Question

QUESTION 1. The QuickBuild Company builds houses (Q) in the Lower Mainland. It combines capital (K) and labour (L) in its production function in the following way: Q(KL) = K1/3L1/3 Labour cost: W = $8 Capital rental cost: V = $8 1A. What do we call this form of production function? 1B. According to its production function, what returns to scale does QuickBuild enjoy? Demonstrate. What happens to output when QuickBuild doubles the amount of capital and labour used? 1C. QuickBuild wants to build 12 homes. What is its cost-minimizing choice of Capital and Labour? (Hint: What is the objective function? What is the constraint?)

I'm stuck on question 1C -

Answer 1

Solution:

Solving only for part 1C, as asked:

Given labor cost of w = $8, and capital rental coat of v = $8, total cost for QuickBuild becomes:

Total cost = w*L + v*K

C = 8L + 8K

We have to minimize this cost function and find optimal values for labor, L and capital, K.

Also, with production function of: Q = K1/3L1/3

As QuickBuild want to build 12 homes, Q = 12, so the constraint becomes: 12 = K1/3L1/3

So, Lagrangian becomes:

Z = (8L + 8K) + d*(12 - K1/3*L1/3); where d is the lagrangian multiplier

So, we can solve for optimal values of L and K using the first order conditions by putting partial derivatives to 0: dZ/dL = 0 and dZ/dK = 0

dZ/dL = 8 - d*(1/3)*K1/3*L1/3 - 1

So, dZ/dL = 0 gives: 8 - d*(1/3)*K1/3L-2/3 = 0

d = 8/((1/3)K1/3L-2/3) ... (i)

Similarly, dZ/dK = 8 - d*(1/3)*K1/3-1L1/3

So, dZ/dK = 0 gives us: 8 - d*(1/3)*K-2/3*L1/3 = 0

d = 8/((1/3)*K-2/3L1/3) ... (ii)

Then, from (i) and (ii), we get the optimality condition:

8/((1/3)*K1/3L-2/3) = 8/((1/3)*K-2/3L1/3)

On simplifying, this gives us K = L

Then, substituting this in the constraint we get:

12 = L^1/3L^1/3

12 = L^2/3

So, L = 12^3/2 = 41.57 approximately.

With K = L, K = 41.57 (approx). Rounding this we get 42. So, cost minimizing choice of labor is 42 units and capital is 42 units.