Question

In: Economics

Find nash equilibrium for t and y utitily functions for rich (r) and poor (p) y...

Find nash equilibrium for t and y

utitily functions for rich (r) and poor (p)

y is income and t is tax.

ur = y(1 − t) − 0.5y 2

up(t, y) = ty

Solutions

Expert Solution

We have the utility function of rich and poor as follows:

The utility function of rich is

The utility of poor is

Here, the two players are rich (r) and poor (p), where t = tax and y = income.

Considering the utility function of rich,

if 'r' chooses (t) then we have to differentiate the utility function of rich with respect to (t), hence we get the following

if 'r' chooses (y) then we have to differentiate the utility function of rich with respect to (y), hence we get the following

The Nash Equilibria occurs at the best response of (t) and (y) at (t*,y*), hence we have the Nash equilibrium of rich as follows

Considering the utility function of poor,

if 'r' chooses (t) then we have to differentiate the utility function of poor with respect to (t), hence we get the following

if 'r' chooses (y) then we have to differentiate the utility function of rich with respect to (y), hence we get the following

The Nash Equilibria occurs at the best response of (t) and (y) at (,), hence we have the Nash equilibrium of poor as follows

Conclusion:

The Nash Equilibrium of rich

The Nash Equilibrium of poor

Rahul Sunny answered 2 years ago
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