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Advanced Microeconomics: Problems Atsushi Kajii Institute of Economic Research, Kyoto University January 25, 2008

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.

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Mathematics 2

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

2

) +(x2 ) ,

= 2, and draw (x1 ; x2 ) 2 R2 : f (x1 ; x2 ) = 3 :

(a) Let

= 1,

(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) ; where Dg (x1 ; x2 ) is the gradient vector

@ @ @x1 g (x1 ; x2 ) ; @x2 g (x1 ; x2 )

(c) Solve minx1 ;x2 f (x1 ; x2 ) subject to g (x1 ; x2 ) 1, using the Kuhn-Tucker method. Explain the …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 condition for 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. n o 2 2 (a) Show that (x1 ; x2 ) 2 R2 : (x1 ) + (x2 ) 1 is 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 any t 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 set (remark. this set is called the epigraph of f ).

R : f (x)

g is a convex

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

1

:

2

(c) Let f be a function on C. Suppose that f(x; ) 2 C that f is a convex function.

R : f (x)

g is a convex set. Verify

(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 ) function.

PK

k=1 gk

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

(b) if f (x1 ; :::; xK ) is quasi concave and homogeneous of degree one, f is concave. 5. [Homogeneity and convexity] A function f : RL if f (tx) = + ! R is homogeneous of degree L t f (x) for any x and t > 0. A set C in R 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 any t 2 [0; 1]; tf (x) + (1 t) f (y) f (tx + (1 t) y). A function f : C ! R is said to be concave if f is convex. A function f : C ! R is said to be quasi-convex if fx : f (x) g is a convex set for any . A function f : C ! R is said to be quasi-concave if fx : f (x) g is a convex set for any . (xK ) (a) Show that f (x1 ; :::; xK ) = (x1 ) 1 (x2 ) 2 2+ K = 1 is homogeneous with degree 1.

K

with

k

> 0, k = 1; :::; K and

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. PK (d) Show that g (x1 ; :::; xL ) 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 quasiconcave 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. 1+ 2+ K

1

(x2 )

2

(xK )

K

is concave if

k

> 0, k = 1; :::; K and

6. A simpler version theorem. Let 4 be the L 1 dimensional simplex n of the Gale Nikaido o PL l 1 L L 0 : l=1 p = 1 , and let f : 4 ! RL be a continuous function in R , i.e., 4 := p ; :::; p 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 : q l = 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.

(b) So by Kakutani’s theorem, there is p 2 every l) must hold.

(p ). Show that f (p )

0 (i.e., f l (p )

0 for

(c) Assume in addition that p f (p) = 0 for any p 2 4, and if pl = 0 for some l, there is l0 0 0 (may be the same as l) such that pl = 0 and f l (p) > 0. Show that a …xed point p found above in fact must satisfy f (p ) = 0. 7. A …xed point theorem. Let 4 be the L 1 dimensional simplex in RL , and let f : 4 ! RL PL be a continuous function such that l=1 f l (p) = 0 for any p 2 4. The Fan-Brouwer theorem 0 says that f has a …xed point if f is inward pointing, that is, if pl = 0 implies f l (p) > 0 for 0 some l0 with pl = 0 (l0 may be same as l). This question asks you to establish this result from the Gale-Nikaido theorem:

3

Microeconomic Theory problems by A. Kajii

(a) Show that if f is inward pointing, then pl = 0 implies f l (p) 0. [Hint: consider a sequence pn 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 : q l = 0 if f l (p) < pl . Show that and its graph is closed (i.e., upper hemicontinuous). (c) So by Kakutani’s theorem there is p 2 4 such that p 2 the assumption that f is inward - pointing.

2

is non-empty convex valued (p). Show that this contradicts

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 ) function.

PK

k=1 gk

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

(b) if f (x1 ; :::; xK ) is quasi concave and homogeneous of degree one, f is concave. 3. For a CES production function with K factors of production f (z) P where ak > 0; k = 1; :::; K; are constants with k ak = 1, and

f z 1 ; z 2 ; :::; z K = 1;

6= 0;

hP

K k=1

ak z k

(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; (b) Dw c (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. nP o K 0 , where ak , 5. Consider a general class of linear activity production set Y = k=1 tk ak : tk 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). Draw production 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)?

i1

;

4

(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 fornall k, then the maximum pro…t is zero, and o the supply correspondence PK is given by y (p) = 0, and tk = 0 if p ak < 0 . k=1 tk ak : tk

6. Consider a competitive …rm with a concave production function f z 1 ; :::; z K ; where z k 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 PK the total cost of inputs is k=1 wk z k . 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.

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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 1 x1 ; 2 x2 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 = 1 x1 + 2 x2 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 v 0 > 0; v 00 < 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.] 5. Let x (p; m) = xk (p; m) (a) Show that (b)

@ xl @pl

PL

k=1

(p; m) =

L k=1

be a demand function.

@ xk = 1. pk @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 @m xl (p; m) 0; good l is said to be normal at (p; m). A good is called a normal good @ l if it is normal at any (p; m), otherwise it is called an inferior good. If @p l x (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 :

Microeconomic Theory problems by A. Kajii

5

(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 1 1 3 (4; 3) (d) 2; 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 = y 1 ; y 2 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 y 1 ; y 2 = (2; 2). De…ne utility function u on R2 by the rule u (x) = mink2fa;b;c;dg f k (pk x) + k g ; 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 5 c = 4 , 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) + k g : Play with d to accommodate (d).]

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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 more u00 (W ) gambles as the ratio r (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 00 )W initial wealth and so he receives (x1 W; x2 W )) and uu0 (W [Arrow-Pratt measure of relative (W ) 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.

6

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

p. Find

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 if Rz R [FX (s) FY (s)] ds 0 for any z 2 [ ; ], and [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: 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 (b) Show that if W and X are independent, we have r Jensen’s inequality]

E [X].

E [X]. [Hint. Use the conditional

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

7

Microeconomic Theory problems by A. Kajii

6. Simple investment problem. Suppose your income today is m0 and tomorrow is m1 . But you can invest part of your income m today, and 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, you would solve max u (m0 x

x) + Eu (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 Y tomorrow. Y may be random whereas you cannot insure against the income risk. But you can save part of your income m today. 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. So, you would solve max u (m x

x) + Ev (x + Y )

where E is the expectation with respect to Y . (a) Write down the …rst order condition for this problem. (b) Let Y = y0 for sure. Show that if your future income y0 decreases, then your saving increases. (c) 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 v 000 > 0 (i.e., v 0 is convex). Interpret. (d) The coe¢ cient of absolute prudence of v is de…ned as the coe¢ cient of absolute risk aversion for v 0 . 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 as risky as X if E [ZjX] = 0. (a) Show that if Y is 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 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 one dollar 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., H u (yH

+ xH ) +

L u (yL

+ xL )

subject to pH xH + pL xL = 0, where u is a concave vNM utility function.

8

(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 (d) If

pH pL

>

H L

pH pL

=

H L

, you will choose xH and xL in such a way yH + xH = yL + xL .

, 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 contingent consumption 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 x1 > > x2 .) (c) Assuming S = 2, and …x x 2 R2 with x1 > x2 > 0. Write the set of contingent consumption plans 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 PS amount of consumption in state s, which maximizes the expected utility s=1 s u (xs ) PS given the budget s=1 ps xs w, where s = S1 for s = 1; :::; S. Denote by x (p; w) := x1 (p; w) ; :::; xS (p; w) the contingent plan which maximizes the expected utility. 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). 0

0

iv. Show that if ps = ps , then xs (p; w) = xs (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 > ps 0 if and only if xs (p; w) < xs (p; w).

0

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. 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 51 red 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.

Microeconomic Theory problems by A. Kajii

5

9

Classical Partial Equilibrium Analysis 1. Consider a quasi-linear utility function u x1 ; x2 = v x1 + x2 , where v 0 > 0; v 00 < 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 v 0 > 0; v 00 < 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, h 0i 1 and hence the supply is characterized by c0j (qj ) = p. Write sj (p) = cj (p), i.e., si (p) is the quantity supplied by the …rm j at price p. By the assumptions, each sj is increasing function, PJ and so is the aggregate supply S (p) := 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. PJ Now consider a …ctitious, “representative …rm”, whose cost function is given by C (q) = 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.) 4. Producer surplus. Consider a …rm with cost function c (q) = f (q) + v (q), where (1) v (0) = 0 and v 0 > 0 and v 00 < 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 v (q) dq. Is the producer surplus the same as pro…ts? 0

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 q 0 (0) < 0: (a) Show graphically that taxation leads to a loss in social welfare (consumer surplus + producer 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.

10

(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,

p p0 q ,

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 are positive 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 is increasing 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. Each small …rm has an identical cost function c (q) = 12 q 2 . 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 the product 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. (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? 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, PL assume that it is given by l=1 v l 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.

11

Microeconomic Theory problems by A. Kajii

(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? 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 goods are traded. The total endowments are e 2 RL ++ , and assume each 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 M aximize

L x1 2RL ++ ;:::;xI 2R++

X

i ui

(xi ) subject to

i

> 0, i = 1; :::; I;

i

X

xi = e;

(1)

i

2) Given ui > 0, i = 2; :::; I, M aximize

L x1 2RL ++ ;:::;xI 2R++

I

u1 (x1 ) subject to ui (xi )

ui , i = 2; :::; I and

X

xi = e

(2)

i

I

I

(b) Show that (xi )i=1 2 RL is Pareto e¢ cient if and only if (xi )i=1 solves problem 2 above ++ with some u. Is this true if ui were not strictly increasing?

12

(c) Suppose every ui is concave, di¤erentiable, and Dui (xi ) >> 0, ui (xi ) > 0 for any xi >> 0. I I I solves problem 1 for some if and only if (xi )i=1 solves problem Show that (xi )i=1 2 RL ++ 2 with some u. (Hint. Both problems are then “concave” so compare the Kuhn-Tucker condition.) 2. Pateto E¢ cient share of random income. Consider an economy with I agents. Each agent’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 I xi (y) the amount agent i receives when aggregate income is y, and we call (xi )i=1 an income PI sharing rule if it is feasible allocation of income, i.e., i=1 xi (y) = y for any y. (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. PI 1 1 t (c) Fix a positive i , for i = 1; :::; I, and let g (t) = i=1 (u0i ) ; where (u0i ) is the i inverse of u0i . Show that if we set xi (y) = (u0i ) sharing rule.

1

g

1

(y)

i

I

, then (xi )i=1 is a Pareto e¢ cient

(d) Assume further ui (z) = 1i exp [ i z], 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) = agent i.

1 1

i

z1

i

where where

i

is a positive constant, for every

3. State and prove the …rst fundamental theorem of welfare economics. Discuss why completeness of markets is important in this result. 4. weak e¢ ciency to e¢ ciency. Say that the budget is tight at prices p and income w for ui if p x0 < w and x0 2 Xi implies that x0 is not a utility maximizing demand vector (i.e., to maximize utility, all the given income must be spent). (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) Argue graphically that the budget might not be tight when ui is not strictly increasing (i.e., a thick indi¤erence curve is possible) I

J

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

J ij )j=1

I

). Show that (x ; y ) is Pareto E¢ cient if the budget is tight at xi (given P prices p and income p !i + j ij p yj ) for ui for all consumer i: i=1

5. 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 all the 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 function 1 2 1 2 u x ; x = x + x with initial endowments (1; 0). Show that a quasi-utility maximizer is not necessarily a utility maximizer under some prices.

Microeconomic Theory problems by A. Kajii

13

(c) In general, suppose that Xi is convex, ui is continuous, and there is x ^ 2 Xi such that p 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 + t^ x with small t > 0]. (d) Comment on the second fundamental theorem of welfare economics discussed in the class. 6. the quasi-linear economy. There are two goods, x and m, and there are n traders. There are 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. (a) Show that a trader may be interpreted as a producer with a convex cost function. [Hint: think of ui as the negative of cost function, and check that utility maximization given budget is equivalent to pro…t maximization.] PI I I (b) Show that (xi ; mi )i=1 is Pareto e¢ cient if and only if (a) (xi )i=1 maximizes i=1 ui (xi ) P P P P subject to i xi = i xi , and (b) i mi = i mi hold. P (c) Show that the (general) equilibrium price p depends on x 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? 7. “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 1 price of good x. Assume that u is increasing and concave. Write = (u0 ) . 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. d (c) Compute dp z (1). Show that if the equilibrium allocation is close to the initial endowments d (1 ; ) and ( ; 1 ), then dp z (1) < 0 (i.e., the index of p = 1 is +1).

(d) Show that

d dp z

(1) > 0 is possible for some . What can you conclude from this observation?

8. There is one consumer and one producer in the economy. The consumer’s preferences are represented by utility function u (x1 ; x2 ) = 21 ln x1 + 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

14

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? (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? 9. In the same setting as in question 8, 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" = " +1 z , 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. 10. 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 1 2 ln x1 + 2 ln x2 2 ln x1 + 3 ln x2 3 ln x1 + ln x2

endowments (6; 4) (8; 4) (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 fundamental theorem of welfare economics? Discuss. 11. Consider two …rms, each producing a consumption good from two inputs, z 1 and z 2 , by pro1 1 duction function f1 z11 ; z12 = z11 z12 and f2 z21 ; z22 = z21 z22 , respectively, and

15

Microeconomic Theory problems by A. Kajii

0 < < < 1. The total amount of inputs available in the economy is z 1 and z 2 , respectively. The price of consumption good is given at p1 and p2 . We can interpret that this is a small country 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 z 1 =z 2 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 z 1 or z 2 . (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 z 1 and z 2 change when z1 increases? State this in terms of factor intensity and . 12. 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. 1

Firm 1 produces good 3 with production function f z 1 ; z 2 = z 1 z2 where z 1 and 2 z are the amount of good 1 and 2 used as inputs. So the conditional factor demand given y are z1 (w; y) = z2 (w; y) =

(1

w2 ) w1

(1

) w1 w2

1

y: y

Firm 2 produces kz 2 units of good 1 from z 2 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 . 13. 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.

16 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 k1 z1 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 k2 z2 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. 1 From now on, write (y) := (u0 ) (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. 14. In Section 6 (8), suppose the …rm behaves as a monopoly …rm, and assume that the consumer p has utility function u (x1 ; x2 ) = 2 x1 + 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. 15. 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 consumer’s preferences are represented by utility function u x0 ; x1 = x0 + 21 ln x1 . The consumer owns one unit of the consumption good in period 0, and nothing in period 1. The …rm is a price p taker (competitive …rm) and it’s production function is x1 = k x0 , (x0 0) where k > 0 is a p constant; that is, it produces k 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.

17

Microeconomic Theory problems by A. Kajii

(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. 16. 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 ; x1i = ln x0i + i ln x1i where i 2 (0; 1) is a constant. Every consumer is endowed with one unit of the good in each period. 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 ; x1i ; si 2 R+ R+ R. Notice that negative saving, i.e., borrowing is allowed. (a) Consumer i faces budget constraint p0 x0i + si = p0 with x0i the second period budget constraint.

0 in the …rst period. Write

(b) Find demand x0i , x1i , and si as functions of p0 , 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? 17. 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 < w implies that x0 is not a utility maximizing demand vector (i.e., to maximize utility, all the given income must be spent). (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) I

J

(c) Let (x ; y ; p ) be a competitive equilibrium of a private ownership economy ((ui )i=1 ; (Yj )j=1 ; !i ; ( Show that (x ; y ) is Pareto E¢ cient if the budget is tight at xi (given prices p and income P p !i + j ij p yj ) for ui for all consumer i:

18. Uniquness of no-trade equilibrium. Consider an exchange economy with I consumers and L 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.

J ij )j=1

I i=1

).

18

19. The Negishi Method. Consider an exchange economy with I nconsumers and L goods with difo PI ferentiable utility functions. Assume that for each 2 4I 1 := ( 1 ; :::; I ) 0 : i=1 i = 1 , there is a unique Pareto E¢ cient allocation x ( ) = ( ; xi ( ) ; ) which is obtained by maxP P imizing the weighted utility sum i i ui (xi ) subject to i (xi !i ) = 0, and the corresponding supporting price vector p ( ) (that is, if ui (xi ) > ui (xi ( )), p ( ) xi > p ( ) xi ( )). Assume further that x ( ) and p ( ) are continuous functions of . Set i ( ) := p ( ) (xi ( ) !i ), I and ( ) := ( i ( ))i=1 . (a) Show that

P

(b) Show that if equilibrium.

i

i

( ) = 0 for any . = 0 (i.e.,

i

= 0 for all i), x

;p

constitutes a competitive

(c) Assume that I = 2. Argue using a graph, that under the standard assumptions, there will be such that = 0. 20. The full insurance theorem. Consider an exchange economy where each household h receives a random endowment vector esh 2 RL if state s occurs. Household h, h = 1; ::; H, has utility PS function s=1 s uh (xsh ) where uh is strictly concave, and xsh is the consumption vector in state s. Note that probability weights s , s = 1; ::; S, do not depend on h. Assume that there is PH no aggregate risk; that is, there is a vector e such that e = h=1 eh (s) for all s. Show that consumption does not depend on states in any Pareto e¢ cient allocation. 21. Insurance Markets. Consider an exchange economy where each household h receives a random endowment vector esh 2 RL if state s occurs. Household h, h = 1; ::; H, has utility function PS s uh (xsh ) where uh is strictly concave, and xsh is the consumption vector in state s. Note s=1 that probability weights s , s = 1; ::; S, do not depend on h. Assume that there is no aggregate PH risk; that is, there is a vector e such that e = h=1 esh for all s. (a) Show that consumption does not depend on states in any Pareto e¢ cient allocation.

(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 zhs 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, 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. 22. Sequential trade and asset structure. Consider the two period sequential trade model of an exchange economy, with one good in each state s = 1; :::; S. Let A be a linear subspace of RS . Consider the following optimization problem: PS s s s 1 1 S S (*) maxx ui (x) subject to p (x ! ) = 0 and x ! ; :::; x ! 2 A: i i i i i i s=0 We say that (p; x) constitute an A-equilibrium if every consumer i solves the problem above at PI xi and i=1 (xsi !is ) for s = 0; 1; :::; S. So an Arrow - Debreu equilibrium is an A-equilibrium where A = RS .

19

Microeconomic Theory problems by A. Kajii

(a) Consider J assets, r1 ; :::; rJ . Show that a rational expectation equilibrium is an Aequilibrium for some linear space A. (b) Establish the following converse implication as well: Given an A-equilibrium, for any vectors r1 ; :::; rJ which span A, in the economy with these assets there is a Radner equilibrium with equilibrium consumption idential to the A-equilibrium consumption. h i ^ (c) Show that if two collections of assets r1 ; :::; rJ and r^1 ; :::; r^J generate the same linear subspace, then the sets of rational expectations equilibrium consumption allocations coincide. 23. 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 PS s i, i = 1; ::; I, has utility function ui (xsi ) where ui is strictly concave, and xsi is the s=1 consumption in state s. Note that probability weights s , s = 1; ::; S, do not depend on i. PI Denote by es := i=1 esi the total resource available in state s. (a) Write the …rst order condition for Pareto e¢ ciency.

0

0

(b) Show that if x is a Pareto e¢ cient allocation, xsi xsi implies xsj xsj for all consumers. That is, at e¢ cient allocation, consumption is monotonic to each other (this property is called comonotonicity). (c) Show that consumption must be monotonic with respect to the total resource in any Pareto 0 0 e¢ cient allocation, i.e., for every i, xsi xsi holds if and only if es esi . (d) 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. (e) Do the results above hold if the beliefs are not common? 24. Gains from trade in general equilibrium. Consider a pure exchange economy with I consumers and L goods. Each consumer i is characterized by utility function ui and endowments 1 L ei = e1i ; :::; eL 0 for consumer i’s consumption, and write x = i . Write xi = xi ; :::; xi 1 L (x1 ; :::; xi ; ::::; xI ). Denote by p = p ; :::; p 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 that ui (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 maximization problem.] 25. An Equilibrium model for interest rate. Consider a two period economy with one perishable 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

20 2 (0; 1). The consumer is endowed with e0 units of the good in period 0, and his endowment of 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 max u e0 x

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) Derive the equilibrium bond price and the interest rate of this economy. (c) When the consumer becomes more patient, i.e., the discount factor increases, what happens to the equilibrium bond price? (d) 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? 26. Equilibrium asset pricing model. Consider an economy with a single (representative) consumer 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 total random 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, 2 0 13 J X max u (x) + E 4u @W + yj Dj A5 subject to (3) x;y

x+

X

j=1

pj yj = w0 :

j

(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 12 x2 , where a is positive. Rewrite 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? 27. Equilibrium asset pricing model. Consider an economy with a single (representative) consumer 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 total random 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

21

Microeconomic Theory problems by A. Kajii

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, 2 0 13 J X max u (x) + E 4u @W + yj Dj A5 subject to (4) x;y

x+

X

j=1

pj yj = w0 :

j

(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 stochastically 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? 28. 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 q1 per unit in this period. The risky asset pays r with probability 1 p and r with probability p, 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 2

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

29. Risk sharing. Consider an economy with I agents. Each agent’s preferences are represented by a vNM utility function ui . Assume that each ui satis…es u0i > 0 and u00i < 0. The aggregate income of this economy is denoted by y > 0, and it will be distributed among the agents. Denote P by xi (y) 0 the income the agent i receives when the aggregate income is y, so 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 PK income is yk (so 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.

22 K

(b) Now assume I > 2; K > 2: So agent i receives a vector of income xi := (xi (yk ))k=1 , which PK yields the expected utility Ui (xi ) := k=1 ui (xi (yk )) p (yk ) K

i. Give the de…nition for a feasible allocation x := ; (xi (yk ))k=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 PI expected utility functions, i=1 i Ui , and derive the …rst order condition for Pareto e¢ ciency for interior points. iii. Assume that ui (z) =

1 1

i

z1

i

where where

K ; (xi (yk ))k=1

i. Show that if x = ; k = 1; :::; K where i > 0 is a constant.

i

is a positive constant, for every agent

is Pareto e¢ cient, then xi (yk ) =

i yk

for

30. 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 ; x1i = ln x0i + i ln x1i where i 2 (0; 1) is a constant. Each consumer is endowed with one unit of the good in each period. 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 ; x1i ; 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. 31. Consider a pure exchange economy with I consumers and L goods. Each consumer i is char1 L acterized by utility function ui and endowments ei = e1i ; :::; eL 0 i . Write xi = xi ; :::; xi 1 L for consumer i’s consumption, and write x = (x1 ; :::; xi ; ::::; xI ). Denote by p = p ; :::; p 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 that ui (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 maximization problem.]

23

Microeconomic Theory problems by A. Kajii

32. [core (non)-convergence] Consider an economy with two goods, two consumers. The consumers 1 and 2 have an identical utility function ui x1i ; x2i = min x1i ; x2i , and their endowments 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? 33. [Replica economy] Consider an economy with two goods, two types of consumers. The type 1 and 2 consumers have an identical utility function ui x1i ; x2i = ln x1i + ln x2i , and their endowments 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. 34. [fair allocation] Consider an exchange economy with I consumers and L goods, where the total endowments of goods is ! >> 0. A feasible allocation x = ( ; xi ; ) is called envy-free if for any i and j, ui (xi ) ui (xj ). A feasible allocation is called a fair allocation if it is Pareto e¢ 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.

7

Strategic Market and Game theory

7.1

Nash Equilibrium of strategic form games

1. A strategy si 2 Si is strictly dominated if there is a strategy s0i 2 Si such that ui (si ; s i ) < ui (s0i ; s i ) for all s i 2 S i . Show that a strictly dominated strategy cannot be a Nash equilibrium strategy. 2. Find all (i.e., pure and mixed) Nash equilibria of the following two person games. (a) L (2; 2) (0; 0)

T B

R (0; 0) (1; 1)

(b) T M B

L (2; 2) (1; 1) (0; 1)

C ( 1; 3) ( 2; 0) (1; 2)

R (1; 0) ( 1; 2) ( 1; 1)

(use the domination argument as much as possible) (c) T B

L (z; 1) ( 1; 1)

R ( 1; 1) ; (1; 1)

24

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 that is not a best response to i .(15 points) Consider a game in strategic form, represented by the following table.

player 1

x y z

player 2 b c ( 1; 1) (1; 0) ( 2; 4) (0; 5) ( 5; 2) ( 5; 3)

a (2; 2) (1; 1) (0; 1)

d ( 1; 1) ( 3; 0) (1; 3)

Answer the following questions about this game. For questions (a) to (g), you do not have to explain your answers. (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, if any, 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 either row 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.

U D

( (

L (1

11 ; y)

(1

21 ; y)

11 ; x)

21 ; x)

( (

R (1

12 ; y)

(1

22 ; y)

12 ; x)

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 11 u (x). (a) Show that there is no pure strategy Nash equilibrium.

Microeconomic Theory problems by A. Kajii

25

(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.

7.2

Simple Models of Strategic Competition

1. Cournot Competition. There are two identical …rms with constant marginal cost of production c. The total demand for the product is given by q = a bp, where a; b > 0 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 two identical …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 production 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. 4. Price competition with capacity constraint. In the setting in (3), we shall assume that each …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

26

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 the marginal 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 = 31 . Find a Nash equilibrium. 5. Monopolistic Competition. There are two …rms, 1 and 2, with constant marginal cost of production 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 + j p 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 individual. Each individual own 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 consumption good privately, f (2 (x1 + x2 )) units of public 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 consumption good and 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.(25 points) 7. Consider a community with 2 individuals. Each individual own 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 consumption good privately, f (2 (x1 + x2 )) units of public good is produced. Assume that f (0) = 0; f 0 > 0; and f 00 < 0, and f 0 (0) > 21 > f 0 (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. (b) Find the …rst order condition which characterizes Pareto e¢ cient allocations.

27

Microeconomic Theory problems by A. Kajii

(c) Consider a game where both individuals simultaneously choose x1 and x2 . Give the de…nition 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. 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 second period, any remaining …sh, y c1 c2 , are divided equally between the players. Player i’s utility is given by y c1 c2 ui (ci ; c i ) = ln ci + ln , 2 where ln x is the natural logarithm of x. By convention, utility is zero in the second period.

1 if consumption of …sh is

(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 , but if 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 P y c1 j6=i cj ui (ci ; c i ) = ln ci + ln N where

P

j6=i cj

is the sum of ‘other’players’…rst period consumptions.

(c) Consider the corresponding N -player game where 0 ci =

(

si y=N

si < y, as before, and

P if i si y 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: What ci ’s will she choose? Compare your answer to part (b) and give an intuition.

P

i

ui .

(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.

28

7.3

Dynamic games and commitment

1. A dynamic oligopoly game. There are three …rms, 1, 2, and 3 producing a consumption good whose demand is given by 1 Q = p. Firm j chooses quantity qj as 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 subgame perfect Nash equilibrium. (c) Suppose that after …rm 1 chooses its quantity, …rms 2 and 3 choose their quantities simultaneously. Thus …rms 2 and 3 knows q1 but not 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 simultaneously. 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. Entry deterrence. Consider two …rms, Incumbent and Entrant. Both …rms can produce a consumption 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 of Entrant 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? 3. Imagine a developer building houses near a lake. There are two periods, t = 1 and 2. In each period, the developer can construct a house per unit cost c > 0. Denote by xt the total unit

Microeconomic Theory problems by A. Kajii

29

constructed 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…ts and times period 2 pro…ts. Because of …nancial reasons, the developer must sell all units of houses in the period they are built. In each period, after houses are constructed, many potential buyers come to see the houses. Each buyer buys at most one unit. If a potential buyer buys a house, his payo¤ (in terms of money) is a (x1 + x2 ), where a is a positive constant with 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. Assume that xt ’s are real numbers to simplify the question. (a) Note that the payo¤ from a house is decreasing in x1 + x2 . Interpret. (b) Suppose that x1 units have already been sold in period 1. So, if x2 units of houses are constructed, each buyer’s payo¤ from a house is a (x1 + x2 ). i. If the developer wishes to sell x2 units in period 2, what p2 will 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, assuming that the developer constructs x2 units of houses in the second period. 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? 4. 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 q1 and q2 the quantity produced towards the domestic market in each country and denote by e1 and 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: (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. Solve this game by …rst …nding the second

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stage (i.e., the game where (t1 ; t2 ) is already selected) equilibrium, and then …nding a Nash equilibrium in tari¤ (t1 ; t2 ). 5. In‡ation target. Consider two players, the public (P) and the monetary authority (M). The 2 payo¤ of P is given by ( e ) , where e is the expected rate of in‡ation held by P and is 2 2 the actual rate of in‡ation. The payo¤ of M is given by (y y) , where y is the 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 e of in‡ation and GDP (i.e., a Phillips curve) given by ( y y) + ( ) = 0 where 0 < 1 is a constant, and this relation is known to the both players. So if < 1, the full employment e GDP can be realized only with a “surprise in‡ation”, i.e., > 0. (a) Suppose P selects game.

e

and M controls

simultaneously. Find a Nash equilibrium of this

(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? 6. Alternative for pro…t maximization. Consider two …rms, 1 and 2, producing an identical good 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 cqj plus the payment to the manager. The manager of …rm j is paid " times j (revenue of …rm j)+ (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 " ( i R + (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 ) is mutually observable)? (b) Suppose that

1

and

2

1;

2)

have been set (and the pair

are chosen simultaneously. What will happen?.

(c) Suppose that …rm 1 choose

1

…rst and then …rm 2 choose

2.

What will happen?

7. Centipede Game. Consider the two player game in the …gure below. (For instance, the game ends 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?

31

Microeconomic Theory problems by A. Kajii

I

C

II

S 1 0

I

C

S

C

II

S

0 2

C

5 3

S 2 4

3 1 Centipede Game

8. 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 is accepted by player 1. If rejected, player 1 makes an o¤er to divide N 2 coins, and so on. The game 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. 9. 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. 10. Consider the game where the following one shot game is repeated twice.

C D M

C 3,3 4,-1 0,0

D -1,4 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?

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(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? 11. Repeated prisoner’s dilemma. Consider two shops. Every day, each can either try maintain a 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 3; 0

pC 0; 3 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

8

Economics of Information 1. Auction. There is a single seller who has one unit of an indivisible good to be sold in a …rst price 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 : 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 can be modeled as a Bayesian game, where strategy of each player i is a function bi from [0; 1] to [0; 1]: For simplicity, assume that bi must be di¤erentiable, and increasing, thus its inverse bi 1 is well de…ned on a suitable domain. 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. (a) Assuming that buyer 2 uses strategy b2 , write the interim payo¤ to bid x1 2 [p; 1] for buyer 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¤ 1.

to strategy b1 for buyer

(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 + vc ): 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 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¤ to 2 player i is (x1 + x2 ) 12 (xi ) , 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.

33

Microeconomic Theory problems by A. Kajii

(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. 3. The Hirshleifer e¤ect. Consider the following game (Nature chooses “left” or “right” with even chance. (10,0) (10,0)

(8,8) C C

C C C

C C

C

C C

C

C C

C

C Cu TT

T

T T

C C

(10,0) (8,8) (0,10) C C

C

C C

C

Cu

C C

C

C C

C

C C

C

(0,10)

(0,10)

L L L L L L L L Lu C C

Cu @ @ @

Player 2

Player 2 @

Tu a aa aa

@ @u ! !!

! !! aa ! au !! Nature

Player 1

Figure 1: Game

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 amount of 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.

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(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. (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 output of low type cost of education for low type wage for educated worker wage for non-educated worker

4 2 1 w 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 a1 or a2 . The types are equally likely. The payo¤s are given in the following table, where is a 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 Player 2’s payo¤s a1 a2 a1 a2 t1 1 0 t1 2 1 t2 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 21 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 2 chooses a1 or a2 with probability 21 whichever signal she receives, and Player 1 sends s1 if his type is t1 and s2 if his type is t2 . Does this constitute a perfect Bayesian equilibrium with appropriate beliefs?

Microeconomic Theory problems by A. Kajii

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(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?