<p>That's a coincidence, because they assume nondepreciating capital. If $\delta&gt;0$ was positive we'd have
\begin{align}
\dot K = Y - C - \delta K
\end{align}
which gives
\begin{align}
\dot K = 0  \quad \Rightarrow\quad C = Y - \delta K.
\end{align}</p>

<p>But the static optimization would still read
\begin{align}
&amp;U(C(t)):=\max_{c(t-n,n)}\int^\infty_0 u(c(t-n,n))dn \\
&amp;s.t.\quad \int^\infty_0 u(c(t-n,n))dn \leq C(t) 
\end{align} 
Actually they define an indirect utility function with subject to a resource constraint. In every period (thus static) the sum of individual consumption cannot exceed aggregate consumption. </p>

CC BY-SA 3.0
2015-08-03T01:38:57.280Z
thanks for your answer. But in this case, I don't understand the fact that authors define the aggregate consumption in page 416. So in this case, why they put a constraint on this first stage on aggregate consumption ?
We know that $c\left(t\right)=\int_{0}^{\infty}z\left(t-n,t\right)P\left(n\right)dn$. So, why the constraint does not vanish ?
optimal control
CC BY-SA 3.0
2015-08-03T07:17:40.643Z
(Haven't read the paper. This is how I get the math). They define aggregate consumption to make the dynamic optimization handy. You can easily see the point if you switch stages. You solve over the  periods $t\in[0,\infty)$ for an optimal level of aggregate consumption $\{C(t)\}_{t\geq 0}$ (dynamic optimization). Then you allocate in every period $C(t)$ over the residents, making $C(t)$ a resource constraint in every period. Cause this holds for every period it boils down to a static problem in every period $t$. Dynamic and static optimization are seperated. There is no cennection between...
clueless
CC BY-SA 3.0
2015-08-03T07:19:00.877Z
...the law of motion $\dot K$ and the static problem. Disclaimer: this is my interpretation. But it makes sense to me.
clueless
CC BY-SA 3.0
2015-08-03T12:46:07.987Z
@cluless Sorry to bother you, I have a last question in order to take your advice. Is it legitmate to put also "another"  constraint as $C(t)\geq Z(t)$ where $Z(t)=\int_{0}^{\infty}z\left(t-\tau,t\right)P\left(\tau\right)d\tau=\rho\int_{T}^{t}e^{-\rho\left(t-s\right)}\underbrace{\left\{ \int_{0}^{\infty}c\left(s-\tau,t\right)P\left(\tau\right)d\tau\right\} }ds$. which is a stock variable. (it is a habit stock, as in Ryder and Heal (1972)) $Z$ is a variable that I define here.

Thanks so much in advance for your help ! :)
optimal control
CC BY-SA 3.0
2015-08-04T13:46:05.930Z
No idea. You may specify the question in a seperate thread. But, why not impose a secod constraint?
clueless
CC BY-SA 3.0
2015-08-02T21:28:01.503Z
2015-08-02T21:46:15.810Z
2015-08-02T21:46:15.810Z
clueless
<p>The interesting paper <a href="https://www.dropbox.com/s/huhv6uzfnlvojbi/Calvo%2C%20Obstfeld%20-%201988%20-%20Optimal%20Time-Consistent%20Fiscal%20Policy%20with%20Finite%20Lifetimes.pdf?dl=0" rel="nofollow">Calvo and Obstfeld (1988)</a> uses two-stage optimization on an OLG model which then reduces to a standard representative agent framework. </p>

<p>First stage optimization consists on a static optimization which makes the optimal allocation between different cohorts vertically(c.f equation (9) in the paper.) Authors solve this first stage problem as:</p>

<p>$$\mathcal{L}=u\left[c\left(t-n,t\right)\right]\Delta\left(n\right)P\left(n\right)e^{\rho n}+\lambda\left[C\left(t\right)-\int_{0}^{\infty}c\left(t-n,t\right)P\left(n\right)d\left(n\right)\right]$$</p>

<p>where $C\left(t\right)=\int_{0}^{\infty}c\left(t-n,t\right)P\left(n\right)dn$. </p>

<p>$n, P(n), \Delta\left(n\right)$ are given in the paper and not relevant for my question at this moment. </p>

<p>Normally, in this paper, the dynamics of capital accumulation are given as follows:</p>

<p>$$\dot{K}\left(t\right)=Y\left[K\left(t\right)\right]-C\left(t\right)$$</p>

<p>In fact, my question is trivial but I could not be sure.</p>

<p>Is the part with bracket in Lagrangian comes from $\dot{K}=0$ which gives us the equality $Y\left[K\left(t\right)\right]=C\left(t\right)=\int_{0}^{\infty}c\left(t-n,t\right)P\left(n\right)dn$ ?</p>

<p>As the Lagrangian is for a "static" problem, I think it makes sense but I can not be sure if this is the case.</p>

<p>Any suggestion or hints are welcome. </p>

<macroeconomics><optimization>
static/dynamic optimization
optimal control
# static/dynamic optimization

(Asked by: username_0 on 2015-08-02)

The interesting paper [Calvo and Obstfeld (1988)](https://www.dropbox.com/s/huhv6uzfnlvojbi/Calvo%2C%20Obstfeld%20-%201988%20-%20Optimal%20Time-Consistent%20Fiscal%20Policy%20with%20Finite%20Lifetimes.pdf?dl=0) uses two-stage optimization on an OLG model which then reduces to a standard representative agent framework.

First stage optimization consists on a static optimization which makes the username_0 allocation between different cohorts vertically(c.f equation (9) in the paper.) Authors solve this first stage problem as:

$$\mathcal{L}=u\left[c\left(t-n,t\right)\right]\Delta\left(n\right)P\left(n\right)e^{\rho n}+\lambda\left[C\left(t\right)-\int\_{0}^{\infty}c\left(t-n,t\right)P\left(n\right)d\left(n\right)\right]$$

where $C\left(t\right)=\int\_{0}^{\infty}c\left(t-n,t\right)P\left(n\right)dn$.

$n, P(n), \Delta\left(n\right)$ are given in the paper and not relevant for my question at this moment.

Normally, in this paper, the dynamics of capital accumulation are given as follows:

$$\dot{K}\left(t\right)=Y\left[K\left(t\right)\right]-C\left(t\right)$$

In fact, my question is trivial but I could not be sure.

Is the part with bracket in Lagrangian comes from $\dot{K}=0$ which gives us the equality $Y\left[K\left(t\right)\right]=C\left(t\right)=\int\_{0}^{\infty}c\left(t-n,t\right)P\left(n\right)dn$ ?

As the Lagrangian is for a "static" problem, I think it makes sense but I can not be sure if this is the case.

Any suggestion or hints are welcome.

**Tags:** macroeconomics, optimization


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## 1 answers

### Answer by username_1 on 2015-08-02. Score: 2 | Accepted answer

That's a coincidence, because they assume nondepreciating capital. If $\delta>0$ was positive we'd have
\begin{align}
\dot K = Y - C - \delta K
\end{align}
which gives
\begin{align}
\dot K = 0 \quad \Rightarrow\quad C = Y - \delta K.
\end{align}

But the static optimization would still read
\begin{align}
&U(C(t)):=\max\_{c(t-n,n)}\int^\infty\_0 u(c(t-n,n))dn \\
&s.t.\quad \int^\infty\_0 u(c(t-n,n))dn \leq C(t)
\end{align}
Actually they define an indirect utility function with subject to a resource constraint. In every period (thus static) the sum of individual consumption cannot exceed aggregate consumption.

> **Comments:**
> **username_0**: thanks for your answer. But in this case, I don't understand the fact that authors define the aggregate consumption in page 416. So in this case, why they put a constraint on this first stage on aggregate consumption ? We know that $c\left(t\right)=\int_{0}^{\infty}z\left(t-n,t\right)P\left(n\right)dn$. So, why the constraint does not vanish ?  
> **username_1**: (Haven't read the paper. This is how I get the math). They define aggregate consumption to make the dynamic optimization handy. You can easily see the point if you switch stages. You solve over the  periods $t\in[0,\infty)$ for an username_0 level of aggregate consumption $\{C(t)\}_{t\geq 0}$ (dynamic optimization). Then you allocate in every period $C(t)$ over the residents, making $C(t)$ a resource constraint in every period. Cause this holds for every period it boils down to a static problem in every period $t$. Dynamic and static optimization are seperated. There is no cennection between...  
> **username_1**: ...the law of motion $\dot K$ and the static problem. Disclaimer: this is my interpretation. But it makes sense to me.  
> **username_0**: @cluless Sorry to bother you, I have a last question in order to take your advice. Is it legitmate to put also "another"  constraint as $C(t)\geq Z(t)$ where $Z(t)=\int_{0}^{\infty}z\left(t-\tau,t\right)P\left(\tau\right)d\tau=\rho\int_{T}^{t}e^{-\rho\left(t-s\right)}\underbrace{\left\{ \int_{0}^{\infty}c\left(s-\tau,t\right)P\left(\tau\right)d\tau\right\} }ds$. which is a stock variable. (it is a habit stock, as in Ryder and Heal (1972)) $Z$ is a variable that I define here.  Thanks so much in advance for your help ! :)  
> **username_1**: No idea. You may specify the question in a seperate thread. But, why not impose a secod constraint?  

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