Problem 3,a. ds 1 (s 2 ) ds 2 < 0. = (1+t)

Transcription

1 Problem Set 3. Pay-off functions are given for the following continuous games, where the players simultaneously choose strategies s and s. Find the players best-response functions and graph them. Find the Nash equilibrium strategies for each player. Do the games exhibit strategic complements or strategic substitutes? If t increases, how do the equilibrium strategies change? Assume t > 0. (a). π = (5 s s )(s +ts ) π = (5 s s )(s +ts ) Problem 3,a Best-response functions π = 5 s s ts = 0 s (s ) = 5 (+t)s s Note the symmetry of the problem: relabeling s as s and s as s lets us go from π to π. Then the computations I just did are going to be the same for the second player, so I get s (s ) = 5 (+t)s You can seeinthepicturethat thereaction curves aredownwardsloping, sothereisstrategic substitutes. Note how when the BR function of one agent shifts up, the equilibrium changes so one player is using a higher strategy in the new equilibrium than the old one, and his opponent is using a lower strategy after the change. The other way to verify the game has strategic substitutes is to check that each player s strategy is decreasing his opponent s strategy: ds (s ) ds = (+t) < 0 Nash Equilibrium Strategies You can solve by substituting the best-response functions into one another, as we did with

2 Cournot. That goes as follows: Insert the best-response function for player into player s best response function: s (s (s )) = 5 +t s (s ) = 5 +t [ 5 +t ] s Solve this equation for s : Other ways to solve: s = 5 +t s = 5 (+t)5 4 [ 4 (+t) ] s 4 [ 5 +t ] s + (+t) s 4 = 0 5(t+t) 4 s = 0 (+t)5 4 (+t) Symmetry The problem is completely symmetric, so there is a symmetric solution where s = s. Take either firm s first-order condition: Insert the symmetric strategy s = s = s: 5 s s ts = 0 5 s s ts = 0 and s = 5 3+t You might say that s not the answer we got above. However, s = 0 (+t)5 4 (+t) = So they re the same answer. 5( t) 4 t t = 5( t) 3 t t = Subtract FOC s The two first order conditions are: Subtract the second from the first: 5 s s ts = 0 5 s s ts = 0 5( t) (3+t)( t) = 5 3+t 5 s s ts [5 s s ts ] = (+t)s (+t)s = 0 Then s = s, so call that special strategy s, and we can use the same approach as above with symmetry. Then if t goes up, so 5/(3+t) goes down, so an increase in t reduces both players strategies in equilibrium.

3 (b). π = (5+ts s )s π = (5+ts s )s Problem 3,b, t > / Problem 3,b, t < / Note that relabeling to and to converts the profit function for player into player s profit function, and vice versa, so the game is symmetric. Best Response Functions Step two: Maximize each player s payoff with respect to what that player controls and solve for best response functions: π = 5+ts s = 0 s (s ) = s 5 s t π = 5+ts s = 0 s (s ) = s 5 s t 3

4 Step three: Solve the best-response functions simultaneously to get the Nash equilibrium: or or s = s 5 5 t t 4t s = s 5 5t s = 5 +t 4t = 5 +t ( t)(+t) = 5 t Since the work will be exactly the same for the second player, the Nash equilibrium is s = s = 5 t This equilibrium is kind of funny, since the strategies are positive if t < /, but negative if t > /. This game has strategic complements, since ds (s ) ds = t > 0 so player s strategy is increasing in player s. (Apologies for the ugly graph) Another way to get the solution is that, since the players payoffs simply swap the s and s around, all the work in computing the equilibrium is the same for the two players, so there will be a symmetric equilibrium where s = s = s. So taking one of the players first order conditions: 5+ts s = 0 5+ts s = 0 s = 5 t If t increases, the denominator becomes smaller, so s goes up. (c). π = ( s +s )s +ts s π = ( s +s )s +ts s 4

5 Problem 3,c Step two: Maximize and solve for best responses: π = s +s +ts s = 0 s (s ) = s +ts This isn t a straight-line; it s a quadratic, since there s a s term. By symmetry, player two has a similar best response function: s (s ) = s +ts Note that the players strategies are increasing in their opponent s strategy, so there are strategic complements (For example, ds (s )/ds = (+ts ) > 0). Looking at the best-responsegraph, you can see there are two Nash equilibria. One is at s = 0,s = 0, and the other it where the curves intersect again. This multiplicity of equilibria can happen when there are strategic complements, but never with strategic substitutes. Step three: Solve for the Nash equilibrium: Solving by substituting best response functions into one another yields an equation that has terms of order s 4, so this looks like a bad idea. Since the intersections of the best response functions occur on the diagonal, where s = s = s, we might guess there s a symmetric equilibrium, solve for it, and then verify (or just argue that symmetric games have symmetric equilibria). Take player one s first-order condition, we substitute in the symmetric strategy: s +s +ts = 0 s +s +t(s ) = 0 This equation has two solutions: s = 0, or if you divide by s 0, ++ts = 0 s = /t So the Nash equilibrium of the game is that both players use s =. This is decreasing in t, t since as t goes up, the denominator becomes larger, and s decreases.. There are two firms who sell similar, but different, products. They simultaneously choose their prices, p,p > 0. Firm has marginal cost c and firm has marginal cost c. The demand 5

6 for each firm s product depends on its price and its opponent s price: D (p,p ) = A p +ep D (p,p ) = A p +ep where > e > 0. So as p goes up, more customers switch to Firm, and vice versa. (a.) Write out the firms profits functions. π (p,p ) = (A p +ep )p c (A p +ep ) π (p,p ) = (A p +ep )p c (A p +ep ) (b.) Solve for and graph the firms best-response functions. Does the game have strategic complements or strategic substitutes? We maximize with respect to what each player controls: π p = A p +ep +c = 0 π p = A p +ep +c = 0 Then solve each equation in terms of that player s strategy: p = A+ep +c p = A+ep +c These are upward sloping, since p is increase in p and p is increasing in p ; so the game has strategic complements. (Graph is essentially the same as for the Hotelling model). (c.) Solve for the Nash equilibrium prices. Let s insert the second equation into the first: or or, multiplying by, p = A+e A+ep +c p ( e /) = A +e +c + e c +c p (4 e ) = A(+e)+ec +c 6

7 or Similar work yields p = A(+e)+ec +c 4 e p = A(+e)+ec +c 4 e These values, (p,p ), are the Nash equilibrium of the game. Note that if c = c, we get p = A(+e)+(+e)c (+e)( e) = A+c e If e = 0, this is the monopoly solution: There is no substitution from one firm s product to the other by consumers in response to price increases, and the two firms are essentially monopolies. (c.) How does a change in c affect the equilibrium prices? Is this similar or different to the Cournot game we studied in class? and If c, p = e c 4 e > 0 p c = 4 e > 0 This is different from the Cournot game, because there, an increase in one firm s marginal cost led to a decrease in that firm s production, and an increase in the other firm s production. Here, both firms increase their price in response to an increase in one firm s cost. (d.) How does a change in e affect the equilibrium prices? Explain the economic intuition for this. and If e, p e = (A+c )(4 e )+e(a(+e)+ec +c ) (4 e ) > 0 p e = (A+c )(4 e )+e(a(+e)+ec +c ) (4 e ) > 0 So if e, both firms increase prices. We can interpret e as something like over the elasticity of demand for that product, since in the Hotelling model the parameter e was about equal to e = /t. If e increases, basically, consumers are more willing to switch products, and less willing to drop out of the market altogether. Knowing this, the firms are more willing to jointly raise prices, since they lose consumers to each other. When elasticity between the two products is high, price increases by each firm balance out in terms of demand stealing, and they can support high prices in the market. 3. Consider a market with N firms where the price is p(q) = A Q, and Q = q +q +...+q N. Each firm i has a cost function C(q i ) = cq i. 7

8 (a.) In a perfectly competitive market, what would the price and quantity be? If N = a monopoly what are the market price and quantity? In a perfectly competitive market, price equals marginal cost, so that A Q = c, or Q pc = A c. In the monopoly market, the monopolist maximizes π(q) = (A Q)Q cq with FONC 0 = A Q c Q M = A c So the monopolist makes about half of what the perfectly competitive market would provide. (b.) Solve for the Nash equilibrium of a Cournot model with two firms, so p(q) = A (q +q ). Calculate industry output, Q = q + q ; is it greater or less than the monopoly and competitive outputs? The Nash equilibrium quantities are q = (A c)/3, so total quantity is Q = (/3)(A c). This is /3 of the perfectly competitive quantity, and 4/3 of the monopoly quantity. (d.) Graph the firms best-response curves. Does the model have strategic complements or strategic substitutes? If one firm s costs increase, how does the equilibrium change? If A increases (there s a shock that increases consumers willingness to pay at all quantities), what happens to the equilibrium quantities? See the slides from class for the graphs. The best response functions are q i (q j ) = A c q j which is decreasing in the opponent s strategy, so the game has strategic substitutes. If one firm s marginal cost increases, it shifts that firm s best-response function down, so that firm produces less and the other one produces more in equilibrium. If A increases, both best-response functions shift up, so the equilibrium quantities are higher. (e.) Solve for the Nash equilibrium of a Cournot model with N firms, so that p(q) = A (q + q +...+q N ). (Hint: Use the symmetry of the problem after taking a derivative) The profit function for firm is π (q,q ) = A q N j= q j q cq 8

9 with FONC A q N q j c = 0 If q =... = q N = q, then we can solve the above equation alone to get j= q = A c N + (f.) Calculate industry output for N firms: Q(N) = q + q q N = Nq (N). As N gets very large (N ), what do Q(N) and p(q(n)) tend to? Check your answer in part a to see the connection between Cournot and perfect competition. Industry output is then equation to Q (N) = Nq = N N + (A c) = +/N (A c) As N, /N 0, and the Cournot market produces the same quantity as a perfectly competitive market. Also, p(q (N)) = A N N + (A c) = +N A+ N c, which converges N + to c, the same as a perfectly competitive market. (g.) The Herfindahl-Hirschmann Index (HHI) is a measure of the concentration in an industry, often used by government agencies like the Department of Justice (DOJ) or Federal Trade Commission to quantify whether or not a merger will greatly reduce competition. It is computed as follows: Let s i be the industry share of firm i, s i = q i Q ; then the HHI is: H = Compute H as a function of N. The 00 Merger Guidelines for the DOJ say that An HHI index below.5 indicates an unconcentrated industry An HHI index between.5 and.5 indicates moderately concentrated industry An HHI index above.5 indicates a highly concentrated industry N i= For what values of N is the industry unconcentrated? Moderately concentrated? Highly concentrated? Give some an example of a highly concentrated industries by these criteria, and explain whether this model and analysis over- or under-state how competitive it really is. s i 9

10 so The share of each firm i in the symmetric Cournot model is s i = q (N) Q (N) = H = N + (A c) = N N + (A c) N i= N = N so that the HHI is just /N. Unconcentrated industries are those with N > 6.66, moderately concentrated are those with N between and 4, and highly concentrated are those with fewer than four industries. An example of a highly concentrated industry might be operating systems. Apple and Microsoft dominate the market, with Linux and other operating systems making up a much smaller proportion. This model assumes the firms are symmetric in costs, which is somewhat misleading: If we did this with asymmetric costs, the answers could be very different. N 4. There are two firms engaged in a research and development (R&D). They each choose an R&D intensity r,r 0, respectively, and the cost of R&D is C(r i,) = cr i. The probability that firm i succeeds in developing the technology is p(r i ) = r i +r i ; notice that this is a number between zero and one, p(0) = 0, and p(r i ) < for all r i > 0. If a firm succeeds in developing the technology alone, it gets a monopoly profits of π; if the other firm also develops the technology, they become Bertrand duopolists and each gets a profit of zero; if a firm fails to develop the technology, it gets profits of zero. Assume π > c. (a.) Write out the firms payoffs. The expected payoffs are, for i =,, ( )( ri π i = r ) ( )( ) j ri π cr i = π cr i +r i +r j +r i +r j (b.) Solve for the firms best-response functions and graph them. Is this a game of strategic complements or substitutes? Do you think research intensity is a strategic substitute or complement in real life? Firm i s best response is given by (+r i ) π c = 0 +r j 0

11 or r i (r j ) = π c +r j This game has strategic substitutes, since r i (r j ) is decreasing in r j. The function starts at π/c, and then decreases as rj increases. It seems like research intensity could be a strategic substitute or strategic complement in real life. For example, if my opponent increases his R&D intensity, I might back off because I feel like I am less likely to win. On the other hand, if my opponent increases his R&D intensity, I might feel like I have to compete harder, like when you re running and hear someone coming up behind you. This seems like it depends on the details of how we model the R&D process itself, which is interesting, no? (c.) Solve for the Nash equilibrium R&D intensities. If r i = r j = r, then we solve the single FONC, (+r ) +r π c = 0 to get or or (+r ) 3 = c π π c = (+r ) 3 ( π ) /3 r = c (d.) Set π = 4 and c =. What is the equilibrium probability that the technology is developed (by either firm singly or both firms together)? Suppose the firms colluded and chose r and r to maximize industry profits. Is the technology more or less likely to be developed? The probability that a single firm succeeds here is r.5874, so the probability it gets developed non-cooperatively is Now, the cartel maximizes =.603 r +r +r π + +r r +r π cr cr with FONCs r (+r ) π +r (+r ) π c = 0 +r

12 r (+r ) π +r (+r ) π c = 0 +r If we assume r = r = r and look for a symmetric solution, then we get a single equation, Notice that we can write this as (+r ) +r π r (+r ) +r π c = 0 (+r ) 3π c }{{} Eqm FONC r (+r ) 3π }{{} Externality where the externality term captures the effect of one firm s R&D efforts on the other. Since this piece is negative, we then know that throwing it away will give = 0 (+r ) 3π c > 0 = (+r ) 3π c and since /(+x) 3 is a decreasing function in x, it turns out that r > r. Consequently, the research joint venture exerts less effort on R&D, and the product is less likely to be developed. Notice how we didn t even have to compute r to answer the question, just use some math? For the record, r.365, so that =.463 which is around 3 percent less likely than the non-cooperative regime. If you are really, really into this problem, you may have noticed that there s another possible solution the cartel could choose: Have one firm do the R&D, and keep the other firm out of the race altogether. Then maximizing the monopoly profits gives π r +r cr π (+r) c = 0 or r = π/c = =, so that the probability of success with one firm is /. This is better than the collusive RJV, but not as good as the competitive one. (e.) The government often allows Research Joint Ventures, in which multiple firms pool their efforts to produce a technology. Is this a good or bad policy? Why or why not? Here, the RJV is a bad idea and reduces the probability of innovation substantially. Some other issues are whether the costs are fixed or marginal: If there are huge costs to entering an R&D race, competition to innovate may suffer, and allowing firms to split some of these costs or initial discoveries may be beneficial. Finally, we assume here that the innovating firm alone makes monopoly profits, while they get zero profits if they jointly develop the product as in Bertrand price competition. If these assumptions were relaxed so that other firms could license the discovery or patent law was less strict, we might get different results.