Introductory Microeconomics: Problem Set Week 5
- Consider the production function Y = L1/3K1/2T1/6,
where L is labour, K is capital, and T is land.
- Show whether this production function has decreasing, constant or increasing returns to scale.
- Derive an expression for the marginal product of labour. Does this production function have
diminishing returns to labour?
- Find an expression relating the marginal and the average product of labour.
- How does an increase in the input of capital or land affect the marginal product
of labour?
- 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.
- A company producing bicycles has two plants, A and B. The numbers of bikes produced per month in the two plants are
YA = 40LA1/2 and
YB = 210LB1/3, where
LA and LB 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:
- total output
- output per worker
- the marginal product of labour
Should the firm consider moving workers from one plant to another? Explain.
- 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.
- Calculate the supply function for each firm, market supply, and the equilibrium price
in the market, p*.
- 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?
-
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.)
- Do Exercise 6.6 ("Lazear's Results") in the CORE text.
-
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.
- 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*)
- 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.
- 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.
- Assuming that at a lower wage w more workers are employed, discuss the welfare
implications of the new monitoring technology.
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