<p>I'm trying to derive the first order condition in a partial equilibrium overlapping generations model. The setup contains both consumption and housing goods.</p>

<p><span class="math-container">$$
\underset{c_{t+i},h_{t+i}}{max}U_{t}^{N}=\sum_{i=0}^{N}\beta^{i}\frac{1}{1-\gamma}\left(h_{t+i}^{\alpha}c_{t+i}^{1-\alpha}\right)^{1-\gamma}
$$</span>
S.t.
<span class="math-container">$$
a_{t+i}=y_{t+i}+\left(1+r_{t+i-1}\right)a_{t+i-1}+q_{t+i}(1-d)h_{t+i-1}-p_{t+i}c_{t+i}-q_{t+i}h_{t+i}
$$</span>
<span class="math-container">$$
a_{t+N}\geq 0
$$</span>
<span class="math-container">$$
a_{t}=0
$$</span></p>

<p>The paper which I'm trying to reproduce finds the following FOCs
<span class="math-container">$$
\frac{h_{t+i}}{c_{t+i}}=\frac{\alpha}{1-\alpha}\frac{p_{t+i}}{R_{t+i}}
$$</span>
and
<span class="math-container">$$
\frac{c_{t+i+1}}{c_{t+i}}=\left(\frac{p_{t+i}/R_{t+i}}{p_{t+i+1}/R_{t+i+1}}\right)^{\frac{\alpha(1-\gamma)}{\gamma}}\left(\beta\left(1+\rho_{t+i}\right)\right)^{1/\gamma}
$$</span>
where
<span class="math-container">$$
R_{t+i}=\frac{q_{t+i}}{\left(1+\rho_{t+i}^{q}\right)}\left(\rho_{t+i}^{q}+d\right)
$$</span>
and
<span class="math-container">$$
1+\rho_{t+i}^{q}=\left(1+r_{t+i}\right)\frac{q_{t+i}}{q_{t+i+1}}
$$</span></p>

<p>In particular I am struggling to setup the lagrangian and dealing with the fact that utility has to be maximized across periods.
Thank you!</p>

<p>-----------edit----------
I've come pretty far in solving the problem. But I would still be grateful for input.</p>

<p>The lagrangian:
<span class="math-container">$$
\mathcal{L}=\sum_{i=0}^{N}\left(\frac{1}{1-\gamma}\beta^{i}\left(h_{t+i}^{\alpha}c_{t+i}^{1-\alpha}\right)^{1-\gamma}+\lambda_{t+i}\left(y_{t+i}+\left(1+r_{t+i-1}\right)a_{t+i-1}+q_{t+i}(1-d)h_{t+i-1}-p_{t+i}c_{t+i}-q_{t+i}h_{t+i}-a_{t+i}\right)\right)
$$</span></p>

<p>FOC
<span class="math-container">$$
\frac{\partial\mathcal{L}}{\partial c_{t+i}}=0
$$</span>
<span class="math-container">$$
\beta^{i}\left(h_{t+i}^{\alpha}c_{t+i}^{1-\alpha}\right)^{-\gamma}h_{t+i}^{\alpha}\left(1-\alpha\right)c_{t+i}^{-\alpha} =\lambda_{t+i}p_{t+i}
$$</span>
<span class="math-container">$$
\frac{\partial\mathcal{L}}{\partial h_{t+i}}=0
$$</span>
<span class="math-container">$$
\Leftrightarrow\beta^{i}\left(h_{t+i}^{\alpha}c_{t+i}^{1-\alpha}\right)^{-\gamma}c_{t+i}^{1-\alpha}\alpha h_{t+i}^{\alpha-1} =\lambda_{t+i}q_{t+i}-\lambda_{t+i+1}q_{t+i+1}(1-d)
$$</span>
<span class="math-container">$$
\frac{\partial\mathcal{L}}{\partial a_{t+i}}=0
$$</span>
<span class="math-container">$$
\lambda_{t+i+1}\left(1+r_{t+i}\right)-\lambda_{t+i} =0
$$</span>
<span class="math-container">$$
\Leftrightarrow\lambda_{t+i+1} =\frac{1}{\left(1+r_{t+i}\right)}\lambda_{t+i}
$$</span>
<span class="math-container">$$
\beta^{i}\left(h_{t+i}^{\alpha}c_{t+i}^{1-\alpha}\right)^{-\gamma}c_{t+i}^{1-\alpha}\alpha h_{t+i}^{\alpha-1}-\lambda_{t+i}q_{t+i}+\frac{1}{\left(1+r_{t+i}\right)}\lambda_{t+i}q_{t+i+1}(1-d)
$$</span>
<span class="math-container">$$
\beta^{i}\left(h_{t+i}^{\alpha}c_{t+i}^{1-\alpha}\right)^{-\gamma}c_{t+i}^{1-\alpha}\alpha h_{t+i}^{\alpha-1}=\lambda_{t+i}\left(q_{t+i}-\frac{1}{\left(1+r_{t+i}\right)}q_{t+i+1}(1-d)\right)
$$</span>
<span class="math-container">$$
\beta^{i}\left(h_{t+i}^{\alpha}c_{t+i}^{1-\alpha}\right)^{-\gamma}h_{t+i}^{\alpha}\left(1-\alpha\right)c_{t+i}^{-\alpha}=\lambda_{t+i}p_{t+i}
$$</span>
<span class="math-container">$$
\frac{h_{t+i}}{c_{t+i}}\frac{1-\alpha}{\alpha}=\frac{p_{t+i}}{q_{t+i}-\frac{q_{t+i+1}(1-d)}{\left(1+r_{t+i}\right)}}
$$</span>
<span class="math-container">$$
\frac{h_{t+i}}{c_{t+i}}=\frac{\alpha}{1-\alpha}\frac{p_{t+i}}{q_{t+i}\left[1-\frac{q_{t+i+1}(1-d)}{q_{t+i}\left(1+r_{t+i}\right)}\right]}
$$</span>
substituting definitions
<span class="math-container">$$
R_{t+i}=\frac{q_{t+i}}{\left(1+r_{t+i}\right)\frac{q_{t+i}}{q_{t+i+1}}}\left(\left(1+r_{t+i}\right)\frac{q_{t+i}}{q_{t+i+1}}-1+d\right)
$$</span>
<span class="math-container">$$
\frac{h_{t+i}}{c_{t+i}}=\frac{\alpha}{1-\alpha}\frac{p_{t+i}}{R_{t+i}}
$$</span></p>

<p>Is this approach correct? Is a global maximum guaranteed when maximizing across two periods?</p>

<p>Also I am still having problems with the second FOC
<span class="math-container">$$
\frac{c_{t+i+1}}{c_{t+i}}=\left(\frac{p_{t+i}/R_{t+i}}{p_{t+i+1}/R_{t+i+1}}\right)^{\frac{\alpha(1-\gamma)}{\gamma}}\left(\beta\left(1+\rho_{t+i}\right)\right)^{1/\gamma}
$$</span></p>

CC BY-SA 4.0
2020-02-20T22:27:24.817Z
Can you give a reference to the paper in question?
Herr K.
CC BY-SA 4.0
2020-02-25T06:08:20.520Z
what are $h_t$, $q_t$. $R_t$, $a_t$? If $d$ is dividends, why is it constant? Also what is $q$?
Grada Gukovic
CC BY-SA 4.0
2020-02-26T10:43:22.767Z
@GradaGukovic h is housing stock, q is the price of housing, R is the nominal user cost of housing, a is net assets.
Mr. A
CC BY-SA 4.0
2020-02-26T10:45:10.107Z
@HerrK. The paper in question: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=597444
Mr. A
<utility><olg>
Deriving FOC in OLG model with Cobb Douglas utility
Mr. A
# Deriving FOC in OLG model with Cobb Douglas utility

(Asked by: username_0 on 2020-02-20)

I'm trying to derive the first order condition in a partial equilibrium overlapping generations model. The setup contains both consumption and housing goods.

$$
\underset{c\_{t+i},h\_{t+i}}{max}U\_{t}^{N}=\sum\_{i=0}^{N}\beta^{i}\frac{1}{1-\gamma}\left(h\_{t+i}^{\alpha}c\_{t+i}^{1-\alpha}\right)^{1-\gamma}
$$
S.t.
$$
a\_{t+i}=y\_{t+i}+\left(1+r\_{t+i-1}\right)a\_{t+i-1}+q\_{t+i}(1-d)h\_{t+i-1}-p\_{t+i}c\_{t+i}-q\_{t+i}h\_{t+i}
$$
$$
a\_{t+N}\geq 0
$$
$$
a\_{t}=0
$$

The paper which I'm trying to reproduce finds the following FOCs
$$
\frac{h\_{t+i}}{c\_{t+i}}=\frac{\alpha}{1-\alpha}\frac{p\_{t+i}}{R\_{t+i}}
$$
and
$$
\frac{c\_{t+i+1}}{c\_{t+i}}=\left(\frac{p\_{t+i}/R\_{t+i}}{p\_{t+i+1}/R\_{t+i+1}}\right)^{\frac{\alpha(1-\gamma)}{\gamma}}\left(\beta\left(1+\rho\_{t+i}\right)\right)^{1/\gamma}
$$
where
$$
R\_{t+i}=\frac{q\_{t+i}}{\left(1+\rho\_{t+i}^{q}\right)}\left(\rho\_{t+i}^{q}+d\right)
$$
and
$$
1+\rho\_{t+i}^{q}=\left(1+r\_{t+i}\right)\frac{q\_{t+i}}{q\_{t+i+1}}
$$

In particular I am struggling to setup the lagrangian and dealing with the fact that utility has to be maximized across periods.
Thank you!

-----------edit----------
I've come pretty far in solving the problem. But I would still be grateful for input.

The lagrangian:
$$
\mathcal{L}=\sum\_{i=0}^{N}\left(\frac{1}{1-\gamma}\beta^{i}\left(h\_{t+i}^{\alpha}c\_{t+i}^{1-\alpha}\right)^{1-\gamma}+\lambda\_{t+i}\left(y\_{t+i}+\left(1+r\_{t+i-1}\right)a\_{t+i-1}+q\_{t+i}(1-d)h\_{t+i-1}-p\_{t+i}c\_{t+i}-q\_{t+i}h\_{t+i}-a\_{t+i}\right)\right)
$$

FOC
$$
\frac{\partial\mathcal{L}}{\partial c\_{t+i}}=0
$$
$$
\beta^{i}\left(h\_{t+i}^{\alpha}c\_{t+i}^{1-\alpha}\right)^{-\gamma}h\_{t+i}^{\alpha}\left(1-\alpha\right)c\_{t+i}^{-\alpha} =\lambda\_{t+i}p\_{t+i}
$$
$$
\frac{\partial\mathcal{L}}{\partial h\_{t+i}}=0
$$
$$
\Leftrightarrow\beta^{i}\left(h\_{t+i}^{\alpha}c\_{t+i}^{1-\alpha}\right)^{-\gamma}c\_{t+i}^{1-\alpha}\alpha h\_{t+i}^{\alpha-1} =\lambda\_{t+i}q\_{t+i}-\lambda\_{t+i+1}q\_{t+i+1}(1-d)
$$
$$
\frac{\partial\mathcal{L}}{\partial a\_{t+i}}=0
$$
$$
\lambda\_{t+i+1}\left(1+r\_{t+i}\right)-\lambda\_{t+i} =0
$$
$$
\Leftrightarrow\lambda\_{t+i+1} =\frac{1}{\left(1+r\_{t+i}\right)}\lambda\_{t+i}
$$
$$
\beta^{i}\left(h\_{t+i}^{\alpha}c\_{t+i}^{1-\alpha}\right)^{-\gamma}c\_{t+i}^{1-\alpha}\alpha h\_{t+i}^{\alpha-1}-\lambda\_{t+i}q\_{t+i}+\frac{1}{\left(1+r\_{t+i}\right)}\lambda\_{t+i}q\_{t+i+1}(1-d)
$$
$$
\beta^{i}\left(h\_{t+i}^{\alpha}c\_{t+i}^{1-\alpha}\right)^{-\gamma}c\_{t+i}^{1-\alpha}\alpha h\_{t+i}^{\alpha-1}=\lambda\_{t+i}\left(q\_{t+i}-\frac{1}{\left(1+r\_{t+i}\right)}q\_{t+i+1}(1-d)\right)
$$
$$
\beta^{i}\left(h\_{t+i}^{\alpha}c\_{t+i}^{1-\alpha}\right)^{-\gamma}h\_{t+i}^{\alpha}\left(1-\alpha\right)c\_{t+i}^{-\alpha}=\lambda\_{t+i}p\_{t+i}
$$
$$
\frac{h\_{t+i}}{c\_{t+i}}\frac{1-\alpha}{\alpha}=\frac{p\_{t+i}}{q\_{t+i}-\frac{q\_{t+i+1}(1-d)}{\left(1+r\_{t+i}\right)}}
$$
$$
\frac{h\_{t+i}}{c\_{t+i}}=\frac{\alpha}{1-\alpha}\frac{p\_{t+i}}{q\_{t+i}\left[1-\frac{q\_{t+i+1}(1-d)}{q\_{t+i}\left(1+r\_{t+i}\right)}\right]}
$$
substituting definitions
$$
R\_{t+i}=\frac{q\_{t+i}}{\left(1+r\_{t+i}\right)\frac{q\_{t+i}}{q\_{t+i+1}}}\left(\left(1+r\_{t+i}\right)\frac{q\_{t+i}}{q\_{t+i+1}}-1+d\right)
$$
$$
\frac{h\_{t+i}}{c\_{t+i}}=\frac{\alpha}{1-\alpha}\frac{p\_{t+i}}{R\_{t+i}}
$$

Is this approach correct? Is a global maximum guaranteed when maximizing across two periods?

Also I am still having problems with the second FOC
$$
\frac{c\_{t+i+1}}{c\_{t+i}}=\left(\frac{p\_{t+i}/R\_{t+i}}{p\_{t+i+1}/R\_{t+i+1}}\right)^{\frac{\alpha(1-\gamma)}{\gamma}}\left(\beta\left(1+\rho\_{t+i}\right)\right)^{1/\gamma}
$$

**Tags:** utility, olg

> **Comments:**
> **username_1**: Can you give a reference to the paper in question?  
> **username_2**: what are $h_t$, $q_t$. $R_t$, $a_t$? If $d$ is dividends, why is it constant? Also what is $q$?  
> **username_0**: @username_2 h is housing stock, q is the price of housing, R is the nominal user cost of housing, a is net assets.  
> **username_0**: @HerrK. The paper in question: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=597444  

---

## 0 answers

