ANSWER KEY University of California, Davis Date: June, 05 Department of Economics Time: 5 hours Microeconomic Theory Reading Time: 0 minutes PRELIMINARY EXAMINATION FOR THE Ph.D. DEGREE Please answer four questions (out of five) Question. We postulate the utility function!u : R L R:!u(x) = a x - [/] x B x, (.) where a = (a,..., a L ) R L ++, and B is an L L symmetric, positive definite matrix... Compute the gradient vector!u(x). ANSWER.!u(x) = a Bx. (A.).. Compute the Hessian matrix D!u(x). ANSWER. D!u(x) = B. (A.).3. Is the preference relation represented by!u continuous? Homothetic? Quasilinear? No need to justify your answers. ANSWER. Continuous. Not homothetic. Not quasilinear..4. Is the preference relation represented by!u convex? Strictly convex? Locally nonsatiated? Explain your answers. ANSWER. From (A.), the Hessian of!u is negative definite and therefore!u is a strictly concave function. It follows that the preference relation represented by!u is strictly convex, and hence convex. From (A.),!u(x) = 0 for x = B a, which together with the concavity of!u implies that!u attains a maximum at x. Thus, the preference relation represented by!u has a (global) satiation point at x and therefore fails to satisfy local nonsatiation.
.5. In order to develop your intuition, graphically represent the indifference map for L =, for the b c special case where a = a a and B = c b Be as precise as possible. ANSWER. Expression (A.) becomes. Separately consider the cases c = 0 and c > 0.!u(x,x )!u x!u x = a a b c c b x x = a bx cx a cx bx Hence, the marginal utilities are positive for small x s and may be negative for large x s. It follows that!u x 0 a bx cx 0 and!u x 0 a cx bx 0. CASE : c = 0. Now:!u x 0 x a b.!u x 0 x a b. Both marginal utilities are zero at (x,x ) = CASE : c > 0.. a b, a b. See Figure.. Now!u 0 x x a c b c x!u, and 0 x x a b c b x. Because the positive definiteness of B implies that b > 0, if c > 0 then the slopes of the lines defined by!u x = 0 and!u x = 0 are both negative. Moreover, the positive definiteness of B also implies that b > c, i. e., b > c, so that the slope of the line defined by!u x = 0 is higher, in absolute value, than that of the line defined by!u x = 0. The vertical intercept of the line defined by!u x = 0 is a/b, same as the
3 horizontal intercept of the line defined by!u x = 0. The marginal utilities are both zero at (x, x ) = a b + c, a b + c. See Figure...6. You can assume that, for (p, w) >> 0, a solution to the UMAX[p, w] problem exists and is unique. Without attempting at this point to explicitly solve the UMAX[p, w], show that for w above a certain level, that you should made explicit, demand is insensitive to increases in wealth, whereas for values of w below this level Walrasian demand is affine in wealth. (Hint: For the second part, use the Implicit Function Theorem.) ANSWER. We are interested in the partial derivatives!x j (p,w), j =,..., L, for a given w arbitrary price vector p >> 0. Write, as before, x for the point where the gradient vanishes, i. e.,!u(x) = 0. Define w(p) := p i x. Clearly, for w w(p) the solution to the UMAX[ p,w] problem is constant and equal to x. Hence, the Engel curve corresponding to the price vector p is flat (!x j (p,w) = 0 ) in the interval (w(p), ). w But Walras Law is satisfied if w < w(p). Thus, using the FOC a Bx λp = 0, (A.3) Walrasian demand is defined there by the following system of L+ equations in the L+ unknowns (x,..., x L,λ), with the L+ parameters (p,..., p L,w), a Bx λp!!!!! p i x + w = 0! 0 R L+. In order to apply the Implicit Function Theorem, fix p = p and define F(x,λ,w) (F (x,λ,w),...,f L (x,λ,w)) = a Bx λp and B(x,λ,w) = p i x + w. Partition the matrix of the partial derivatives of F and B as follows:
4 F! F F x x L λ " F w!!!! "! F L! F L F L F " L x x L λ w B! B x x L B λ " B w = b! b L p " 0!!!! "! b L! b LL p L " 0 p! p L 0 ". The IFTh then gives!x w "!x L w! λ w = b " b L p " " " " b " b p L p " p L 0 Note that the entries in the right-hand side matrices do not involve w. Hence, in the interval (0,w(p)), the Engel curve for good j has a constant slope and is, therefore, an affine function of w ( j =,, L). 0 0 " Alternatively, for the case w < w(p) we can solve (A.3) to obtain. x = B a λb p, (A.4) and p x = p B a λp B p = w (by Walras Law), i.e., which substituted back into (A.4) yields!x(p,w) = B a p B a w p B p an expression affine in w. λ = p B a w p B p, B p = B a p B a p B p p + w p B p B p,
5.7. We now consider a society with I consumers, each endowed with the preference relation represented by!u in (.). We denote by P := R L+I ++ the domain of prices and individual wealth vectors ( p;w,...,w I ). Is there a subset of P for which a positive representative consumer exists? Explain. ANSWER. Yes. As just seen, for( p;w,...,w I ) {( p;w,...,w I ) R L+I ++ : w i < w( p)}the Engel curves of all consumers are affine with the same slope, hence by the Gorman Theorem a positive representative consumer exists..8. We return to the one-consumer case. Solve the UMAX[p, w] problem for good in the case where L =, a = a = a and B = b c c b, for c > 0. Can one of the goods be inferior at a point of the domain of the Walrasian demand function? (Hint. Consider the wealth expansion paths in (x, x ) space.) point (x, x ) = ANSWER. As noted in the Answer Key to.6 above, he consumer chooses her satiation a b + c, a b + c if it is attainable, i. e., if p x + p x ) w. Otherwise, her Walrasian demand is given by the solution to the system of three equations in three unknowns: a bx cx = λp, a cx bx = λp, p x + p x = w, which in particular implies that at least one good is normal in the price-wealth domain satisfying p x + p x ) < w. From the first two equations, we obtain: a bx cx a cx bx = p p, (A.5) which can be used to obtain the wealth expansion path as follows: (A.5) can be written p [a bx cx ] = p [a cx bx ], i. e., x [ p c + p b] = [ p p ]a [ p c p b]x, i. e., x = [ p p ]a p b p c + p b p c p b p c x. i. e., x = [ p p ]a p b p c + p b p c p b p c x. (A.6)
6 First note that, if c < 0, then the slope p b p c of the wealth expansion path (A.6) is p b p c positive, which implies that both goods are normal, and therefore ordinary, at any point in the domain where p x + p x ) > w : this is depicted in Figure.. But if c > 0 (Figure.), then the slope of the wealth expansion path may be negative, with one good normal and the other good inferior. For instance, if p < c p b, then the denominator of p b p c p b p c (because b/c > c/b), yielding a negatively sloped wealth expansion path. and its numerator is positive To explicitly solve for the Walrasian demand for good ( w < w ), we use Walras Law and (A.6) to obtain p x + p [ p p ]a p b p c + p b p c p b p c x = w, p i. e., x p + p b p c p b p c = w p i. e., x p [ p b p c]+ p [ p b p c] [ p p ]a p b p c [ ] = [ p b p c]w p [ p p ]a, i. e.,!x (p,w) = [ p b p c]w p [ p p ]a b [ p ] + [ p ] p p c..9. We now consider a consumer who faces uncertainty. Her ex ante preference relation satisfies the expected utility hypothesis with von Neumann-Morgenstern-Bernoulli utility function u :[0,a / b ] R, where a and b are positive parameters, and with coefficient of absolute risk aversion equal to b. She is facing the contingent-consumption optimization problem of a bx maximizing her expected utility subject to a budget constraint. Let there be two states of the world, s and s, with probabilities π and π, respectively.9.. Comment on her ex ante preferences. ANSWER. We observe that her coefficient of ARA is increasing in x, which implies that her absolute risk aversion increases with wealth. This is empirically awkward: for many people their absolute risk tolerance increases with wealth. However, this type of preference relation is often postulated for convenience, because it implies that the ex ante utility of a risky asset depends only on its mean and variance.
7.9.. To what extent is her problem formally a special case of the UMAX problem of.6 above? Explain. ANSWER. From her coefficient of ARA, we see that her vnmb utility function is the quadratic u :[0,a / b ] R :u(x) = ax bx. Hence, when facing the prospect that gives x with probability π and x with probability π ( x a / b, x a / b ), her ex ante utility is U(x, x ;π) = πax π[/ ]b [ x ] + [ π]ax [ π][/ ]b [ x ] = [πa, [ π]a] x x [x x ] πb 0 0 [ πb a special case of (.) for L = and the off-diagonal entries of the matrix B equal to zero. For fixed π we can compute: U(x, x ) = (π[a bx ],[ π][a bx ]) which vanishes at (x, x ) = (a / b,a / b ) so that the indifference map in contingent commodity space looks like Figure. above..9.3. Draw a plausible wealth expansion path in contingent commodity space. As her wealth increases, does the consumer bear absolutely more or less risk? Relatively more or less risk? ANSWER. See Figure.3. As her wealth increases, she bears absolutely less risk and, a fortiori, relatively less risk. x x,
ANSWER KEY Question. Let there be two goods: good is an input, and good is an output. We set at p = the price of the input and denote by p p / p the relative price of the output. Throughout this question we postulate an industry comprised of two firms, named Firm and Firm. We denote by z > 0 (resp. q > 0) the amount of input used by (resp. output produced by) Firm, and similarly we denote by z > 0 (resp. q > 0) the amount of input used by (resp. output produced by) Firm. We assume that both firms are price takers in the input market. Buyers of output are assumed to be price-taking consumers. The aggregate demand-foroutput function in this market is given by y = - p, where y is the aggregate quantity of output demanded by consumers, assumed to be independent from their wealth levels. We will be concerned about productive efficiency. For the purposes of this question, we say that the vector(z,q ;z,q ) R 4 + is productively efficient if with the total amount of input z + z it is not possible to produce an aggregate amount of output higher than q + q. We consider three alternative models, named Market A, Market B and Market C. The information provided above applies to all three markets. A. MARKET A: PRICE TAKING, NO EXTERNALITIES Both firms are price takers in the output market. Firm s direct production function is f :R + R + : f (z ) = z. (.) A.. Obtain Firm s cost function, and graph its average cost and its marginal cost. Obtain Firm s supply-of-output correspondence (or function). ANSWER. See Figure A. below. Cost function: c (q ) = q. Average cost = Marginal cost = Domain of the supply-of-output correspondence: P = {(, p) R ++ : p }. Supply-of-output correspondence: Q : P R + :Q (, p) = {0}, if p <, R +, if p =.
In output space (i. e., with q on the horizontal axis and p on the vertical axis), Firm s supply correspondence has two segments: one follows the vertical axis from zero to ; the other one is a horizontal line at height. A.. Firm s direct production function is f :R + R + : f (z ) = z. (.) Obtain Firm s cost function, and graph its average cost and its marginal cost. Obtain Firm s supply-of-output correspondence (or function). ANSWER. See Figure A.. Cost function: c (q ) = q Average cost = q Marginal cost = q. Domain of the supply-of-output function : P := {(, p) R : p R ++ }. Supply function: The supply of output is determined by the condition price = marginal cost, i. e., p = q. In output space, Firm s supply function coincides with the marginal cost curve with q on the horizontal axis and p on the vertical axis. A.3. Find the vector (!z,!q ;!z,!q ) of input and output quantities for the two firms at the equilibrium of Market A. ANSWER. See Figure A.3 below. First, the profit maximization condition for Firm implies that the equilibrium market price cannot exceed p =, and hence the quantity demanded by consumers must be at least. If p <, then the supply of output by Firm is zero, and that of Firm is less than ½. Hence the equilibrium price is, with Firm supplying ½ unit of output, and Firm supplying ½ unit of output. It follows that (!z,!q ;!z,!q ) = (, ; 4, ). A.4. Is (!z,!q ;!z,!q ) productively efficient? Argue your answer. ANSWER. Yes. We can compare the marginal costs of firms and at the equilibrium values. From A., the marginal cost of Firm is everywhere. From A., the marginal cost of Firm is q, which
3 if evaluated at q = ½ also becomes. Thus, the marginal costs are equalized across both firms, and aggregate output cannot be increased by transferring input from one firm to the other one. Alternatively, we can explicitly solve an optimization problem. The total use of input at (!z,!q ;!z,!q ) equals ¾. Aggregate output when aggregate input is ¾ and z is allowed to vary in the interval [0, ¾] is given by 3 4 z + + z = 0, z, with FOC for maximization i. e., z =, or z = 4, which in turn yields z = 3 4 4 =. Hence, with the total amount of input 3 4 it is not possible to produce an aggregate amount of output higher than. Of course, productive efficiency is implied by Pareto efficiency, which in the case is guaranteed by the First Fundamental Theorem of Welfare Economics. B. MARKET B: DUOPOLY, NO EXTERNALITIES We now consider Market B, with two firms, again named Firm and Firm with the same technologies as above, i. e., with the technology given by (.) for Firm, and that of (.) for Firm. The difference is that now the two firms behave as a Cournot duopoly. B.. Find the vector (z,q ;z,q ) of input and output quantities for the two firms at the Cournot Equilibrium of Market B. ANSWER. Firm takes the output, q, of Firm as given, and chooses q in order to maximize [ q q ] q q with FOC [ q q ] - q = 0, i. e., - q = q, or q = q (A.)
4 Similarly, Firm takes the output, q, of Firm as given, and chooses q in order to maximize [ q q ] q [q ] with FOC [ q q ] q q = 0, i. e., q = 4 q, or q = q 4 Putting (A.) and (A.) together, we obtain (A.) q = q 4, i. e., 4q = 4 + q,i. e., 8q = 3+ q, or q = 3 7 <, (A.3) which in turn yields q = 3 7 = 7 3 4, i. e., q == 4 4 with z = 4 4 and z = 3 7 B.. Is (z,q ;z,q )productively efficient? Argue your answer. ANSWER. No. We compare the marginal costs of the two firms. For Firm it is always. For Firm, it is q, which if evaluated at q = 3 7 < (see (A.3)) becomes q = 3 <. Hence transferring some 7 input from Firm to Firm would increase aggregate output.
5 C. MARKET C: PRICE TAKING, EXTERNALITIES We now consider Market C, with two firms, again named Firm and Firm. The technology of Firm is as given by (.) above, but that of Firm involves a positive production externality generated by Firm. More precisely, the output of Firm depends on the level of output of Firm, in addition to depending on Firm s use of the input, z, according to the expression q = q z. (.3) (For instance, increasing the output of Firm generates technical knowledge that spills over to Firm.) As in Market A, both firms operate independently as price takers. C.. Find the vector(ẑ, ˆq ;ẑ, ˆq ) of input and output quantities for the two firms at the equilibrium of Market C. ANSWER. Because of the profit maximization condition for Firm,we must have p < (as in Market A). But if p <, then the supply of output by Firm would be zero, which in turn would imply zero output for Firm, given the technology. Hence, the equilibrium price must be with associated aggregate consumer demand of. From (.3), the cost function of Firm is given by: z = q q i. e., c (q q ) z = q, q with marginal cost c q = q q. Thus, the equilibrium quantities of output (q, q ) must satisfy: (i) Price = Marginal cost in Firm : q q = ; (ii) Aggregate supply = Aggregate demand: q + q =. Solving this system of two equations in two unknowns, we find: q + q =, i. e.,
6 ( ˆq, ˆq ) = 3, 3 with ẑ = 3 and ẑ 3 = 3 See Figure C. = 9 3 = 6. C.. Is (ẑ, ˆq ;ẑ, ˆq )productively efficient? Argue your answer. ANSWER. No. Aggregate output can be written as a function of z and z as z + z z. Suppose that, starting from (z, z ), we transfer ε units of input from Firm to Firm. Aggregate output is then ϕ(ε) := z + ε + z + ε z ε, with ϕ'(ε) := + z ε z + ε z + ε z ε, which if evaluated at z = 3,z =,ε = 0, yields 6 ϕ'(0) := + 6 3 3 6 = + 3 3 3 3 = + = 4 > 0. Hence, if we start from the competitive equilibrium input demands (ẑ,ẑ ) = 3, 6 and we transfer a small positive amount of input from Firm to Firm, then aggregate output increases. Intuitively, the competitive equilibrium does not take account of the positive externality that Firm generates on Firm, and hence it allocates to Firm an amount of input that is too small from the viewpoint of productive efficiency. D. Briefly comment on the productive efficiency in the three markets studied. ANSWER. The familiar property of productive efficiency that obtains under price taking in the absence of externalities (implied by Pareto efficiency, in turn guaranteed by the First Fundamental Theorem of Welfare Economics) does not extend to non-price taking behavior or to production externalities. Under price taking in the absence of externalities, aggregate production is
7 always efficient: social marginal costs are always equalized among firms, and the merging of firms does not affect aggregate equilibrium quantities of outputs and inputs. The marginal costs of the various firms do not have to be equalized at a market -power equilibrium. Hence, productive inefficiency may arise. Conceivably, the productive inefficiency (but not Pareto efficiency) would disappear if all firms merged. With production externalities, price taking guarantees the equality of marginal private costs among firms, but this does not suffice for productive efficiency because of the disparity between private and social costs caused by the externality. Again, the productive inefficiency would disappear if all firms merged, but the merged firm would have to be a price taker (perhaps an unrealistic assumption here) would have to be a price taker to be consistent with Pareto efficiency.
Answer key Question 4 June 05
Micro Prelim June 05 Answer Keys for Question 5 (a) The idea is to construct a distribution where there is correlation between low prices and low consumption and between high prices and high consumption, so that charging a higher price attracts a pool of customers more concentrated on high consumption. For example the following (0,) (0, ) (,) (, ) distribution: 49 49. In this case if the restaurant charges p = 0 then 00 00 00 00 everybody shows up, but only 60 consumers can be served; the expected number of dishes consumed per served customer is + =.5 so that the expected profit is π (0, c) = 60(0.5 c) = 600 90c. If the restaurant charges p = then only 60 consumers show up and they can all be served; the expected number of dishes consumed per customer is 49 + =.98 so that the expected profit is π (, c) = 60(.98 c) = 660 8.8c. If 50 50 c >.0834 then π (0, c) > π (, c). For example, if c = 5 then π (0,5) = 50 > π (,5) = 66. An even simpler example would be j= (0,) (0, ) (,) (, ) 50% 0 0 50%. (b) As remarked previously, there should be correlation between low prices and low consumption and between high prices and high consumption, so that charging a higher price attracts a worse pool of customers, that is, one with a higher proportion of high-consumption individuals. This is the phenomenon of adverse selection. m (c) Let g( i) = F( i, j) ; then g( i) = a m for every i =,, n. In the case under consideration we have a uniform distribution over W. Thus, no matter what price the restaurant charges, the restaurant faces a lottery where the distribution of consumption levels (that is, the distribution over D = {,..., m} ) is m+ uniform. Hence the average consumption per customer is equal to and thus the expected cost per customer is m + m+ c (which is less than or equal to, since c ). Hence, no matter what K is, the + restaurant s profit per customer is π ( p) = p m c and its total profit is Kπ ( p) (since capacity is such that even at the highest price there are enough consumers to fill the restaurant); thus the profitmaximizing price is the largest price, namely n. (d) Also in this case we have that g( i) = a m, for every i =,, n, so that if the restaurant charges the highest price, p = n, demand is equal to capacity (since K = a m ) and if the restaurant charges a price p n then there is excess demand. (d.) When the firm charges p = n 3 then all the consumers with reservation price greater than or equal to n 3 will show up (thus a number exceeding capacity) and K of them will be served. The total number of customers who show up is 4 a m and, of these, 3 a m would consume dish while the remaining a m (those whose reservation price is n) would consume m dishes. Thus the expected consumption per customer is 3 m+ + m = 3 and thus the expected cost per m 3 customer is c 4 4 4 4 +. Thus the expected total profits are ( 3 m + 3 m+ K n ) a m ( n 3 3 c) =. 4 4 (d.) First we compute π ( n k) (the profit per customer) for an arbitrary k and then show that
π ( n k) is decreasing in k. When p = n k (with k ) all the consumers with a reservation price greater than or equal to n k will show up; their total number is ( k + ) m a and of these k m a have a consumption level of and m a have a consumption level of m, so that the expected consumption of a served customer is k k + + m = m. Hence total profits are k + k + k + k + m Kπ ( n k) where π ( n k) = n k k + c. Next we show that π ( n k) is decreasing in k: k + m k + m+ m π n ( k ) π ( n k) + c = + c ; this expression is decreasing in k ( + ) = ( ) k + k + ( k + )( k + ) and increasing in c and thus achieves it maximum value at k = c = ; the maximum value is + m, which is negative as long as m 6. 6 (d.3) By part (d.), the restaurant will only consider charging either p = n or p = n. If p = n then all the consumers that show up consume m dishes, so that the cost per served customer is m c Π ( n) = K n m c. If p = n, then half of the consumers who and total profit is ( ) show up ( a m out of a m ) consume dish and the other half consume m dishes, so that the expected cost per served customer is m + c and total profit is ( ) ( m + Π n = K n c). If Π( n ) > Π ( n) then the restaurant will not want to charge p = n and thus will create excess demand. Π( n ) > Π ( n) if and only if m + n m c > n c if and only if m > +. c