Question 5 Cost Function Suppose a firm is a pricetaker and

Question 5: Cost Function Suppose a firm is a price-taker and can sell the good it produces at price p. Suppose the firm has cost function: 2 e(q) = a + (a) What are the firm\'s fixed costs, and what are the variable costs? b) How much will the firm produce? (c) Calculate the firm\'s elasticity of supply with respect to price. (d) How does the elasticity of supply vary with a and b? Explain. (e) If a increases, how does this affect the quantity the firm produces? (f) Suppose the firm can recoup its fixed cost by shutting down (i.e. the fixed cost is not sunk). For which values of a will the firm shut down\'? (g) Suppose the fixed cost is sunk. For which values of a will the firm shut down?

Solution

a) Fixed cost is the constant value in cost function that does not have quantity associated with it. It is therefore FC = \'a\'. Variable cost varies with output and so it is (1/2)bq^2

b) Currently the price is p. This should be equated against MC. Here MC = dC/dq = (1/2)b*(2q) or bq. Hence when MC = p, we have bq = p. So the profit maximizing output is q* = p/b.

c) Elasticity of supply is es = slope of supply function x price /quantity. Supply function is the marginal cost function (beyond minimum of AVC). Hence supply fucntion is MC = p or q = p/b. Slope of the supply function is (1/b) while price is p and quantity is q = p/b.

Elasticity = (1/b)*(p)/(p/b) = 1

d) Elasticity is currently 1 so it does not vary with either a or b

e) Increase in a implies increase in fixed cost which does not affect the production level. It reduces the profit though.

f) Shutting down decision does not depend on the value of fixed cost. Whether the fixed cost is high or low, shut down occurs when price falls below the minimum of AVC which is zero in this case. Hence for any value of a, shut down will not take place.

g) When fixed cost is sunk cost, there is no need to shut down because fixed cost cannot be eliminated.