Best-Response Curves and Continuous Decision Choices

Best-Response Curves Best-response curves are used to analyze simultaneous decisions when the decision choices are continuous rather than discrete. A continuous choice variable can take on any value between two points. A discrete choice variable can only take on a specific value. A firm’s best-response curve indicates the best strategy the firm can choose, given the strategy the firm expects its rival to use. The best strategy maximizes profit.

Best-Response Curves Application Demand functions Q A = 4,000 – 25P A + 12P B Q B = 3,000 – 20P B + 10P A Q A & Q B are the number of round- trip tickets sold by each airline P A & P B are the prices charged The managers of two competing airlines, Arrow Airlines and Bravo Airways, must set round- trip ticket prices from Lincoln, Nebraska to Colorado Springs, Colorado for a four day Christmas holiday. The airlines’ products are differentiated, because Arrow has newer planes than Bravo. The demand functions for both airlines are known by both airlines.

Best-Response Curves Application The managers must choose prices months in advance and will not be able to change their prices once they are set. At the time of the pricing decision, all costs are variable. Since Arrow has newer, more fuel- efficient planes than Bravo, Arrow has lower costs. LAC A = LMC A = $160 LAC B = LMC B = $180 In order to choose the profit maximizing price, each airline needs to know its best price for any price its rival might charge. That is, both managers need to know their own best-response curves as well as the best-response curves of their competitor. To construct these best- response curves, the managers must know, or estimate, their own and their competitors demand and cost conditions.

Construction of Best-Response Curves To construct its best-response curve, Arrow must anticipate Bravo’s likely price choice. Suppose Arrow thinks Bravo will set a price of $100. The demand facing Arrow at that price is found by plugging Bravo’s price into Arrow’s demand Q A = 4,000 – 25P A + 12P B Q A = 4,000 – 25P A + 12*100 Q A = 5,200 – 25P A By solving for P A, Arrow can obtain its inverse demand and marginal revenue functions P A = (5,200 – Q A )/25 = 208 – 0.04Q A MR A = 208 – 0.08Q A Now Arrow sets MR A = LMC A to find its profit maximizing output when Bravo’s price = $100. LMC A = 160, so MR A = 208 – 0.08Q A = 160 = LMC A 208 – 160 = 0.08 Q A 48/0.08 = Q A Q A * = 600

Construction of Best-Response Curves The profit-maximizing price for Arrow when Bravo charges $100 per ticket is found by substituting Q A * = 600 into Arrow’s inverse demand function. P A = 208 – 0.04Q A P A = 208 – 0.04*600 P A * = $184 Now that the profit-maximizing price and output are determined for Arrow when Bravo is charging $100 per ticket, only one other best price response is needed to finish constructing Arrow’s best-response curve. Only one other point is needed is because the best-response curve is a straight line when the demand and marginal costs curves are both linear. By drawing a line through the two points, all other best prices are determined and Arrow’s best-response curve is complete.

Construction of Best-Response Curves Suppose Arrow chooses $200 as Bravo’s price. Plug this price into Arrow’s demand equation Q A = 4,000 – 25P A + 12*200 Q A = 6,400 – 25P A Solve for inverse demand and marginal revenue functions P A = 256 – 0.04Q A MR A = 256 – 0.08Q A Set MR A equal to LMC A and solve for Q A 256 – 0.08Q A = 160 Q A * = 1,200 Plug Q A * into Arrow’s inverse demand function to solve for Arrow’s profit- maximizing price P A = 256 – 0.04*1,200 P A = 208 Arrow’s Best Response Curve

Best-Response Curves & Nash Equilibrium If we went through the same exercise for Bravo, we could find its best response curve This is shown in the graph Managers at both airlines are likely to set prices at the intersection of the best-response curves This occurs at point N, where P A = $212 and P B = $218 At N, each firm’s price is a best response to the price set by the other At point N, neither airline can increase its individual profit by changing its own price alone Point N is a Nash Equilibrium Best-Response Curves for Both Firms

Best-Response Curves & Nash Equilibrium The number of tickets each airline sells when setting prices in Nash equilibrium can be found by substituting those prices back into the original demand functions. Q A = 4,000 – 25* *218 Q A = 1,316 Q B = 3,000 – 20* *212 Q B = 760 At Nash equilibrium, profits for each firm are (P – LMC)*Q Arrow = ( )*1,316 Arrow Profits = $68,432 Bravo = ( )*760 Bravo Profits = $28,880 Best-Response Curves for Both Firms

Best-Response Curves & Nash Equilibrium At point N, neither firm is making as much profit as is possible if both airlines cooperated and set higher prices. Point N is similar to the Nash equilibrium in a Prisoners’ Dilemma Game. Point C, for example, gives both firms higher profits than at point N Getting to point C requires cooperation Cooperation is risky and unreliable due to the incentives of each airline to cheat. If one firm thinks the other will price at C, then that firm prefers a lower price Point C is unstable due to these incentives Best-Response Curves for Both Firms

Finding Best Response Curves Directly Wouldn’t it be easier to derive Best Response Curves mathematically, then solve for the Nash Equilibrium? Yes, but………this requires some calculus. Form the profit functions for both firms π A = (P A - LMC A )Q A = 8000P A - 25P A P B P A – 640, P B π B = (P B - LMC B )Q B = 6600P B - 20P B P B P A – 540, P A Take the derivative of profit with respect to price and set = 0 ∂π A /∂P A = 8000 – 50P A + 12P B = 0 ∂π B /∂P B = 6600 – 40P B + 10P B = 0 These are the first order conditions for a maximum

Nash Equilibrium Prices Solve the f.o.c. for each price P A = P B Arrow’s Best Response Function P B = P A Bravo’s Best Response Function Substitute each BRF into the other to find the best prices P A = ( P A ) = P A P A = 199.6/0.94 = $ P B = ( P B ) = P B P B = 205/0.94 = $ These are the Nash Equilibrium prices for each airline