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The consumer at each period maximizes \begin{equation} \displaystyle\sum_{i=1}^{I}\beta^{i-1}U(c^i_{t+i},l^i_{t+i}) , t=0,1,2,3,... \end{equation} subject to \begin{equation} (1+\eta_t)c^{i}_t+a^{i+1}_{t+1}\leq (1-\tau_t)w_t\epsilon^il_{t}^{i}+(1+r_t(1-\theta_t))a^{i}_t, 1\leq i \leq I, t=0,1,2,.. \end{equation} \begin{equation} a^{1}_{t}=0, 0\leq l_t^{i}\leq 1, c^{i}_{t},a^{I+1}_t\geq 0, t=0,1,2,... \end{equation}

Where

$\bullet$ $r_t$ denotes the interest rate net of depreciation and $w_t$ is the wage rate per efficiency unit of labor.

$\bullet$ $l_{t}^i$: hours worked, $c_{t}^i$ consumption of an individual of age $i$ at time $t$, $\epsilon^i$ denote her efficiency units.

$\bullet$ The goverment in this economy finances an exogenous sequence of expenditure $\{G\}_{t=0}^\infty$ using proportional capital taxes $\theta_t$, consumption taxes $\eta_t$, labor taxes $\tau_t$ an debt $D_t$

$\bullet$ Each generation is born with no assets, and can accumulaate wealth $a^{i+1}_{t+1}$ by buying one-period goverment debt and lending to firms.

Lagrangian for this problem is

\begin{equation} L=\displaystyle\sum_{i=1}^{I}\beta^{i-1}\{U(c^i_{t+i},l^i_{t+i})-\lambda_t[(1+\eta_t)c^{i}_t+a^{i+1}_{t+1}-(1-\tau_t)w_t\epsilon^il_{t}^{i}-(1+r_t(1-\theta_t))a^{i}_{t}]\} \end{equation}

The necessary conditions with respect to $c_{t+i}^i, l_{t+i}^i, a^{i+1}_{t+1}$ are respectively

$\star$ deriving with respect to $c_{t+i}^i$ and equaling zero \begin{equation} \beta^{i-1}U_c(c^i_{t+i},l^i_{t+i})-\beta^{i-1}\lambda_{t+i}(1+\eta_t)=0 \Longleftrightarrow U_c(c^i_{t+i},l^i_{t+i})=\lambda_{t+i}(1+\eta_{t+i}) \end{equation}

$\star \star$ deriving with respect to $l_{t+i}^i$ and equaling zero \begin{equation} \beta^{i-1}U_l(c^i_{t+i},l^i_{t+i})+\beta^{i-1}\lambda_{t+i}(1-\tau_{t+i})w_{t+i}\epsilon^i=0 \Longleftrightarrow U_l(c^i_{t+i},l^i_{t+i})=-\lambda_{t+i}(1-\tau_{t+i})w_{t+i}\epsilon^i \end{equation}

$\star \star \star$ deriving with respect to $a_{t+1}^{i+1}$ and equaling zero \begin{equation} -\beta^{i-1}\lambda_{t}+\beta^{i-1}\lambda_{t+1}(1+r_{t+1}(1-\theta_{t+1}))=0\Longleftrightarrow \lambda_{t}\beta^{i-1}=\beta^{i-1}\lambda_{t+1}(1+r_{t+1}(1-\theta_{t+1})) \Longleftrightarrow \lambda_{t}=\lambda_{t+1}(1+r_{t+1}(1-\theta_{t+1})) \end{equation}

Questions: .

_Is the Lagrangian correct? It guided me from another document

_ Are the first order conditions correct? In other books I see that in ($\star \star$) is with negative sign $ \beta^{i-1}U_l(c^i_{t+i},l^i_{t+i})$.

Any help or recommendation I will be very grateful. Thanks

Paper https://journals.sagepub.com/doi/full/10.1177/1091142117735601

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  • $\begingroup$ Your FOC for labor is correct. LHS is the marginal disutility from labor, which should be negative. So the negative sign on the RHS balances the signs. You are missing a discount factor on your Euler - so might want to check the discounts on your Lagrangian. $\endgroup$
    – erik
    Dec 12 '21 at 17:35
  • $\begingroup$ Maybe you forgot some of the indices for time. For example when time moves forward by one period $\lambda$ should change by a period - your Lagrangian does not have that though you do move the multiplier forward in the FOC. And I think you maybe forgot to move the exponent of $\beta$ forward by one period - which gets rid of the discount factor in your multiplier equation. $\endgroup$
    – erik
    Dec 12 '21 at 17:50
  • $\begingroup$ This is a cool way of writing OLG. I learned something new. Welcome to Economics StackExchange :) $\endgroup$
    – erik
    Dec 12 '21 at 17:51
  • $\begingroup$ @erik please turn your comments into an answer $\endgroup$
    – 1muflon1
    Dec 13 '21 at 18:19

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