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Best-Response Curves and Continuous Decision Choices

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.

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Best-Response Curves and Continuous Decision Choices

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  1. Best-Response Curves and Continuous Decision Choices

  2. 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.

  3. Best-Response Curves Application • Demand functions • QA = 4,000 – 25PA + 12PB • QB = 3,000 – 20PB + 10PA QA & QB are the number of round-trip tickets sold by each airline PA & PB are the prices charged • The managers of two competing airlines, Arrow Airlines and Bravo Airways, must set round-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.

  4. 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. • LACA = LMCA = $160 • LACB = LMCB = $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.

  5. 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 • QA = 4,000 – 25PA + 12PB • QA = 4,000 – 25PA + 12*100 • QA = 5,200 – 25PA • By solving for PA, Arrow can obtain its inverse demand and marginal revenue functions • PA = (5,200 – QA)/25 = 208 – 0.04QA • MRA = 208 – 0.08QA • Now Arrow sets MRA = LMCA to find its profit maximizing output when Bravo’s price = $100. • LMCA = 160, so • MRA = 208 – 0.08QA = 160 = LMCA • 208 – 160 = 0.08 QA • 48/0.08 = QA • QA* = 600

  6. Construction of Best-Response Curves • The profit-maximizing price for Arrow when Bravo charges $100 per ticket is found by substituting QA* = 600 into Arrow’s inverse demand function. • PA = 208 – 0.04QA • PA = 208 – 0.04*600 • PA* = $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.

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

  8. Best-Response Curves & Nash Equilibrium Best-Response Curves for Both Firms • 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 PA = $212 and PB = $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

  9. Best-Response Curves & Nash Equilibrium Best-Response Curves for Both Firms • 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. • QA = 4,000 – 25*212 + 12*218 • QA = 1,316 • QB = 3,000 – 20*218 + 10*212 • QB = 760 • At Nash equilibrium, profits for each firm are (P – LMC)*Q • Arrow = (212-160)*1,316 • Arrow Profits = $68,432 • Bravo = (218-180)*760 • Bravo Profits = $28,880

  10. Best-Response Curves & Nash Equilibrium Best-Response Curves for Both Firms • 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

  11. 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 • πA = (PA - LMCA)QA = 8000PA - 25PA2 + 12PBPA – 640,000 - 1920PB • πB = (PB - LMCB)QB = 6600PB - 20PB2 + 10PBPA – 540,000 - 1800PA • Take the derivative of profit with respect to price and set = 0 • ∂πA/∂PA = 8000 – 50PA + 12PB = 0 • ∂πB/∂PB = 6600 – 40PB + 10PB = 0 • These are the first order conditions for a maximum

  12. Nash Equilibrium Prices • Solve the f.o.c. for each price • PA = 160 + 0.24PB Arrow’s Best Response Function • PB = 165 + 0.25PA Bravo’s Best Response Function • Substitute each BRF into the other to find the best prices • PA = 160 + 0.24(165 + 0.25PA) = 199.6 + 0.06PA • PA = 199.6/0.94 = $212.34 • PB = 165 + 0.25(160 + 0.24PB) = 205 + 0.06PB • PB = 205/0.94 = $218.085 • These are the Nash Equilibrium prices for each airline

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