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On solving equations for variables included in functions R^n to R

  • To: mathgroup at smc.vnet.net
  • Subject: [mg102375] On solving equations for variables included in functions R^n to R
  • From: Mauricio Esteban Cuak <cuak2000 at gmail.com>
  • Date: Fri, 7 Aug 2009 05:30:19 -0400 (EDT)

Hello. This question has to do half with mathematics and half with
mathematica, so I apologise beforehand.
I'm trying to symbollicaly maximize the following equation with respect to
t1 and t2:

obj = (1 - p1) (1 - p2)*U[ L[t1, t2] + iuno[t1, t2] + idos[t1, t2] ] +
  (1 - p1) (p2)*U[L[t1, t2] + iuno[t1, t2] + idos[t1, t2] - F ] +
  (p1) (1 - p2)*U[L[t1, t2] + iuno[t1, t2] + idos[t1, t2] - F ] +
  (p1)*(p2)*U[L[t1, t2] + iuno[t1, t2] + idos[t1, t2] - 2 F ];

As you can see, there are five undefined functions (  U, L, F, iuno, idos ),
two parameters (p1, p2) and two variables (t1,t2). I know that the five
functions have continuous and monotonic partial derivatives and that they
are all concave.
Given those assumptions, my first approach was to get the first-order
conditions:

eq1= Simplify [    D[obj,t1] ];
eq2 = Simplify [    D[obj,t2]  ];

Which give the following output:

eq1 = (       ((p1 - 1)*(p2 - 1)*Derivative[1][U][idos[t1, t2] + iuno[t1,
t2] +
       L[t1, t2]] + p1*p2*Derivative[1][U][-2*F + idos[t1, t2] +
       iuno[t1, t2] + L[t1, t2]] + (-2*p2*p1 + p1 + p2)*
     Derivative[1][U][-F + idos[t1, t2] + iuno[t1, t2] + L[t1, t2]])*
   (Derivative[1, 0][idos][t1, t2] + Derivative[1, 0][iuno][t1, t2] +
    Derivative[1, 0][L][t1, t2]) == 0      )

and

eq2 = (        ((-1 + p1)*(-1 + p2)*Derivative[1][U][idos[t1, t2] + iuno[t1,
t2] +
      L[t1, t2]] + p1*p2*Derivative[1][U][-2*F + idos[t1, t2] +
      iuno[t1, t2] + L[t1, t2]] + (p1 + p2 - 2*p1*p2)*
    Derivative[1][U][-F + idos[t1, t2] + iuno[t1, t2] + L[t1, t2]])*
  (Derivative[0, 1][idos][t1, t2] + Derivative[0, 1][iuno][t1, t2] +
   Derivative[0, 1][L][t1, t2]) == 0      )

What I need now is to somehow solve the system of equations for t1 and t2,
but as there
are functions with two arguments, I don't know how to use inverse for them.
Is there anyway to tell mathematica to construct a third function to express
the inverse?
If it can't be done, is there any way to obtain the symbolic maximum
arguments t1 and t2?

Thanks for your time!

ep



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