Question

Problem 3: A firm has the following production function: ?(?1, ?2 ) = ?1 + 4?2 A) Does this firm’s technology exhibit constant, increasing, or decreasing returns to scale? B) Suppose the firm wants to produce exactly ? units and that input 1 costs $?1 per unit and input 2 costs $?2 per unit. What are the firm’s conditional input demand functions? C) Write down the formula for the firm’s total cost function as a function of ?1, ?2, and ?. D) If ?1 = 1, ?2 = 2, what is the cost minimizing choice of ?1 and ?2 for Bob to produce 100 units of output? E) If ?1 = 1, ?2 = 2, what is the minimum cost of producing one unit of output?

Answer 1

A) y = f(x1, x2) = x1 + 4x2
Let x1 = tx1 and x2 = tx2 where t > 1
So, y' = (tx1) + 4(tx2) = t(x1 + 4x2) = ty
So, there are constant returns to scale because increasing the inputs buy a multiple t will increase the output in the same proportion as power of t is 1.

b) The given production function shows that the two goods are perfect substitutes so their usage depends on the relationship between MRTS and cost ratio.
MRTS = (Marginal product of input 1, MP1)/(Marginal product of input 2, MP2)
MP1 = dy/dx1 = 1
MP2 = dy/dx2 = 4
So, MRTS = 1/4 = 0.25
Cost ratio = w1/w2

If MRTS > w1/w2 then only input 1 will be used.
So, x2 = 0 and x1 = y
If MRTS < w1/w2 then only input 2 will be used.
So, x1 = 0 and y = 4x2
So, x2 = y/4

c) If MRTS > w1/w2
Cost, C = w1x1 + w2x2 = w1y
If MRTS < w1/w2
C = w1x1 + w2x2 = w₂(y/4)

d) w1/w2 = 1/2 = 0.5
So, MRTS < w1/w2
Cost minimizing choice is x1 = 0 and x2 = y/4 = 100/4 = 25
Thus, x1 = 0 and x2 = 25

e) Minimum cost of producing y = 1 is
C = w₂(y/4) = 2*(1/4) = 0.5