Relative prices and marginal rate of substitution

Question

Ive been taking this microeconomics class this semester and Ive always just memorized that MRS= dy/dx=-px/py, but something is lacking to my understanding of this. I get that dy/dx represents how many units of Y a consumer would sacrifice for an additional unit of x, but what I find really confusing is, why at the optimal, this has to equal px/py.

Im lacking I think the source of my confusion is this; we learned that px/py is the cost, in units of y, of an additional unit of x. Now, to me that’s just the interpretation of dy/dx, and I cant seem to get an intuition of why px/py is the same thing. Why would px/py even be in units of y? When we want something in units of y, the y should be in the numerator, like in dy/dx, no? This is the one piece of misunderstanding that has been in the back of my mind since I started this course, and would like to finally clarify this. Thanks for any help!

In: 1

Answer 1

Just plug in some numbers. If px = 1 and py = 2 then the rate of substitution is 1/2; The consumer sacrifices half a unit of y to gain a unit of x which matches up with the relative costs, x being half the cost of y.