Help with a simple problem.

• To: mathgroup at smc.vnet.net
• Subject: [mg28526] Help with a simple problem.
• From: Eric <hearts121 at deleteme.hotmail.com>
• Date: Wed, 25 Apr 2001 01:30:30 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```Hi everyone,

I have a simple econ problem that I want to solve in Mathematica, but I
don't know how to input the problem correctly in Mathematica. The
following is the problem and what I tried to do in Mathematica. Any help
is greatly appreciated.

Eric

***********************************************************************

First, let Ex and Ey be some constant and Ex > Ey > 0.

Then

Xa + Xb = Ex and
Ya + Yb = Ey

Second, let functions

Ua(Xa , Ya) = V(Xa) + V(Ya)
Ub(Xb , Yb) = V(Xb) + V(Yb)

where V(.) is any strictly concave, increasing and differentiable function.

If we know that the following condition with the partial derivatives are
true:

D[Ua , Xa]     D[Ub , Xb]
---------- ==  ----------
D[Ua , Ya]     D[Ub , Yb]

prove that we must have Xa >= Ya and Xb >=Yb.

What I tried to do was solve for Xa in terms of Ya and Ex so I can
compare Xa and Ya.

*********************************************************************
Here is what I tried to do in Mathematica. But I get an error message.

Ex = Xa + Xb

Ey = Ya + Yb

U[x_, y_] := V[x] + V[y]

Solve[(D[U[Xa, Ya], Xa]/D[U[Xa, Ya], Ya]) == (D[U[Xb, Yb], Xb]/D[U[Xb,
Yb], Yb]) , {Xa, Ya, Xb, Yb}]

```

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