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Allisa [31]
3 years ago
6

Let S represent the amount of steel produced (in tons). Steel production is related to the amount of labor used (L) and the amou

nt of capital used (C) by the following function: S = 15 L0.2 C 0.8 In this formula L represents the units of labor input and C the units of capital input. Each unit of labor costs $50, and each unit of capital costs $100. (a) Formulate an optimization problem that will determine how much labor and capital are needed in order to produce 50,000 tons of steel at minimum cost. Min L + C L0.2 C 0.8 L, C (b) Solve the optimization problem you formulated in part a. Hint: When using Excel Solver, start with an initial L > 0 and C > 0. If required, round your answers to two decimal places. L = $ C = $ Cost = $
Business
1 answer:
Phantasy [73]3 years ago
6 0

Solution

S = 15 x L^{0.2} x C^{0.8}

Total cost, T = wL + rC = 50L + 100C

Total revenue, R = Output price (P) x Quantity = P x 15 x L^{0.2}x C^{0.8}

(a)

Optimization problem will be:

Max R = P x 15 x L^{0.2} x C^{0.8}

Subject to T = 50L + 100C

(b) When S = 50,000

Cost is minimized when (MPL / MPC) = w / r

MPL = \partialR / \partialL = P x 15 x 0.2 x (C / L)^{0.8} = P x 3 x (C / L)^{0.8}

MPC = \partialR / \partialC = P x 15 x 0.8 x (L / C)^{0.2} = P x 12 x (L / C)^{0.2}

MPL / MPC = (3/12) x (C / L) = 50/100

C / 4L = 1/2

4L = 2C

2L = C

Substituting in production function,

15 x L^{0.2} x C^{0.8} = S

15 xL^{0.2} x (2L)^{0.8} = 50,000

15 x 2^{0.8} x L^{0.2} x L^{0.8} = 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.

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