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

In a Cournot market with two firms, given the demand function \(P = 1 - Q\) and constant marginal costs \(c_1\) and \(c_2\) for firm 1 and firm 2 respectively, we first determine the Nash equilibrium quantities for both firms, which are \(q_1^* = \frac{2 - c_1 - c_2}{3}\) and \(q_2^* = \frac{2 - c_1 - c_2}{3}\). From this, we can calculate market output, market price, firm and industry profits, consumer surplus, and total welfare. After plotting the best-response functions for both firms on a diagram, we can observe the effect of a reduction in firm 1's cost on equilibrium quantity. A decrease in firm 1's cost leads to an upward shift in its best-response function, resulting in a higher Nash equilibrium quantity for firm 1 and a lower Nash equilibrium quantity for firm 2. By drawing an isoprofit curve for firm 1, we can also see that in a Cournot model, firm 1's profit increases with both a higher Nash equilibrium quantity for firm 1 and a lower Nash equilibrium quantity for firm 2.

Step-by-step solution

Determine the inverse demand function

Given the demand function is \(P = 1 - Q\), where \(Q\) is the total quantity supplied.

Define firm 1 and firm 2 quantities, cost functions, and reaction functions

Let \(q_1\) be the quantity supplied by firm 1 and \(q_2\) be the quantity supplied by firm 2. Total output in the market is \(Q = q_1 + q_2\). Now, let the constant marginal and average costs for the two firms be denoted as \(c_1\) and \(c_2\). The profit function for firm i can be denoted as \(\pi_i = P \times q_i - c_i \times q_i\). To derive the reaction functions for both firms, we need to find the maximization of their profit functions with respect to their quantity. First, profit maximization for firm 1: \(\pi_1 = (1 - Q)q_1 - c_1q_1\) Now, we derive the first-order condition with respect to \(q_1\): \(\frac{\partial \pi_1}{\partial q_1} = 1 - 2q_1 - q_2 - c_1 = 0\) This leads to the reaction function for firm 1: \(q_1^* = \frac{1 - c_1 - q_2}{2}\) Similarly, we derive the reaction function for firm 2: \(q_2^* = \frac{1 - c_2 - q_1}{2}\)

Solve for Nash Equilibrium quantities

To find the Nash equilibrium, we need to solve the reaction functions simultaneously. Substitute the reaction function for firm 1 into the reaction function of firm 2: \(q_2^* = \frac{1 - c_2 - (\frac{1 - c_1 - q_2}{2})}{2}\) By solving this quadratic equation, we obtain the Nash equilibrium quantities for both firms: \(q_1^* = \frac{2 - c_1 - c_2}{3}\) \(q_2^* = \frac{2 - c_1 - c_2}{3}\)

Calculate market output, market price, and firm and industry profits

Market output: \(Q = q_1 + q_2 = \frac{4 - 2(c_1+c_2)}{3}\) Market price: \(P = 1 - Q = \frac{1 + c_1 + c_2}{3}\) Firm 1 profit: \(\pi_1 = P \times q_1 - c_1 \times q_1 = (\frac{1 + c_1 + c_2}{3})(\frac{2 - c_1 - c_2}{3}) - c_1(\frac{2 - c_1 - c_2}{3})\) Firm 2 profit: \(\pi_2 = P \times q_2 - c_2 \times q_2 = (\frac{1 + c_1 + c_2}{3})(\frac{2 - c_1 - c_2}{3}) - c_2(\frac{2 - c_1 - c_2}{3})\) Industry profit: \(\pi_{industry} = \pi_1 + \pi_2\)

Calculate consumer surplus and total welfare

Consumer surplus: \(CS = \frac{1}{2}(1 - P)(Q)\) Total welfare: \(TW = CS + \pi_{industry}\) #Part B: Nash Equilibrium Representation on Best-Response Diagram#

Plot best-response functions

On a best-response diagram, plot the reaction functions for firm 1 and firm 2 (\(q_1^* = \frac{1 - c_1 - q_2}{2}\) and \(q_2^* = \frac{1 - c_2 - q_1}{2}\)) with \(q_1\) on the x-axis and \(q_2\) on the y-axis. The intersection point of these reaction functions is the Nash equilibrium we computed earlier.

Analyze the effect of a reduction in firm 1's cost on equilibrium

If firm 1's cost \(c_1\) decreases, the best-response function for firm 1 shifts upward, with the slope still at -0.5. The new equilibrium occurs at the new intersection of the best-response functions, resulting in a higher Nash equilibrium quantity for firm 1 (\(q_1^*\)) and a lower Nash equilibrium quantity for firm 2 (\(q_2^*\)).

Draw representative isoprofit for firm 1

On the same best-response diagram, draw an isoprofit curve for firm 1 (indicating the set of points where firm 1 makes the same profit). The curve should be upward-sloping, with the slope greater than 0. This is because, in a Cournot model, firm 1's profit increases with both a higher \(q_1^*\) and a lower \(q_2^*\).