Question

Consider two companies producing identical products that are perfect substitutes, and these products are produced with a constant marginal cost c. Each company i needs to choose the selling price pi ? [0,1]. Assume the demand curve is linear so that the total demand for this product at the price p is Q(p) = 1 - p, where p = min (pi,p2)- In other words, the demand for this project will be fully directed to the company with the smaller price. If both prices are equal, then the demand of this product will be equally shared between the two companies. Let qi be the demand for the company then the demand for company 1 can be stated as the following picture. q1(p1, p2) = { 1 - pi if p1 < p2, 0 if p1 > p2, (1 - p1)/2 if p1 = p2. Assume both companies are rational and they both want to maximize their project. For example, the utility for the company i is: ui(p1, p2) = piqi - cqi i ? {1, 2}. (a) Suppose that company 1 is alone in the market, what is the optimal price pi, and what is the quantity purchased qi? (b) What is the best responses of company 1 to a price p2 of company 2? (c) What is the Nash equilibrium?

          Consider two companies producing identical products that are perfect substitutes, and these products are produced with a constant marginal cost c. Each company i needs to choose the selling price pi ? [0,1]. Assume the demand curve is linear so that the total demand for this product at the price p is Q(p) = 1 - p, where p = min (pi,p2)- In other words, the demand for this project will be fully directed to the company with the smaller price. If both prices are equal, then the demand of this product will be equally shared between the two companies. Let qi be the demand for the company then the demand for company 1 can be stated as the following picture. q1(p1, p2) = { 1 - pi if p1 < p2, 0 if p1 > p2, (1 - p1)/2 if p1 = p2. Assume both companies are rational and they both want to maximize their project. For example, the utility for the company i is: ui(p1, p2) = piqi - cqi i ? {1, 2}. (a) Suppose that company 1 is alone in the market, what is the optimal price pi, and what is the quantity purchased qi? (b) What is the best responses of company 1 to a price p2 of company 2? (c) What is the Nash equilibrium?

Consider two companies producing identical products that are perfect substitutes, and these products are produced with a constant marginal cost c. Each company i needs to choose the selling price pi ? [0,1]. Assume the demand curve is linear so that the total demand for this product at the price p is Q(p) = 1 - p, where p = min (pi,p2)- In other words, the demand for this project will be fully directed to the company with the smaller price. If both prices are equal, then the demand of this product will be equally shared between the two companies. Let qi be the demand for the company then the demand for company 1 can be stated as the following picture. q1(p1, p2) = 1 - pi if p1 < p2, 0 if p1 > p2, (1 - p1)/2 if p1 = p2. Assume both companies are rational and they both want to maximize their project. For example, the utility for the company i is: ui(p1, p2) = piqi - cqi i ? 1, 2. (a) Suppose that company 1 is alone in the market, what is the optimal price pi, and what is the quantity purchased qi? (b) What is the best responses of company 1 to a price p2 of company 2? (c) What is the Nash equilibrium?

Added by Thomas F.

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