<p>The consumer at each period maximizes
<span class="math-container">\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}</span>
subject to
<span class="math-container">\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}</span>
<span class="math-container">\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}</span></p>
<p>Where</p>
<p><span class="math-container">$\bullet$</span> <span class="math-container">$r_t$</span> denotes the interest rate net of depreciation and <span class="math-container">$w_t$</span> is the wage rate per efficiency unit of labor.</p>
<p><span class="math-container">$\bullet$</span> <span class="math-container">$l_{t}^i$</span>: hours worked, <span class="math-container">$c_{t}^i$</span> consumption of an individual of age <span class="math-container">$i$</span> at time <span class="math-container">$t$</span>, <span class="math-container">$\epsilon^i$</span> denote her efficiency units.</p>
<p><span class="math-container">$\bullet$</span>  The goverment in this economy finances an exogenous sequence of expenditure <span class="math-container">$\{G\}_{t=0}^\infty$</span> using proportional capital taxes <span class="math-container">$\theta_t$</span>, consumption taxes <span class="math-container">$\eta_t$</span>, labor taxes <span class="math-container">$\tau_t$</span> an debt <span class="math-container">$D_t$</span></p>
<p><span class="math-container">$\bullet$</span> Each generation is born with no assets, and can accumulaate wealth <span class="math-container">$a^{i+1}_{t+1}$</span> by buying one-period goverment debt and lending to firms.</p>
<p>Lagrangian for this problem is</p>
<p><span class="math-container">\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}</span></p>
<p>The necessary conditions with respect to <span class="math-container">$c_{t+i}^i, l_{t+i}^i, a^{i+1}_{t+1}$</span> are respectively</p>
<p><span class="math-container">$\star$</span> deriving with respect to <span class="math-container">$c_{t+i}^i$</span> and equaling zero <span class="math-container">\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}</span></p>
<p><span class="math-container">$\star \star$</span> deriving with respect to <span class="math-container">$l_{t+i}^i$</span> and equaling zero <span class="math-container">\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}</span></p>
<p><span class="math-container">$\star \star \star$</span> deriving with respect to <span class="math-container">$a_{t+1}^{i+1}$</span> and equaling zero
<span class="math-container">\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}</span></p>
<p>Questions:
.</p>
<p>_Is the Lagrangian correct? It guided me from another document</p>
<p>_ Are the first order conditions correct? In other books I see that in (<span class="math-container">$\star \star$</span>) is with negative sign <span class="math-container">$ \beta^{i-1}U_l(c^i_{t+i},l^i_{t+i})$</span>.</p>
<p>Any help or recommendation I will be very grateful.
Thanks</p>
<p>Paper <a href="https://journals.sagepub.com/doi/full/10.1177/1091142117735601" rel="nofollow noreferrer">https://journals.sagepub.com/doi/full/10.1177/1091142117735601</a></p>

CC BY-SA 4.0
2021-12-12T17:35:44.987Z
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.
erik
CC BY-SA 4.0
2021-12-12T17:50:32.550Z
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.
erik
CC BY-SA 4.0
2021-12-12T17:51:44.343Z
This is a cool way of writing OLG. I learned something new. Welcome to Economics StackExchange :)
erik
CC BY-SA 4.0
2021-12-13T18:19:04.380Z
@erik please turn your comments into an answer
1muflon1
<macroeconomics><optimization><dynamic-optimization><optimal-taxation><olg>
Necessary conditions in overlapping generations model (OLG)
Franciscolli
# Necessary conditions in overlapping generations model (OLG)

(Asked by: username_0 on 2021-12-12)

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>

**Tags:** macroeconomics, optimization, dynamic-optimization, optimal-taxation, olg
> **Comments:**
> **username_1**: 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.  
> **username_1**: 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.  
> **username_1**: This is a cool way of writing OLG. I learned something new. Welcome to Economics StackExchange :)  
> **username_2**: @erik please turn your comments into an answer  

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## 0 answers

