Answer:
The equilibrium combination of K and L is 7, 28. That is, 7 K’s and 28 L’s.
Explanation:
This can be determined as follows:
Q = 3L^2K
s.t.
400L + 800K -16800
Using a Langrangian multiplier function G with ∅ as the multiplier, we have:
G = 3L^2K – ∅(400L + 800K -16800) ……………… (1)
Partially differentiate G with respect to L, K and ∅, we have:
∂G / ∂L = 6LK – ∅400 = 0 …………….. (2)
∂G / ∂K = 3L^2 – ∅800 = 0 ……………. (3)
∂G / ∂∅ = 400L + 800K – 16800 = 0 …………… (4)
From equation (2), we have:
6LK = ∅400
∅ = 6LK / 400
∅ = 0.015LK …………………….. (5)
From equation (3), we have:
3L^2 = ∅800
∅ = 3L^2 / 800
∅ = 0.00375L^2 …………… (6)
Equating (5) and (6) and solve for L, we have:
0.00375L^2 = 0.015LK
L^2 / L = 0.015K / 0.00375
L = 4K ……….. (7)
Substituting L = 4K into equation (4) and solve K, we have:
400(4K) + 800K – 16800 = 0
1600K + 800K = 16800
2400K = 16800
K = 16800 / 2400
K = 7
Substitute K = 7 into equation (7), we have:
L = 4 * 7
L = 28
Therefore, the equilibrium combination of K and L is 7, 28. That is, 7 K’s and 28 L’s.
