Question

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.

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

e)

The firm's objective is to minimize cost subject to the technology:

At equilibrium, the MRTS is equal to the ratio of the factor prices,

The production function becomes:

The cost function of the firm is:

The average cost is given by:

The marginal cost is given by:

Plotting the three curves:

The marginal cost is equal to average cost at (0,0), the total cost is equal to average cost at (1,0.22), and the total cost is equal to marginal cost at (2,0.89).

f) The graph indicates increasing returns to scale. When l is increased from 0 to 1, the total cost increases in a lower proportion. This is true for all points on the graph. This can also be shown by observing that the slope of the marginal cost curve is less than one, that is, a unit increase in factors of production increases costs by less than one unit.

g) Since the firm is in a perfectly competitive market, the profit is given by:

Plotting the firm's supply;

h) Given price = 4, quantity is:

For the firm to supply 18 units: