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Question assumptions:

Consider the effect of a capital tax on the OLG model. The government imposes a capital tax rate at the rate $\tau\in[0,1)$ and pays all the tax revenues back to the old in a lump sum T. Assuming log utility, the maximization problem for a household born at the beginning of t is:

$\max lnc_{1t}+\beta lnc_{2t+1}$ s.t. $c_{1t}=w_t-a_{t+1}$ and $c_{2t+1}=(1-\tau)(1+r_{t+1})a_{t+1}+T_{t+1}$

(a) derive the savings of a young consumer of generation t for a given $w_t$, $r_{t+1}$, and $T_{t+1}$.

My solution attempt:

We know that the utility of consumers born at the beginning of t is:

$u(c_{1t},c_{2t+1})=u(c_{1t})+\beta u(c_{2t+1})$ (1)

Where $\beta\in(0,1)$ is the discount factor, c is consumption when young (1t) and old (2t+1)

Our budget constraint is:

$c_{1t}= w_t-a_{t+1}$ (2)

$c_{2t+1}= (1+r_{t+1})a_{t+1}$ (3)

Eliminating $a_{t+1}$ we arrive at our lifetime budget constrain which is:

$c_{1t}+\frac{c_{2t+1}}{1+r_{t+1}}=w_t$ (4)

$(w_t-a_{t+1})+\frac{[(1-\tau)(1+r_{t+1})a_{t+1}+T_{t+1}}{1+r_{t+1}}$

Our log utility function is:

$\frac{1}{c_t}=\beta (1+r_{t+1})\frac{1}{c_{2t+1}}$

with the FOC being: $c_{2t+1}=\beta (1+r_{t+1})c_{1t}$

Comment: I am unsure if this is the appropriate budget constraint and utility function based on my problem set. Furthermore, when they ask me to derive the savings do they mean the law of motion of savings? Any help would be appreciated.

Notice: the problem is unfinished as I am still working out the solution, and I am just unsure if the basic materials needed to derive the solution are correct. Thank you to anyone who provides a solution.

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