<p>I am trying to derive an expression for the optimal level of consumption in the  basic problem:</p>
<p><span class="math-container">$ max \hspace{1cm} U_t = E_t \left[\sum_{s=t}^{\infty}  \beta^{s-t} \left( C_s - a\frac{Cs}{2}^2 \right) \right] $</span></p>
<p><span class="math-container">$ s.t \hspace{1cm} A_{s + 1} = A_s(1+r) + Y_s - C_s  $</span></p>
<p>The problem leads to the Euler Equation:</p>
<p><span class="math-container">$  E_t \left[ \frac{u'(C_s)}{u'(C_{s+1})} \right] = \beta (1+r)$</span></p>
<p>Assuming <span class="math-container">$\beta = (1+r)^{-1}$</span> we have:</p>
<p><span class="math-container">$  E_t \left[ \frac{u'(C_s)}{u'(C_{s+1})} \right] = 1$</span></p>
<p>Which leads to:</p>
<p><span class="math-container">$ C_s = E_t[C_{s+1}]$</span>.</p>
<p>Iterating forward the BC and using the trasversality condition I derived the intertemporal BC:</p>
<p><span class="math-container">$ \sum_{s=t}^{\infty} \frac{C_s}{(1+r)^{s-t}} = A_t(1+r) + \sum_{s=t}^{\infty} \frac{Y_s}{(1+r)^{s-t}} $</span></p>
<p>Now, how can I find an expression for <span class="math-container">$C_t$</span>? If time were finite intuitively (I think) I would have:</p>
<p><span class="math-container">$ C_t =  \frac{1}{T - t}(A_t + \sum_{s=t}^{T} E_t [Y_s])  $</span></p>
<p>How do I deal with the infinite horizon to find and expression for <span class="math-container">$C_t$</span>?</p>
<p>I was wondering if I can use the geometric series, something like:</p>
<p><span class="math-container">$C_t \sum_{s = t}^{\infty} \left( \frac{1}{1+r} \right)^{s-t} = C_t \frac{1}{1 -  \frac{1}{1+r}} = C_t\frac{1+r}{r}  $</span></p>

CC BY-SA 4.0
2021-08-31T15:49:07.960Z
Are you not missing expectations operators in the second equation from the bottom?
BrsG
CC BY-SA 4.0
2021-08-31T17:58:15.557Z
Yes, I could take expectation of both sides and get:  $C_t +  \sum_{s=t+1}^{\infty}E_t\left[ \frac{C_s}{(1+r)^{s-t}} \right] = A_t(1+r) + Y_t +  \sum_{s=t+1}^{\infty}E_t\left[ \frac{Y_s}{(1+r)^{s-t}} \right]$
Giorgetto
CC BY-SA 4.0
2021-08-31T18:10:23.887Z
hence: $ C_t = A_t(1+r) + Y_t + \sum_{s=t+1}^{\infty} E_t \left[ \frac{Y_s - C_s}{(1+r)^{s-t}}\right]$
Giorgetto
CC BY-SA 4.0
2021-08-31T21:25:32.103Z
@Giorgetto is right. If you want to proceed, you would need an expression for $Y_s$ (or the difference). An easy way forward is to consider $Y_t$ a constant, typically $Y_t=1$, making $C_t=f(A_t)$ and having all variables in term of the economy's product for the period.
one_teach_wonder
<macroeconomics><consumption><dynamic-optimization>
Optimal level of consumption in discrete time model with quadratic preferences and infinite horizon
Giorgetto
# Optimal level of consumption in discrete time model with quadratic preferences and infinite horizon

(Asked by: username_0 on 2021-08-31)

I am trying to derive an expression for the optimal level of consumption in the basic problem:

$ max \hspace{1cm} U\_t = E\_t \left[\sum\_{s=t}^{\infty} \beta^{s-t} \left( C\_s - a\frac{Cs}{2}^2 \right) \right] $

$ s.t \hspace{1cm} A\_{s + 1} = A\_s(1+r) + Y\_s - C\_s $

The problem leads to the Euler Equation:

$ E\_t \left[ \frac{u'(C\_s)}{u'(C\_{s+1})} \right] = \beta (1+r)$

Assuming $\beta = (1+r)^{-1}$ we have:

$ E\_t \left[ \frac{u'(C\_s)}{u'(C\_{s+1})} \right] = 1$

Which leads to:

$ C\_s = E\_t[C\_{s+1}]$.

Iterating forward the BC and using the trasversality condition I derived the intertemporal BC:

$ \sum\_{s=t}^{\infty} \frac{C\_s}{(1+r)^{s-t}} = A\_t(1+r) + \sum\_{s=t}^{\infty} \frac{Y\_s}{(1+r)^{s-t}} $

Now, how can I find an expression for $C\_t$? If time were finite intuitively (I think) I would have:

$ C\_t = \frac{1}{T - t}(A\_t + \sum\_{s=t}^{T} E\_t [Y\_s]) $

How do I deal with the infinite horizon to find and expression for $C\_t$?

I was wondering if I can use the geometric series, something like:

$C\_t \sum\_{s = t}^{\infty} \left( \frac{1}{1+r} \right)^{s-t} = C\_t \frac{1}{1 - \frac{1}{1+r}} = C\_t\frac{1+r}{r} $

**Tags:** macroeconomics, consumption, dynamic-optimization

> **Comments:**
> **username_1**: Are you not missing expectations operators in the second equation from the bottom?  
> **username_0**: Yes, I could take expectation of both sides and get:  $C_t +  \sum_{s=t+1}^{\infty}E_t\left[ \frac{C_s}{(1+r)^{s-t}} \right] = A_t(1+r) + Y_t +  \sum_{s=t+1}^{\infty}E_t\left[ \frac{Y_s}{(1+r)^{s-t}} \right]$  
> **username_0**: hence: $ C_t = A_t(1+r) + Y_t + \sum_{s=t+1}^{\infty} E_t \left[ \frac{Y_s - C_s}{(1+r)^{s-t}}\right]$  
> **username_2**: @username_0 is right. If you want to proceed, you would need an expression for $Y_s$ (or the difference). An easy way forward is to consider $Y_t$ a constant, typically $Y_t=1$, making $C_t=f(A_t)$ and having all variables in term of the economy's product for the period.  

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## 0 answers

