<p>The household's utility function is
<span class="math-container">$$
v(c_1, c_2) := \frac{1}{\beta(c_1)} \log c_1 + \log c_2 - A(c_1, c_2)
$$</span>
where
<span class="math-container">$$
\beta(c_1) = \begin{cases} 
\beta_1 &gt; 0 &amp;\text{ and } A(c_1, c_2) = 0 &amp; \text{ if } c_1 &lt; \bar{c} \\
\beta_2 &gt; \beta_1 &amp;\text{ and } A(c_1, c_2) = \frac{\beta_2-\beta_1}{\beta_1\beta_2} \log c_2 &amp; \text{ if } c_1 &gt; \bar{c}
\end{cases}
$$</span></p>
<p>We can rewrite the household's utility function as:
<span class="math-container">$$v(c_1, c_2) :=\begin{cases} 
\frac{1}{\beta_1} \log c_1 + \log c_2 &amp; \text{ if } c_1 &lt; \bar{c} \\
\frac{1}{\beta_2} \log c_1 + \log c_2 - \frac{\beta_2-\beta_1}{\beta_1\beta_2} \log c_2 &amp; \text{ if } c_1 \geq \bar{c}
\end{cases}$$</span></p>
<p>In the paper, it is given that <span class="math-container">$v$</span> is a continuous, piecewise differentiable utility function. However, we can see that <span class="math-container">$v$</span> as described above is not continuous at <span class="math-container">$c_1=\bar{c}$</span>. In fact, <span class="math-container">$v$</span> is not monotonic either.</p>

CC BY-SA 4.0
2023-11-01T07:17:14.260Z
2023-11-01T07:17:14.260Z
Amit
<p>The household maximizes his lifetime utility function according to
<span class="math-container">$$
\max_{c_1, c_2} v(c_1, c_2) := \frac{1}{\beta(c_1)} \log c_1 + \log c_2 - A(c_1, c_2)
$$</span>
subject to
<span class="math-container">$$
\beta(c_1) = \begin{cases} 
\beta_1 &gt; 0 &amp;\text{ and } A(c_1, c_2) = 0 &amp; \text{ if } c_1 &lt; \bar{c} \\
\beta_2 &gt; \beta_1 &amp;\text{ and } A(c_1, c_2) = \frac{\beta_2-\beta_1}{\beta_1\beta_2} \log c_2 &amp; \text{ if } c_1 &gt; \bar{c}
\end{cases}
$$</span>
Budget constraint is
<span class="math-container">$$
c_1 + \frac{c_2}{R_{t+1}} \leq w_t
$$</span></p>
<ol>
<li>How did he derive the savings as follows
<span class="math-container">$$
s_t = 
\begin{cases}
\frac{\beta_1}{1+\beta_1} w_t &amp; \text{ if } w_t \leq (1+\beta_1) \bar{c}, \\
w_t - \bar{c} &amp; \text{ if } w_t \in \left[ (1+\beta_1)\bar{c}, (1+\beta_2)\bar{c} \right], \\
\frac{\beta_2}{1+\beta_2} w_t &amp; \text{ if } w_t &gt; (1+\beta_2) \bar{c}
\end{cases}
$$</span></li>
<li>Is there any particular reason why he assumes <span class="math-container">$A = \frac{\beta_2-\beta_1}{\beta_1\beta_2} \log c_2$</span> when <span class="math-container">$c_1 &gt; \bar{c}$</span>?</li>
</ol>
<p>This is section 3.1 of <a href="https://link.springer.com/content/pdf/10.1007/BF00150197.pdf" rel="nofollow noreferrer">Azariadis (1996): The Economics of Poverty Traps Part One: Complete Markets</a> in JEG.</p>

<macroeconomics><utility><economic-growth>
How the saving rates were derived in Azariadis (1996)'s Impatience Trap?
teddi
# How the saving rates were derived in Azariadis (1996)'s Impatience Trap?

(Asked by: username_0 on 2023-10-31)

The household maximizes his lifetime utility function according to
$$
\max\_{c\_1, c\_2} v(c\_1, c\_2) := \frac{1}{\beta(c\_1)} \log c\_1 + \log c\_2 - A(c\_1, c\_2)
$$
subject to
$$
\beta(c\_1) = \begin{cases}
\beta\_1 > 0 &\text{ and } A(c\_1, c\_2) = 0 & \text{ if } c\_1 < \bar{c} \\
\beta\_2 > \beta\_1 &\text{ and } A(c\_1, c\_2) = \frac{\beta\_2-\beta\_1}{\beta\_1\beta\_2} \log c\_2 & \text{ if } c\_1 > \bar{c}
\end{cases}
$$
Budget constraint is
$$
c\_1 + \frac{c\_2}{R\_{t+1}} \leq w\_t
$$

1. How did he derive the savings as follows
   $$
   s\_t =
   \begin{cases}
   \frac{\beta\_1}{1+\beta\_1} w\_t & \text{ if } w\_t \leq (1+\beta\_1) \bar{c}, \\
   w\_t - \bar{c} & \text{ if } w\_t \in \left[ (1+\beta\_1)\bar{c}, (1+\beta\_2)\bar{c} \right], \\
   \frac{\beta\_2}{1+\beta\_2} w\_t & \text{ if } w\_t > (1+\beta\_2) \bar{c}
   \end{cases}
   $$
2. Is there any particular reason why he assumes $A = \frac{\beta\_2-\beta\_1}{\beta\_1\beta\_2} \log c\_2$ when $c\_1 > \bar{c}$?

This is section 3.1 of [Azariadis (1996): The Economics of Poverty Traps Part One: Complete Markets](https://link.springer.com/content/pdf/10.1007/BF00150197.pdf) in JEG.

**Tags:** macroeconomics, utility, economic-growth


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## 1 answers

### Answer by username_1 on 2023-11-01. Score: 2

The household's utility function is
$$
v(c\_1, c\_2) := \frac{1}{\beta(c\_1)} \log c\_1 + \log c\_2 - A(c\_1, c\_2)
$$
where
$$
\beta(c\_1) = \begin{cases}
\beta\_1 > 0 &\text{ and } A(c\_1, c\_2) = 0 & \text{ if } c\_1 < \bar{c} \\
\beta\_2 > \beta\_1 &\text{ and } A(c\_1, c\_2) = \frac{\beta\_2-\beta\_1}{\beta\_1\beta\_2} \log c\_2 & \text{ if } c\_1 > \bar{c}
\end{cases}
$$

We can rewrite the household's utility function as:
$$v(c\_1, c\_2) :=\begin{cases}
\frac{1}{\beta\_1} \log c\_1 + \log c\_2 & \text{ if } c\_1 < \bar{c} \\
\frac{1}{\beta\_2} \log c\_1 + \log c\_2 - \frac{\beta\_2-\beta\_1}{\beta\_1\beta\_2} \log c\_2 & \text{ if } c\_1 \geq \bar{c}
\end{cases}$$

In the paper, it is given that $v$ is a continuous, piecewise differentiable utility function. However, we can see that $v$ as described above is not continuous at $c\_1=\bar{c}$. In fact, $v$ is not monotonic either.


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