Necessary conditions in overlapping generations model (OLG)

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

• 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
Dec 12 '21 at 17:35
• 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
Dec 12 '21 at 17:50
• This is a cool way of writing OLG. I learned something new. Welcome to Economics StackExchange :)
– erik
Dec 12 '21 at 17:51