Society has 20 units of capital and 20 units of labor. The production function for X can be expressed as X = K1/2L1/2, and the production for Y can be expressed as Y = K + L. Which of the following input allocations is not efficient?

Question

Society has 20 units of capital and 20 units of labor. The production function for X can be expressed as X = K1/2L1/2, and the production for Y can be expressed as Y = K + L. Which of the following input allocations is not efficient?

Solution:
Industry X uses 13.33 units of capital and 6.67 units of labor. Industry Y uses 6.67 units of capital and 13.33 units of labor.

Explanation:
Efficient input allocation requires that the MRTS is equal across all production processes. The MRTS for X equals K/L. The MRTS for Y equals 1. Efficient input allocation would require that each industry uses equal amounts of capital and labor. An inefficient allocation would be one where an industry's capital did not equal its labor.

**Don't understand how MRTS is calculated and why? Any help would be greatly appreciated!

Answer 1

The marginal rate of substitution, in this context, is the amount of additional K you need to hold production constant in because of a decrease in L. (or vice versa)

MRS = -MPk/MPl.
In the production of X, MPk = (1/2)K^(-1/2)*L^(1/2)
MPl = 1/2)L^(-1/2)*K^(1/2)

So, MPk/MPl = -L^(1/2)/K^(1/2)

In the production of Y, MPk=1 and MPl=1

The industry is not efficient, you could give Y 10 units of K and 10 units of L which would hold Y production constant, however the remaning portions of 10K and 10L would increase the production of X.

Answer 2

The Marginal Rate of Technical Substitution (MRTS) is a measure of the rate at which one factor of production can be substituted for another while keeping the level of output constant. In this case, the MRTS for X is equal to the ratio of the marginal product of labor (MPL) to the marginal product of capital (MPK), which is K/L.

To calculate the MRTS for X, we take the partial derivative of the X production function with respect to labor (L) and divide it by the partial derivative of the X production function with respect to capital (K):

MRTS_X = (∂X/∂L) / (∂X/∂K)

Taking the partial derivatives of the X production function, we have:

∂X/∂L = (1/2)(K^(1/2))(L^(-1/2))

∂X/∂K = (1/2)(L^(1/2))(K^(-1/2))

Now we can calculate the MRTS for X:

MRTS_X = ((1/2)(K^(1/2))(L^(-1/2))) / ((1/2)(L^(1/2))(K^(-1/2)))

Simplifying the expression, we find:

MRTS_X = (K^(1/2))(L^(-1/2)) / (L^(1/2))(K^(-1/2))

MRTS_X = K/L

Similarly, we can calculate the MRTS for Y. The Y production function can be written as Y = K + L. Taking the partial derivatives, we find:

∂Y/∂L = 1

∂Y/∂K = 1

Thus, the MRTS for Y is equal to 1.

Now, in order to have efficient input allocation, the MRTS should be equal across industries. In other words, the MRTS for X should be equal to the MRTS for Y. Since MRTS_X equals K/L and MRTS_Y equals 1, efficient input allocation would require that K/L equals 1.

In the given choices, if an industry's capital does not equal its labor, for example, if industry X uses 13.33 units of capital and 6.67 units of labor, it would result in an inefficient allocation of inputs.