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Chapter 12: Problem 1

Suppose there are 100 identical firms in a perfectly competitive industry. Each firm has a short-run total cost function of the form \\[ C(q)=\frac{1}{300} q^{3}+0.2 q^{2}+4 q+10 \\] a. Calculate the firm's short-run supply curve with \(q\) as a function of market price (P). b. On the assumption that there are no interaction effects among costs of the firms in the industry, calculate the short-run industry supply curve. c. Suppose market demand is given by \(Q=-200 P+8,000 .\) What will be the short-run equilibrium price-quantity combination?

Short Answer

Expert verified
In a perfectly competitive industry with 100 identical firms each having a total cost function of: \\[ C(q)=\frac{1}{300} q^{3}+0.2 q^{2}+4 q+10 \\] The marginal cost function (MC) of an individual firm is: \\[ MC(q) = \frac{3}{300} q^{2} + 0.4 q+ 4 \\] The individual firm's short-run supply curve is: \\[ q(P) = \frac{300(P - 4) - 120 q}{3} \\] The industry short-run supply curve is: \\[ Q(P) = 10000P - 40000 - 4000q \\] The short-run equilibrium price-quantity combination is P = $ \frac{25}{4} and Q = 6875.

Step by step solution

01

- Find the marginal cost function of an individual firm

The first step is to find the marginal cost function (MC) of an individual firm by taking the derivative of the total cost function (C(q)) with respect to the quantity (q). The given total cost function is: \\[ C(q)=\frac{1}{300} q^{3}+0.2 q^{2}+4 q+10 \\] To find the MC function, take the derivative of the total cost function with respect to q: \\[ MC(q) = \frac{dC(q)}{dq} = \frac{3}{300} q^{2} + 0.4 q+ 4 \\]
02

- Find the individual firm's short-run supply curve

In a perfectly competitive market, each firm's supply curve is determined by equating marginal cost (MC) with the market price (P). Setting MC(q) equal to P, we have: \\[ P = \frac{3}{300} q^{2} + 0.4 q+ 4 \\] Reorder the equation to solve for q: \\[ q(P) = \frac{300(P - 4) - 120 q}{3} \\] Now we have the short-run supply curve for an individual firm: \\[ q(P) = \frac{300(P - 4) - 120 q}{3} \\]
03

- Find the industry short-run supply curve

To find the short-run industry supply curve, multiply the individual firm's supply curve (q(P)) by the number of firms (100), since there are no interaction effects among the firms' costs. \\[ Q(P) = 100 \cdot q(P) = 100 \cdot \frac{300(P - 4) - 120 q}{3} \\] Expanding the equation, we get: \\[ Q(P) = 10000P - 40000 - 4000q \\]
04

- Find the short-run equilibrium price-quantity combination

To find the short-run equilibrium price-quantity combination, we will equate the industry supply curve (Q(P)) with the demand curve (Q) given by Q = -200P + 8000. So, set Q(P) equal to the given demand curve: \\[ 10000P - 40000 - 4000q = -200P + 8000 \\] Solve for P: \\[ P = \frac{7500}{10200} = \frac{25}{4} \\] Now that we have the equilibrium price, we can find the equilibrium quantity Q by substituting the value of P into the demand or supply curve (we will use the demand curve). \\[ Q = -200P + 8000 = -200\left(\frac{25}{4}\right) + 8000 = 6875 \\] So, the short-run equilibrium price-quantity combination is P = $ \frac{25}{4} and Q = 6875.

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Most popular questions from this chapter

A perfectly competitive market has 1,000 firms. In the very short run, each of the firms has a fixed supply of 100 units. The market demand is given by \\[ Q=160,000-10,000 P \\] a. Calculate the equilibrium price in the very short run. b. Calculate the demand schedule facing any one firm in this industry. c. Calculate what the equilibrium price would be if one of the sellers decided to sell nothing or if one seller decided to sell 200 units. d. At the original equilibrium point, calculate the elasticity of the industry demand curve and the elasticity of the demand curve facing any one seller. Suppose now that, in the short run, each firm has a supply curve that shows the quantity the firm will supply \(\left(q_{i}\right)\) as a function of market price. The specific form of this supply curve is given by \\[ q_{i}=-200+50 P \\] Using this short-run supply response, supply revised answers to (a)-(d).

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