Introductory Microeconomics: Problem Set Week 5

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 perfectly competitive industry with 24 producers, each with the same cost function C(y) = 16 + 4y². Market demand is given by Y = 150 - 2p.
   
   
   1. Calculate the supply function for each firm, market supply, and the equilibrium price in the market, p*.
   2. In the long run, firms can leave the industry, or new ones (with the same cost functions give above) can enter. In the long-run equilibrium, what is the price, the output of each firm, and the number of firms?


4. Is the market for short-distance personal transport (taxis) perfectly competitive? Would it be better if it were? (Your answer should be a short, exam-style essay of a paragraph or two. Any research or citations are welcome but not necessary.)


5. Do Exercise 6.6 ("Lazear's Results") in the CORE text.


6. A firm's workers dislike effort. They respond to any wage w > 0 by putting in effort
   
     e = 1 - (R/w)^α,
   
   where R is their reservation wage and α is a parameter.
   
   
   1. Plot a worker's best response curve, and on the same axes, draw the firm's isocost lines. Explain why the firm will offer the worker a wage w* such that e(w*)/w* = e'(w*)
   2. Find the firm's optimal wage w*, in terms of R and α. What effort e* will a worker put in if offered this wage?
   
   Suppose the reservation wage depends on unemployment benefits U, how long the worker can expect to be unemployed before finding a new job, and the wage they can expect when they do:
   
     R = λU + (1-λ)w,
   
   where λ is the unemployment rate.
   
   
   3. If all employers set the wage w* you found in part (b), then find (in terms of R and U) the fraction λ of people in the community who will be unemployed. Show that λ = 0 is not possible.
   
   Now imagine that a new and more intrusive efficient way of monitoring worker activity is deployed. All other things being equal, workers must work harder to avoid losing their jobs. If parameter α was 1 originally, it now takes the value 2. That is, worker effort is now given by
   
     e = 1 - (R/w)².
   
   
   4. Assuming that at a lower wage w more workers are employed, discuss the welfare implications of the new monitoring technology.

Back to micro home page