Calculate the Marginal Utilities and Marginal Rate of Substitution for each of the Utility functions below. (Remember, marginal utility of a given good gives us the impact of an additional unit of that good on the total utility.)

The Marginal Rate of Substitution looks at the balance in changes of good 1 and good 2 required for the consumer to be indifferent between his/her consumption bundles before and after trade. But what does indifference mean? It means that utility for both bundles is exactly equal. Therefore, ΔU=0

There is some (negative) change in utility resulting from giving up a little bit of good 2, and as we saw in the previous section, this change equals MU₂Δx₂, Similarly, there is some (positive) change in utility from getting a little more of good 1, MU₁Δx₁ which equals Since we want to be indifferent before and after the trade, it must be that the sum of these changes equals zero. That is, U=MU₁Δx₁ + MU₂Δx₂ = 0, now the MRS can be find from this equation of marginal utilities!

First, MU₁Δx₁ from both sides. Then, MU₂Δx₂ = - MU₁Δx₁ Next, divide both sides by x₁ and MU₂. The result is Δх₁/Δх = - MU₁ / MU₂ The left hand side is just the MRS, and the right hand side is the negative ratio of marginal utilities. In the MRS section, we learned why the left hand side would automatically be negative.

Explanation:

The right hand side needs the negative sign because marginal utility is positive for goods, so the ratio of marginal utilities is always positive.