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.
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!
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