a. This is an inequality constrained maximization problem. The Lagrangian is µ 1 q 1 = 0 µ 2 q 2 = 0 1 (q 1 ) φ φ c 2 (q 2) p 1 c 1(q 1 ) = 0

Transcription

1 Practice Exam Questions 1 1. There is a firm that produces quantities of two goods, q 1 and q 2. The price of good 1 is p 1 and the price of good 2 is p 2. The firm s cost structure is C(q 1,q 2 ) = c 1 (q 1 )+c 2 (q 2 ) φq 1 q 2 where C(q 1,q 2 ) > 0 for all q 1,q 2 0; c 1 (q 1 ) and c 2 (q 2 ) are positive, increasing, convex, differentiable functions with c 1 (0) = c 2 (0) = 0; and φ 0. The parameter φ measures the spillovers or learning by doing that arise from producing more of both q 1 and q 2 for example, workers become more familiar with both products, and consequently become more efficient at producing each. a. Characterize the set of profit-maximizing quantities for all values of φ. b. Suppose that q 1 and q 2 are both finite and strictly positive. Compute the change in q 1 with respect to a small change in φ, and the change in q 2 with respect to a small change in p 1. c. Suppose the firm chooses φ, facing a positive, increasing, convex, differentiable cost k(φ). Characterize the optimal level of φ as completely as possible. a. This is an inequality constrained maximization problem. The Lagrangian is L = p 1 q 1 +p 2 q 1 c 1 (q 1 ) c 2 (q 2 )+φq 1 q 2 +µ 1 q 1 +µ 2 q 2 The FONCs and complementary slackness conditions are p 1 c 1 (q 1)+φq 2 +µ 1 = 0 p 2 c 2 (q 2)+φq 1 +µ 2 = 0 µ 1 q 1 = 0 µ 2 q 2 = 0 At an interior solution, the SOSC s would be [ c 1 (q 1 ) φ φ c 2 (q 2) be negative definite, or that c i (q i ) > 0 and c 1 (q 1)c 2 (q 2) > φ 2. On the corners, the tangency condition must fail, so that for µ 1 = 0 but µ 2 0 we have so that p 1 c 1(q 1 ) = 0 p 2 c 2 (0)+φq 1 +µ 2 = 0 p 2 c 2 (0)+φq 1 < 0 so that producing any q 2 would have a negative impact on profits, and c 1(q 1 ) > 0, so that producing any more or less of q 1 would not be a profitable adjustment. ] 1

2 b. Use the IFT with the FONC s for an interior solution. c. Use the envelope theorem so that V(φ) = p 1 q 1 +p 2 q 1 c 1 (q 1 ) c 2 (q 2 )+φq 1 q 2 +µ 1 q 1 +µ 2 q 2 and so that the FONC in φ would be V (φ) = q 1 q 2 maxv(φ) k(φ) 0 = q 1 (φ)q 2 (φ) k (φ) φ Note that φ 0 only if q 1 q 2 > 0, or that we are at an interior solution. Otherwise, cost-reducing technology is unprofitable. The SOSC is then q 1 (φ)q 2(φ)+q 1 (φ)q 2 (φ) k (φ) < 0 so that k(φ) must be sufficiently convex in φ so that k (φ) is greater than the first two, positive terms (for example, a linear k(φ) = kφ wouldn t work, since k (φ) = 0 and the inequality above would fail). 2. An agent consumes in period one and period two, and then dies. His utility function over bundles (c 1,c 2 ) is u(c 1 )+δu(c 2 ) where u(c) is positive and increasing in both arguments, and 0 < δ < 1. His initial wealth is w and he has no income in period two. The price of consumption in period one is 1, and the price of consumption in period two is p. The interest rate on savings is r, and for convenience, let R = (1+r). Assume u (c) > 0 and u (c) < 0. a. Characterize the optimal bundle, (c 1,c 2 ) for all R and p. For each solution, check secondorder sufficient conditions. b. Compute the change in the optimal consumption in period 2 with respect to w. c. If r increases, how does the agent s payoff change? How does the optimal consumption in period 1 respond to a change in r? d. How does optimal consumption in period 1 and period 2 vary in p? a. The maximization problem is maxu(c 1 )+δu(c 2 ) c 1,c 2 where pc 1 +s = w and Rs = c 2. Then the problem is maxu((w s)/p)+δu(rs) s 2

3 with FONC 1 p u ((w s)/p)+δru (Rs) which is automatically satisfied. 1 p 2u ((w s)/p)+δr 2 u (Rs) < 0 b. The sign of the change in s with respect to w is given by u ((w s)/p)/p 2 > 0 whichispositive, sosavingincreases inw. Sincec 2 = Rs, thisimpliesc 2 isalsoincreasing in w. c. Well, R = 1+r, so if r increases R increases. Let s see what happens when R increases. The envelope theorem implies V (R) = δru (Rs ) > 0 so that the agent s payoff goes up if r goes up. How does c 1 change? If R goes up, using the IFT on the FONC implies the sign of s / R is δu (Rs)+δR 2 u (Rs) which is ambiguous in sign, so that we can t tell if c 1 goes up or down when R goes up (there s an income effect and a substitution effect that potentially go in opposite directions). d. If p goes up, the IFT implies that the change in s is signed by (w s) p 3 u ((w s)/p) < 0 so that savings is decreasing in first period price. Therefore c 1 increases but c 2 decreases unambiguously. 3. There is a perfectly competitive market in which a consumer with quasi-linear utility u(q,m,g) = v(q)+m+h(g) faces budget constraint w = (p +t)q + m, where p is the price of the good, t is a tax, and g is a public service that all agents take as given. The firm has cost function C(q) = c 2 q2. Assume v(0) = 0, v (0) > 1, and v (0) 0; and h(0) = 0, h (0) > 1, and h (0) < 0. i. Characterize the perfectly competitive equilibrium price and quantity. How do these vary in t? How does the firm s profits vary in t? 3

4 ii. Suppose the government raises money through taxes to fund provision of the public good g. It seeks to maximize social welfare, v(q (t))+m (t)+h(g)+p (t)q (t) C(q (t)) where m (t) = w (p (t) + t)q (t), less the total cost of providing the public good, kg, subject to a budget-balance constraint that kg = tq (t). Formulate this as an optimization problem, solve for the optimal level of public good provision or the optimal tax, and provide the second-order sufficient conditions that ensure your solution is a global maximum. iii. If c increases, how does the optimal tax from part ii change? i. In a perfectly competitive equilibrium, households maximize utility taking price as given, firms maximizing profits taking price as given, and markets clear so that supply equals demand. Then the household solves yielding the FONC and the firm solves with FONC maxv(q)+w (p+t)q +h(g) q v (q ) (p +t) = 0 v (q ) < 0 maxpq c q 2 q2 p cq = 0 c < 0 The the market-clearing quantity is given by v (q ) t cq = 0 If we take the system of equations v (q ) (p +t) = 0 p cq = 0 and totally differentiate with respect to t, we get [ ][ v (q ) 1 q / t c 1 p / t ] [ 1 = 0 ] 4

5 And using Cramer s rule gives q t = 1 v (q ) c p t = c v (q ) c Isolated and not in equilibrium, the firm s profits satisfy π(t) = maxpq c q 2 q2 and π (t) = 0, since t appears nowhere in the firm s maximization problem. However, in equilibrium, the firm s profits are a function and differentiating gives π(t) = p (t)q (t) c 2 q (t) 2 π (t) = p (t)q (t)+(p (t) cq (t))q (t) = so that equilibrium profits are decreasing in taxes. ii. The government s problem is maxv(q (t))+w+h(g) C(q (t)) kg g,t c v (q ) c q (t) < 0 where kg = tq (t). Call g(t) = tq (t)/k. Substituting this into the objective yields The FONC then is maxv(q (t)) tq (t)+w+h(g(t)) C(q (t)) kg(t) t q (t)+h (g(t))g (t) kg (t) = 0 So there is a welfare loss from increasing the tax because of the market distortion (first term), and then the marginal benefit of the public good should be set equal to the marginal cost of providing it (second two terms). The SOSCs are that q (t)+h (g(t))g (t) 2 +h (g(t))g (t) kg (t) < 0 Since q (t) > 0 and h (g(t)) < 0, these conditions are inherently ambiguous. iii. In the FONC for the market-clearing quantity, c appears in the implicit solution for q (t,c), so that increasing c reduces q. Differentiating the FONC s with respect to c (to use the implicit function theorem) yields Since q c +h (g ) g g c t +h (g ) 2 g g c t k 2 c t g (c) = tq (t,c)/k the sign of g / c is the same as q / c, which is negative. This implies the first two terms have opposite signs, since the first is positive while the second is negative, and the comparative static is inherently ambiguous. 5