Question

Suppose marginal product of labor (MPL) = 0.3*K and wage = $ 2 while the marginal product of capital (MP_K) = 0.7*L and price of capital = $ 1.

(a) What the marginal Rate of technical substitution between labor and capital at L = 50 and K = 100?

(b) What is the relative wage ratio?

(c) Is the input allocation L=50 and K=100 optimal?

Answer 1

Let me correct one thing

The Marginal Product of Labor is

MPL = 0.3K.......(1)

The Marginal Product of Capital is

MPK = 0.7L.......(2)

(a) The Marginal Rate of Technical Substitution or MRTS between labor and capital is defined as the ratio of MPL and MPK. Hence,

MRTS = MPL/MPK

or, MRTS = 0.3K/0.7L

or, MRTS = 3K/7L..........(3)

Now, at L=50 and K=100

MRTS = (3×100)/(7×50)

or, MRTS = 6/7.......(4)

Hencez the marginal rate of technical substitution between labor and capital at L=50 and K=100 is (6/7).

(b) Now, it is given that, Wage (w) is $2. Hence,

w = 2

And, Price of capital (r) is $1. Hence,

r = 1

Hence, the relative wage ratio is defined as the ratio of wage to price of capital. Hence,

Relative Wage Ratio = w/r = 2/1

or, (w/r) = 2........(4)

Hence, the relative wage ratio is 2.

(c) Now, the producer maximizes his profit at that input combination (L,K), for which the Marginal Rate of Technical Substitution equals the Relative Wage Ratio. In other words,

The input allocation is optimal when

MRTS = w/r

Hence, we will check for (L,K) = (50,100) that, whether MRTS equals the relative wage ratio (w/r) or not.

From part (a) we got,

​​​​​​When, (L,K) = (50,100), the MRTS is

MRTS = 6/7

Also, from part (b) we got,

Relative Wage Ratio = w/r = 2.

Hence, we can clearly see that,

MRTS = 6/7 < 2 = w/r

Hence, MRTS is not equal to Relative Wage Ratio when L=50 and K=100. Hence, L=50 and K=100 is not optimal.

Hence, the input allocation L=50 and K=100 is not optimal.

Hope the solution is clear to you my friend.