1 / 12

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.

amara
Download Presentation

Best-Response Curves and Continuous Decision Choices

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related