<p>I think Barro means in the footnote that Giovanni and Weil find the same equation, $U_t=\Phi C^{1-\gamma}$, but using the optimal path of $C_t$.
In Barro's paper, the approach is different given that the dynamics of $C_t$ is exogenous: $C_t=Y_t$ by assumption.</p>

<p>Barro uses the limit case when the length of a period gets close to 0. Maybe what may bother the reader is that the model is defined as discrete.</p>

<h2>Rewrite the model</h2>

<p>First, we can rewrite the model with a length of period $\delta$ and then use $\delta\to 0$.
The GDP dynamics write
$$\log(Y_{t+\delta})=\log(Y_t)+g\delta+u_{t+\delta}+v_ {t+\delta}$$ 
with $u_{t+\delta}\sim \mathcal{N}(0,\delta\sigma^2)$, and $v_{t+\delta}=0$ with probability $1-p\delta$ and $\log(1-b)$ with probability $p\delta$.
The utility satisfies
$$
U_t=\frac{1}{1-\gamma}\left\lbrace C_t^{1-\theta}+\frac{1}{1+\rho\delta}\left[(1-\gamma)E_tU_{t+\delta}\right]^\frac{1-\theta}{1-\gamma}\right\rbrace^\frac{1-\gamma}{1-\theta}.
$$</p>

<h2>1) Find $\Phi$ as a function of $E_t\left[\left(\frac{C_{t+\delta}}{C_t}\right)^{1-\gamma}\right]$</h2>

<p>From now suppose there is a $\Phi$ such that $U_t=\Phi C^{1-\gamma}$ (note that $\Phi$ depends on $\delta$ a priori).
Define $H(U)=[(1-\gamma)U]^\frac{1-\theta}{1-\gamma}$, the utility satisfies
\begin{align}
H(U_t)= C_t^{1-\theta}+\frac{1}{1+\rho\delta}H(E_tU_{t+\delta}).
\end{align}
We substitute $U_t$:
\begin{align}
H(\Phi)C_t^{1-\theta}= C_t^{1-\theta}+\frac{1}{1+\rho\delta}H(\Phi)\left(E_t\left[C_{t+\delta}^{1-\gamma}\right]\right)^\frac{1-\theta}{1-\gamma}.
\end{align}
Hence, we obtain for $C_t\neq 0$,
\begin{align}
\frac{1}{H(\Phi)}= 1-\frac{1}{1+\rho\delta}\left(E_t\left[\left(\frac{C_{t+\delta}}{C_t}\right)^{1-\gamma}\right]\right)^\frac{1-\theta}{1-\gamma}.
\end{align}</p>

<h2>2) Find $E_t\left[\left(\frac{C_{t+\delta}}{C_t}\right)^{1-\gamma}\right]$ fromp the GDP dynamics</h2>

<p>The trick is to find the expectation in the right-hand side from the GDP dynamics.
\begin{align}
\left(\frac{Y_{t+\delta}}{Y_t}\right)^{1-\gamma}= \exp\left((1-\gamma)g\delta\right).\exp\left((1-\gamma)u_{t+\delta}\right).\exp\left((1-\gamma)v_ {t+\delta}\right).
\end{align}
Taking the expectation and using the independence between $u_{t+1}$ and $v_{t+1}$, it follows
\begin{align}
E_t\left(\frac{Y_{t+\delta}}{Y_t}\right)^{1-\gamma}= \exp\left((1-\gamma)g\delta\right).E_t\exp\left((1-\gamma)u_{t+\delta}\right).E_t\exp\left((1-\gamma)v_ {t+\delta}\right).
\end{align}
The expectation of $\exp(X)$ where $X$ follows $\mathcal{N}(0,\sigma^2)$ is $\exp(\sigma^2/2)$. $\exp\left((1-\gamma)v_ {t+\delta}\right)$ is a random variable equal to $1$ with probability $1-p\delta$ and $(1-b)^{1-\gamma}$ with probability $p\delta$.
We substitute the expectation operator:
\begin{align}
E_t\left(\frac{Y_{t+\delta}}{Y_t}\right)^{1-\gamma}= \exp\left((1-\gamma)g\delta\right).\exp\left(\frac{(1-\gamma)^2\sigma^2\delta}{2}\right).\left(1-p\delta+pE[(1-b)^{1-\gamma}]\delta\right).
\end{align}
Finally, we use $C_t=Y_t$ to compute an equation for $\Phi$:
\begin{align}
\frac{1}{H(\Phi)}&amp;= 1-\frac{1}{1+\rho\delta}\left\lbrace\exp\left((1-\theta)g\delta\right).\exp\left(\frac{(1-\gamma)(1-\theta)\sigma^2\delta}{2}\right)\right.\\
 &amp;\left. .\left(1-p\delta+pE[(1-b)^{1-\gamma}]\delta\right)^\frac{1-\theta}{1-\gamma}\right\rbrace.
\end{align}</p>

<h2>3) Take the approximation $\delta\to 0$</h2>

<p>The last step consists in taking a first-order approximation (I abusively keep the equal symbol):
\begin{align}
\frac{1}{H(\Phi)}&amp;= 1-(1-\rho\delta). \left(1+(1-\theta)g\delta\right).\left(1+\frac{(1-\gamma)(1-\theta)\sigma^2\delta}{2}\right)\\
 &amp; .\left(1-\frac{1-\theta}{1-\gamma}p\delta+\frac{1-\theta}{1-\gamma}pE[(1-b)^{1-\gamma}]\delta\right).
\end{align}
Pursuing the first-order apprixmation (all the $\delta^i$ with $i&gt;1$ can be neglected), we have
\begin{align}
\frac{1}{H(\Phi)}&amp;= \rho\delta -(1-\theta)g\delta-\frac{(1-\gamma)(1-\theta)\sigma^2\delta}{2}\\
 &amp; +\frac{1-\theta}{1-\gamma}p\delta-\frac{1-\theta}{1-\gamma}pE[(1-b)^{1-\gamma}]\delta.
\end{align}
Substitute $g$ using $g^*=g+\frac{\sigma^2}{2}-pEb$, 
\begin{align}
\frac{1}{H(\Phi)}&amp;= \rho\delta -(1-\theta)g^*\delta+(1-\theta)\frac{\sigma^2}{2}\delta -(1-\theta)pEb\delta -\frac{(1-\gamma)(1-\theta)\sigma^2\delta}{2}\\
 &amp; +\frac{1-\theta}{1-\gamma}p\delta-\frac{1-\theta}{1-\gamma}pE[(1-b)^{1-\gamma}]\delta.
\end{align}
We take $\delta=1$ and invert function $H$ to find the solution in the footnote 7 of the paper. The right-hand side of this equation "simplifies" to the within braces in the formula.</p>

CC BY-SA 3.0
2017-10-23T17:35:23.860Z
2017-10-23T17:35:23.860Z
GuiWil
<p>In Barro (2009) <em><a href="http://piketty.pse.ens.fr/files/Barro2009.pdf" rel="noreferrer">Rare disasters, asset prices and welfare costs</a></em> Barro develops a Lucas tree model with Epstein-Zin preferences. </p>

<p>My question concerns the paper's equation (10). In this equation Barro states that under the optimal solution utility $U_t$ is proportional to consumption $C_t$ rased to the power of $1-\gamma$, where $\gamma$ is the coefficient of relative risk aversion, i.e. </p>

<p>$U_t=\Phi C_t^{1-\gamma}$ </p>

<p>While I understand the logic of this result, I do not understand how he derives the constant $\Phi$, which is shown in footnote 7 of the mentioned paper:</p>

<blockquote>
  <p>Alberto Giovannini and Philippe Weil (1989, appendix) show that, with
  the utility function in equation (9), attained utility, $U_t$ , is
  proportional to wealth raised to the power $1-\gamma$. The form in
  equation (10) follows because $C_t$ is optimally chosen as a constant
  ratio to wealth in the i.i.d. case. The formula for $\Phi$ is, if
  $\gamma \neq 1$ $\theta \neq 1$, $$\Phi  =
 (\frac{1}{1-\gamma})\{\rho+(\theta-1)g^* - (1/2)\gamma(\theta
 -1)\sigma^2 - (\frac{\theta-1}{\gamma-1})p[E(1-b)^{1-\gamma} - 1 - (\gamma  - 1)Eb] \}^{(\gamma-1)/(1-\theta)}$$</p>
</blockquote>

<p>Barro quotes the 1989 NBER paper by Giovannini and Weil. In this paper I can derive the constant. However, it looks completely different than Barro's version, because I end up with an expression that includes $E[R_t^{1-\gamma}]$, where $R_t$ is the return on equity. I believe Barro has replaced $E[R_t^{1-\gamma}]$ with the equilibrium solution of  $R_t$. However, his expression does not include any logs or exp expressions. </p>

<p>I would be grateful for a solution or any hints to the solution.  </p>

CC BY-SA 3.0
2017-10-24T22:03:08.450Z
This looks great!
Thanks for your effort. It will take me a couple of days to review 
part 2 and 3 of your answer, but it looks very intuitive.
drcms02
<macroeconomics><finance><academic-graduate>
Barro's (2009) rare disaster model in the AER: How to derive equation (10)?
drcms02
# Barro's (2009) rare disaster model in the AER: How to derive equation (10)?

(Asked by: username_0 on 2016-01-18)

In Barro (2009) *[Rare disasters, asset prices and welfare costs](http://piketty.pse.ens.fr/files/Barro2009.pdf)* Barro develops a Lucas tree model with Epstein-Zin preferences.

My question concerns the paper's equation (10). In this equation Barro states that under the optimal solution utility $U\_t$ is proportional to consumption $C\_t$ rased to the power of $1-\gamma$, where $\gamma$ is the coefficient of relative risk aversion, i.e.

$U\_t=\Phi C\_t^{1-\gamma}$

While I understand the logic of this result, I do not understand how he derives the constant $\Phi$, which is shown in footnote 7 of the mentioned paper:

> Alberto Giovannini and Philippe Weil (1989, appendix) show that, with
> the utility function in equation (9), attained utility, $U\_t$ , is
> proportional to wealth raised to the power $1-\gamma$. The form in
> equation (10) follows because $C\_t$ is optimally chosen as a constant
> ratio to wealth in the i.i.d. case. The formula for $\Phi$ is, if
> $\gamma \neq 1$ $\theta \neq 1$, $$\Phi =
> (\frac{1}{1-\gamma})\{\rho+(\theta-1)g^\* - (1/2)\gamma(\theta
> -1)\sigma^2 - (\frac{\theta-1}{\gamma-1})p[E(1-b)^{1-\gamma} - 1 - (\gamma - 1)Eb] \}^{(\gamma-1)/(1-\theta)}$$

Barro quotes the 1989 NBER paper by Giovannini and Weil. In this paper I can derive the constant. However, it looks completely different than Barro's version, because I end up with an expression that includes $E[R\_t^{1-\gamma}]$, where $R\_t$ is the return on equity. I believe Barro has replaced $E[R\_t^{1-\gamma}]$ with the equilibrium solution of $R\_t$. However, his expression does not include any logs or exp expressions.

I would be grateful for a solution or any hints to the solution.

**Tags:** macroeconomics, finance, academic-graduate
> **Comments:**
> **username_0**: This looks great! Thanks for your effort. It will take me a couple of days to review  part 2 and 3 of your answer, but it looks very intuitive.  

---

## 1 answers

### Answer by username_1 on 2017-10-23. Score: 4 | Accepted answer

I think Barro means in the footnote that Giovanni and Weil find the same equation, $U\_t=\Phi C^{1-\gamma}$, but using the optimal path of $C\_t$.
In Barro's paper, the approach is different given that the dynamics of $C\_t$ is exogenous: $C\_t=Y\_t$ by assumption.

Barro uses the limit case when the length of a period gets close to 0. Maybe what may bother the reader is that the model is defined as discrete.

## Rewrite the model

First, we can rewrite the model with a length of period $\delta$ and then use $\delta\to 0$.
The GDP dynamics write
$$\log(Y\_{t+\delta})=\log(Y\_t)+g\delta+u\_{t+\delta}+v\_ {t+\delta}$$
with $u\_{t+\delta}\sim \mathcal{N}(0,\delta\sigma^2)$, and $v\_{t+\delta}=0$ with probability $1-p\delta$ and $\log(1-b)$ with probability $p\delta$.
The utility satisfies
$$
U\_t=\frac{1}{1-\gamma}\left\lbrace C\_t^{1-\theta}+\frac{1}{1+\rho\delta}\left[(1-\gamma)E\_tU\_{t+\delta}\right]^\frac{1-\theta}{1-\gamma}\right\rbrace^\frac{1-\gamma}{1-\theta}.
$$

## 1) Find $\Phi$ as a function of $E\_t\left[\left(\frac{C\_{t+\delta}}{C\_t}\right)^{1-\gamma}\right]$

From now suppose there is a $\Phi$ such that $U\_t=\Phi C^{1-\gamma}$ (note that $\Phi$ depends on $\delta$ a priori).
Define $H(U)=[(1-\gamma)U]^\frac{1-\theta}{1-\gamma}$, the utility satisfies
\begin{align}
H(U\_t)= C\_t^{1-\theta}+\frac{1}{1+\rho\delta}H(E\_tU\_{t+\delta}).
\end{align}
We substitute $U\_t$:
\begin{align}
H(\Phi)C\_t^{1-\theta}= C\_t^{1-\theta}+\frac{1}{1+\rho\delta}H(\Phi)\left(E\_t\left[C\_{t+\delta}^{1-\gamma}\right]\right)^\frac{1-\theta}{1-\gamma}.
\end{align}
Hence, we obtain for $C\_t\neq 0$,
\begin{align}
\frac{1}{H(\Phi)}= 1-\frac{1}{1+\rho\delta}\left(E\_t\left[\left(\frac{C\_{t+\delta}}{C\_t}\right)^{1-\gamma}\right]\right)^\frac{1-\theta}{1-\gamma}.
\end{align}

## 2) Find $E\_t\left[\left(\frac{C\_{t+\delta}}{C\_t}\right)^{1-\gamma}\right]$ fromp the GDP dynamics

The trick is to find the expectation in the right-hand side from the GDP dynamics.
\begin{align}
\left(\frac{Y\_{t+\delta}}{Y\_t}\right)^{1-\gamma}= \exp\left((1-\gamma)g\delta\right).\exp\left((1-\gamma)u\_{t+\delta}\right).\exp\left((1-\gamma)v\_ {t+\delta}\right).
\end{align}
Taking the expectation and using the independence between $u\_{t+1}$ and $v\_{t+1}$, it follows
\begin{align}
E\_t\left(\frac{Y\_{t+\delta}}{Y\_t}\right)^{1-\gamma}= \exp\left((1-\gamma)g\delta\right).E\_t\exp\left((1-\gamma)u\_{t+\delta}\right).E\_t\exp\left((1-\gamma)v\_ {t+\delta}\right).
\end{align}
The expectation of $\exp(X)$ where $X$ follows $\mathcal{N}(0,\sigma^2)$ is $\exp(\sigma^2/2)$. $\exp\left((1-\gamma)v\_ {t+\delta}\right)$ is a random variable equal to $1$ with probability $1-p\delta$ and $(1-b)^{1-\gamma}$ with probability $p\delta$.
We substitute the expectation operator:
\begin{align}
E\_t\left(\frac{Y\_{t+\delta}}{Y\_t}\right)^{1-\gamma}= \exp\left((1-\gamma)g\delta\right).\exp\left(\frac{(1-\gamma)^2\sigma^2\delta}{2}\right).\left(1-p\delta+pE[(1-b)^{1-\gamma}]\delta\right).
\end{align}
Finally, we use $C\_t=Y\_t$ to compute an equation for $\Phi$:
\begin{align}
\frac{1}{H(\Phi)}&= 1-\frac{1}{1+\rho\delta}\left\lbrace\exp\left((1-\theta)g\delta\right).\exp\left(\frac{(1-\gamma)(1-\theta)\sigma^2\delta}{2}\right)\right.\\
&\left. .\left(1-p\delta+pE[(1-b)^{1-\gamma}]\delta\right)^\frac{1-\theta}{1-\gamma}\right\rbrace.
\end{align}

## 3) Take the approximation $\delta\to 0$

The last step consists in taking a first-order approximation (I abusively keep the equal symbol):
\begin{align}
\frac{1}{H(\Phi)}&= 1-(1-\rho\delta). \left(1+(1-\theta)g\delta\right).\left(1+\frac{(1-\gamma)(1-\theta)\sigma^2\delta}{2}\right)\\
& .\left(1-\frac{1-\theta}{1-\gamma}p\delta+\frac{1-\theta}{1-\gamma}pE[(1-b)^{1-\gamma}]\delta\right).
\end{align}
Pursuing the first-order apprixmation (all the $\delta^i$ with $i>1$ can be neglected), we have
\begin{align}
\frac{1}{H(\Phi)}&= \rho\delta -(1-\theta)g\delta-\frac{(1-\gamma)(1-\theta)\sigma^2\delta}{2}\\
& +\frac{1-\theta}{1-\gamma}p\delta-\frac{1-\theta}{1-\gamma}pE[(1-b)^{1-\gamma}]\delta.
\end{align}
Substitute $g$ using $g^\*=g+\frac{\sigma^2}{2}-pEb$,
\begin{align}
\frac{1}{H(\Phi)}&= \rho\delta -(1-\theta)g^\*\delta+(1-\theta)\frac{\sigma^2}{2}\delta -(1-\theta)pEb\delta -\frac{(1-\gamma)(1-\theta)\sigma^2\delta}{2}\\
& +\frac{1-\theta}{1-\gamma}p\delta-\frac{1-\theta}{1-\gamma}pE[(1-b)^{1-\gamma}]\delta.
\end{align}
We take $\delta=1$ and invert function $H$ to find the solution in the footnote 7 of the paper. The right-hand side of this equation "simplifies" to the within braces in the formula.


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