Exercise 8.8 Ljungqvist and Sargent, Corner Solutions

Question

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

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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 $\alpha=\mu (1+\beta^{-1})$.

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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 $$\frac{\alpha\beta}{\mu}>\frac{\mu (1+\beta^{-1})\beta}{\mu}=1+\beta >1$$

Answer 1

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.

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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, $p_1 = 1$.