(a) Does this production function exhibit constant, increasing or decreasing returns to scale? Explain how you would be able to see this on an isoquant graph. (b) Set out the unconstrained profit maximisation problem (assuming input prices are w for labour and r for capital and the output price is p) and then solve for the unconditional input demand functions for capital and labour and find the firm's supply function. (c) Now set up the cost minimisation problem and solve for the conditional labour and capital demands. What is the difference between the unconditional input demand functions from part (b) and the conditional input demand functions you have just derived? (d) Give an expression for the total cost function (no need to simplify) and explain how the average and marginal cost functions are related to total cost. (You do not need to derive a function for AC or MC). How would you derive the supply function from this point and how would you expect it to be related to your answer from part (c)? (e) If we were to consider the short run problem for a firm, rather than the long run problem, would we expect to see their total costs being higher in the short run or in the long run? Explain your answer using a diagram to help you.
(a) Does this production function exhibit constant, increasing or decreasing returns to scale? Explain how you would be able to see this on an isoquant graph. (b) Set out the unconstrained profit maximisation problem (assuming input prices are w for labour and r for capital and the output price is p) and then solve for the unconditional input demand functions for capital and labour and find the firm's supply function. (c) Now set up the cost minimisation problem and solve for the conditional labour and capital demands. What is the difference between the unconditional input demand functions from part (b) and the conditional input demand functions you have just derived? (d) Give an expression for the total cost function (no need to simplify) and explain how the average and marginal cost functions are related to total cost. (You do not need to derive a function for AC or MC). How would you derive the supply function from this point and how would you expect it to be related to your answer from part (c)? (e) If we were to consider the short run problem for a firm, rather than the long run problem, would we expect to see their total costs being higher in the short run or in the long run? Explain your answer using a diagram to help you.
Chapter9: Production Functions
Section: Chapter Questions
Problem 9.1P
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solve d and(e)
Transcribed Image Text:Suppose a firm has a production technology given by the Cobb-Douglas production
function: Q=f(L,K) = AL¤ KB with a + ß < 1 and for all A, a, ß > 0, where L and K
represent labour and capital respectively and A is technology.
(a) Does this production function exhibit constant, increasing or decreasing returns to
scale? Explain how you would be able to see this on an isoquant graph.
(b) Set out the unconstrained profit maximisation problem (assuming input prices are
w for labour and r for capital and the output price is p) and then solve for the
unconditional input demand functions for capital and labour and find the firm's
supply function.
(c) Now set up the cost minimisation problem and solve for the conditional labour and
capital demands. What is the difference between the unconditional input demand
functions from part (b) and the conditional input demand functions you have just
derived?
(d) Give an expression for the total cost function (no need to simplify) and explain how
the average and marginal cost functions are related to total cost. (You do not need
to derive a function for AC or MC). How would you derive the supply function from
this point and how would you expect it to be related to your answer from part (c)?
(e) If we were to consider the short run problem for a firm, rather than the long run
problem, would we expect to see their total costs being higher in the short run or in
the long run? Explain your answer using a diagram to help you.
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