Missed the day this was talked baout in class. All problems on this homework ass
ID: 1209430 • Letter: M
Question
Missed the day this was talked baout in class.
All problems on this homework assignment will refer to a firm that uses the production function f(x_1, x_2) = x^1/3_1 x^2/3_2. Assume that the prices of the inputs are given by W_1 = 1 and W_4 = 4. Note that many of the answers to these questions are not algebraically "pretty." Find c(y), the firm's long-run total cost function. Explain why we should not be surprised that this cost function is linear in output. Suppose the quantity of input 2 is fixed in the short run. Briefly explain why this might happen; why can't the firm change x2 in the short run? How long is the short run? Find c_s(y, x-_2), the firm's short-run total cost function if input 2 is fixed at some arbitrary level X-_2.Explanation / Answer
A) The production function is given as f = x11/3x22/3. The marginal product function of x1 given as MP(x1) = 1/3(x22/3/ x12/3) and MP(x2) = 2/3(x11/3/ x21/3).
Note that rate of technical substitution RTS is the ratio of marginal products of two factors
At equilibrium level of output, RTS is equal to the price ratio of two inputs w1/w2
MP(x1)/MP(x2) = w1/w2
1/3(x22/3/ x12/3) /2/3(x11/3/ x21/3) = w1/ w2
x2/2x1 = 1/4
x2= (1/2)x1
Given the cost, the required number of units of x2= (1/2)x1
Let the long run cost function be described as C(y) = w1x1 + w2x2
Substitute the values in the equation so that we have:
C(y) = (1)x1 + (4)(1/2)x1
C(y) = x1 + 2x1
C(y) = 3x1
Hence the long run cost function is C = 3x1
B) The cost function is linear since it exhibits the budget cost for the firm with only one input in use.
C) Input 2 can be a machine, plant size or capital equiment that cannot be changed in the short-run, Hence it is fixed. A long run has all the factors variable including the capital so short run is long as long as their are both variable factors and fixed factors present.
D) When x2 = fixed, firm's cost function changes to C(y) = x1 + 4x2_bar