Question:
Suppose Mr X maximises his utility for two periods, his total utility is given by log $c_1$ + $\beta$ log $c_2$ where $\beta$ $\in$ ( 0,1) and $c_1$ and $c_2$ are his consumption for period 1 and period 2 respectively. Suppose he earns a wage only in period 1 and it is given by W. He saves for the second period on which he enjoys a gross return of (1+r) where r > 0 is the net interest rate. Suppose the government implements a scheme where T $\geq$ 0 is collected from agents ( this is also from Mr. X) in the first period and returns the same amount T back in the second period. What is the optimum T for which his total utility is maximised?.
My working: I tried putting $c_2= [(M - c_1- T ) ( 1 + r )] + T$ and plugged in the values that helps the consumer achieve optimum utility for the given budget constraint. Using the lagrangean method of optimisation I got the following $$\frac{c_2}{\beta c_1} = 1+r$$ Solving the budget constraint using this, I got $$ c_2=\frac{\beta[ M ( 1 + r) - Tr ]}{\beta+ 1}$$Next I tried differentiating $c_2$ with respect to T so as to find the optimum T that maximises $c_2$ and in turn the utility function. This method doesn't seem to work. Could somebody correct me ?. Thank you.
