Cobb-Douglas Production Function - Finding units of labour to maximise production
Given production function $ f(L,K)=16L^\frac{1}{4}K^\frac{3}{4} $ , where each unit of labour costs £50 and each unit of capital costs £100 and you have a budget of £500,000. Find the number of units of labour to maximise production.
We were given this question in a maths class, so I am slightly unsure of my solution, as follows:
We have $ Y = 16L^\frac{1}{4}K^\frac{3}{4} + \lambda(500,000-50L-100K) $
$ \frac{\partial Y}{\partial L}=4L^\frac{-3}{4}K^\frac{3}{4}-50\lambda =0 $
$ \frac{\partial Y}{\partial K}=12L^\frac{1}{4}K^\frac{-1}{4}-100\lambda =0 $
$ \frac{\partial Y}{\partial \lambda}=500,000 - 50L - 100K =0 $
Then, $ \frac{(\frac{\partial Y}{\partial L})}{(\frac{\partial Y}{\partial K})} = \frac{4L^\frac{-3}{4}K^\frac{3}{4}}{12L^\frac{1}{4}K^\frac{-1}{4}} = \frac{50 \lambda}{100 \lambda} $
$ \implies \frac{K}{3L}=\frac{1}{2} \implies k = \frac{3}{2}L $
Then inputting into $ \frac{\partial y}{\partial \lambda}, 500,000 = 50L + 100(\frac{3}{2}L) = 200L $
$ \implies L = 2,500 $
Therefore, at a budget of £500,000, 2,500 units of labour maximises production?
If your aim is to maximize the production then your approach is correct.
But if the aim is to find the optimal number of units of labor, then you should solve it as a profit maximization problem with a budget constraint.
Then the problem should be as following
Maximize the profit such that the total cost spent on production does not exceed the budget $$ Max_{L,K}\ Profit(K,L)=P(16L^\frac{1}{4}K^\frac{3}{4}) - 50L - 100K $$ $$ st. 50L + 100K \le 500000 $$
Then the Lagrange function would be: $$ \mathcal{L}(K,L,\lambda) = (P(16L^\frac{1}{4}K^\frac{3}{4}) - 50L - 100K) + \lambda(500,000-50L-100K) $$
FOC:
$ \frac{\partial \mathcal{L}}{\partial L}=4PL^\frac{-3}{4}K^\frac{3}{4} - 50-50\lambda =0 $
$ \frac{\partial \mathcal{L}}{\partial K}=12PL^\frac{1}{4}K^\frac{-1}{4} - 100-100\lambda =0 $
$ \frac{\partial \mathcal{L}}{\partial \lambda}=500,000 - 50L - 100K =0 $
Solving the system of FOC equations will yield $ L = 2500, K = 3750 $ I used Wolfram to get the solution, but you can solve that using the approach you used in the question.
Wolfram code:
Z = Lambda
Solve[{4P (L^(-3/4))(K^(3/4)) - 50 - 50Z==0, 12P (L^(1/4))(K^(-1/4)) - 100 - 100Z==0, 500000-50L-100K==0},{L,K,Z}]