Answer to Question #216384 in Microeconomics for Ida Mohamma

Question

Suppose that your company is offered a contract to produce 1,000 units of a good X per day. Your company uses labor and capital to produce good X. The quantity of labor used, L, is expressed by hours of work, and the wage rate is $10 per hour. The quantity of capital utilized, K, is expressed in machine-hours, and the rental rate per machine hour is $40.

Thus total cost of production is TC = 10L + 40K. The production function is given by Q = 20L ^0.5 K ^0.5. In the contract, your company is required to produce 1,000 of good X.

a)    Using the Lagrange function, determine the amount of labor and capital to use to minimize the cost of producing 1,000 units of good X.

(14 marks)

b)    Interpret lambda and what it means to your company.

(6 marks)

Answer 1

A)

The given cost function : $C = 10 L + 40 K$

Given production function : $Q = 20\times K^{0.5}\times L^{0.5}$

The firm need to produce 1000 units of good.

The minimization problem is :

Min $: 10L + 40 K$

Subject to : $20\times K^{0.5}\times L^{0.5}=1000$

$S = 10L + 40 K + C (20\times K^{0.5}\times L^{0.5} - 1000)$ -- Langrange function , Let C = Lamda

Partially differentiate the langrange function w.r.t 'L' & equating with 0 we get  :

$\frac{d S}{ d L} = 10 + C(20\times 0.5\times L^{-.5}\times K^{0.5}) = 0 ---(1)$

Partially differentiate the langrange function w.r.t 'K' & equating with 0 we get  :$\frac{d S}{d K} = 40 + C(20\times 0.5\times L.^{5}\times K^{-0.5} ) = 0 --- (2)$

Solving (1) & (2) simultaniously , we get :

$(\frac{10}{40}) = \frac{C(20\times 0.5\times L-^{.5}\times K^{0.5})} { C(20\times 0.5\times L.^{5}\times K^{-0.5} )} => (\frac{1}{4}) = \frac{K}{L} => L = 4K$ -- Substituting this in the given constraint , we get :

$=> 20\times K^{0.5}\times L^{0.5} = 1000\\ => 20\times K^{0.5}(4K)^{0.5} = 1000\\ => 2K = 50 => K = 25 \space and \space as L = 4K , \space therefore\space L = 100\\ Hence K = 25 \space and \space L =100$

B) the objective function is the minimum cost function & the exogenous variable is the output level.

Now if we substitute K = 25 & L = 100 in (2) we get : C (Lamda) = -2

Hence we can say that as the output level increase by 1 unit the minimum total cost will decrease by 2 units which is the value of lambda.