Solving Nash bargaining model

Hi everyone,

I need to solve a Nash bargaining model (a bit complex one), a need help on how to do this in matlab (I have never done this before!!). My function looks like this:

(x1 - b1(x1-s1)^(2))^(a)(x2 - b2(x2-s2)^(2))^(1-a) subject to: x1 + x2 + x3 = 1

When doing this "by hand" I first insert for x2, and then I want to maximize it with respect to x1 and x3 and solve for these two variables.

Hence I have:

(x1 - b1(x1-s1)^(2))^(a)(1-x1-x3 - b2(1-x1-x3-s2)^(2))^(1-a)

Since it is often simpler to do this is log, this turns out to be:

a ln(x1 - b1(x1-s1)^(2)) + (1-a) ln(1-x1-x3 - b2(1-x1-x3-s2)^(2))

Can anyone help me with where to start? I want x3 as a function of a (or (1-a)).

[Walter Roberson]

What are the constraints on the ranges of values? For example is everything real-valued and non-negative? Is 0 < a < 1 ?

[Torsten]

I don't see x3 in the objective function.

Is your problem equivalent to

max: (x1 - b1(x1-s1)^(2))^(a)(x2 - b2(x2-s2)^(2))^(1-a) 
subject to
x1,x2 >=0
x1+x2 <=1

?

Best wishes

Torsten.

Answer 1

[Alan Weiss]

I think that you might have to resign yourself to obtaining numerical solutions, rather than analytic solutions in closed form. If you have Optimization Toolbox you can use fmincon to minimize an objective function subject to various constraints. See the Getting Started example, or the examples of constrained nonlinear minimization.

If you have Symbolic Math Toolbox you might be able to get an analytic solution. See the documentation.

By the way, do you mean to assume that x1, x2, and x3 are each positive? If so, you need to put that in the problem formulation.

Good luck,

Alan Weiss

MATLAB mathematical toolbox documentation

[charlotte88]

Thanks Alan! Yes x1, x2 and x3 are all non-negative (can be zero). I will look into the optimization toolbox and see if I manage! Thanks again!