Assignment 3

Notes: Show all steps.

Consider the following system of equations, where x and y are endogenous variables and c is an exogenous variable:

(1)     f(x,y) = 0
(2)     g(x,y) = c

Assume the following: f₁ > 0, f₂ < 0, g₁ > 0, g₂ > 0.

Suppose x* and y* satisfy (1) and (2). How will a small change in c affect x* and y*? Follow these steps:

1. Differentiate equations (1) and (2) totally.
   
   
2. Write the resulting system of equations in matrix form AX = B.
   
   
3. What condition involving f₁, f₂, g₁ and g₂ must be met to ensure that a unique solution for the preceding system exists?
   
   
4. Assuming that the condition in Q. 3 is met, use Cramer's Rule to obtain the following:
   dx* / dc = _________________
   dy* / dc = _________________
   
   
5. Noting the signs of the derivatives above, explain how an increase in c affects x* and y*.