Microeconomic Theory Problem Set: Answer Key
Economics 370 Microeconomic Theory Problem Set 6 Answer Key
1) Describe the effects on output and welfare if the government regulates a monopoly so that it may not charge a price above p, which lies between the unregulated monopoly price and the optimally regulate price (determined by the intersection of the firm’s marginal cost and the market demand curve).
As usual, the monopoly determines its optimal output on the basis of MR = MC. Here, however, it cannot charge a price in excess of p*. So, for any output less than Q(p*) (where Q(p) is the demand function) its marginal revenue is p*. On the graph below that gives:
pm p* MR
MC
Demand q m q
*
2) The inverse demand curve a monopoly faces is p=10Q-1/2. The firm’s cost curve is c(Q) = 10 + 5Q. Find the profit maximizing price and quantity, and economic profit for the monopoly.
Revenue = pQ = Q(10Q-1/2) = 10Q1/2 MR = 5Q-1/2 MC = 5 Profit maximization implies MR = MC, so 5Q-1/2 = 5, or Q* = 1; p* = 10. Economic Profit = Revenue – Cost = Q × p – c(Q) = 1(10) – (10 + 5Q) Economic Profit = 10 – 15 = -5. So, the monopoly will not produce at all, and will have a profit of zero.
3) The inverse demand curve a monopoly faces is p = 100 – Q. Find the profit maximizing price and quantity, and economic profit if: a) The total cost curve is c(Q) = 10 + 5Q.
p = 100 – Q, R = p × Q = (100 – Q) × Q, so MR = 100 – 2Q. C(Q) = 10 + 5Q, therefore MC = 5. The profit-maximizing rule is MR = MC. 100 – 2Q = 5 ⇒ Q* = 47.5, p* = 100 – Q* = 52.5
So the profit-maximizing quantity is 47.5 units. The firm will charge $52.5 per unit. Economic Profit = Revenue – Cost = Q × p – c(Q) = Q(100 – Q) – (10 + 5Q) Economic Profit = 47.5(52.5) – (10 + 5(47.5)) = $2,246.25
b) The total cost curve is c(Q) = 100 + 5Q. How is this similar/different from that found in part a?
The optimal price and quantity are the same, because the marginal cost doesn’t change. The marginal cost is constant at $5 as before. By setting MR =
1) Describe the effects on output and welfare if the government regulates a monopoly so that it may not charge a price above p, which lies between the unregulated monopoly price and the optimally regulate price (determined by the intersection of the firm’s marginal cost and the market demand curve).
As usual, the monopoly determines its optimal output on the basis of MR = MC. Here, however, it cannot charge a price in excess of p*. So, for any output less than Q(p*) (where Q(p) is the demand function) its marginal revenue is p*. On the graph below that gives:
pm p* MR
MC
Demand q m q
*
2) The inverse demand curve a monopoly faces is p=10Q-1/2. The firm’s cost curve is c(Q) = 10 + 5Q. Find the profit maximizing price and quantity, and economic profit for the monopoly.
Revenue = pQ = Q(10Q-1/2) = 10Q1/2 MR = 5Q-1/2 MC = 5 Profit maximization implies MR = MC, so 5Q-1/2 = 5, or Q* = 1; p* = 10. Economic Profit = Revenue – Cost = Q × p – c(Q) = 1(10) – (10 + 5Q) Economic Profit = 10 – 15 = -5. So, the monopoly will not produce at all, and will have a profit of zero.
3) The inverse demand curve a monopoly faces is p = 100 – Q. Find the profit maximizing price and quantity, and economic profit if: a) The total cost curve is c(Q) = 10 + 5Q.
p = 100 – Q, R = p × Q = (100 – Q) × Q, so MR = 100 – 2Q. C(Q) = 10 + 5Q, therefore MC = 5. The profit-maximizing rule is MR = MC. 100 – 2Q = 5 ⇒ Q* = 47.5, p* = 100 – Q* = 52.5
So the profit-maximizing quantity is 47.5 units. The firm will charge $52.5 per unit. Economic Profit = Revenue – Cost = Q × p – c(Q) = Q(100 – Q) – (10 + 5Q) Economic Profit = 47.5(52.5) – (10 + 5(47.5)) = $2,246.25
b) The total cost curve is c(Q) = 100 + 5Q. How is this similar/different from that found in part a?
The optimal price and quantity are the same, because the marginal cost doesn’t change. The marginal cost is constant at $5 as before. By setting MR =