Question

Answers for e, f, g and h please.

A competitive firm uses two inputs, capital (?) and labour (?), to produce one output, (?). The price of capital, ??, is $1 per unit and the price of labor, ?? , is $1 per unit. The firm operates in competitive markets for outputs and inputs, so takes the prices as given. The production function is ?(?, ?) = 3? 0.25? 0.25. The maximum amount of output produced for a given amount of inputs is ? = ?(?, ?) units.

a) Use the method of Lagrange to find the conditional factor demands for cost minimization. [8 marks]

b) Find the firm’s cost function. [2 marks]

c) Would you call this a short-run cost function or a long-run cost function? Explain why. [1 mark]

d) Write the equations for the firm’s average cost function and marginal cost function. [2 marks]

e) Draw the firm’s total cost function, average cost function, and marginal cost function on a diagram. Clearly label the axes, the curves, and any key points on the graph (eg., axis intercepts, curve intersections, and minimums) with the numbers specifying the exact prices and quantities at these points. What are the coordinates of the points where the average cost curve and marginal cost curve intersect with the total cost curve? [6 marks]

f) Does your graph indicate increasing, decreasing, or constant returns to scale? Explain. [1 mark] Hint: Think about the relationship between the total cost function and returns to scale.

g) Show the firm’s long-run supply function on your diagram and write a supply function for the firm. [2 marks]

h) Using your supply function, find the profit maximising quantity if the price of output ? = 4. What price would be needed for the firm to supply 18 units of output? [2 marks]

Answer 1

The production function is , and the cost of production is , as the prices are 1. The cost minimization would be where or or . Putting this in the production function, we have or or , and hence . Hence, the cost function would be . Also, the average cost would be , and the marginal cost would be .

(e) The graph is as below.

The point O is the minimum of all three curves, and is the point where MC cuts the AC.

(f) The graph indicates decreasing returns to scale, as the total cost and hence AC is constantly increases. Under decreasing returns to scale, increasing the inputs by a-times would increase the output by less than a-times. This means that as the quantity is increased, it is costlier to produce the next quantity.

(g) The firm's long run supply function would be the MC above the AC. As in the positive quadrant, MC>AC, the whole MC is the supply function, as P=MC or . The graph is as below.

(h) The graph for both cases is as below.

For P=4, the profit maximizing quantity would be 9 units. The price needed for 18 units of output would be $8.