Proving sign of cross partial

• To: mathgroup at smc.vnet.net
• Subject: [mg70774] Proving sign of cross partial
• From: Patrik <hosanagar at gmail.com>
• Date: Fri, 27 Oct 2006 00:28:12 -0400 (EDT)

```I have the following functions:

w = (1 - y/x)^(1/(1 - alpha))
Term1 = w /. {y -> L2, x -> n}
Term2 = w/.{y -> (R2-L2)*(R2 +(R1 (n + L1 - R1) - L1 L2)/((n-R1)*(n + L1 - L2 - R1)), x->n}

U = Term1*D*(1/(n-L2))*(R1 - ((L1 L2)/(n+L1-R1)) + Term2*(1-(1/(n-L2))*(R1 - ((L1 L2)/(n+L1-R1)))

All parameters except L2 are constants with 0< alpha<1, 0<D<1, 0<=L1<=R1, R1 and R2 are positive, and n is very large compared to R1 and R2. L2 is a variable with  0<=L2<=R2. L2 is chosen to minimize U.

Numerically, I find that the cross partial of U with respect to L1 and L2 is always >=0. However, analytically, This seems harder to show. Any thoughts on how to prove this analytically.

Note: I tried the binomial expansion of the two w(x,y) terms and ignored the quadratic and higher order terms because n is large. The cross-partial is positive, however the solution for minimize U by choosing L2 returns a complex L2. So, that simplification is not useful.