Question

Price taking firm with a P = $5 with the following production function

Q = f(L,K) = 20 x L^0.25 x K^0.5

w = $20

r = $10

What is the profit maximising input combination?

With step by step working please

Answer 1

Given:

w = $20, r = $10 and P = $5

So, for this firm, the total revenue is given by:

=P x Q

Total cost is given by:

=wL + rK

where L = Labor and K = Capital

=20L + 10K

So, the profit is given by:

Profit(π ) = Total Revenue - Total Cost

So, in order to maximize this profit function, we'll derive the first order conditions which are:

∂π / ∂L = 0

∂π / ∂K = 0

So, we'll calculate the partial derivatives of profit function w.r.t L and K:

∂π / ∂L

Similarly,

∂π / ∂K

Now, substituting these values in the first order conditions we have:

=> ---------------------(i)

=> ---------------------(ii)

Dividing eq(ii) by eq(i), we have:

Substituting the value of K in eq(i), we have:

L = 39.0625

Since, K = 4L , K = 4 x 39.0625 = 156.25

Hence, the profit maximizing input combination is given by: (K*, L*) = (156.25, 39.0625).