assume a firm has the production function q=k^1/4L^1/4, where k represents capital, L represents labor, r...

assume a firm has the production function q=k^1/4L^1/4, where k represents capital, L represents labor, r represents the price of capital and w represents the price of labor.

Using the Lagrange method, derive the optimal quantities of k and l as a function of q,r,w

Solutions

Expert Solution

q=k^1/4 L^1/4

Total cost= wL+rK

Lagrangian function= L''= wL+rK+λ(q-k^1/4 L^1/4 )

dL''/dK= r-(1/4)λk^-3/4 L^1/4 = 0

λ = r/(1/4)k^-3/4 L^1/4 = 4r/k^-3/4 L^1/4 Equation 1

Put equation 1 and 2 equals to each other:

dL''/dL= w-(1/4)λk^1/4 L^-3/4 = 0

λ =4w/k^1/4 L^-3/4 Equation 2

4r/k^-3/4 L^1/4 = 4w/k^1/4 L^-3/4

K/L = w/r

K= Lw/r Equation 3

dL''/dλ = q-k^1/4 L^1/4 =0

k^1/4 L^1/4 =q

From equation 3:

(Lw/r)^1/4 L^1/4 =q

L^2/4 = q(r/w)^1/4

Squaring both sides:

L= q² (r/w)^1/2 Optimal quantity of L

Use L in equation 3:

K= Lw/r = [q² (r/w)^1/2 ] (w/r)= q² (w/r)^1/2 Optimal quantity of K

Rahul Sunny answered 1 year ago

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