Question

This homework assignment compares a competitive market with a monopolistic market. The market demand curve is P 122-¼Q. For each firm, marginal oosts are 20 + qi50 and fixed costs are 1 00. We assume first that the market is competitive. Module 8explains the competitive pricing procedure. Wederive the long-run price from the firms' cost curve competitive firms price at long-run minimum average costs. Question: Why is this relation true? Answer: Decreasing marginal utility implies an upward sloping marginal cost curve in the short run; the absence of barriers toentryimplies that new entrants take market share awayfrom anyfirm that prices above the minimum. A. what is the total cost curve for each firm? (Integrate the marginal oost curve to get the total variable cost curve. Variable costs are zero at an output of zero, so the constant of integration is zero. Add the fixed costs to get total costs.) what is the average cost curve for each firm? (Divide by quantity) What quantity minimizes the average cost? (Set the partial derivative with respect to quantity equal to zero B. C. D. What is the price of the good? (The average total cost at that quantity which equals the marginal cost E. F. at that quantity.) What is the industry quantity at this price? (Use the market demand curve.) How many firms compete in this industry? (Divide the industry quantity by each firm's quantity.) Supposethe firms merge into a profit-maximizing monopolywith a marginal cost of 20+ q/200. The marginal costs are the same as before, with a slight adjustment for discrete numbers. If four firms produced one unit each before the merger, the marginal costfor each unit was 20+1/50. Now one firm produces four units, and the marginal cost is 20 + 4/200 = 20 + 1/50·The merger creates no synergies or cost reductions; the only effect is market power. G. Whatis the total revenue curve? Totalrevenue is Px Q. Express total revenue as a function of Q alone, by writing Pas a function of Q; use the demand curve for this.) H. What is the marginal revenue curve? (Marginal revenue is the partial derivative of total revenue with respect to quantity.) I. What is the monopoly quantity? (Equate marginal revenue and marginal cost.) J. Whatis the monopoly price? (Use the demand curve, not the marginal cost curve or the marginal revenue curve. The monopolist asks how much consumers will pay for this output, not how much the output costs to produce.) How might one decide if the monopolization is socially beneficial? K. (Did quantity increase or decrease?)

Answer 1

A.

Marginal cost of a firm is defined as the cost of producing one additional unit of output. Mathematically, it is given as -

$MC = \frac{\mathrm{d} TC}{\mathrm{d} q}$

where MC = marginal cost = 20 + q/50 and

TC = total cost = fixed cost + variable cost = 100 + VC

From the definition of marginal cost, we can find total cost by integrating on both sides -

$TC = \int dTC = \int MCdq = \int (20 + \frac{q}{50})dq = 20\int dq + \frac{1}{50}\int qdq$

$= 20q + \frac{q^2}{100} + c$

where c is the cost of integration.

We know that at q = 0, variable cost VC = 0,

thus TC = fixed cost = 100.

Substituting this in equation of TC, we get:

100 = 0 + 0 + c => c = 100

$\therefore TC = 20q + \frac{q^2}{100} + 100$

B.

Average cost of the firm is given by -

$AC = \frac{TC}{q} = 20 + \frac{q}{100} + \frac{100}{q}$

C.

To find the quantity, that minimizes the AC, we have to use first and second order conditions i.e.

$\frac{\partial AC}{\partial q} = 0 \Rightarrow \frac{1}{100} - \frac{100}{q^2} = 0 \Rightarrow \frac{100}{q^2} = \frac{1}{100} \Rightarrow q^2 = 100 * 100$

$\Rightarrow q^* = 100$

To verify that this is indeed the average cost minimizing quantity, let us verify the second order conditions -

$\frac{\partial^2 AC}{\partial q^2} = \frac{200}{q^3} > 0$ at q = q* = 100

Hence q* = 100 is a minima of AC.

D.

By the competitive pricing procedure,

P = long run minimum average cost

= AC at q = 100

Substituting q = 100 in equation of either AC or MC, we get:

P = 20+ 100/100 + 100/100 = 22

E.

Industry quantity at this price is given by the market demand curve -

P = 122 - Q/4 => 22 = 122 - Q/4

=> Q/4 = 122 - 22 = 100 => Q = 400

F.

Quantity produced by each firm is calculated by profit maximization of each firm.

This is solved using -

P = MC => 22 = 20 + q/50

=> q/50 = 2 => q = 100

No. of firms in the industry = Total market demand / Quantity of each firm

= 400 / 100 = 4