Question

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 firms" output decisions do not affect their costs, 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

Based on the given information about a perfectly competitive market with 100 identical firms, the short-run equilibrium price-quantity combination is (4.71, 7,100). This is determined by first calculating the firm's marginal cost and equating it to the market price, leading to the firm's short-run supply curve. Afterward, the short-run industry supply curve is found by multiplying the firm's supply curve by the number of firms in the industry. Finally, the equilibrium is found by equating the industry supply curve to the market demand curve.

Step-by-step solution

Calculate the marginal cost of the firm

To find the marginal cost of the firm, we need to differentiate the total cost function with respect to the quantity \((q)\). The total cost function is given as follows: \[C(q)=\frac{1}{300} q^{3}+0.2 q^{2}+4 q+10\] Now we find the derivative of this function with respect to \(q\): \[\frac{dC}{dq} = MC(q) = \frac{1}{100} q^{2} + 0.4 q + 4\]

Calculate the firm's short-run supply curve

In a perfectly competitive market, the firms are price takers, meaning that they equate the marginal cost to the market price \((P)\). So, we will set the marginal cost function equal to the market price to find the firm's short-run supply curve: \[P = \frac{1}{100} q^{2} + 0.4 q + 4\] Now, we can rearrange this equation to make \(q\) a function of \(P\): \[q(P) = 100(P - 4) - 0.4q\] So, the firm's short-run supply curve is given by: \[q(P) = 100(P - 4)\]

Calculate the short-run industry supply curve

To find the short-run industry supply curve, we need to multiply the firm's supply curve by the number of firms in the industry, which in this case is 100: \[Q_s(P) = 100 \cdot q(P) = 100 \cdot 100(P - 4)\] Therefore, the short-run industry supply curve is: \[Q_s(P) = 10,000(P - 4)\]

Find the short-run equilibrium price-quantity combination

To find the equilibrium price and quantity, we need to equate the industry supply curve to the market demand curve, which is given as \(Q=(-200P+8,000)\). When we set these two functions equal, we have: \[10,000(P - 4) = -200P + 8,000\] Solving for the market price \(P\): \[10,000P - 40,000 = -200P + 8,000\] \[10,200P = 48,000\] \[P = \frac{48,000}{10,200}\] \[P = 4.71\] Now that we have the equilibrium price, we can find the equilibrium quantity by plugging the equilibrium price back into either the industry supply curve or the market demand curve. We will use the industry supply curve: \[Q_s(4.71) = 10,000(4.71 - 4)\] \[Q_s = 10,000(0.71)\] \[Q_s = 7,100\] So, the short-run equilibrium price-quantity combination is \((P, Q) = (4.71, 7,100)\).