A certain production process employs two inputs labor (L) and raw materials (R).

Question

A certain production process employs two inputs labor (L) and raw materials (R). Output (Q) is a
function of these two inputs and is given by the following relationship:
Q = 6L2 R2 - 0.10L3 R3
Assume that raw materials (input R) are fixed at 10 units.
(a) Determine the total product function (TPL) for input L.

(b) Determine the marginal product function for input L.

(c) Determine the average product function for input L.

(d) Find the number of units of input L that maximizes the total product function.

(e) Find the number of units of input L that maximizes the marginal product function.

(f) Find the number of units of input L that maximizes the average product function.

(g) Determine the boundaries for the three stages of production.

Explanation / Answer

We have:

Q= 6L^2R^2 – 0.10 L^3 R^3

Plugging R= 10, we get:

Q= 6L^2 x (10^2) – 0.10 L^3 x (10^3)

Q= 600 L^2 -100 L^3

a)

Total product function for input L is :

Q= 600 L^2 -100 L^3

b)

To compute marginal product function, we need to differentiate total product function with respect to L:

MP = 1200 L – 300 L^2

C)

Average product would be total product divided by no. of units of L:

Average Product = (600 L^2 -100 L^3)/ L

                                = 600 L – 100 L^2

D)

To compute no. of units of L, we will plug MP= 0 and solve for L:

0 = 1200 L -300 L^2

300 L = 1200

L = 4

So at 4 units of Labor, total product would be maximum.

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