Question

1. A firm has the production function Q= 4LK. The marginal products are given by MPL = 4K and MPK= 4L.

1. Suppose that the prices of labour and capital are given by w and r. Solve for the quantities of L and K that minimize the cost of producing Q units of output. Provide an expression for the long run total cost function.
2. What returns to scale are exhibited by this production function? What economies of scale are exhibited? Show the algebraic work for both.
3. Now suppose Q = 4L^0.5K^0.5, where MPL = 2K^0.5L^-0.5 and MPK = 2L^0.5K^-0.5. Repeat (a).
4. With the production function in (c), repeat (b). Based on your findings in (b) and (c), what can you conclude?

Answer 1

(a)

Cost is minimized when MPL/MPK = w/r

MPL/MPK = K/L = w/r

K = L x (w/r)

Plugging in production function,

4 x L x L x (w/r) = Q

L² = (Q/4) x (r/w)

L = 0.5 x Q^0.5 x (r/w)^0.5

K = 0.5 x Q^0.5 x (r/w)^0.5 x (w/r) = 0.5 x Q^0.5 x (w/r)^0.5

Total cost (TC) = wL + rK

TC = w x [0.5 x Q^0.5 x (r/w)^0.5] + r x [0.5 x Q^0.5 x (w/r)^0.5]

TC = [0.5 x Q^0.5 x (wr)^0.5] + [0.5 x Q^0.5 x (wr)^0.5]

TC = 2 x 0.5 x Q^0.5 x (wr)^0.5

TC = Q^0.5 x (wr)^0.5

(b)

(I) Doubling both inputs, new production function is

Q* = 4 x (2L) x (2K) = 4 x 4 x LK = 4 x (4LK) = 4 x Q

Q*/Q = 4 > 2

Since doubling both inputs more than doubles output, there are increasing returns to scale.

(II) ATC = TC/Q = (wr)^0.5 / (Q)^0.5

As Q increases, [(Q)^0.5] increases, hence ATC decreases and so there are economies of scale.

(c)

Cost is minimized when MPL/MPK = w/r

MPL/MPK = K/L = w/r

K = L x (w/r)

Plugging in production function,

4 x L^0.5 x L^0.5 x (w/r)^0.5 = Q

L = (Q/4) x (r/w)^0.5

K = (Q/4) x (r/w)^0.5 x (w/r) = (Q/4) x (w/r)^0.5

Total cost (TC) = wL + rK

TC = w x [(Q/4) x (r/w)^0.5] + r x [(Q/4) x (w/r)^0.5]

TC = [(Q/4) x (wr)^0.5] + [(Q/4) x (wr)^0.5]

TC = 2 x (Q/4) x (wr)^0.5

TC = (Q/2) x (wr)^0.5

(d)

(I) Doubling both inputs, new production function is

Q* = 4 x (2L)^0.5 x (2K)^0.5 = 4 x 2^0.5 x 2^0.5 x L^0.5K^0.5 = 2 x 4 x L^0.5K^0.5 = 2 x Q

Q*/Q = 2

Since doubling both inputs exactly doubles output, there are constant returns to scale.

(II) ATC = TC/Q = (1/2) x (wr)^0.5

Since ATC is independent of Q, there are neither economies of scale nor diseconomies of scale.

(III)

It can be concluded that if sum of coefficients of L and K in production function is higher than (lower than) 1, there are increasing (decreasing) returns to scale, and cost function shows economies (diseconomies) of scale. If sum of coefficients of L and K in production function is equal to 1, there are constant returns to scale, and cost function shows neither economies nor diseconomies of scale.