The board of directors of two companies determines the salary of its CEO according to the following reaction functions:
S1 = 100,000 + (1/2) S2
S2 = 70,000 + (2/3) S1
Where Si is salary of company i's CEO (i= 1,2) Use the inverse matrix method to find the nash equilibrium in this problem
If I want to turn this into a matrix function from the start, I have to move the left values to the right, correct? making them.
0 = 100,000 + (1/2) S2 - S1
0 = 70,000 + (2/3) S1 - S2
Giving me the matrix function, will this work :/ ? Thanks
You have
Now the $S_1,S_2$-terms have to be put to LHS:
$ - (1/2) S_2 + S_1 = 100,000 $
$ -(2/3) S_1 + S_2 = 70,000 $
Now rearrange the $S_1,S_2$-terms of the first equation.
$S_1 - (1/2) S_2 = 100,000 $
$ -(2/3) S_1 + S_2 = 70,000 $
This linear equation can written in matrix form:
$\begin{pmatrix}1 & -\frac{1}{2} \\ -\frac{2}{3} & 1 \end{pmatrix} \cdot \begin{pmatrix}S_1 \\ S_2 \end{pmatrix}=\begin{pmatrix} 100,000 \\ 70,000 \end{pmatrix}$
Thus $\begin{pmatrix}S_1 \\ S_2 \end{pmatrix}=\begin{pmatrix}1 & -\frac{1}{2} \\ -\frac{2}{3} & 1 \end{pmatrix} ^{-1} \cdot\begin{pmatrix} 100,000 \\ 70,000 \end{pmatrix}$
The inverse of $\begin{pmatrix}1 & -\frac{1}{2} \\ -\frac{2}{3} & 1 \end{pmatrix}$ is $\begin{pmatrix}\frac{3}{2} & \frac{3}{4} \\ 1 & \frac{3}{2} \end{pmatrix}\Rightarrow \begin{pmatrix}S_1 \\ S_2 \end{pmatrix}=\begin{pmatrix}\frac{3}{2} & \frac{3}{4} \\ 1 & \frac{3}{2} \end{pmatrix} \cdot\begin{pmatrix} 100,000 \\ 70,000 \end{pmatrix} $
Therefore $S_1=100,000\cdot \frac{3}{2}+70,000\cdot \frac{3}{4}$
and $S_2=100,000+ 70,000\cdot \frac{3}{2}$