<p>Below is the sketch of the solution.</p>
<p>From the Lagrangian <span class="math-container">$$\max _{\left\{c_{t}, s_{t+1}\right\}} \Pi_{t=0}^{\infty} c_{t}^{\beta^{t}} - \Pi_{t=0}^{\infty}\lambda_t(c_{t}+s_{t+1}-y_{t}-(1+r) s_{t}) \\ \text{s.t.} \ s_0 = 0,  c_t &gt; 0 $$</span>, you can get the Euler equation <span class="math-container">$$\frac{\beta_{t+1}c_{t+1}^{\beta_{t+1}-1}}{\beta_{t}c_{t}^{\beta_{t}-1}}= \frac{\beta_{t+1}\left[y_{t+1}+(1+r)s_{t+1}-s_{t+2}\right]^{\beta_{t+1}-1}}{\beta_{t}\left[y_{t}+(1+r)s_{t}-s_{t+1}\right]^{\beta_{t}-1}} = \frac{1}{1+r}$$</span>, and then you can write <span class="math-container">$s_{t}^{*} \forall t&gt;1$</span> as a function of <span class="math-container">$s_1$</span> with the initial condition <span class="math-container">$s_0$</span>. By using the transversality condition as a terminal condition: <span class="math-container">$$\lim _{t \rightarrow \infty} \lambda_ts_{t+1}= \lim _{t \rightarrow \infty} {\beta_{t}\left[y_{t}+(1+r)s_{t}-s_{t+1}\right]^{\beta_{t}-1}}s_{t+1} =0$$</span>, you can solve  for <span class="math-container">$\left\{s_{t}\right\}_{t=1}^{\infty}$</span> forwards by guessing an initial value for <span class="math-container">$s_1$</span>  and yielding the sequence that does not violate the TVC or the nonnegativity constraint on consumption. And once you get <span class="math-container">$s_{t}^{*} \forall t$</span> you get <span class="math-container">$c_{t}^{*} \forall t$</span>.</p>

CC BY-SA 4.0
2021-09-15T13:56:04.107Z
Sorry, I do not understand this answer. Are you saying $s_0 = 0$ implies that the agent doesn't save any money in $t=0$? That is, $c_0 + s_1 = c_0 = y_0$?
Arturo Sbr
CC BY-SA 4.0
2021-09-16T01:00:08.553Z
@ArturoSbr Sorry I made some mistakes in the initial answer. I think I have corrected it.
Alalalalaki
CC BY-SA 4.0
2021-09-15T00:45:30.993Z
2021-09-16T00:57:05.497Z
2021-09-16T00:57:05.497Z
Alalalalaki
<p>I have the following optimization problem:</p>
<blockquote>
<p><span class="math-container">$\max_{\{c_t, s_{t+1}\}} \Pi_{t=0}^\infty c_t^{\beta^t}$</span></p>
<p><span class="math-container">$\text{subject to } \space c_t + s_{t+1} = y_t + (1 + r) s_t \text{ and } s_0 = 0$</span></p>
</blockquote>
<p>How do I find the optimal consumption level for each time period <span class="math-container">$t$</span>?</p>
<p><strong>Here's what I have so far:</strong></p>
<p>Let <span class="math-container">$p_t = \frac{1}{(1 + r)^t}$</span></p>
<p>Then, from the budget constraint <span class="math-container">$p_{t+1} s_{t+1} - p_{t+1} s_t = p_{t+1}(y_t - c_t)$</span>.</p>
<p>If we sum over <span class="math-container">$T$</span> periods:
<span class="math-container">$(p_1 s_1 - p_0 0) + (p_2 s_2 - p_1 s_1) + ... (p_{T+1} s_{T+1} - p_T s_T) = \sum_{t=0}^\infty p_{t+1}(y_t - c_t)$</span></p>
<p>If I use a natural logarithm transformation on the utility function and assume <span class="math-container">$\lim_{T\to\infty} p_T s_T = 0$</span> (no ponzi schemes allowed), I can rewrite the optimization problem as:</p>
<p><span class="math-container">$\max_{\{c_t\}} \sum_{t=0}^\infty \beta^t\ln(c_t)$</span></p>
<p><span class="math-container">$\text{subject to} \sum_{t=0}^\infty p_{t+1} c_t = \sum_{t=0}^\infty p_{t+1} y_t$</span></p>
<p>Hence, my lagrangian function should look like (unless I'm missing something):</p>
<p><span class="math-container">$\mathcal{L} = \sum \beta^t\ln(c_t) - \lambda(\sum_{t=0}^\infty p_{t+1} c_t - \sum_{t=0}^\infty p_{t+1} y_t) + \mu c_t$</span></p>
<p>How do I get <span class="math-container">$c_t^* \space \forall \space t$</span>?</p>

<macroeconomics><consumer-theory><optimization><intertemporal-choice>
Optimal consumption for infinite number of periods and exogenous income
Arturo Sbr
# Optimal consumption for infinite number of periods and exogenous income

(Asked by: username_0 on 2021-09-14)

I have the following optimization problem:

> $\max\_{\{c\_t, s\_{t+1}\}} \Pi\_{t=0}^\infty c\_t^{\beta^t}$
>
> $\text{subject to } \space c\_t + s\_{t+1} = y\_t + (1 + r) s\_t \text{ and } s\_0 = 0$

How do I find the optimal consumption level for each time period $t$?

**Here's what I have so far:**

Let $p\_t = \frac{1}{(1 + r)^t}$

Then, from the budget constraint $p\_{t+1} s\_{t+1} - p\_{t+1} s\_t = p\_{t+1}(y\_t - c\_t)$.

If we sum over $T$ periods:
$(p\_1 s\_1 - p\_0 0) + (p\_2 s\_2 - p\_1 s\_1) + ... (p\_{T+1} s\_{T+1} - p\_T s\_T) = \sum\_{t=0}^\infty p\_{t+1}(y\_t - c\_t)$

If I use a natural logarithm transformation on the utility function and assume $\lim\_{T\to\infty} p\_T s\_T = 0$ (no ponzi schemes allowed), I can rewrite the optimization problem as:

$\max\_{\{c\_t\}} \sum\_{t=0}^\infty \beta^t\ln(c\_t)$

$\text{subject to} \sum\_{t=0}^\infty p\_{t+1} c\_t = \sum\_{t=0}^\infty p\_{t+1} y\_t$

Hence, my lagrangian function should look like (unless I'm missing something):

$\mathcal{L} = \sum \beta^t\ln(c\_t) - \lambda(\sum\_{t=0}^\infty p\_{t+1} c\_t - \sum\_{t=0}^\infty p\_{t+1} y\_t) + \mu c\_t$

How do I get $c\_t^\* \space \forall \space t$?

**Tags:** macroeconomics, consumer-theory, optimization, intertemporal-choice


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## 1 answers

### Answer by username_1 on 2021-09-15. Score: 3

Below is the sketch of the solution.

From the Lagrangian $$\max \_{\left\{c\_{t}, s\_{t+1}\right\}} \Pi\_{t=0}^{\infty} c\_{t}^{\beta^{t}} - \Pi\_{t=0}^{\infty}\lambda\_t(c\_{t}+s\_{t+1}-y\_{t}-(1+r) s\_{t}) \\ \text{s.t.} \ s\_0 = 0, c\_t > 0 $$, you can get the Euler equation $$\frac{\beta\_{t+1}c\_{t+1}^{\beta\_{t+1}-1}}{\beta\_{t}c\_{t}^{\beta\_{t}-1}}= \frac{\beta\_{t+1}\left[y\_{t+1}+(1+r)s\_{t+1}-s\_{t+2}\right]^{\beta\_{t+1}-1}}{\beta\_{t}\left[y\_{t}+(1+r)s\_{t}-s\_{t+1}\right]^{\beta\_{t}-1}} = \frac{1}{1+r}$$, and then you can write $s\_{t}^{\*} \forall t>1$ as a function of $s\_1$ with the initial condition $s\_0$. By using the transversality condition as a terminal condition: $$\lim \_{t \rightarrow \infty} \lambda\_ts\_{t+1}= \lim \_{t \rightarrow \infty} {\beta\_{t}\left[y\_{t}+(1+r)s\_{t}-s\_{t+1}\right]^{\beta\_{t}-1}}s\_{t+1} =0$$, you can solve for $\left\{s\_{t}\right\}\_{t=1}^{\infty}$ forwards by guessing an initial value for $s\_1$ and yielding the sequence that does not violate the TVC or the nonnegativity constraint on consumption. And once you get $s\_{t}^{\*} \forall t$ you get $c\_{t}^{\*} \forall t$.

> **Comments:**
> **username_0**: Sorry, I do not understand this answer. Are you saying $s_0 = 0$ implies that the agent doesn't save any money in $t=0$? That is, $c_0 + s_1 = c_0 = y_0$?  
> **username_1**: @ArturoSbr Sorry I made some mistakes in the initial answer. I think I have corrected it.  

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