A manufacturer estimates the daily output at a certain factory is given by Q(K,L

Question

A manufacturer estimates the daily output at a certain factory is given by Q(K,L) = 27K^(1/3)L^(2/3) where K is the capital investment in units of $1,000 and L is the size of the labor force in worker-hours. Suppose currently the manufacturer has a capital investment of $125,000 and the labor level is 900 worker-hours.

a.) What is the annual output at this factory under the current conditions?

b.) Find the marginal productivities of capital and labor, QK and QL . Evaluate each partial derivative at the current values of K and L.

c.) Using your work in part (b), interpret the meaning of Qk(125,900) and Ql (125,900). Use complete sentences in your explanation.

d.) Use marginal analysis to ESTIMATE the change in daily production if the amount of the capital expenditures decrease by $2500 but the labor force increases by 5 worker hours.

Explanation / Answer

Solution

(a) Q(K,L) = 27K^(1/3)L^(2/3)

Q(125,900) = 27(125)1/3(900)2/3 = 12,584.29=12,584 approx

(b) Q'k = 27(1/3)(k)-2/3(L)2/3 = 9(L/K)2/3 i.e Q'(125,900) = 33.56 Qk(125,900)

Q'l = 27(k)1/3(2/3)(L)-1/3 = 18(K/L)1/3 i.e = 9.32 = 9 Ql (125,900)

(c) The marginal product of capital (MPK) is the additional output resulting from the use of an additional unit of capital i.e if additonal $1,000 is invested the additional output will be 34 units.

The marginal product of labor (MPL) is the additonal output resulting from the use of anadditonal unit of labor i.e. if additonal 1 labor is being employed output will increase by 9 units.

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