Question

1. Suppose in the short-run a firm has fixed cost equal to 10 and variable cost equal to q

2 . Assume that 60% of fixed costs are sunk. (a.) Write down the firm's short-fun cost function. (b.) Derive the firm's marginal cost function. (c.) Derive the firm's short-run average cost function. (d.) For what values of q>0 will the firm have economies of scale? /Hint: Recall that we have economies of scale when average cost is decreasing with quantity, and diseconomies of scale when average cost is increasing./ (c.) Derive the firm's short-run average avoidable cost function. (f.) Find the shutdown price. (g.) Assume the firm is a price-taker and derive its short-run supply function.

Answer 1

In the short run, the firm's fixed cost is $10 and variable cost is q^2. The firm's short-run cost function is Cost = 10 + q^2. The firm's marginal cost function is MC = 2q.

Step-by-step explanation:

(a.) The firm's short-run cost function can be written as:

Cost = Fixed Cost + Variable Cost

Given that fixed cost is $10 and variable cost is q^2, the firm's short-run cost function is:

Cost = 10 + q^2

(b.) To derive the firm's marginal cost function, we need to take the derivative of the cost function with respect to quantity (q). The derivative of q^2 is 2q. Therefore, the firm's marginal cost function is:

MC = 2q

(c.) To derive the firm's short-run average cost function, we need to divide the cost function by quantity (q). Therefore, the firm's short-run average cost function is:

AC = (10 + q^2)/q

(d.) The firm will have economies of scale when the short-run average cost is decreasing with quantity. In other words, when the derivative of the short-run average cost function with respect to quantity (q) is negative. To find the values of q for which this is true, we need to find the critical points of the short-run average cost function. Taking the derivative of the short-run average cost function with respect to quantity (q), we get:

dAC/dq = (2q - 10)/q^2

Solving for q when dAC/dq = 0, we find that q = 5. Therefore, the firm will have economies of scale for q > 5.

(e.) The firm's short-run average avoidable cost function can be derived by subtracting the fixed cost from the short-run average cost function. The firm's short-run average avoidable cost function is:

AVC = (q^2)/q = q

(f.) The shutdown price is the price at which the firm will choose to shut down in the short run. The shutdown price is equal to the minimum of the firm's short-run average variable cost. Therefore, the shutdown price is q = 0.

(g.) As a price-taker, the firm's short-run supply function is determined by its short-run marginal cost. Therefore, the firm's short-run supply function is:

Qs = MC