Question

Assume for simplicity that a monopolist has no costs of production and faces a demand curve given by $Q=150-P$ a. Calculate the profit-maximizing price-quantity combination for this monopolist. Also calculate the monopolist's profit. b. Suppose instead that there are two firms in the market facing the demand and cost conditions just described for their identical products. Firms choose quantities simultaneously as in the Cournot model. Compute the outputs in the Nash equilibrium. Also compute market output, price, and firm profits. c. Suppose the two firms choose prices simultaneously as in the Bertrand model. Compute the prices in the Nash equilibrium. Also compute firm output and profit as well as market output. d. Graph the demand curve and indicate where the market price-quantity combinations from parts (a)-(c) appear on the curve.

Short answer

Summary of Results: a. Monopoly: - Profit-maximizing price: P = 75 - Profit-maximizing quantity: Q = 75 - Monopolist's profit: π = 5625 b. Cournot model: - Nash equilibrium: - Firm 1 output: q1 = 50 - Firm 2 output: q2 = 50 - Market output: Q = 100 - Market price: P = 50 - Firm profits: π1 = π2 = 2500 c. Bertrand model: - Nash equilibrium: - Firm 1 output: q1 = 37.5 - Firm 2 output: q2 = 37.5 - Market output: Q = 75 - Market price: P = 75 - Firm profits: π1 = π2 = 2812.5 d. Graph the demand curve with points A (Monopoly), B (Cournot), and C (Bertrand) marked as described above.

Step-by-step solution

Obtain the revenue function

Since the demand curve is given by Q = 150 - P, we want to express price (P) in terms of quantity (Q): P = 150 - Q. The revenue function is obtained by multiplying price and quantity: R(Q) = P × Q. Thus, R(Q) = (150 - Q) × Q = 150Q - Q^2.

Find the profit function and maximize it

Since there are no costs of production, the profit function will be equal to the revenue function: π(Q) = R(Q) = 150Q - Q^2. To maximize profits, we need to find the value of Q that maximizes π(Q). We do that by finding the derivative of π(Q) and setting it to 0: dπ(Q)/dQ = 150 - 2Q. Thus, 150 - 2Q = 0, so Q = 75.

Calculate price, quantity, and profit

Now that we have the monopoly quantity (Q = 75), we can substitute it back into the demand curve (P = 150 - Q) to find the price: P = 150 - 75 = 75. The monopolist's profit is calculated by substituting the quantities into the profit function: π(75) = 150 × 75 - 75^2 = 5625. b. Cournot model:

Set up reaction functions for both firms

Let's denote firm i's output by q_i and firm j's output by q_j (where i ≠ j). From the demand curve, we have P = 150 - (q_i + q_j). The revenue for firm i is R_i = P × q_i, so R_i = (150 - (q_i + q_j)) × q_i = 150q_i - q_i^2 - q_iq_j. Since there are no production costs, π_i = R_i.

Obtain best responses and find Nash equilibrium

To find firm i's best response, we take the derivative of its profit function with respect to q_i and set it to 0: dπ_i/dq_i = 150 - 2q_i - q_j = 0. This would give us q_i = (150 - q_j) / 2 as firm i's best response. Since both firms are identical, these best response functions can be used to find the Nash equilibrium: q_i = (150 - q_j) / 2 and q_j = (150 - q_i) / 2. By solving this system of equations, we have q_i = q_j = 50. Thus, the output levels in the Cournot Nash equilibrium are both 50.

Calculate market output, price, and firm profits

Since both firms produce 50 units, the total market output Q = q_i + q_j = 50 + 50 = 100. The market price can be obtained from the demand curve: P = 150 - 100 = 50. Firm profits (for both firms) can be found by substituting their quantities in their respective profit functions: π_i = π_j = 150 × 50 - 50^2 - 50 × 50 = 2500. c. Bertrand model:

No incentive to undercut given zero costs

In the Bertrand model, firms choose prices simultaneously. Here, since there are zero costs of production, there is no incentive for either firm to undercut the other. The Nash equilibrium in this case is simply charging the same price and sharing the market demand.

Calculate prices, output, and profits

Since there is no incentive to change prices, the firms will charge the same price. They will share the market demand, leading to each firm producing half the total demand, i.e., q_i = q_j = Q/2. As for monopoly profit maximization, the monopolist quantity equals 75. Now, firms sell half of that, so q_i = q_j = 75/2 = 37.5. The market price remains the same as in the monopoly case, P = 75. Substitute these values into the profit function for each firm: π_i = π_j = 150 × 37.5 - 37.5^2 = 2812.5. d. Graph the demand curve: The demand curve can be graphed with the quantity on the x-axis and the price on the y-axis. Additionally, each price-quantity combination can be marked on the graph as follows: - Monopoly: P = 75, Q = 75 (point A) - Cournot model: P = 50, Q = 100 (point B) - Bertrand model: P = 75, Q = 75 (point C, same as the monopoly case) Please note that you will need to draw the graph by yourself, as I am unable to provide graphical illustrations.