# MRS of good X for good Y

I'm having some confusion with using the terminology of MRS. For example, let's take up this question:

If good 1 is a “neutral,” what is its marginal rate of substitution for good 2?

Zero—if you take away some of good 1, the consumer needs zero units of good 2 to compensate him for his loss.

The question is asking for the MRS of good 2. The answer interprets this question as "how much of good 2 will one need to give up for a loss in good 1?". In that case, it is, of course, 0. But, you can just as easily interpret the question as: "How much of good 1 will one need to give up for a loss in good 2?" -- in which case, the answer is infinity -- the consumer would not want to give up any amount of good 1 for a loss in good 2.

In a different question, this was asked: "What is your marginal rate of substitution of \$1 bills for \$5 bills?". And the answer was "5". So, in this question, the good after the word "of" (\$1 bills) is what you're giving up. In other words, the answer is the answer to the question: "How many \$1 bills would you give up for a \\$5 bill?" So this answer interprets the question in completely the opposite way.

To phrase it more clearly, when someone asks:

What is the MRS of good 1 for good 2?

How should I interpret it? Is it "how much good 2 you would give up for an increase in good 1?" or "How much good 1 you would give up for an increase in good 2?"

$$\frac{\partial u(x^*)/\partial x_\ell}{\partial u(x^*)/\partial_k}=\frac{p_\ell}{p_k}\tag{3.D.5}$$ The expression on the left of $$\text{(3.D.5)}$$ is the marginal rate of substitution of good $$\ell$$ for good $$k$$ at $$x^*$$, $$MRS_{\ell k}(x^*)$$; it tells us the amount of good $$k$$ that the consumer must be given to compensate her for a one-unit marginal reduction in her consumption of good $$\ell$$.