The individual lives for two periods. He has a utility function $u(c_{1} )+ bu(c_2)$. His budget constraint requires that his period I consumption be his period I endowment minus any savings, $c_1 = w_1 - s$ . In the second period, his consumption will be $c_2 = w_2 + (1 + r)qs + a$. The government has taxed savings at the rate $1 - q$, and uses it to finance a lump-sum transfer of $a$. The individual takes $q$ and $a$ as given, but for the government the government’s budget to balance, we must have $a = (1 - q)(1 + r)s$ . Find $\frac{\partial s}{\partial q}$
I had thought of this problem for a while, but I still could not set up the utility maximization problem in order to apply the 1st-order condition and Implicit Function Theorem. Can anyone please help me with the setup?
You can write the individual's utility as a function of savings $s$ only: $$u(w_1-s)+bu(w_2+(1+r)qs+a)$$ The first order condition for choosing $s$ optimally is: $$-u'(w_1-s)+b(1+r)qu'(w_2+(1+r)qs+a)=0$$ We treat $a$ as a constant because from the individual's point of view it is a lump-sum transfer. Next, you replace $a$ using the government budget constraint. Doing this, and simplifying, you get the first order condition: $$-u'(w_1-s)+b(1+r)qu'(w_2+(1+r)s)=0$$ This equation, implicitly, under appropriate conditions, defines the optimal $s$ as a function of $q$, and you can find $\partial s/\partial q$ using the implicit function theorem.