# Computation of Different Axiomatic Barganing Solutions

I need help with the following question.

Take a two-agent bargaining problem. d=(0,0). Pick a strictly concave function for the bargaining frontier. Fix the intersection of the frontier with the y-axis to be at 1. Vary the intersection of the frontier with the x-axis between 0.1 and 10 with increments of 0.1. Graph the points proposed by the following solution concepts.

So y^2=1-x^2/0.1 can be the function. Does anyone know how can it be computed in matlab or a similar program?

• Not sure if there's an existing MATLAB program for it, but it should be pretty easy to code one yourself. Have you tried this at all? Mar 11 '20 at 3:43

The frontier functions you seem to propose are of the form $$y_n(x)=\sqrt{1-\frac{100x^2}{n^2}}$$, where $$n\in\{1,2,\ldots,100\}$$.
For the Nash solution you maximize $$xy_n(x)$$ and after substituting for $$y_n(x)$$ and setting the derivative to zero you will find $$x_n=\frac{n}{\sqrt{200}}$$ and $$y_n=\frac{1}{\sqrt{2}}$$. For the KS solution you set $$y_n(x)=\frac{10x}{n}$$ and after substituting you will again find $$x_n=\frac{n}{\sqrt{200}}$$ and $$y_n=\frac{1}{\sqrt{2}}$$. Graphing these solutions is easy.
Remark: That the solutions coincide is not a coincidence. The different frontier functions are constructed in such a way that they are all just horizontally "compressed" or "stretched" versions of the frontier function $$y_{10}(x)=\sqrt{1-x^2}$$. This means that they are really all equivalent in the sense that they result from each other after some rescaling of $$x$$ (the utility of agent 1). But the frontier function $$y_{10}$$ just describes the quarter unit circle in the positive orthant, which is symmetric w.r.t. the agents. In symmetric feasible sets the Nash and the KM solutions are known to coincide.