# Multivariable Calculus

1. Homework Statement

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. Homework 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.