1.   (12 pts) Consider the short-run production function f (L, K ) = 6L2 K .

a.   What is the marginal product of labor?

b.   Is the marginal product of labor diminishing?

c.   What is the average product of labor?

2.    (16 pts) Consider the production function f (L, K ) = LaK  + Ln  where a, ,n > 0 .

a.   Iff(L, K) is homogeneous of degree 2 what’s the most we can say about a,  andY ?

b.   Iff(L, K) exhibits increasing returns to scale what’s the most we can say about a,  andY ?

3.   (24 pts) Consider the short-run production function f (L, K) = min 5L,8K where K is fixed at 10. The wage rate is w = 10 and the rental rate is r = 20.

a.   What’s the maximum output that can be produced?

For parts (b) – (e) assume output is less than your answer to (a).

b.   Give the equation for the short-run total cost function.

c.   Find the short-run marginal cost function.

d.   Find the short-run average cost function.

e.   Give the equation for the short-run total cost function if the production function changes to f (L, K ) = min 5L ,8K .

4.   (16 pts) The isoquant wheref(L, K) = 12 is graphed below. The wage rate of labor is w = 9 and the rental rate of capital is r = 9. You need to show enough work on the graph for us to follow your logic in order to receive full credit.

K

6

5

4

3

2

1

0              1          2           3          4           5          6          7        L

a.   What is the slope of an isocost curve?

b.   Using the graph above, identify the bundle of inputs (L*, K*) that minimizes the cost of producing 12 units of output.

c.   What is the (total) cost of producing 12 units of output?

d.   Suppose someone used our Lagrangean technique to identify potential  solutions to the cost minimization problem described in (b). Would this technique identify any other candidate solutions? If so, label them with asterisks (*). If not, briefly explain why not.

Note: These graphs are here just in case you need spares for question #4. You       don’t need to use them if you clearly answered everything asked in question #4 on the previous page.

K

6

5

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2

1

K

6

5

4

3

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1

 0

1          2           3          4           5          6          7        L

1          2           3          4           5          6          7        L

5.   (32 pts) Consider the production function f (x, y, z ) = 3xy z  where 0 <  <  . The price of x is px = 1, the price ofy is py = 8 and the price of z is pz = 4. Parts     (a) – (e) refer to the short run where the level of z is fixed at z = 1. In the short run both x and y can vary. Part (f) refers to the long run where all three inputs can       vary.

a.   Write out the short-run production functionf(x, y, 1) and simplify the    expression. Which of our common types of production functions does it look like?

b.   Find the cost minimizing choice of inputs for producing 96 units of output.