Suppose a firm's production function is y = 10K^1/2*L^1/2. The cost of a unit of labor is $20 and the cost of a unit of capital is $80. They have no other inputs or costs.

(a) The firm is currently producing 100 units of output and has determined that the cost-minimizing quantities of labor and capital are 20 and 5, respectively. Graphically illustrate this using isocost lines and isoquants and write a description of what your graph is showing. Show at least two isocost lines as you explain what the firm has chosen.

(b) The firm now wants to increase production to 140 units. If capital is fixed in the short run, how much labor will the firm require? On a graph showing labor and ouptut (y), show this as movement along the firm's production function (remember capital is fixed, so the production function is only in terms of labor here).

(c) What is the change in the firm's total cost?

(d) On your graph from part a), graphically show the optimal level of capital and labor in the long run

if the firm wants to produce 140 units of output.

(e) Use the tangency condition to solve for numerical values of the optimal quantities of capital and labor for this level of production.

(f) Now use the Lagrangian to solve for the optimal input demand functions for a general level of production y-bar. Confirm that these give you the

same answer you got above for y = 140.

(g) Write down and graph on the same set of axes the total cost function, the average cost function, and the marginal cost function for this firm and assuming the prices given above.