Varian Microeconomic Analysis (3rd ed) SOLUTIONS

Ch. 14 MONOPOLY 39

14.16.b For the special case p(y; t) = a(y) + b(t), the second term on the right-hand side is zero, so that @p=@t = @b=@t.

14.17.a Di erentiating the rst-order conditions in the usual way gives

14.17.b The appropriate welfare function is W = u1(x1)+u2(x2)−c1(x1)− c2(x2). The total di erential is

dW = (u01 − c01)dx1 + (u02 − c02)dx2:

14.17.c Somewhat surprisingly, we should tax the competitive industry and subsidize the monopoly! To see this, combine the answers to the rst two questions to get the change in welfare from a tax policy (t1; t2).

The change in welfare from a small tax or subsidy on the competitive industry is zero, since price equals marginal cost. But for the monopolized industry, price exceeds marginal cost, so we want the last term to be positive. But this can only happen if dt2 is negative|i.e., we subsidize industry 2.

14.18.a The pro t maximization problem is

max r1 + r2 such that a1x1 − r1 0

a2x2 − r2 0

a1x1 − r1 a1x2 − r2 a2x2 − r2 a2x1 − r1

x1 + x2 10:

14.18.b The binding constraints will be a1x1 = r1 and a2x2−r2 = a2x1−r1; and x1 + x2 = 10.

14.18.c The expression is a2x2 + (2a1 − a2)x1:

40 ANSWERS

14.18.d Formally, our problem is to solve

max a2x2 + (2a1 − a2)x1

subject to the constraint that x1 + x2 = 10. Solve the constraint for x2 = 10 − x1 and substitute into the objective function to get the problem

max 10a2 + 2(a1 − a2)x1:

x1

Since a2 > a1 the coe cient on the second term is negative, which means that x1 = 0 and, therefore, x2 = 10. Since x2 = 10, we must have r2 = 10a2. Since x1 = 0, we must have r1 = 0.

14.19.a The pro t-maximizing choices of p1 and p2 are

p1 = a1=2b1 p2 = a2=2b2:

These will be equal when a1=b1 = a2=b2.

14.19.b We must have p1(1 − 1=b1) = c = p2(1 − 1=b2). Hence p1 = p2 if and only if b1 = b2.

14.20.a The rst-order condition is (1 − t)[p(x) + p0(x)x] = c0(x), or p(x) + p0(x)x = c0(x)=(1 − t). This expression shows that the revenue tax is equivalent to an increase in the cost function, which can easily be shown to reduce output.

14.20.b The consumer's maximization problem is maxx u(x) − m − px + tpx = maxx u(x) − m − (1 − t)px. Hence the inverse demand function satis es u0(x) − (1 − t)p(x); or p(x) = u0(x)=(1 − t).

14.20.c Substituting the inverse demand function into the monopolist's objective function, we have

(1 − t)p(x)x − c(x) = (1 − t)u0(x)x=(1 − t) − c(x) = u0(x)x − c(x):

Since this is independent of the tax rate, the monopolist's behavior is the same with or without the tax.

Ch. 15 GAME THEORY 41

Solve each equation for PD, set the results equal to each other, and solve for t to nd

14.22.a Note that his revenue is equal to 100 for any price less than or equal to 20. Hence the monopolist will want to produce as little output as possible in order to keep its costs down. Setting p = 20 and solving for demand, we nd that D(20) = 5.

14.22.b They should set price equal to marginal cost, so p = 1.

14.22.c D(1) = 100.

14.23.a If c < 1, then pro ts are maximized at p = 3=2 + c=2 and the monopolist sells to both types of consumers. The best he can do if he sells only to Type A consumers is to sell at a price of 2 + c=2. He will do this if c 1.

14.23.b If a consumer has utility ax1 −x21=2+x2, then she will choose to pay k if (a−p)2=2 > k. If she buys, she will buy a−p units. So if k < (2−p)2=2, then demand is N(4 − p) + N(2 − p). If (2 − p)2 < k < (4 − p)2=2, then demand is N(4 − p). If k > (4 − p)2=2, then demand is zero.

14.23.c Set p = c and k = (4 − c)2=2. The pro t will be N(4 − c)2=2.

14.23.d In this case, if both types of consumers buy the good, then the pro t-maximizing prices will have the Type B consumers just indi erent between buying and not buying. Therefore k = (2 − p)2=2. Total pro ts will then be N((6 − 2p)(p − c) + (2 − p)2=2). This is maximized when p = 2(c + 2)=3.

Chapter 15. Game Theory

15.1There are no pure strategy equilibria and the unique mixed strategy equilibrium is for each player to choose Head or Tails with probability 1=2.

15.2Simply note that the dominant strategy on the last move is to defect. Given that this is so, the dominant strategy on the next to the last move is to defect, and so on.

15.3The unique equilibrium that remains after eliminating weakly dominant strategies is (Bottom, Right).

15.4Since each player bids v=2, he has probability v of getting the item, giving him an expected payo of v2=2.

Ch. 16 OLIGOPOLY 43

16.2@F (p; u)=@u = 1 − r=p. Since r is the largest possible price, this expression will be nonpositive. Hence, increasing the ratio of uninformed consumers decreases the probability that low prices will be charged, and increases the probability that high prices will be charged.

16.3Let = 1 2 − γ2. Then by direct calculation: ai = ( i j − jγ)= , bi = j = , and c = γ= .

16.4The calculations are straightforward and may be found in Singh & Vives (1984). Let = 4 1 2 − γ2, and D = 4b1b2 − c2. Then it turns out

that pci − pbi = iγ2= and qib − qic = aic2=D, where superscripts refer to Bertrand and Cournot.

16.5The argument is analogous to the argument given on page 297.

16.6The problem is that the thought experiment is phrased wrong. Firms in a competitive market would like to reduce joint output, not increase it. A conjectural variation of −1 means that when one rm reduces its output by one unit, it believes that the other rm will increase its output by one unit, thereby keeping joint output|and the market price|unchanged.

16.7In a cartel the rms must equate the marginal costs. Due to the assumption about marginal costs, such an equality can only be established

when y1 > y2.

16.8Constant market share means that y1=(y1 + y2) = 1=2, or y1 = y2. Hence the conjectural variation is 1. We have seen that the conjectural variation that supports the cartel solution is y2=y1. In the case of identicalrms, this is equal to 1. Hence, if each rm believes that the other will attempt to maintain a constant market share, the collusive outcome is \stable."

16.9In the Prisoner's Dilemma, (Defect, Defect) is a dominant strategy equilibrium. In the Cournot game, the Cournot equilibrium is only a Nash equilibrium.

16.10.a Y = 100

16.10.b y1 = (100 − y2)=2

16.10.c y = 100=3) 16.10.d Y = 50

16.10.e y1 = 25, y2 = 50

16.11.a P (Y ) + P 0(Y )yi = c + ti

44 ANSWERS

16.11.b Sum the rst order conditions to get nP (Y ) + P 0(Y )Y = nc +

Pn

i=1 ti, and note that industry output Y can only depend on the sum of the taxes.

16.11.c Since total output doesn't change, yi must satisfy

P (Y ) + P 0(Y )[yi + yi] = c + ti + ti:

Using the original rst order condition, this becomes P 0(Y ) yi = ti, oryi = ti=P 0(Y ).

16.12.a y = p

16.12.b y = 50p

16.12.c Dm(p) = 1000 − 100p

16.12.d ym = 500

16.12.e p = 5

16.12.f yc = 50 5 = 250

16.12.g Y = ym + yc = 750.

Chapter 17. Exchange

17.1In the proof of the theorem, we established that xi i x0i. If xi and x0i were distinct, a convex combination of the two bundles would be feasible and strictly preferred by every agent. This contradicts the assumption that x is Pareto e cient.

17.2The easiest example is to use Leontief indi erence curves so that there are an in nite number of prices that support a given optimum.

17.3Agent 2 holds zero of good 2.

17.4 x1A = ay=p1 = ap2=p1; x1B = x2B so from budget constraint, (p1 + p2)x1B = p1, so x1B = p1=(p1 + p2). Choose p1 = 1 an numeraire and solve ap2 + 1=(1 + p2) = 1.

17.5 There is no way to make one person better o without hurting someone else.

Ch. 18 PRODUCTION 45

17.7 The Slutsky equation for consumer i is

@xi = @hi : @pj @pj

17.8The strong Pareto set consists of 2 allocations: in one person A gets all of good 1 and person B gets all of good 2. The other Pareto e cient allocation is exactly the reverse of this. The weak Pareto set consists of all allocations where one of the consumers has 1 unit of good 1 and the other consumer has at least 1 unit of good units of good 2.

17.9In equilibrium we must have p2=p1 = x23=x13 = 5=10 = 1=2.

17.10Note that the application of Walras' law in the proof still works. 17.11.a The diagram is omitted.

17.11.b We must have p1 = p2.

17.11.c The equilibrium allocation must give one agent all of one good and the other agent all of the other good.

Chapter 18. Production

18.1.a Consider the following two possibilities. (i) Land is in excess supply. (ii) All land is used. If land is in excess supply, then the price of land is zero. Constant returns requires zero pro ts in both the apple and the bandanna industry. This means that pA = pB = 1 in equilibrium. Every consumer will have income of 15. Each will choose to consume 15c units of apples and 15(1 − c) units of bandannas. Total demand for land will be 15cN. Total demand for labor will be 15N. There will be excess supply of land if c < 2=3. So if c < 2=3, this is a competitive equilibrium.

If all land is used, then the total outputs must be 10 units of apples and 5 units of bandannas. The price of bandannas must equal the wage which is 1. The price of apples will be 1 + r where r is the price of land. Since preferences are homothetic and identical, it will have to be that each person consumes twice as many apples as bandannas. People will want to consume twice as much apples as bandannas if pA=pB = (1−c c) (1=2). Then it also must be that in equilibrium, r = (pA=pB) −1 0. This last inequality will hold if and only if c 2=3. This characterizes equilibrium for c 2=3.

18.1.b For c < 2=3.

18.1.c For c < 2=3.

Ch. 22 WELFARE 47

Chapter 20. Asset Markets

20.1 The easiest way to show this is to write the rst-order conditions as

0 ~ ~ 0 ~

Eu (C)Ra = Eu (C)R0

0 ~ ~ 0 ~

Eu (C)Rb = Eu (C)R0

and subtract.

20.2 Dividing both sides of the equation by pa and using the de nition

~ ~

Ra = Va=pa, we have

− ~ ~

Ra = R0 R0cov(F (C); Ra):

Chapter 21. Equilibrium Analysis

21.1The core is simply the initial endowment.

21.2Since the income e ects are zero, the matrix of derivatives of the Marshallian demand function is equal to the matrix of derivatives of the Hicksian demand function. It follows from the discussion in the text that the index of every equilibrium must be +1, which means there can be only one equilibrium.

21.3Di erentiating V (p), we have

dV (p)

dt

= −2z(p)Dz(p)p

=−2z(p)Dz(p)Dz(p)−1z(p)

=−2z(p)z(p) < 0:

Chapter 22. Welfare

22.1 We have the equation

xi = Xk tj @hj :

j=1 @pi

Multiply both sides of this equation by ti and sum to get

48 ANSWERS

The right-hand side of expression is nonpositive (and typically negative) since the Slutsky matrix is negative semide nite. Hence has the same sign as R.

22.2 The problem is

max v(p; m)

Xk

such that (pi − ci)xi(pi) = F:

i=1

This is almost the same as the optimal tax problem, where pi − ci plays the role of ti. Applying the inverse elasticity rule gives us the result.

Chapter 23. Public goods

23.1Suppose that it is e cient to provide the public good together, but neither agent wants to provide it alone. Then any set of bids such that b1 + b2 = c and bi ri is an equilibrium to the game. However, there are also many ine cient equilibria, such as b1 = b2 = 0.

23.2If utility is homothetic, the the consumption of each good will be proportional to wealth. Let the demand function for the public good be

given by

Then the equilibrium amount of the public good is the same as in the Cobb-Douglas example given in the text.

23.3 Agent 1 will contribute g1 = w1. Agent 2's reaction function is f2(w2 + g1) = maxf (w2 + g1) − g1; 0g. Solving f2(w2 + w1) = 0 yields w2 = (1 − )w1.

23.4The total amount of the public good with k contributors must satisfy

G = wk + Gk :

Solving for G, we have G = w=(k − ). As k increases, the amount of wealth becomes more equally distributed and the amount of the privately provided public good decreases.

23.5 The allocation is not in general Pareto e cient, since for some patterns of preferences some of the private good must be thrown away. However,

the amount of the public good provided will be the Pareto e cient amount:

P

1 unit if i ri > c, and 0 units otherwise.