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# ECON 2100 Assignment 2 – A competitive firm uses two inputs

Subject: Economics / Macroeconomics
Question
GL/ECON 2100: Assignment 2
2017

Question 1 A competitive firm uses two inputs, capital k and labour l, in its production
process, in order to produce output y. Its production technology is given by: y = l1/2 k 1/4 .
Consider here the firm’s long-run perspective.
a) Show algebraically that the firm’s technology exhibits decreasing returns to scale.
b) If the rate of return on capital is r = 8 and the wage rate is w = 1, represent
graphically the firm’s cost minimization problem for y = 1. Then, find the firm’s cost
function, i.e C(y) ?y ? 0.
c) Find an algebraic expression for the firm’s average cost function, AC(y), and for its
marginal cost function, M C(y).
d) Find an algebraic expression for the firm’s long-run supply function, y(p).

Question 2 A competitive firm faces a two-input production function given by y = 3×1 +
5×2 . Input prices are given by w1 = 10 and w2 = 6.
a) Represent graphically the firm’s cost minimization problem, and then find algebraic
expressions for its conditional factor demand functions, x1 (w1 , w2 , y) and x2 (w1 , w2 , y), as
well as its cost function, C(w1 , w2 , y), for all possible input prices.
b) Given the above input prices, find the firm’s long-run supply function, y(p).
c)
Suppose that the firm cannot vary input 1 in the short run, which remains set
at x1 = 5. Find an algebraic expression for the firm’s short-run total cost function, and
decompose it between variable and fixed costs. Then, find an expression for the firm’s shortrun supply function, y S (p). How does it compare with the firm’s long-run supply function?
1 d) What would happen if input prices were w1 = 6 and w2 = 10 instead? Find the
firm’s new long-run cost function, and an expression for the firm’s long-run supply function,
y(p). 2