Let a utility function for a consumer be defined as $u(x_{1},x_{2})=x_{1}^{1/2} x_{2}^{1/2}$. With the budget $x_{1}p_{1}+x_{2}p_{2}=m$. With values $p_1=p_2=1, m=32$. The state now adds a tax of unit 3 on $p_{1}$ (pr. Unit $x_1$)
How does it effect utility? What does the state earn?
- I got the utility before to be 16 and after to be 8 with taxes correlating to 12 pr unit $x_1$
The state now considers an income tax such that the income is now $m-T$ How much will the state earn with the new system whilst keeping the consumer indifferent? Which system is better?
- I figured that i solve for the Tax in the utility function under optimal demand conditions so that i kepy utility equal to 8. This gave me 16 units of income tax.
How does one do the last part mathematically. I figure the income tax is better for the consumer but how can i show it mathematically?