Suppose there are 100 identical firms in a perfectly competitive industry. Each firm has a short-run total cost curve of the form \\[ c-3^{\wedge} 09^{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 \(Q=-200 \mathrm{P}+8,000\). What will be the short-run equilibrium price-quantity combination?

First, we need to find the marginal cost of the firm. Marginal cost (MC) is the derivative of the total cost (TC) function with respect to the quantity (q). The total cost function is given as \(c-3^{\wedge} 09^{3}+0.2 q^{2}+4 q+10\). Let's find the derivative: \(MC(q) = \frac{dTC(q)}{dq} = 0.4q + 4\)

In a perfectly competitive market, each firm will supply goods until the marginal cost (MC) is equal to the market price (P). Therefore, we can find the short-run supply curve for one firm by setting MC equal to P: \(0.4q + 4 = P\) We can solve for q to find the supply curve: \(q(P) = \frac{P - 4}{0.4}\)

There are 100 identical firms in the industry, so the total quantity supplied in the industry (Q) is equal to the sum of the quantities supplied by each firm (q). The short-run industry supply curve can be found by multiplying the individual supply curve by the number of firms: \(Q(P) = 100 * \frac{P - 4}{0.4}\)

To find the short-run equilibrium price and quantity, we need to equate the industry supply curve with the market demand curve, which is given by \(Q=-200P+8,000\). Set the industry supply and demand curves equal to each other: \(100 * \frac{P - 4}{0.4} = -200P + 8,000\) Let's solve for the equilibrium price, P: \(100 * \frac{P - 4}{0.4} + 200P = 8,000\) \(\frac{100}{0.4}(P - 4) + 200P = 8000\) \(250(P - 4) + 200P = 8000\) \(250P - 1000 + 200P = 8000\) \(450P = 9000\) \(P = \frac{9000}{450} = 20\) Now, let's solve for the equilibrium quantity, Q: \(Q = 100 * \frac{20 - 4}{0.4}\) \(Q = 100 * (\frac{16}{0.4})\) \(Q = 100 * 40\) \(Q = 4000\) The short-run equilibrium price-quantity combination is (P, Q) = (20, 4000).