Econ 519: David Reiley
Due Saturday, 5 August 2006

## Problem Set #5

The multi-equation systems can be harder to grasp than the single-equation problems. It is essential that you master the single-equation concepts. If you're struggling to keep up, leave Questions 14.21 and 15.18 through 15.32 to the end, because they deal with multi-equation systems.

1. Problem 14.18.
2. Problem 14.21.
3. Problem 14.23.
4. Problem 14.26.
5. Problem 14.27.
6. Problem 15.5.
7. Problem 15.6.
8. Problem 15.8.
9. Problem 15.9.
10. Problem 15.10a.
11. Problem 15.18.
12. Problem 15.19.
13. Problem 15.21.
14. Problem 15.27.
15. Problem 15.28. You could plug the results of (48) in and show that the equations in (47) are satisfied... but this wouldn't show that the solution to (47) is unique. To prove that (48) is the unique solution to (47), please proceed as follows.
1. First, prove that x1=y1. Do this by contradiction.
1. Assume that x1>y1, and use the concavity of u1 and u2 to show that the equations (47) then imply a contradiction. You might find it easiest to define x1=y1+c, with c>0, in order to keep track of which quantities are bigger than other.
2. Do the same thing for the assumption that x1<y1.
2. Once you know that x1=y1, the rest of the results in (48) should follow from solving the equations (47). (Though the model doesn't explicitly say this, you can assume that the first derivatives of of u1 and u2 are both positive; that is, the consumers always have positive marginal utility for each good.)
16. Problem 15.30. My notes on this application should help. The trickiest part is understanding that the quantities r1, r2, R1, R2, and D are all parameters that depend on the first and second derivatives of u1 and u2.
17. Problem 15.32. Note that I want you to assume that e1=e2=1, so that the system of equations is just (48). Also, you can ignore the variables y1 and y2,because with this assumption they are always equal to x1and x2, respectively. So you have three equations in three endogenous variables (x1, x2, p) and one exogenous parameter (alpha). Find how the endogenous variables change with the exogenous parameter. You can do this using the Implicit Function Theorem, or it might be easier to do it by finding explicit functions for the endogenous variables in terms of alpha.