Question

NVidia, the pioneering computer graphics chip maker has a line of production of a semiconductor component of one of its chips. The production process represents number of units produced per day (Q) as a function of labor (L), which costs $400 per unit, and capital (K), which costs $800 per unit. The following production function describes the process:

Q= 3L^2K

Required:
If cost were limited to $16,800, the equilibrium combination of K and L is:_______

Answer 1

The equilibrium combination of K and L is 7, 28. That is, 7 K’s and 28 L’s.

Step-by-step 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.