MS&E HW #1 Solutions

Transcription

1 MS&E HW #1 Solutions 1) a) Because supply and demand are smooth, the supply curve for one competitive firm is determined by equality between marginal production costs and price. Hence, C y p y p. b) Let y c be the total output from the competitive sector. Since the firms are identical, we know that y c 50y. Hence, y c 50 p y c 50p. c) At a price p, demand will be p. Production by the competitive sector will be 50 p. Let y m be the monopoly quantity. Assert that the monopolist will suppy all demand in excess of the competitive supply, or y m D y c. Hence, y m p 50p y m p. d & e) The monopolist attempts to maximize profit, Π m. Π m r m c m p y m p c m m p p p c m m p 1000p 100p 2 c m m p Because demand and supply are smooth, we can take the first-order conditions, equating marginal costs with marginal revenue, to find the global optimum. Π m p p 0 p 5 y m 500. f) The competitive sector will provide y c 50p 250. g) Total output will be y y c y m ) a) In a competitive equilibrium, it will be the case that p MC. Thus, 100 Y 0 Y 100. b) If each firm behaves as a cournot competitor, it will solve: max yi 100 y i y i i holding y i constant. For firm 1 this yields FOC: 100 2y 1 y c) To find the cournot equilibrium output, we solve both firms FOCs simultaneously: 100 2y 1 y y 1 2y 2 0 c 1 y c

2 d) If the two work as a cartel, they are solving together the problem Notice that this is the same as the monopolists problem: So we know that p ε Y ε 0, where ε Y Y m 50. max y1,y y 1 y y 1 y 2 2 max Y 100 Y Y This equation is solved when Y 50. So, the cartel output is y 1 y 2 e) We begin with firm 1. Suppose firm 2 has already choosen to produce amount y 2. Then firm 1 will produce y 1 2 as derived in part b) because y 1 2 is the optimal amount for firm 1 to produce given that firm 2 is producing y 2. Firm 2 is able to anticipate what firm 1 will do, and takes this into consideration when choosing y 2. Firm 2 solves. max y2 100 y 1 2 y 2 2 FOC: y s 2 50, y s 1 y 1 s ) The British firm is a profit maximizer operating with a fixed price, p*: the market is competitive. The British firm solves the maximization max y Π p y c w, r, y a) With an import tax and export subsidy, the maximization is now, max y Π p s t c w, r, y The optimal y will remain unchanged from the no-intervention case if t s. b) With the capital subsidy, the optimization becomes max y Π p t c w, r s, y Take the first-order conditions: Π p t c w,r s,y 0 p t c w,r s,y c) Now we are going to look at the change in t with changes in the subsidy level. Take the derivative of the equality in (b) with respect to s. t s c w, r s, y 2 s c w, r s, y K w, r s, y r s Note that we used Shepherd s lemma in the final step. d) With constant returns to scal e, K(w,r-s,y) K(w,r-s,1)y. Hence, t s K w, r s, 1. e) If the factor of production is inferior, then K be the American import tax. < 0. Hence, the higher is the British capital subsidy, the lower need 4) a & b) First, look at the equality between the unconstrained monopoly and licensing solutions. Say that p m is the monopoly price associated with marginal cost m. The monopolist can arrive at the solution by optimizing over price or over quantity. Optimizing over price gives the first-order conditions 2

3 (1) q p m p m m q p m 0. If the firm licenses, the market price is given by m L, assuming that m L < M. The firm s profit per unit is given by L. Total profit is given by (2) L q m L The optimal solution is given by the first-order conditions, (3) q m L L q m L 0. Substitutiong p m L gives (4) q p p m q p 0, which is identical to (1), the monopolist first-order conditions. Hence, the unconstrained optimal solution with licensing is identical to that without licensing, or p m m L. The problem splits into two possible scenarios. Either p m > M or p m < M. Let p c be the competitive price associated with the pre-innovation marginal cost, M. Case 1: p m > M (c onstrained) Without licensing, the optimal solution is given by p M - Ε, where Ε is small. The innovator can t price above M because the competitors will undercut the innovator. Since the unconstrained monopoly price is greater than M and the profit function is concave, the innovator wants to price as close to M as possible. a price below M will have lower profits than p M (or Ε below M, to be precise). Hence the optimal profit is given by (5) M Ε q M Ε m q M Ε M Ε m q M Ε The profit function with licensing is also concave with the optimal unconstrained license price lying above M - m. Hence, the constrained optimal license price is given by L M m. Profit by licensing is given by (6) L q m L M m q M Hence, profits are at least as high with licensing as when the innovator produces his or herself. Case 2: p m < M (unconstrained) First show that if m is small enough p m < M. Note that a monopolist maximizes p q q m q, using quantity as the variable of choice now. If marginal cost is m, the first-order conditons give p q m q m p q m m. We know that p q c q c p q c > 0, where q c is the competitive quantity with marginal costm. Set m 0, and p q c q c p q c > 0 m, which implies that p m < p c. (1) and (4) show that the optimal solution is the same with and without licensing in the unconstrained case. 5) The following results were generated using Mathematica c. a) In this case Acme faces the demand function : In[1]:= w p 6 p 1.5 p 3 p 9 p 36 which is 4 i 1 w i p. The firm will choose p in order maximize profit. (Notice that we make use of the duality between price and quantity choices to simplify the mathematics. If we tried to model the monopolist as making a quantity choice, we would have a very difficult time characterizing demand in closed form!) In[2]:= S p w p w p solutions Solve D S, p 0, p 3

4 Out[2]= p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p There are many potential solutions to this problem, but only one of them is real. Thus, we take the uniform price to be: In[3]:= pstar solutions Out[3]= The number of widgets sold at this price is: In[4]:= w pstar Out[4]= b) In this case Acme charges each type a different price. The firm will solve a monopolist s problem for each type of consumer. Thus, for each type we know that p i k ε ε 1. From this equation we can solve for p i and plug result back into the demand to get w i. In[5]:= w1 p 6p 1.5 p1star N w1star w1 p1star Out[5]= 3. Out[5]= In[6]:= w2 p 6p 3 p2star N w2star w2 p2star Out[6]= 1.5 Out[6]= In[7]:= w3 p 6p 9 p3star N w3star w3 p3star Out[7]= Out[7]=

5 In[8]:= w4 p 6p 36 p4star N w4star w4 p4star Out[8]= Out[8]= c) To see who does better and who does worse is straightforward; we simply compute the welfare for each party in each case and compare them. For Acme LLC, we compare profits: In[9]:= N pstar w pstar pstar N p1star w1 p1star w1 p1star p2star w1 p2star w2 p2star p3star w1 p3star w3 p3star p4star w4 p4star w4 p4star Out[9]= Out[9]= As we expected, Acme does significantly better when price discriminating. For each of the consumer types, we comare the consumer surplus in each case. But we have to be careful consumer surplus is most naturally found by integrating the inverse demand function p i w i! Thus the consumer surplus in each case is w p 0 i w w p i w w for appropriate value of w. In[10]:= p1 w w Integrate p1 w, w, 0, w1 pstar pstar w1 pstar Integrate p1 w, w, 0, w1star p1star w1star Out[10]= Out[10]= So type 1 consumers are worse off when Acme price discriminates. In[11]:= p2 w w Integrate p2 w, w, 0, w2 pstar pstar w2 pstar Integrate p2 w, w, 0, w2star p2star w2star Out[11]= Out[11]= In[12]:= p3 w w Integrate p3 w, w, 0, w3 pstar pstar w3 pstar Integrate p3 w, w, 0, w3star p3star w3star Out[12]= Out[12]=

6 In[13]:= p4 w w Integrate p4 w, w, 0, w4 pstar pstar w4 pstar Integrate p4 w, w, 0, w4star p4star w4star Out[13]= Out[13]= But type 2, 3, and 4 consumers are better off when Acme price discriminates. 6) a) In a perfectly competitive market p MC, so q A B q m A m B and p m. b) In a cournot equilibrium, each producer i solves max qi A B Q q i m q i where Q n i 1 q i and holding q j constant j i. This yields the n FOCs: A B Q m B q i, i q i q j q o, i, j substituting we get c) In the limit these values are: A B n q o m B q o q o A m n 1 B Q n A m n 1 B p A n A m n 1 lim n Q A m B Q lim n p A A m m p 6