Stephen Hutton ECONOMICS 600 August 2005 Office: Tyd 5128c Mathematical Economics: Problem Set One Solutions

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1 COLLEGE PARK THE UNIVERSITY OF MARYLAND MD Stephen Hutton ECONOMICS 600 August 2005 Office: Tyd 5128c Mathematical Economics: Problem Set One Solutions 1. The point of this problem is to help you visualize the fact that logical arguments, set theoretic arguments, and arguments using necessary and sufficient conditions are essentially the same. i) C T F. ii) Concavity is sufficient for continuity since C T means that anything that is concave is also continuous. Continuity is necessary for concavity since C T means that anything that if a function is not continuous it cannot be concave. This type of proof uses what is known as the contrapositive if A implies B then (not B) implies (not A). iii) Use the above argument to note that since g is clearly not continuous, it cannot be concave. Diagramatically, F g 1 (x) C T C g 2(x) C

2 Since g 1 (x) lies outside the T set, it must lie outside the C set. iv) In this case, g 2 (x) is in fact concave, but we cannot use the argument above to prove it. I drew in a possible configuration where a nonconcave but continuous g 2 (x) could lie. 2. Consider the following market. There are two firms, 1 and 2 who sell the same product. The firms charge prices p 1,p 2 respectively. Total market demand curve is given by Q = A - B p where p = min{p 1,p 2 }. The demand for each firm s product is given by q i = 0, if p i > p j, q i = Q/2, if p i = p j, q i = Q, if p i < p j,. That is, if a firm prices strictly below its rival, it captures all the market, if it prices the same as its rival, they share the market. The firms have identical constant marginal costs, c. In this question, suppose that firm 2's price is fixed and known at some level, p 2. [Note that this setup is typical of Bertrand competition.] i) Characterize firm 1's profit maximization problem. The problem is max p1 q 1 (p 1,p 2 )(p-c) where q 1 (p,p 2 ) is given in the problem. Note that q 1 (p,p 2 ) is not a continuous function so the objective function is not continuous. ii) Show that there exists a number, M such that if p 2 > M, there is a unique solution to firm 1's profit maximization problem. Suppose that firm 2 was not around. The firm s problem would be simply like that of a monopolist. max p (A-Bp)(p-c) The derivative of the objective function is da/dp = A+Bc-2Bp This is decreasing in p (so the objective function is concave.) It is equal to zero at p=(a+bc)/(2b). The profit function looks like

3 c (A+Bc)/(2B) p Notice profits are increasing in p until p=p*=(a+bc)/(2b) then decreasing. That is, if the firm selected a price strictly below p* it would want to raise the price. If it selected a price strictly above, it would want to lower price. Now suppose that firm 2 is present. Suppose that p 2 > p*. The profit function looks like this: Observe that the firm s optimal solution stays the same. Thus a candidate for M is (A+Bc)/(2B). iii) Show that there exists a number, m such that if p 2 m, there is a solution to firm 1's profit maximization problem. Is it unique?

4 Suppose p 2 c, then if the firm makes any sales, it can only be if p 1 p 2 c which implies nonpositive profits. However, setting p 1 = c now IS a solution to the problem. If p 2 < c, then any p 1 > p 2 is also a solution since it yields zero sales and zero profits. You should be able to argue that the profit function is upper semicontinuous for p 2 < c. Why? iv) Show that if m < p 2 < M, there is no solution to firm 1 s profit-maximization problem. For p 2 (c,(a+bc)/(2b)], the profit function is strictly increasing for p 1 < p 2 but strictly falls at p 1 = p 2 so the objective function is not continuous. v) What is the mathematical issue that is responsible for the lack of a solution? See above answer. 3. Consider a monopolist luxury auto market. The monopolist produces cars at a constant marginal cost of $20,000. Inverse market demand is given by P c = 50,000 - Q where P c is the price paid by consumers. (Not necessarily the price received by the monopolist which is P M.) i) Suppose there is no tax so P c =P M. Compute the profit-maximizing price. The objective function is (50,000-20,000-Q)Q. This is strictly concave in Q (and in P) and its derivative is 30,000-2Q which gives a solution at zero of Q*=15,000 or P M = 35,000. ii) Suppose (as has happened in the U.S.) there is a luxury tax. For any car sold at a consumer price above $30,000, a tax of t is imposed on the amount above $30,000. a) Determine the relationship between P c and P M. P M =P c if P c < 30,000 P M =30,000+ (1-t)*(P c -30,000) if P c 30,000. =t30,000+ (1-t)*P c if P c 30,000. b) Use this to determine the inverse demand curve as a function of P M and the profit-maximizing price at t=.5.

5 P 50K Demand with no tax 30K Demand with tax And see below. Q c) Why can t you use the same approach as in i)? The profit function is continuous but not everywhere differentiable. The function is given by so A(P C ) = (P C -20K)(50K-P C ) if P C < 30K = ((1-t)P C +t30k-20k)(50k-p C ) if P C > 30K da/dp C =70K-2P C if P C < 30K = -(1-t)P C -t30k -20K)+(1-t)(50K-P C ) if P C > 30K = (1-t)50K -t30k +20K -2(1-t)P C if P C > 30K = 70K -t80k -2(1-t)P C if P C > 30K = 30K - P C if P C > 30K This function is positive for P C < 30K and strictly negative for P C > 30K. This means the solution is at P C = 30K but the derivative is not zero. iii) Now suppose there is no luxury tax. In addition to the inverse market demand given above (generated by overpaid lawyers), there is another component of demand (generated by poor economists) that is given by

6 P c = 32,000 - Q e. a) Graph the market demand curve and compute the profit maximizing price. P For P > 32000, the demand is as before. For P < we need to add in the demand of the economists. Q e = P, Q S = P, adding the two together gives total market demand which is Q = P or an inverse demand curve of P= Q for P < If P is restricted to be above 30000, we already know the optimal price is If P is restricted to be below 32000, the optimal price solves max P (41K-.5Q-20K)Q I get a solution of Q = 21 so P = 30.5K < 32K. Now we just compare profits at the two levels. The high price gives profits of 225. The low price gives profits of so it looks like the high price does better. Who gets shut out of the market? b) What sort of problems arise using the calculus approach here? Q

7 Here the issue is not really the non-differentiability but the fact that there are two local maxima to the profit function. 4.Suppose that x=(1,0,0). i) Find a vector which is orthogonal to x. Both (0,1,0) and (0,0,1) are orthogonal. ii) Find a vector which forms an acute angle with x. Try the vector (1,1,1). iii) Show that if y is orthogonal to x, then ay is also orthogonal to x. (ay) x=a(y) x=0. iv) Show that there are two linearly independent vectors which are orthogonal to x.see i) above. 5. Some standard problems relating calculus to economics: i) Let C(x) be the cost of producing output x. Suppose it is continuously differentiable and strictly increasing. a) What is the marginal cost of producing x? Marginal cost is the cost of producing one extra (small) unit so MC(x)=C (x). b) What is the average cost of producing x? AC(x)=C(x)/x. c) Show that the average cost of producing x is increasing if and only if the marginal cost exceeds the average cost. Using the quotient rule, d(ac(x))/dx=(c (x)x-c(x))/x 2 =(C (x)- C(x)/x)/x= (MC(x)-AC(x))/x. The conclusion follows. d) Show that the marginal cost of producing x is increasing if and only if the cost function is concave. From a), MC(x) is increasing if C (x) is increasing or if C (x) is positive which implies C(x) is concave. ii) The demand curve for a monopolist is Q(P), a decreasing function. The elasticity of demand at price P is defined as η(p)=-q (P)*P/Q(P) a) Prove that a necessary condition for P to maximize profits is η(p) >1. Note that monopolist profits are Q(P)*P-C(Q(P)) and we can assume that C is non-decreasing. Consider the revenue part of this term. d(q(p)*p)/dp = Q (P)P+Q(P)=(Q (P)P/Q(P)+1)Q(P)=(- η(p)+1)q(p). This is positive if and only if η(p) < 1 so revenue is increasing in P if and only if η(p) < 1. Suppose that η(p) < 1. Then by raising P, revenue rises and costs fall (why?) so profits must go up. b) When is this sufficient? Suppose C (x)=0. Then maximizing profits is the same as maximizing revenues.