Solution
S = 15 x [tex]L^{0.2}[/tex] x [tex]C^{0.8}[/tex]
Total cost, T = wL + rC = 50L + 100C
Total revenue, R = Output price (P) x Quantity = P x 15 x [tex]L^{0.2}[/tex]x [tex]C^{0.8}[/tex]
(a)
Optimization problem will be:
Max R = P x 15 x [tex]L^{0.2}[/tex] x [tex]C^{0.8}[/tex]
Subject to T = 50L + 100C
(b) When S = 50,000
Cost is minimized when (MPL / MPC) = w / r
MPL = [tex]\partial[/tex]R / [tex]\partial[/tex]L = P x 15 x 0.2 x [tex](C / L)^{0.8}[/tex] = P x 3 x [tex](C / L)^{0.8}[/tex]
MPC = [tex]\partial[/tex]R / [tex]\partial[/tex]C = P x 15 x 0.8 x [tex](L / C)^{0.2}[/tex] = P x 12 x [tex](L / C)^{0.2}[/tex]
MPL / MPC = (3/12) x (C / L) = 50/100
C / 4L = 1/2
4L = 2C
2L = C
Substituting in production function,
15 x [tex]L^{0.2}[/tex] x [tex]C^{0.8}[/tex] = S
15 x[tex]L^{0.2}[/tex] x [tex](2L)^{0.8}[/tex] = 50,000
15 x [tex]2^{0.8}[/tex] x [tex]L^{0.2}[/tex] x [tex]L^{0.8}[/tex] = 50,000
L = 50,000 / (15 x 20.8)
L = 1,914.50
C = 2L = 3,829.00
Total cost ($) = 50 x 1,914.50 + 100 x 3,829.00 = 95,725.00 + 382,900 = 478,625.00
Note: This optimization problem can be solved without using Solver too, as shown here.
