Problem 1 - Duopoly ModelsTwo firms produce a homogeneous product. Let p denote the product's price.The output level of firm 1 is denoted by q1, and the output level of firm 2 by92. The aggregate industry output is denoted by Q = 41 + 92. The aggregateindustry demand curve for this product is given byр 3 200— 4QAssume that the marginal and average cost for each firm is MC = AC = 20.(a)Solve for the Cournot (Nash) equilibrium. Find the output level ofeach firm and the market price.(Ь)Solve for the Bertrand (Nash) equilibrium. Find the output level ofeach firm and the market price.(c)Solve for the Stackelberg (sequential-moves) equilibrium where firmone chooses quantity in the first stage and firm 2 chooses quantityin the second stage. Find the output of each firm and the marketprice.

The equilibrium price(P) and quantity(Q) can be estimated by using the following steps: First: Find out the profit(π) functions for firm 1 & 2 Profit (Firm: 1) = Total Revenue - Total Cost =P×Q1 - C×Q1 ...... { P - price and C - marginal cost}=200-4Q1+Q2×Q1 - 20Q1=200Q1-4Q12-4Q1Q2-20Q1=180Q1-4Q12-4Q1Q2Profit (Firm: 2) = 180Q2-4Q1Q2-4Q22 Second: Take the partial derivate of profit(π) function for firm 1 with respect to Q1 and similarly the derivate of profit(π) function for firm 2 with respect to Q2. Equate the equations derived after taking derivate to 0: ∂π1∂Q1=180-8Q1-4Q2∂π1∂Q1=0180-4Q2=8Q1Q1=180-4Q28=22.5-0.5Q2 ...... {equation - 1} Similarly, for firm 2 also: ∂π2∂Q2=180-4Q1-8Q2=0180-4Q18=Q2Q2=22.5-0.5Q1 ..... {equation - 2} Substituting the value of Q2 from the equation – 2 into the value of Q1 or in the equation – 1: Q1=22.5-0.5Q2=22.5-0.522.5-0.5Q1=22.5-11.25+0.25Q10.75Q1=11.25Q1=15 Now, substituting back Q1 = 15 units back into equation – 2 to get the value of Q2. Therefore, we can estimate the Q = Q1 + Q2: Q2=22.5-0.5Q1=22.5-0.5×15=15Q=Q1+Q2⇒15+15=30Q=30 Now the price(P) is equal to: P=200-4Q=200-4×30⇒80 In the Bertrand model, Nash-equilibrium is a point where the price(P) charged by each firm is similar to marginal cost(MC). Therefore, P1 = P2 = MC= 20. Now, putting P = 20, we can estimate the quantity(Q): P=200-4Q20=200-4Q4Q=180Q=90 Each firm will sell 45 units each. In this duopoly model – the sales get evenly divided between two firms at the price(P) that is similar to MC. Follow the steps to get the equilibrium price(P) and quantity(Q) produced under the Stackelberg model: First: Best response function of the firm: 2 is estimated below: Profit function (Firm: 2)π2=200-4Q1+Q2×Q2-20Q2=200Q2-4Q1Q2-4Q22-20Q2=180Q2-4Q1Q2-4Q22∂π2∂Q2=180-4Q1-8Q2=0Q2=180-4Q18=22.5-0.5Q1 ...... {Best response function} Second: Estimate the profit(π) function for the firm - 1 and put the value of Q2 from the best response function into the profit(π) function of the firm – 1: Profit function(Firm: 1)π1=200-4Q1+Q2×Q1-20Q1=200Q1-4Q12-4Q1Q2-20Q1=180Q1-4Q12-4Q1Q2=180Q1-4Q12-4Q1×22.5-0.5Q1=180Q1-4Q12-90Q1+2Q12=90Q1-2Q12 Now take the derivative with respect to Q1, then equate it to 0: ∂π1∂Q1=90-4Q1=04Q1=90Q1=22.5 Putting the value of Q1 = 22.5 units back into the best response function: Q2=22.5-0.5Q1=22.5-11.25⇒11.25Q=Q1+Q2=22.5+11.25=33.75 The price(P) charged is equal to: P=200-4Q=200-4×33.75=65

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Problem 1 - Duopoly Models Two firms produce a homogeneous product. Let p denote the product's price. The output level of firm 1 is denoted by q1, and the output level of firm 2 by 42. The aggregate industry output is denoted by Q = 41 + 92. The aggregate industry demand curve for this product is given by р 3 200- 4Q Assume that the marginal and average cost for each firm is MC = AC = 20. %3D

Problem 1 - Duopoly Models Two firms produce a homogeneous product. Let p denote the product's price. The output level of firm 1 is denoted by q1, and the output level of firm 2 by 42. The aggregate industry output is denoted by Q = 41 + 92. The aggregate industry demand curve for this product is given by р 3 200- 4Q Assume that the marginal and average cost for each firm is MC = AC = 20. %3D

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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