1.     Given the production function: Q = A L½ K½

What returns to scale, does this function exhibit? Why?

2.      Prove that   MRTS = MP L / MP K

1. The function exhibits a constant returns to scale. The reason is that the power of t is equal to 1.

2. Since MRST = MPL / MPK = - ΔK / ΔL, it is therefore proved that:

MRST = MPL / MPK

Step-by-step explanation:

1. What returns to scale, does this function exhibit? Why?

Note: The production function is properly written before answering the question as follows:

Q = A L^(½) K^(½) ……………….. (1)

To determine the type of return to scale, the input usages K and L are scaled by the multiplicative factor t, and substituting it into equation (1) and then solve as follows:

Q = A ((Lt)^(½) (Kt)^½)

Q = AL^(1/2)t^(1/2)K^(1/2)t^(1/2)

Q = AL^(1/2)^(1/2)t^(1/2)t^(1/2)

Q = AL^(1/2)^(1/2)t^(1/2 + 1/2)

Q = AL^(1/2)^(1/2)t^1

The function exhibits a constant returns to scale.

The reason is that the power of t is equal to 1.

2. Prove that MRTS = MPL / MPK

Given the production function:

Q = LK (1)

We can obtain the change in Q as follows:

ΔQ = MPL * ΔL + MPK * ΔK (2)

Equating equation (2) to zero and solve for MPL/MPK, we have:

0 = MPL * ΔL + MPK * ΔK

MPL * ΔL = - MPK * ΔK

MPL / MPK = - ΔK / ΔL (3)

The equation (3) above is the marginal product of technical substitution (MRST) as follows:

MRST = MPL / MPK = - ΔK / ΔL

It is therefore proved that:

MRST = MPL / MPK

y > 1/3x

y < = -x + 4

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