ECON7030 Microeconomic Analysis

Tutorial Exercise 9 (Week 10)

Question 1

A firm has the production function:

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F (L, K) = L 3 K 3

The firm faces wage rate w and rental rate r, and wishes to produce y units of output.

(a) Find the Marginal Rate of Technical Substitution.

(b) Write down the tangency condition and the isoquant constraint, and use them to

derive the conditional factor demands Lc (w, r, y) and K c (w, r, y).

(c) Substitute your conditional factor demands into C = wL + rK to derive the total

cost function C ∗ (w, r, y).

(d) Derive AC(y) and M C(y).

(e) Now suppose w = 2 and r = 4.

(i) Write down T C(y), AC(y), and M C(y) under these factor prices.

(ii) At what value of y does T C = 48? At that output level, what are AC and

M C?

(iii) Suppose output increases from y = 8 to y = 27. Calculate the change in total

cost, and verify this is consistent with your M C expression.

Question 2

A firm has the production function:

F (L, K) = min{2L, K}

The firm faces wage rate w = 3 and rental rate r = 2.

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(a) Explain briefly why the tangency condition cannot be used to solve this cost min-

imisation problem, and derive the cost-minimising input combination as a function

of output y.

(b) Derive C ∗ (y), AC(y), and M C(y).

(c) What returns to scale does this production function exhibit? Verify that your

expressions for T C, AC and M C in (b) are each consistent with this.

(d) Using your expressions from (b):

(i) Calculate T C, AC and M C at y = 10 and y = 20.

(ii) A firm currently produces y = 10 and is considering expanding to y = 15.

What is the additional cost of doing so? How does this compare to M C × ∆y?

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