Econ 519: David Reiley

Due Saturday, 5 August 2006

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.

- Problem 14.18.
- Problem 14.21.
- Problem 14.23.
- Problem 14.26.
- Problem 14.27.
- Problem 15.5.
- Problem 15.6.
- Problem 15.8.
- Problem 15.9.
- Problem 15.10a.
- Problem 15.18.
- Problem 15.19.
- Problem 15.21.
- Problem 15.27.
- 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.- First, prove that x
_{1}=y_{1}. Do this by contradiction.- Assume that x
_{1}>y_{1}, and use the concavity of u_{1}and u_{2}to show that the equations (47) then imply a contradiction. You might find it easiest to define x_{1}=y_{1}+c, with c>0, in order to keep track of which quantities are bigger than other. - Do the same thing for the assumption that x
_{1}<y_{1}.

- Assume that x
- Once you know that x
_{1}=y_{1}, 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 u_{1}and u_{2}are both positive; that is, the consumers always have positive marginal utility for each good.)

- First, prove that x
- Problem 15.30. My notes on this application
should help. The trickiest part is understanding that the quantities r
_{1}, r_{2}, R_{1}, R_{2}, and D are all parameters that depend on the first and second derivatives of u_{1}and u_{2}. - Problem 15.32. Note that I want you to assume that e
_{1}=e_{2}=1, so that the system of equations is just (48). Also, you can ignore the variables y_{1 }and y_{2,}because with this assumption they are always equal to x_{1}and x_{2}, respectively. So you have three equations in three endogenous variables (x_{1}, x_{2}, 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.