Question

A monopolist faces the following demand curve: Q=144p −2. Its average variable cost is AVC= Q and its fixed cost is 5 , i.e. cost function is c(Q)=Q3/2

+5. (a) What are its profit-maximizing price and quantity? What is the resulting profit? (b) Suppose the government regulates the price to be no greater than $4 per unit. How much will the monopolist produce? What will its profit be? (c) Suppose the government wants to set a ceiling price that induces the monopolist to produce the largest possible output. What price will accomplish this goal? II. (optional) A monopolist engages in uniform pricing, charging the same price from all of its customers. It has a cost function c(q)=cq and faces an inverse demand function p=a−bq, where a,b and c are positive constants, with a>c. Suppose the government imposes a ceiling price, that the price it (the monopolist) charge not exceed
pˉ , where c< pˉ

Answer 1

A monopolist can determine its profit-maximizing price and quantity by using the demand curve and cost functions. By setting marginal revenue equal to marginal cost, the monopolist can find the profit-maximizing quantity. The corresponding price can be determined by drawing a line from the profit-maximizing quantity to the demand curve.

Step-by-step explanation:

A monopolist can determine its profit-maximizing price and quantity by using the demand curve and cost functions. In this case, the demand curve is represented by Q = 144p - 2, where Q is the quantity and p is the price. The average variable cost (AVC) is Q, and the fixed cost (FC) is 5. The profit-maximizing quantity can be found by setting marginal revenue (MR) equal to marginal cost (MC). The monopolist will determine the corresponding price by drawing a line straight up from the profit-maximizing quantity to the demand curve.

(a) To find the profit-maximizing quantity, we set MR = MC. The marginal revenue can be obtained by taking the derivative of the demand curve, which is MR = 144 - 4Q. The marginal cost is Q. By setting MR equal to MC, we have 144 - 4Q = Q. Solving for Q gives Q = 36. Then, we can substitute this value back into the demand curve to find the corresponding price: P = (144 * 36) - 2 = 5182. The profit can be calculated as follows: Total Revenue (TR) = P * Q = 5182 * 36 = 186,712; Total Cost (TC) = FC + AVC * Q = 5 + 36 * 36 = 5 + 1296 = 1301; Profit (π) = TR - TC = 186,712 - 1301 = 185,411.

(b) If the government regulates the price to be no greater than $4 per unit, the monopolist will produce the quantity at which the market demand intersects the price ceiling. The price ceiling is $4, so we substitute this value into the demand curve: 4 = (144 * Q) - 2. Solving for Q gives Q = 26. The profit can be calculated using the same method as in (a). Total Revenue (TR) = P * Q = 4 * 26 = 104; Total Cost (TC) = FC + AVC * Q = 5 + 26 * 26 = 5 + 676 = 681; Profit (π) = TR - TC = 104 - 681 = -577. Therefore, the monopolist would incur a loss of $577 under this price regulation.

(c) To set a ceiling price that induces the monopolist to produce the largest possible output, we need to find the price at which the monopolist's marginal revenue equals zero (MR = 0). The marginal revenue can be obtained by taking the derivative of the demand curve and setting it equal to zero. The derivative of the demand curve is dQ/dp = 144, so MR = 144 - 4Q = 0. Solving for Q gives Q = 36. Substituting this value into the demand curve gives P = (144 * 36) - 2 = 5182. Therefore, a ceiling price of $5182 would induce the monopolist to produce the largest possible output.