance, the average of si is12 , and the average of � is 1. To help your computation, you may use the following results:R 10

�(x+ y)

2�dx = y2 + y + 1

3 andR 10

�R 10

�(x+ y)

2�dx�dy = 7

6 .

(a) (5 points) Suppose that investment decision must be done simultaneously before observing

the private signal, and thus each investor is interested in maximizing the (ex ante) expect

payo¤ (=R 10

R 10

�(s1 + s2) (x1 + x2)� 1

2 (xi)2�ds1ds2). Find a Nash equilibrium level of

investment.

(b) (5 points) Suppose that each player can observe the other�s private signal as well, thus each

player already knows � at the time of investment decision. Suppose investment decisions

are done simultaneously. Find a Nash equilibrium after � is known.

(c) (10 points) Suppose that each player cannot observe the other�s signal: an investment

decision is made after si is learned but before s�i is revealed.

i. Denote by f2(s2) the amount player 2 will invests if he observes s2. Write the interim

payo¤maximization problem of player 1, that is, the problem to maximize the expected

returns after s1 is observed.

ii. Find a Bayesian Nash equilibrium.

(d) (5 points) Compare the ex ante expected equilibrium payo¤s of each player, i.e., the ex-

pectation taken before the random signals are drawn, for the three cases considered above.

Interpret your �ndings.

4. The Hirshleifer e¤ect. Consider the following game (Nature chooses �left�or �right�witheven chance.

1. (a) Does Player 2 observes Player 1�s choice of action? Does the players observe Nature�s move?

(b) Show that both players choosing �left� at every information set constitutes a perfect

Bayesian equilibrium.

(c) Suppose instead Nature�s move is observable to both players. Write this environment in an

extensive form game. Find the unique subgame perfect equilibrium.

(d) Compare ex ante utility levels of players in (1b) and (1c). Comment.

2. A risk neutral principal and a risk averse agent. The agent chooses e¤ort level e in [0; 1], which

is not observable. The agent�s concave vNM utility function is u (x)� e, where x is the amountof wage received. The output may be yH or yL, with yH > yL. The conditional probability of

achieving yH given e is denoted by � (e), where �0 > 0, �00 < 0. The realized output is observable

and veri�able. Wage schedule is written as (wH ; wL), i.e., when the observed output is high, the

agent receives wH , otherwise wL. The reservation utility level for the agent is normalized to be

0. Assume enough degree of di¤erentiability in the following.

(a) Suppose e were observable. Write down the �rst order condition that characterize the

optimal e¤ort level for the principle. Find the wage level that implements the optimal.

From now on, assume that e is not observable.

44

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CCCCCCCCCCCCCC

CCCCCCCCC

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CCCCCCCCCCCCCC

LLLLLLLLL

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(8,8)

Nature

Player 1

Player 2

Player 2

(8,8)(10,0) (10,0) (10,0) (0,10) (0,10) (0,10)

Figure 1: Game

(b) Write down the agent�s utility maximization problem, given (wH ; wL).

(c) Write down the principle�s problem. Show that at the optimum:

i. The �rst best e¤ort level you found in 2a cannot be achieved.

ii. The individual rationality constraint must be binding;

iii. wH > wL.

3. Consider the Spence job market signaling model, with the following parameters:

output of high type 4

output of low type 2

cost of education for low type 1

wage for educated worker w

wage for non-educated worker 1

Assume high type and low type are equally likely. The wage for non-educated worker is �xed at

1 by law, and the reservation payo¤ for workers is 0.

(a) Suppose w > 1 has been �xed. Formulate the problem as in a simple game as in the

previous question. Find an equilibrium which yields the best outcome for the employer.

(b) Suppose the employer �rst pick w, then the game considered above takes place. Find an

equilibrium that yields the best outcome for the employer.

(c) Suppose that w is chosen after education is �nished, but before the productivity is revealed.

Find an equilibrium that yields the best outcome for the employer.

4. Consider a two player game as follows. Player 1 (Sender) observes his type �rst, and then send

a message to Player 2. Player 2 (Receiver) chooses an action after observing the message. The

Player 1�s types are, t1, or t2, and possible messages are s1 or s2, and Player 2�s actions are a1or a2. The types are equally likely. The payo¤s are given in the following table, where � is a

Microeconomic Theory problems by A. Kajii 45

constant. Notice that the payo¤s are independent of the choice of messages. In words, there is

no cost for sending a message, i.e., player 1 is �just talking�.

Player 1�s payo¤s

a1 a2

t1 1 0

t2 � 1

Player 2�s payo¤s

a1 a2

t1 2 1

t2 1 2

(a) Write this game in extensive form.

(b) Consider the following strategy pro�le: Player 2 chooses a1 or a2 with probability 12

whichever signal she receives (i.e., she �ignores� the message), and Player 1 sends s1 or

s2 with equal chance whichever the type is. Show that this pro�le constitutes a perfect

Bayesian equilibrium with appropriate beliefs.

(c) Consider the following strategy pro�le: Player 1 sends s1 if his type is t1 and s2 if his type

is t2, and Player 2 chooses a1 or a2 with probability 12 whichever signal she receives. Does

this constitute a perfect Bayesian equilibrium with appropriate beliefs?

(d) Consider the following strategy pro�le: Player 1 sends s1 if his type is t1, and s2 if t2.

Player 2 chooses a1 if the message is s1, chooses a2 if the message is s2. Show that this

constitutes a perfect Bayesian equilibrium with appropriate beliefs, if � < 1. (So, in this

case, the message conveys some meaning.)

(e) Is there a perfect Bayesian equilibrium where the message conveys some meaning when

� > 1?

5. Imagine a two player game of sequential moves where one of the player moves �rst and the other

follows. Player i�s payo¤ is given by ��ixi � 12 (xi)

2+ �ijxixj where j 6= i and �ij � 0 and

�� := �12�21 <12 . Assume that ��i > 0 is large so that the payo¤ function is increasing in the

relevant range for i = 1; 2.

(a) Suppose that � is a known parameter to both players.

i. Find a subgame perfect equilibrium where player i moves �rst and player j follows.

ii. Suppose you are interested in maximizing the sum x1+x2 in equilibrium. Which player

do you want to move �rst?

(b) Suppose that player 1 moves �rst, and � is unknown parameter continuously distributed

on [�0; �1] with mean ��.

i. Find a subgame perfect equilibrium when neither player learns � (thus the payo¤s are

e¤ectively ���ixi � 12 (xi)

2+ �ijxixj)

ii. Find a fully revealing Bayesian equilibrium when player learns � before he makes his

move.