Cost Minimization for a Firm’s Production

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Suppose that a firm’s production function is given by Q = K^0.5L ^0.5 (where MPL = 0.5[K/L]^½And K MP K = 0.5[K/L]^½ ). Q is the quantity of output of the good, L denotes the quantity of labor used in the production of the good, and k denotes the quantity of capital used in the production of the good. Also, suppose that the current wage rate (w) is $4 per unit of labor and the rental rate of capital (r) is $8 per unit. The firm is interested in producing 100 units of output.

Suppose that the firm is currently in the short-run and has a fixed level of K at 25 units.

The firm’s cost-minimizing input combination and the corresponding level of minimum cost

Q = K^0.5* L^0.5

Where Q= 100 units, K= 25 units

100= 25^0.5* L^0.5

100= 5* L^0.5

L =(20)²

L = 400 units

Corresponding minimum cost

= labour cost + capital cost

= 400*4 + 25*8

= 1600 + 200

= $ 1800

(As illustrated in Perloff 2012).

The input combination the firm will use in the long run and the corresponding level of minimum cost

Q = K^0.5L^0.5

In the long run:

100= K^0.5L^0.5

∆Q= ∆K^0.5L^0.5

Changes in both capital and labor will result in the following two equations:

100=0.5K^‑0.5L^0.5………..(i)*1

100=K^0.50.5L^-0..5……….(ii)*0.5

100=0.5K^-0.5L^0.5

– 50=0.5K^0.50.5L^-0.5

50= 0 + 0.5L

L= 50/0.5 = 100 units

Substituting 100 units of labour in equation (i)

100=0.5K^-0.5L^0.5

100=0.5K^-0.5(100)^0.5

100=5K^-0.5

K^-0.5= 100/5 = (20)^0.5 = 5 units

(As illustrated in Sullivan & Sheffrin 2003).

Can this firm experience economy of scale in the long run?

Economies of scale = total cost/quantities

In the short run

Total cost = labour cost + capital cost

= $ 1800

Quantities produced = 100 units

Therefore

EOS= $1800/100 = $18

In the long run

Total cost= labour cost*capital cost

=(100*4) + (5*8)

= 400 + 40

=$ 440

Therefore

EOS = $440/100=$4.4

The firm is going to experience economies of scale because there is an overall reduction in total cost from $1800 in the short run to $440 in the short run. In addition to this, the average cost per unit produced will have a difference of $13.6 in the long run (Perloff 2012).

References

Carbaugh, R 2006, Contemporary economics: an applications approach, Cengage Learning, New York.

Gilboa, I 2009, Theory of decision under uncertainty, Cambridge University Press, Cambridge.

Mankiw, G 2008, Principle of economics, Cengage Learning, New York.

Mas-colell, A Whinston, M & Green, J 2010. Microeconomic theory, Oxford University Press, New York.

Perloff, JM 2012, Microeconomics: global edition, Pearson Education, London.

Rubin, H & Capra, C 2011, The evolutionary psychology of economics, Oxford University Press, Oxford.

Sullivan, A & Sheffrin, M 2003, Economics: Principles in action, Pearson Prentice Hall, Upper Saddle River, New Jersey.

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