## Multivariable Calculus

1. The problem statement, all variables and given/known data

Since consumers cannot be outside the set of affordable bundles, we get the rate of exchange that consumers face provided the spend all of their budget

dy/dx = -Px/Py (where Px is the price of good x, PY is the price of good y)

by totally differentiating the budget constraint and requiring that dm = 0, where m is total income. Suppose now that the consumer is a price taker in the x market but not a price taker in the y market. What is the rate of exchange that the market offers the consumer?

2. Relevant equations

dy/dx = -Px/Py
m = Px(X) + Py(Y), where X and Y represent total number of goods X and Y, respectively.

3. The attempt at a solution

I know I have to totally differentiate the budget constraint. In other words, I take the derivative of the equation m = Px(X) + Py(Y). Then I have to figure out how altering the quantity of Y with affect Py, the price of Y.

By rearranging the equation m = Px(X) + Py(Y), I get Y = (-Px/Py)(X) + m/Py, and since dm= 0, we arrive at the equation dy = (-Px/Py)dx.