Suppose that Harry and Steven make their living selling contraband at opposite ends of a town that is 1 mile long. Because it's a crowded city, the citizens use taxi-cabs for transportation. For traveling a distance j the cab fare is 3j. The marginal cost per gram of their produce is c = 6. Each resident of the linear city has a sufficiently high willingness-to-pay for the good V that they will all buy from either Steven or Harry.
(a) Suppose that Harry sets his price at p_{h} = 9 while Steven considers his price p_{s}.
(i) Assuming that Harry is at the zero point, find the indifferent consumer i in terms of p_{s}.
(ii) If Steven sets his price at p_{s} = 15 how much quantity will each of them sell?
(iii) Use the position of the indifferent consumer to find the profit-maximizing output for Steven. (iv) Solve for the Price p_{s} Steven must set to achieve this output.
(v) How much profit do they each earn?
(b) Suppose now that Harry and Steven are simultaneously choosing prices; solve for and express their
joint Nash Equilibrium prices.
(c) Suppose that successful research/development has enabled Steven to decrease his marginal cost to
c_{s} = 4. Harry continues to operate at c_{h} = 6.
(i) What will be Steven's new best response function to an arbitrary price p_{h}?
(ii) Without solving anything, will Harry's equilibrium price go up or down? How do you know? (iii) Solve for the new equilibrium in prices.
(iv) What will Steven's market share be?
Q. Suppose that you manage a firm that gives hot-air balloon tours over the Ohio river. You serve a market
that has a demand curve of Q^{M kt} = 100 -^{ }2. You have a cost function of TC1 = 12q. There is a second company in your market that has a cost function of T C_{2} = 8q + 1, 200; this company rents the balloons to you, and-at the end of each tour-you and they split the money paid by your customers. You meet to determine how to divide the price per ride (i.e. P = P_{1} + P_{2}). Both you and the other firm are painfully aware that you are the only providers of your service in the area, meaning q_{1} = q_{2}.
(a) Find the Best Response function of firm 2 as follows:
(i) Write the profit function for firm 2 (don't forget to include the fixed cost)
(ii) take the derivative of the profit function to write the first-order condition (you should be able to multiply everything out, or recall that D_{x}f (x)g(x) = g(x)D_{x}f (x) + f (x)D_{x}g(x)).
(iii) Set that condition equal to zero and solve for the profit-maximizing output of the other firm P_{2 }that is a best response for your output P_{1}.
(iv) Suppose that corporate spies report to you that the other firm is pricing at one half of the market 2 = 55 (you have good reason to trust the spies...you paid them!):
A. how much will firm 1 produce as a best response?
B. what will be the market quantity?
C. what will be the total profit of each firm?
(b) Finally, consider the Nash Equilibrium of these two firms:
(i) Is this a "Bertrand" game or a "Cournôt" game? How do you know?
(ii) What will be your best response output P 2of your firm expressed in terms of the other firm's price P_{1}?
(iii) What will be the Nash Equilibrium prices for both firms?
(iv) What will be the market quantity under this equilibrium?
(v) How much profit will each earn under this equilibrium?
Q. Suppose that there are two competing firms in a market; they compete according to all the standard assumptions (no entry/exit, homogenous goods, and a static model). The market demand curve is given by P =1- 1 . Both firms have a marginal cost of c = 0.28.
1000 Q
(a) If both firms compete by setting prices,
(i) What will be each firm's Bertrand-Nash Equilibrium price? (ii) What will be the Market Price?
(iii) What will be the market Quantity?
(iv) How much consumer surplus is there in this equilibrium?
(b) Now, suppose that each firm is capacity-constrained to have a maximum output of 360.
(i) If Firm 1 prices at the price found above (let's call this price P^{ B} ), then what is the residual demand curve of firm 2? (hint: write the curve in terms of quantity and then subtract firm 1's capacity from that quantity)
(ii) using the residual demand, find the profit-maximizing price for firm 2. This is the best response of firm 2 to firm 1's price.
(iii) How much quantity will firm 2 produce at this price?
(iv) Is pricing at P^{ B} a best response of firm 1 to this new price for firm 2. If not, give a price that firm 1 could move to unilaterally for more profit.
(c) Now, suppose that firm 1 finds a new technology that decreases their marginal cost to 0.25. This new technology has no capacity constraint.
(i) How much quantity does each firm produce in a Cournot equilibrium?
(ii) If firm 1 sets P_{1} = 0.25, name as many best response prices for firm 2 as you can. Do any of these pairs constitute a Bertrand-Nash Equilibrium?
(iii) How do firm profits compare in the equilibria you found?
Q. Consider a Stackelberg game of quantity competition between two firms with marginal cost c = 180. Market Demand is given by the function P = 12, 000 - 60Q. Under the Cournot Oligopoly model, firm 1 has a best
response function of q_{1} =^{ }127 -^{ }^{2} .
(a) What is firm 2's Best Response Function (hint: you should be able to write this down doing no work; this is also known as a reaction curve)?
(b) Suppose that firm 1 is allowed to announce his output quantity before firm 2. This announcement is completely binding on firm 1.
(i) What will be firm 2's best response quantity q^{2} as a function of q_{1}?
(ii) Firm 1 will be able to anticipate this response of firm 2: write the demand curve for firm 1, and impose the assumption that firm 2 maximizes profits (i.e. impose q_{2} = q^{2}).
(iii) Now write the marginal revenue function.
(iv) Write the first order condition, and solve for the profit-maximizing quantity q^{1}.
(v) Return to firm 2's best response function to find the optimal response q^{2}. (vi) What will be the market price?
(vii) Find the profit earned by each firm in this equilibrium.
(c) Finally, assume that firm 2 has an engineer who finds a way to reduce costs, but he leaves firm 2 and starts his own, competing company with the same technology, so that c_{2} = c_{3} = 150; unfortunately, the delays in start-up mean that firms 2 and 3 will have to choose a production schedule until after firms 1 does. c_{1} = 180.
(i) Write the profit function for firm 1 for any anticipated quantities produced by the other two firms.
(ii) Suppose that firm 1 hires a consultant to predict the strategies of the other two firms:
A. if the consultant predicts that the other two firms will produce output at the competitive quantity, what quantity will they each choose taking the output of firm 1 as given?
B. write the profit function for firm 1, and substitute in q_{3} = q^{3} and q_{2} = q^{2} to find the profit function for firm 1 in this case.
(iii) A. if the consultant predicts that the other two firms will compete to a Cournot-Nash Equilibrium, what quantity will they each choose taking the output of firm 1 as given?
B. write the profit function for firm 1, and substitute in q_{3} = q^{3} and q_{2} = q^{2} to find the profit function for firm 1 in this case.