Introductory Microeconomics: Problem Set 2

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. You will probably want to do the maths chapter on differentiation first.
  1. Consider the production function Y =  L1/3K1/2T1/6, where L is labour, K is capital, and T is land.

    1. Show whether this production function have 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 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:

    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 the following production function: f(x1x2) =  x11/3x21/2. The prices of both factors equal 1: w1 = w2  = 1.

    1. Are the isoquants of this technology well-behaved? How do you know?
    2. Show in a diagram how the firm will minimise the cost of producing a given amount y of output.
    3. Show that at the optimum production point we have x2 = (3/2)x1.
    4. Find the minumum cost of producing y =  (3/2)1/2 of output.
    5. How does your solution method or answer illustrate conditional factor demands?

  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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