Introductory Microeconomics: Problem Set 4

In this week's problem set you may be asked to find expressions for marginal costs or marginal products. These will be the derivatives of the cost or production function with respect to the relevant variable. Just like marginal utility, with which you are already familiar.

1. Consider the production function Y =  L^1/3K^1/2T^1/6, where L is labour, K is capital, and T is land.
   


1. Show whether this production function has decreasing, constant or increasing returns to scale.
2. Derive an expression for the marginal product of labour. Does this production function have diminishing returns to labour?
3. Find an expression relating the marginal and the average product of labour.
4. How does an increase in the input of capital or land affect the marginal product of labour?
5. Assuming T = 1, draw the isoquants for K and L for this production function. What happens to the MRTS as K increases? Interpret this result.


2. A company producing bicycles has two plants, A and B. The numbers of bikes produced per month in the two plants are Y_A = 40L_A^1/2 and Y_B = 210L_B^1/3, where L_A and L_B are the numbers of workers employed. Suppose that 400 workers are currently employed in plant A, and 1000 in plant B. For each of Plant A and Plant B, calculate:
   
   
1. total output
2. output per worker
3. the marginal product of labour

Should the firm consider moving workers from one plant to another? Explain.


3. Consider a firm with the production function y = K^1/2 L^1/2, where K and L denote the inputs of capital and labour. Let r be the cost of capital and w be the cost of labour.
   
   
   1. Show that the firm's conditional demands for labour and capital are
      L = y (r/w)^1/2 and K = y (w/r)^1/2 .
   2. Suppose that the firm has fixed capital inputs (denoted K ). Derive its short run cost function.
   3. Using an appropriate diagram compare the firm's short and long run cost functions and compare its returns to scale in the short and long run.
   4. Derive and graph the following cost functions of the firm:
      
      1. short run average cost function
      2. short run average variable cost function
      3. short-run marginal cost function
      4. long run average cost function
   5. Derive the firm's short run supply curve.
   6. Suppose that w = 4 and r = 1 and that the firm needs to produce 6 units of output. Calculate and compare the firm's long run and short run labour productivity when in the short run its capital is fixed at K = 4 and provide an explanation for the difference you find.
   
   (This problem appeared on the 2016 prelims exam question paper.)


4. Consider the cost function C(y) = 5y² + 10. Derive expressions for:


1. Total variable costs VC(y)
2. Total fixed costs FC(y)
3. Average variable costs AVC(y)
4. Average fixed costs AFC(y)
5. Average total costs ATC(y)
6. Marginal costs MC(y)


5. What is the link between the average cost function and returns to scale? What is the economic rationale for the conventional assumption that the average cost function is U- shaped? Why are average costs higher in the short-run?

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