Suppose that a firm’s production function is given by Q = K0.5L 0.5 (where MPL = 0.5[K/L]½ And 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 = K0.5 * L0.5
Where Q= 100 units, K= 25 units
100= 250.5* 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 = K0.5L0.5
In the long run:
100= K0.5L0.5
∆Q= ∆K0.5L0.5
Changes in both capital and labor will result in the following two equations:
100=0.5K‑0.5L0.5………..(i)*1
100=K0.5 0.5L-0..5……….(ii)*0.5
100=0.5K-0.5L0.5
– 50=0.5K0.50.5L-0.5
50= 0 + 0.5L
L= 50/0.5 = 100 units
Substituting 100 units of labour in equation (i)
100=0.5K-0.5L0.5
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.