Solving Nash bargaining model

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charlotte88
charlotte88 2016-1-20
编辑： Torsten 2016-1-21
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)).
  2 个评论
Walter Roberson
Walter Roberson 2016-1-21
What are the constraints on the ranges of values? For example is everything real-valued and non-negative? Is 0 < a < 1 ?
Torsten
Torsten 2016-1-21
编辑：Torsten 2016-1-21
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.

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回答（1 个）

Alan Weiss
Alan Weiss 2016-1-20
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
  1 个评论
charlotte88
charlotte88 2016-1-21
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!

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