Practice Exam Questions 2 1. There is a household who maximizes discounted utility u(c 1 )+δu(c 2 ) and faces budget constraints, w = L+s+c 1 and rl+s = c 2, where c 1 is consumption in period 1 and c 2 is consumption in period 2, where w is wealth, and L is an amount of money lent. Assume u(c) 0, u (c) 0, and u (c) < 0. The amounts saved and lent must both be positive, s,l 0. a. Characterize the profit maximizing bundle (c 1,c 2 ) by providing first-order necessary conditions and second-order sufficient conditions (focus on the amounts saved and lent). How do the optimal decisions vary in r and w? How does the household s payoff vary in r and w? An entrepreneur is trying to raise money for his business. The revenues produced are π(l), where L is the total amount borrowed. The entrepreneur takes the interest rate r as given and maximizes profit, π(l) rl Assume π (L) 0 and π (L) 0. b. Characterize the profit-maximizing amount for the firm to borrow in terms of r by providing first-order necessary and second-order sufficient conditions. How does the optimal amount borrowed depend on r? c. In a competitive lending market, the supply of funds from the household will equal the demand for funds from the entrepreneur. Characterize a perfectly competitive equilibrium. How does the amount of funds lent to the entrepreneur depend on w and p? a. The Lagrangian is yielding FONC s and complementary slackness conditions L = u(w s L)+δu(s+rL)+µ s s+µ L L u (w s L)+δu (s+rl)+µ s = 0 u (w s L)+δru (s+rl)+µ L = 0 µ L L = 0,µ s s = 0 Note that since lending and saving are perfect substitutes, so that the household is willing to lend to both if r = 1 for the total amount of savings σ that satisfies u (w σ)+δu (σ) = u (w σ)+δru (rσ) = 0 If r < 1, the agent only chooses savings, and if r > 1, the agent only chooses to lend. b. The firm maximizes max B π(l) rl 1
yielding FONC and SOSC π (L) r = 0 π (L) < 0 characterizing the demand for funds. The amount of funds desired is decreasing in r since π (L) < 0. c. Parts a and b give the supply and demand. Market clearing implies that π (L) = r, so that the market clears at u (w L)+δπ (L)u (π (L)L) = 0 if π (L) 1, and L = 0 if π (L) is everywhere less than 1. As long as funds are lent, the optimal amount is increasing in w, since the IFT and the FONC imply that it has the same sign as u (w L ). There is no p, so L doesn t depend on p (my apologies). 2. There is a firm that uses capital, K, and labor, L, to produce quantity according to production function F(K,L) = q, where F(K,L) is weakly increasing in each argument. The price of hiring capital is r and the price of labor is w. Consider the cost minimization problem min rk wl K,L subject to the constraints F(K,L) = q, K 0, and L 0. Assume F(K,L) 0, F(K,L) 0 for all (K,L) 0, and F(0,0) = 0. a. Characterize the set of solutions of the optimization problem using first-order necessary conditions and second-order sufficient conditions. b. How does the cost-minimizing K vary in w and q? c. Let c(q,r,w) be the optimized value of the cost function. If the firm gets a price p for its good, what is the optimal quantity to produce? How do firm profits vary in r? How do the profit-maximizing choices of K and L depend on p? a. similar to homework. b. similar to homework, depends on regime c. Then the firm solves which has FONC and SOSC maxpq c(q,r,w) q p c q (q,r,w) = 0 c qq (q,r,w) < 0 2
Using the value function from the constrained problem yields and Using the IFT, we can solve for λ / q: c q (q,r,w) = λ (q,r,w) c qq (q,r,w) = λ q λ q = λ2 (F KK F LL F KL F LK ) deth so that F KK F LL F LK F KL < 0 so that the SOSC is satisfied (or that F(K,L)is concave). < 0 To see how firm profits vary in r, use the envelope theorem on to get K < 0. V(r) = maxpq c(q,r,w) q To see how the optimal K and L depend on p, let s look at the FONCs Then q is increasing in p, since p c q (q,r,w) = 0 q p = 1 c qq (q,r,w) but then we return to our original problem, we can then see that K p = K q q p and using the IFT and likewise for L. K q = λ (F K F LL F L F LK ) deth 3. An agent consumes two market goods, q 1 and q 2, and a composite other good, m. The agent s utility function is u(q 1,q 2,m) = v(q 1,q 2 )+m where v(q 1,q 2 ) 0, and he faces a budget constraint (p 1 +t 1 )q 1 +(p 2 +t 2 )q 2 +m = w, and non-negativity constraints q 1,q 2 0. a. Characterize the utility-maximizing bundles by providing first-order necessary and secondorder sufficient conditions. When are the second-order sufficient conditions satisfied? 3
b. How does the utility-maximizing m respond to a change in p 2? How does the agent s payoff vary in p 2? c. Suppose q 1,q 2 > 0. Will raising t 1 or t 2 by a small amount raise more tax revenue? Explain your answer clearly. a. Similar to class/homework b. Comparative statics: Use the IFT for the interior solution. Otherwise, do comparative statics holding the set of active constraints constant. c. Let Then Re-arranging yields and similarly Let R = t 1 q 1(t 1,t 2 )+t 2 q 2(t 1,t 2 ) = q1 t +t q1 q2 1 +t 2 1 t 1 t 1 = q t 1(1+ 11 )+ t 2 1 1 t 1 q2 21 = q2 t (1+ 22)+ t 1 1 2 t 2 q1 12 ij = q i t j t j q i be the cross-price elasticity of q i to t j. Then good 1 raises more revenue if q1 (1+ 11)+ t 2 1 t 1 q2 21 > q2 (1+ 22)+ t 1 1 t 2 q1 12 This condition accounts for the fact that while one good might be less elastic with respect to its own-price, it might be very elastic with respect to the other good s price. Depending on this correction that is proportional to the ratio of the taxes, t i 1 t j qj ji we can have either good raise more revenue. If we assume that q 1 = q 2 = q and t 1 = t 2 = t, then or q (1+ 11 )+ 1 q 21 > q (1+ 22 )+ 1 q 12 q 11 + 1 q 21 > q 22 + 1 q 12 or, increasing good 1 s tax raises more revenue if 11 22 > 1 q 2( 12 21 ) 4
so that, for example, if 1 is more own-price sensitive and has a lower cross-price elasticity, it will surely raise more revenue. On the other hand, just because 11 > 22, it does not follow that raising the tax on 1 will raise more revenue, since the right-hand side might be even larger. 5