Question

Let \(c_{i}\) be the constant marginal and average cost for firm \(i\) (so that firms may have different marginal costs). Suppose demand is given by \(P=1-Q\) a. Calculate the Nash equilibrium quantities assuming there are two firms in a Cournot market. Also compute market output, market price, firm profits, industry profits, consumer surplus, and total welfare. b. Represent the Nash equilibrium on a best-response function diagram. Show how a reduction in firm 1 's cost would change the equilibrium. Draw a representative isoprofit for firm 1

Short answer

Compute market output, market price, firm profits, industry profits, consumer surplus, and total welfare. Answer: Using the solution steps, first compute the reaction functions for both firms: \(R_1(q_2) = \frac{1 - 0.25 - 2q_2}{2} = \frac{0.75 - 2q_2}{2}\) \(R_2(q_1) = \frac{1 - 0.3 - 2q_1}{2} = \frac{0.7 - 2q_1}{2}\) To find the equilibrium quantities, substitute the reaction functions into each other and solve for \(q_1^*\) and \(q_2^*\): \(q_1^* = \frac{0.75 - 2(\frac{0.7 - 2q_1^*}{2})}{2}\) \(q_2^* = \frac{0.7 - 2(\frac{0.75 - 2q_2^*}{2})}{2}\) Solving the system of equations, we get \(q_1^* = 0.125\) and \(q_2^* = 0.1\). The market output (\(Q^*\)) is 0.225. The market price (\(P^*\)) is 0.775. Firm profits are: \(\pi_1^* = (0.775 - 0.25)(0.125) = 0.065625\) \(\pi_2^* = (0.775 - 0.3)(0.1) = 0.0475\) Industry profits are 0.113125. Consumer surplus is 0.086875. Total welfare is 0.2.

Step-by-step solution

Define the problem and setup the reaction functions

For Cournot competition, we first need to set up the reaction functions for both firms. The inverse demand function is given by \(P = 1 - Q\), where \(Q = q_1 + q_2\). The profit function for each firm is given by: \(\pi_i = (P - c_i) q_i = (1 - Q - c_i)q_i = (1 - (q_1 + q_2) - c_i)q_i , i = 1, 2\) To find the reaction functions, we will take the first-order derivative of the profit function with respect to each firm's output and set it to zero.

Calculate the reaction functions

Find the first-order conditions by taking the partial derivatives of \(\pi_i\) with respect to each firm's output \(q_i\) and set it equal to zero: \(\frac{\partial \pi_1}{\partial q_1} = 1 - q_1 - c_1 - 2q_2 = 0\) \(\frac{\partial \pi_2}{\partial q_2} = 1 - q_2 - c_2 - 2q_1 = 0\) Now, solve these equations to find each firm's reaction function, \(R_1(q_2)\) and \(R_2(q_1)\): \(R_1(q_2) = \frac{1 - c_1 - 2q_2}{2}\) \(R_2(q_1) = \frac{1 - c_2 - 2q_1}{2}\)

Find Cournot equilibrium quantities

To find the Nash equilibrium output levels, we will solve the system of reaction functions: \(q_1^* = R_1(q_2^*) = \frac{1 - c_1 - 2q_2^*}{2}\) \(q_2^* = R_2(q_1^*) = \frac{1 - c_2 - 2q_1^*}{2}\) Solving this system of equations will give us the equilibrium quantities \(q_1^*\) and \(q_2^*\).

Calculate the market output, market price, and profits

Market output (\(Q^*\)) is the sum of \(q_1^*\) and \(q_2^*\): \(Q^* = q_1^* + q_2^*\) Market price (\(P^*\)) can be found using the inverse demand function: \(P^* = 1 - Q^*\) Firm profits (\(\pi_i^*\)) can be calculated using the profit function: \(\pi_i^* = (P^* - c_i)q_i^*\) Industry profits are the sum of the two firms' profits: \(P_{industry}^* = \pi_1^* + \pi_2^*\)

Compute consumer surplus and total welfare

Consumer surplus (CS) is given by the area between the demand curve and the market price: \(CS = \frac{1}{2} (1 - P^*)Q^*\) Total welfare (TW) is the sum of consumer surplus and industry profits: \(TW = CS + P_{industry}^*\)

Represent the Nash equilibrium on a best-response function diagram

On a diagram with \(q_1\) on the x-axis and \(q_2\) on the y-axis, plot both reaction functions \(R_1(q_2)\) and \(R_2(q_1)\). The Nash equilibrium corresponds to the point where the reaction functions intersect. To see how a reduction in firm 1's cost would change the equilibrium, reduce the value of \(c_1\) in the reaction function for firm 1 and recalibrate its reaction function, finding the new intersection.

Draw a representative isoprofit curve for firm 1

An isoprofit curve for firm 1 represents all the output combinations \((q_1, q_2)\) that generate the same level of profit for firm 1. Using the profit function for firm 1, set up an equation for the isoprofit curve: \(\pi_1 = (1 - (q_1 + q_2) - c_1)q_i , \text{where \)q_1\( represents the output level on the x-axis}\) Plot the isoprofit curve on the diagram with the reaction functions. The isoprofit curve should be tangent to the reaction function of firm 1 at the Nash equilibrium point. With the solution steps provided, you can find the Nash equilibrium quantities, market output, market price, firm profits, industry profits, consumer surplus, and total welfare and represent the Nash equilibrium and isoprofit curve on a diagram.