Man 503 Homework VIII Fall 2009
1) A firm faces the following average revenue (demand) curve:
where Q is weekly production and P is price, measured in cents per unit. The firm’s cost function is given by C = 60Q + 25,000. Assume that the firm maximizes profits.
a. What is the level of production, price, and total profit per week?
The profit-maximizing output is found by setting marginal revenue equal to marginal cost. Given a linear demand curve in inverse form, P = 120 - 0.02Q, we know that the marginal revenue curve will have twice the slope of the demand curve. Thus, the marginal revenue curve for the firm is MR = 120 - 0.04Q. Marginal cost is simply the slope of the total cost curve. The slope of TC = 60Q + 25,000 is 60, so MC equals 60. Setting MR = MC to determine the profit-maximizing quantity:
120 - 0.04Q = 60, or
Q = 1,500.
Substituting the profit-maximizing quantity into the inverse demand function to determine the price:
P = 120 - (0.02)(1,500) = 90 cents.
Profit equals total revenue minus total cost:
p = (90)(1,500) - (25,000 + (60)(1,500)), or
p = $200 per week.
b. If the government decides to levy a tax of 14 cents per unit on this product, what will be the new level of production, price, and profit?
Suppose initially that the consumers must pay the tax to the government. Since the total price (including the tax) consumers would be willing to pay remains unchanged, we know that the demand function is
P* + T = 120 - 0.02Q, or
P* = 120 - 0.02Q - T,
where P* is the price received by the suppliers. Because the tax increases the price of each unit, total revenue for the monopolist decreases by TQ, and marginal revenue, the revenue on each additional unit, decreases by T:
MR = 120 - 0.04Q - T
where T = 14 cents. To determine the profit-maximizing level of output with the tax, equate marginal revenue with marginal cost:
120 - 0.04Q - 14 = 60, or
Q = 1,150 units.
Substituting Q into the demand function to determine price:
P* = 120 - (0.02)(1,150) - 14 = 83 cents.
Profit is total revenue minus total cost:
$14.50 per week.
Note: The price facing the consumer after the imposition of the tax is 97 cents. The monopolist receives 83 cents. Therefore, the consumer and the monopolist each pay 7 cents of the tax.
If the monopolist had to pay the tax instead of the consumer, we would arrive at the same result. The monopolist’s cost function would then be
TC = 60Q + 25,000 + TQ = (60 + T)Q + 25,000.
The slope of the cost function is (60 + T), so MC = 60 + T. We set this MC to the marginal revenue function from part (a):
120 - 0.04Q = 60 + 14, or
Q = 1,150.
Thus, it does not matter who sends the tax payment to the government. The burden of the tax is reflected in the price of the good.
2) Suppose that an industry is characterized as follows:
a. If there is only one firm in the industry, find the monopoly price, quantity, and level of profit.
If there is only one firm in the industry, then the firm will act like a monopolist and produce at the point where marginal revenue is equal to marginal cost:
For a quantity of 11.25, the firm will charge a price P=90-2*11.25=$67.50. The level of profit is $67.50*11.25-100-2*11.25*11.25=$406.25.
b. Find the price, quantity, and level of profit if the industry is competitive.
If the industry is competitive then price is equal to marginal cost, so that 90-2Q=4Q, or Q=15. At a quantity of 15 price is equal to 60. The level of profit is therefore 60*15-100-2*15*15=$350.
c. Graphically illustrate the demand curve, marginal revenue curve, marginal cost curve, and average cost curve. Identify the difference between the profit level of the monopoly and the profit level of the competitive industry in two different ways. Verify that the two are numerically equivalent.
The graph below illustrates the demand curve, marginal revenue curve, and marginal cost curve. The average cost curve hits the marginal cost curve at a quantity of approximately 7, and is increasing thereafter (this is not shown in the graph below). The profit that is lost by having the firm produce at the competitive solution as compared to the monopoly solution is given by the difference of the two profit levels as calculated in parts a and b above, or $406.25-$350=$56.25. On the graph below, this difference is represented by the lost profit area, which is the triangle below the marginal cost curve and above the marginal revenue curve, between the quantities of 11.25 and 15. This is lost profit because for each of these 3.75 units extra revenue earned was less than extra cost incurred. This area can be calculated as 0.5*(60-45)*3.75+0.5*(45-30)*3.75=$56.25. The second method of graphically illustrating the difference in the two profit levels is to draw in the average cost curve and identify the two profit boxes. The profit box is the difference between the total revenue box (price times quantity) and the total cost box (average cost times quantity). The monopolist will gain two areas and lose one area as compared to the competitive firm, and these areas will sum to $56.25.
3) A monopolist faces the following demand curve
Q = 16 - P
Where Q is the quantity demanded and P is price. Its average variable cost is
AVC = 3Q,
And its fixed cost is $5. What is the loss of efficiency (dead weight loss) due to the monopoly?
P = 16- Q
TR = 16Q – Q2
MR = 16 – 2Q
TC = 5 + 3Q2
MC = 6Q
Monopolist output ŕ MC = MR ŕ Q = 2
Perfectly competitive output ŕ MC = P ŕ Q = 16/7
DWL = (14-12)*(16/7 – 2)*0.5 = 2.29