<p>Let <span class="math-container">$p_t$</span> be the price of the Arrow-Debreu security at period <span class="math-container">$t$</span>.</p>
<p>Let's first look at an equilibrium where consumer 1 always has positive consumption: <span class="math-container">$q^1_t &gt; 0$</span> for all <span class="math-container">$t$</span>. In this case, the first order conditions for consumer 1 give:
<span class="math-container">$$
p_t = \beta^t p_0
$$</span>
and the first order conditions for consumer 2 give (consumer 2 will always consume positive amounts):
<span class="math-container">$$
q^2_t = \beta^t q^2_0 \frac{p_0}{p_t} = q^2_0.
$$</span></p>
<p>The total resources for individual 2 are:
<span class="math-container">$$
\begin{align*}
\sum_{t \text{ odd }} \alpha p_t &amp; = \alpha p_0 \sum_{t \text{ odd}} \beta^t,\\
&amp;= \alpha p_0 \frac{\beta}{1 - \beta^2}.
\end{align*}
$$</span>
Total expenditures are:
<span class="math-container">$$
\sum_t q^2_0 p_t = q^2_0 p_0 \frac{1}{1 - \beta}.
$$</span>
Equating the two gives
<span class="math-container">$$
q^2_t = q^2_0 = \alpha  \frac{\beta}{1 + \beta}.
$$</span>
Then,
<span class="math-container">$$
q^1_t = \begin{cases} \mu - \alpha\frac{\beta}{1 + \beta} &amp;\text{ if $t$ is even}\\
\mu + \frac{\alpha}{1 + \beta} &amp;\text{ if $t$ is even}.
\end{cases}.
$$</span>
Finally prices are such that <span class="math-container">$p_t = \beta^{t} p_0$</span> and we can normalize prices such that, for example, <span class="math-container">$p_0 = 1$</span>.</p>
<p>Now, we see that <span class="math-container">$q^1_t \ge 0$</span> if and only if <span class="math-container">$\alpha \le \mu\frac{1 + \beta}{\beta}$</span>.</p>
<p>If <span class="math-container">$\alpha &gt; \mu\frac{1 + \beta}{\beta}$</span> we need to look for a different equilibrium. Intuitively, this will be one where <span class="math-container">$q^1_t = 0$</span> when <span class="math-container">$t$</span> is even.</p>
<hr />
<p>Consider a case where <span class="math-container">$q^1_t = 0$</span> if <span class="math-container">$t$</span> is even and <span class="math-container">$q^1_t &gt; 0$</span> if <span class="math-container">$t$</span> is odd.</p>
<p>Then if <span class="math-container">$t$</span> and <span class="math-container">$j$</span> are odd, we should have (by the first order conditions of consumer 1:
<span class="math-container">$$
p_t = \beta^{t -j} p_j
$$</span>
In particular period 1 is odd, so:
<span class="math-container">$$
p_t = \beta^{t-1} p_1.
$$</span>
Next, if <span class="math-container">$t$</span> is even then prices should be to high, so:
<span class="math-container">$$
p_t \ge \beta^{t-1} p_1.
$$</span></p>
<p>Next, by the first order condition of consumer 2, we have that if <span class="math-container">$t$</span> is odd, then:
<span class="math-container">$$
q^2_t = \beta^{t-1} p_1 q^2_1 \frac{p_1}{p_t} = q^2_1.
$$</span>
If <span class="math-container">$t$</span> is even then <span class="math-container">$q^2_t = \mu$</span> (as <span class="math-container">$q^1_t = 0$</span>).
So, by the same first order condition:
<span class="math-container">$$
\mu = \beta^{t-1} q^2_1 \frac{p_1}{p_t} \to p_t = \beta^{t-1} p_1 \frac{q^2_1}{\mu}.
$$</span></p>
<p>The total revenue of consumer 2 is
<span class="math-container">$$
\sum_{t \text{ is odd}} \alpha p_t = \alpha \sum_{t \text{ is odd}} \beta^{t-1} p_1 = \alpha p_1 \frac{1}{1 - \beta^2}
$$</span>
The total expenditure is:
<span class="math-container">$$
\begin{align*}
&amp;\sum_{t \text{ is even }} p_t q_2^t + \sum_{t \text{ is odd}} p_t q^2_t,\\
&amp;= \sum_{t \text{ is even}}\beta^{t-1} p_1 \frac{q^2_1}{\mu} \mu + \sum_{t \text{ is odd}} \beta^{t-1} p_1 q^2_1,\\
&amp;=p_1 q^2_1 \sum_{t \text{ is even}} \beta^{t-1} + p_1 q^2_1 \sum_{t \text{ is odd}} \beta^{t-1},\\
&amp;=p_1 q^2_1 \left(\frac{\beta^{-1}}{1 - \beta^2} + \frac{1}{1 - \beta^2}\right),\\
&amp;=p_1 q^2_1 \frac{1}{1 - \beta^2}\frac{1 + \beta}{\beta}.
\end{align*}
$$</span>
So:
<span class="math-container">$$
q^2_1 = {\alpha} \frac{\beta}{1 + \beta}.
$$</span>
This gives:
<span class="math-container">$$
q^2_t = \begin{cases} \mu &amp;\text{ if $t$ is even}\\
\alpha \frac{\beta}{1 + \beta} &amp;\text{ if $t$ is odd}\end{cases}
$$</span></p>
<p>Then using the resource constraints:
<span class="math-container">$$
q^1_t = \begin{cases} 0 &amp;\text{ if $t$ is even}\\
\mu + \frac{\alpha}{1 + \beta} &amp; \text{ if $t$ is odd} \end{cases}
$$</span></p>
<p>Finally,
<span class="math-container">$$
p_t = \begin{cases} \beta^{t-1} p_1 &amp;\text{ if $t$ is odd}\\
\beta^{t-1} p_1 \frac{\alpha \beta}{(1+\beta)\mu} &amp;\text{ if $t$ is even} \end{cases}
$$</span>
Prices in even periods are higher than in odd period.
Note that prices can be normalized so that, for example, <span class="math-container">$p_1 = 1$</span>.</p>

CC BY-SA 4.0
2024-11-25T17:35:48.863Z
Thank you for your answer! I haven’t had time to look at it properly yet, but from a glance I think you’re assuming both types of consumer have the logarithmic utility. However one has a “risk-neutral” intra-period utility function and one has logarithmic. I think the exercise is meant to focus on the impact of corner solutions on interest rates which are generated by violations of the Inada conditions.
Joseph Basford
CC BY-SA 4.0
2024-11-26T16:07:31.430Z
Ok, I made some changes. Sorry for the confusion. Anyway, the model needs to be closed using the budget constraints of the consumers.
tdm
CC BY-SA 4.0
2024-12-02T13:32:18.607Z
Very nice. Thank you for taking the time to write it out, it makes complete sense. Just to highlight a small notational difference, where you've written $q$ for consumption, I had $c$ for consumption and $q$ for the price of Arrow-Debreu securities, but it doesn't matter as it's clear anyway. Thank you again!
Joseph Basford
CC BY-SA 4.0
2024-11-25T16:47:10.790Z
2024-11-26T16:06:33.467Z
2024-11-26T16:06:33.467Z
tdm
<p>This question is exercise 8.8 from Ljungqvist and Sargent &quot;Recursive Macroeconomic Theory&quot; 3rd edition.</p>
<hr />
<p>I repeat the question here for clarity.</p>
<p>There is a pure endowment economy and two types of consumer, <span class="math-container">$i=1,2$</span>. The type-<span class="math-container">$1$</span> consumers have utility <span class="math-container">$$\sum_{t=0}^{\infty} \beta^t c_t^1$$</span> and the type-2 consumers have utility <span class="math-container">$$\sum_{t=0}^{\infty} \beta \ln c_t^2$$</span> We enforce that <span class="math-container">$c_t^i\geq 0$</span> and <span class="math-container">$\beta\in (0,1)$</span>. The type-1 consumer has endowment stream <span class="math-container">$$y_t^1=\mu&gt;0 \qquad \forall t\geq 0$$</span> and the type-2 consumer has endowment stream <span class="math-container">$$y_t^2=\begin{cases}
    0 &amp; \text{  if } t\geq 0 \text{ is even} \\
    \alpha &amp; \text{ if } t\geq 0 \text{ is odd}
\end{cases}$$</span></p>
<p>We assume the consumption good is tradable but not storable. Agents have access to a complete set of Arrow-Debreu securities which are traded only at date <span class="math-container">$t=0$</span>.</p>
<p>The question asks to solve for the competitive equilibrium in the case when <span class="math-container">$\alpha=\mu (1+\beta^{-1})$</span> and when <span class="math-container">$\alpha&gt;\mu (1+\beta^{-1})$</span>. It focuses on the implied one-period gross interest rates, asking whether they change between even and odd periods, and whether they are higher or lower when <span class="math-container">$\alpha&gt;\mu (1+\beta^{-1})$</span> versus when <span class="math-container">$\alpha=\mu (1+\beta^{-1})$</span>.</p>
<hr />
<p>I first solve when <span class="math-container">$\alpha= \mu(1+\beta^{-1})$</span>.</p>
<p>The Lagrangian for agent <span class="math-container">$i$</span>'s problem at date <span class="math-container">$0$</span> is <span class="math-container">$$\mathcal{L}^i=\sum_{t=0}^{\infty} \beta^t u^i(c_t^i) -\lambda^i\left[\sum_{t=0}^{\infty} q_t^0(c_t^i-y_t^i)\right]+\sum_{t=0}^{\infty} \mu_t^i c_t^i$$</span> where <span class="math-container">$q_t^0$</span> is the price on the Arrow-Debreu security at date <span class="math-container">$0$</span> for one unit of the consumption good at date <span class="math-container">$t$</span>. This has FOC <span class="math-container">$$0=\beta^t (u^i)'(c_t^i)-\lambda^i q_t^0 +\mu_t^i$$</span> as <span class="math-container">$\mu_t^i\geq 0$</span> and is strict if and only if <span class="math-container">$c_t^i&gt;0$</span>, our FOC is <span class="math-container">$$\beta^t(u^i)'(c_t^i)\leq \lambda^i q_t^0$$</span> For the two types this is <span class="math-container">\begin{align*}
    \beta^t &amp;\leq \lambda^1 q_t^0 \qquad (i=1) \\
    \beta^t\frac{1}{c_t^2} &amp;\leq \lambda^2 q_t^0 \qquad (i=2)
\end{align*}</span></p>
<p>Because <span class="math-container">$\lim_{c_t^2\to 0}\frac{1}{c_t^2}\to \infty$</span>, the type-2 consumer will never set <span class="math-container">$c_t^2=0$</span>, so their FOC holds with equality. For the type-1 consumer, if <span class="math-container">$\beta^t &lt; \lambda^1 q_t^0$</span> then <span class="math-container">$c_t^1=0$</span> and if <span class="math-container">$\beta^t = \lambda^1 q_t^0$</span>, <span class="math-container">$c_t^1$</span> may take any value in <span class="math-container">$[0,\infty )$</span> as the type-1 consumer. So, our FOCs become (with some abuse of notation), <span class="math-container">\begin{align*}
    c_t^1&amp;=\begin{cases}
        0 &amp; \text{ if } \beta^t &lt; \lambda^1 q_t^0  \\
        [0,\infty) &amp; \text{ if } \beta^t = \lambda^1 q_t^0 
    \end{cases} \\
    c_t^2&amp;=\frac{\beta^t}{q_t^0}\frac{1}{\lambda^2}
\end{align*}</span> I make the conjecture that <span class="math-container">$$q_t^0=\beta^t \text{  and   } \lambda^2=\frac{1}{\mu}$$</span> Thus, by the per-period endowment constraint <span class="math-container">\begin{align*}
    c_{2t+1}^1 &amp;=\mu (1+\beta^{-1}) \\
    c_{2t}^1 &amp;= 0 \\
    c_t^2 &amp;= \mu
\end{align*}</span></p>
<p>The interest rate is then <span class="math-container">$-1+\frac{q_t^0}{p_{t+1}}=\beta^{-1}-1$</span>. I think this result is intuitive as the type-<span class="math-container">$2$</span> consumer wants to perfectly consumption smooth since their intertemporal elasticity is <span class="math-container">$1$</span> and the type-<span class="math-container">$1$</span> consumer will let them do that since they are risk-neutral. The type-<span class="math-container">$1$</span> consumer is compensated with the interest rate <span class="math-container">$(1+\beta^{-1})$</span> on the amount they lend in even periods, <span class="math-container">$\mu$</span>.</p>
<p>Is there a better way to solve this than by conjecture? I think the conjecture seems natural, but is there a more direct approach?</p>
<p>Now, when <span class="math-container">$\alpha&gt;\mu (1+\beta^{-1})$</span>, I suspect the interest rate will rise in even periods but decrease in odd periods as the type-2 consumer demands more assets in even periods and is willing and able to supply more assets in odd periods.</p>
<p>However, I'm not sure how to make progress. If I make the same conjecture about prices being <span class="math-container">$q_t^0=\beta^t$</span> and <span class="math-container">$\lambda^2=\frac{1}{\mu}$</span>, I find that <span class="math-container">\begin{align*}
    c_{2t+1}^1 &amp;=\alpha \\
    c_{2t}^1 &amp;= 0 \\
    c_t^2 &amp;= \mu
\end{align*}</span> And the gross interest rate is still <span class="math-container">$\beta^{-1}-1$</span>. But this feels wrong since the type <span class="math-container">$1$</span> consumer is lending out <span class="math-container">$\mu$</span> in even periods and getting back <span class="math-container">$\alpha -\mu &gt; \beta^{-1}\mu$</span> in the subsequent even periods. What I think I should be finding is that the interest rate is <span class="math-container">$$\frac{q^0_{2t}}{q^0_{2t+1}}=\frac{\alpha}{\mu}-1$$</span> so that consumption is as outline above. However, I'm not sure how to derive this and am not sure it is correct. If I conjecture that <span class="math-container">$q_t^0=\left[\frac{\mu}{\alpha}\right]^t$</span> then the interest rate is as I want it to be, but then <span class="math-container">$$c_t^2=\frac{\beta^t}{q_t^0}\frac{1}{\lambda^2} =\frac{1}{\lambda^2} \left[ \frac{\alpha \beta}{\mu}\right]^t$$</span> which blows up as <span class="math-container">$t\to \infty$</span> because <span class="math-container">$$\frac{\alpha\beta}{\mu}&gt;\frac{\mu (1+\beta^{-1})\beta}{\mu}=1+\beta &gt;1$$</span></p>

<macroeconomics><mathematical-economics><general-equilibrium>
Exercise 8.8 Ljungqvist and Sargent, Corner Solutions
Joseph Basford
# Exercise 8.8 Ljungqvist and Sargent, Corner Solutions

(Asked by: username_0 on 2024-11-25)

This question is exercise 8.8 from Ljungqvist and Sargent "Recursive Macroeconomic Theory" 3rd edition.

---

I repeat the question here for clarity.

There is a pure endowment economy and two types of consumer, $i=1,2$. The type-$1$ consumers have utility $$\sum\_{t=0}^{\infty} \beta^t c\_t^1$$ and the type-2 consumers have utility $$\sum\_{t=0}^{\infty} \beta \ln c\_t^2$$ We enforce that $c\_t^i\geq 0$ and $\beta\in (0,1)$. The type-1 consumer has endowment stream $$y\_t^1=\mu>0 \qquad \forall t\geq 0$$ and the type-2 consumer has endowment stream $$y\_t^2=\begin{cases}
0 & \text{ if } t\geq 0 \text{ is even} \\
\alpha & \text{ if } t\geq 0 \text{ is odd}
\end{cases}$$

We assume the consumption good is tradable but not storable. Agents have access to a complete set of Arrow-Debreu securities which are traded only at date $t=0$.

The question asks to solve for the competitive equilibrium in the case when $\alpha=\mu (1+\beta^{-1})$ and when $\alpha>\mu (1+\beta^{-1})$. It focuses on the implied one-period gross interest rates, asking whether they change between even and odd periods, and whether they are higher or lower when $\alpha>\mu (1+\beta^{-1})$ versus when $\alpha=\mu (1+\beta^{-1})$.

---

I first solve when $\alpha= \mu(1+\beta^{-1})$.

The Lagrangian for agent $i$'s problem at date $0$ is $$\mathcal{L}^i=\sum\_{t=0}^{\infty} \beta^t u^i(c\_t^i) -\lambda^i\left[\sum\_{t=0}^{\infty} q\_t^0(c\_t^i-y\_t^i)\right]+\sum\_{t=0}^{\infty} \mu\_t^i c\_t^i$$ where $q\_t^0$ is the price on the Arrow-Debreu security at date $0$ for one unit of the consumption good at date $t$. This has FOC $$0=\beta^t (u^i)'(c\_t^i)-\lambda^i q\_t^0 +\mu\_t^i$$ as $\mu\_t^i\geq 0$ and is strict if and only if $c\_t^i>0$, our FOC is $$\beta^t(u^i)'(c\_t^i)\leq \lambda^i q\_t^0$$ For the two types this is \begin{align\*}
\beta^t &\leq \lambda^1 q\_t^0 \qquad (i=1) \\
\beta^t\frac{1}{c\_t^2} &\leq \lambda^2 q\_t^0 \qquad (i=2)
\end{align\*}

Because $\lim\_{c\_t^2\to 0}\frac{1}{c\_t^2}\to \infty$, the type-2 consumer will never set $c\_t^2=0$, so their FOC holds with equality. For the type-1 consumer, if $\beta^t < \lambda^1 q\_t^0$ then $c\_t^1=0$ and if $\beta^t = \lambda^1 q\_t^0$, $c\_t^1$ may take any value in $[0,\infty )$ as the type-1 consumer. So, our FOCs become (with some abuse of notation), \begin{align\*}
c\_t^1&=\begin{cases}
0 & \text{ if } \beta^t < \lambda^1 q\_t^0 \\
[0,\infty) & \text{ if } \beta^t = \lambda^1 q\_t^0
\end{cases} \\
c\_t^2&=\frac{\beta^t}{q\_t^0}\frac{1}{\lambda^2}
\end{align\*} I make the conjecture that $$q\_t^0=\beta^t \text{ and } \lambda^2=\frac{1}{\mu}$$ Thus, by the per-period endowment constraint \begin{align\*}
c\_{2t+1}^1 &=\mu (1+\beta^{-1}) \\
c\_{2t}^1 &= 0 \\
c\_t^2 &= \mu
\end{align\*}

The interest rate is then $-1+\frac{q\_t^0}{p\_{t+1}}=\beta^{-1}-1$. I think this result is intuitive as the type-$2$ consumer wants to perfectly consumption smooth since their intertemporal elasticity is $1$ and the type-$1$ consumer will let them do that since they are risk-neutral. The type-$1$ consumer is compensated with the interest rate $(1+\beta^{-1})$ on the amount they lend in even periods, $\mu$.

Is there a better way to solve this than by conjecture? I think the conjecture seems natural, but is there a more direct approach?

Now, when $\alpha>\mu (1+\beta^{-1})$, I suspect the interest rate will rise in even periods but decrease in odd periods as the type-2 consumer demands more assets in even periods and is willing and able to supply more assets in odd periods.

However, I'm not sure how to make progress. If I make the same conjecture about prices being $q\_t^0=\beta^t$ and $\lambda^2=\frac{1}{\mu}$, I find that \begin{align\*}
c\_{2t+1}^1 &=\alpha \\
c\_{2t}^1 &= 0 \\
c\_t^2 &= \mu
\end{align\*} And the gross interest rate is still $\beta^{-1}-1$. But this feels wrong since the type $1$ consumer is lending out $\mu$ in even periods and getting back $\alpha -\mu > \beta^{-1}\mu$ in the subsequent even periods. What I think I should be finding is that the interest rate is $$\frac{q^0\_{2t}}{q^0\_{2t+1}}=\frac{\alpha}{\mu}-1$$ so that consumption is as outline above. However, I'm not sure how to derive this and am not sure it is correct. If I conjecture that $q\_t^0=\left[\frac{\mu}{\alpha}\right]^t$ then the interest rate is as I want it to be, but then $$c\_t^2=\frac{\beta^t}{q\_t^0}\frac{1}{\lambda^2} =\frac{1}{\lambda^2} \left[ \frac{\alpha \beta}{\mu}\right]^t$$ which blows up as $t\to \infty$ because $$\frac{\alpha\beta}{\mu}>\frac{\mu (1+\beta^{-1})\beta}{\mu}=1+\beta >1$$

**Tags:** macroeconomics, mathematical-economics, general-equilibrium

---

## 1 answers

### Answer by username_1 on 2024-11-25. Score: 1 | Accepted answer

Let $p\_t$ be the price of the Arrow-Debreu security at period $t$.

Let's first look at an equilibrium where consumer 1 always has positive consumption: $q^1\_t > 0$ for all $t$. In this case, the first order conditions for consumer 1 give:
$$
p\_t = \beta^t p\_0
$$
and the first order conditions for consumer 2 give (consumer 2 will always consume positive amounts):
$$
q^2\_t = \beta^t q^2\_0 \frac{p\_0}{p\_t} = q^2\_0.
$$

The total resources for individual 2 are:
$$
\begin{align\*}
\sum\_{t \text{ odd }} \alpha p\_t & = \alpha p\_0 \sum\_{t \text{ odd}} \beta^t,\\
&= \alpha p\_0 \frac{\beta}{1 - \beta^2}.
\end{align\*}
$$
Total expenditures are:
$$
\sum\_t q^2\_0 p\_t = q^2\_0 p\_0 \frac{1}{1 - \beta}.
$$
Equating the two gives
$$
q^2\_t = q^2\_0 = \alpha \frac{\beta}{1 + \beta}.
$$
Then,
$$
q^1\_t = \begin{cases} \mu - \alpha\frac{\beta}{1 + \beta} &\text{ if $t$ is even}\\
\mu + \frac{\alpha}{1 + \beta} &\text{ if $t$ is even}.
\end{cases}.
$$
Finally prices are such that $p\_t = \beta^{t} p\_0$ and we can normalize prices such that, for example, $p\_0 = 1$.

Now, we see that $q^1\_t \ge 0$ if and only if $\alpha \le \mu\frac{1 + \beta}{\beta}$.

If $\alpha > \mu\frac{1 + \beta}{\beta}$ we need to look for a different equilibrium. Intuitively, this will be one where $q^1\_t = 0$ when $t$ is even.

---

Consider a case where $q^1\_t = 0$ if $t$ is even and $q^1\_t > 0$ if $t$ is odd.

Then if $t$ and $j$ are odd, we should have (by the first order conditions of consumer 1:
$$
p\_t = \beta^{t -j} p\_j
$$
In particular period 1 is odd, so:
$$
p\_t = \beta^{t-1} p\_1.
$$
Next, if $t$ is even then prices should be to high, so:
$$
p\_t \ge \beta^{t-1} p\_1.
$$

Next, by the first order condition of consumer 2, we have that if $t$ is odd, then:
$$
q^2\_t = \beta^{t-1} p\_1 q^2\_1 \frac{p\_1}{p\_t} = q^2\_1.
$$
If $t$ is even then $q^2\_t = \mu$ (as $q^1\_t = 0$).
So, by the same first order condition:
$$
\mu = \beta^{t-1} q^2\_1 \frac{p\_1}{p\_t} \to p\_t = \beta^{t-1} p\_1 \frac{q^2\_1}{\mu}.
$$

The total revenue of consumer 2 is
$$
\sum\_{t \text{ is odd}} \alpha p\_t = \alpha \sum\_{t \text{ is odd}} \beta^{t-1} p\_1 = \alpha p\_1 \frac{1}{1 - \beta^2}
$$
The total expenditure is:
$$
\begin{align\*}
&\sum\_{t \text{ is even }} p\_t q\_2^t + \sum\_{t \text{ is odd}} p\_t q^2\_t,\\
&= \sum\_{t \text{ is even}}\beta^{t-1} p\_1 \frac{q^2\_1}{\mu} \mu + \sum\_{t \text{ is odd}} \beta^{t-1} p\_1 q^2\_1,\\
&=p\_1 q^2\_1 \sum\_{t \text{ is even}} \beta^{t-1} + p\_1 q^2\_1 \sum\_{t \text{ is odd}} \beta^{t-1},\\
&=p\_1 q^2\_1 \left(\frac{\beta^{-1}}{1 - \beta^2} + \frac{1}{1 - \beta^2}\right),\\
&=p\_1 q^2\_1 \frac{1}{1 - \beta^2}\frac{1 + \beta}{\beta}.
\end{align\*}
$$
So:
$$
q^2\_1 = {\alpha} \frac{\beta}{1 + \beta}.
$$
This gives:
$$
q^2\_t = \begin{cases} \mu &\text{ if $t$ is even}\\
\alpha \frac{\beta}{1 + \beta} &\text{ if $t$ is odd}\end{cases}
$$

Then using the resource constraints:
$$
q^1\_t = \begin{cases} 0 &\text{ if $t$ is even}\\
\mu + \frac{\alpha}{1 + \beta} & \text{ if $t$ is odd} \end{cases}
$$

Finally,
$$
p\_t = \begin{cases} \beta^{t-1} p\_1 &\text{ if $t$ is odd}\\
\beta^{t-1} p\_1 \frac{\alpha \beta}{(1+\beta)\mu} &\text{ if $t$ is even} \end{cases}
$$
Prices in even periods are higher than in odd period.
Note that prices can be normalized so that, for example, $p\_1 = 1$.

> **Comments:**
> **username_0**: Thank you for your answer! I haven’t had time to look at it properly yet, but from a glance I think you’re assuming both types of consumer have the logarithmic utility. However one has a “risk-neutral” intra-period utility function and one has logarithmic. I think the exercise is meant to focus on the impact of corner solutions on interest rates which are generated by violations of the Inada conditions.  
> **username_1**: Ok, I made some changes. Sorry for the confusion. Anyway, the model needs to be closed using the budget constraints of the consumers.  
> **username_0**: Very nice. Thank you for taking the time to write it out, it makes complete sense. Just to highlight a small notational difference, where you've written $q$ for consumption, I had $c$ for consumption and $q$ for the price of Arrow-Debreu securities, but it doesn't matter as it's clear anyway. Thank you again!  

---

