<p>Certainty equivalence in the context of the Permanent Income Hypothesis implies that <span class="math-container">$u'(E_t[c_{t+1}]) = E_t[u'(c_{t+1})]$</span>, which only holds if marginal utility <span class="math-container">$u'(\cdot)$</span> is linear (by linearity of expectations) and thus only if actual utility <span class="math-container">$u(\cdot)$</span> is quadratic. </p>

<p>To see where this comes from, consider the utility maximization problem of a representative household over an infinite horizon where consumption stream <span class="math-container">$\{c_t\}$</span> and asset holdings stream <span class="math-container">$\{a_{t+1}\}$</span> are chosen, taken as given the real interest rate, fixed at <span class="math-container">$r$</span>, initial wealth <span class="math-container">$a_0$</span> and a stream of income <span class="math-container">$\{y_t\}$</span>. Lastly, take utility to be quadratic over consumption: <span class="math-container">$u(c_t) = \alpha c_t - \frac{1}{2}c^2_t$</span> with sufficiently large <span class="math-container">$\alpha$</span>. </p>

<p>So the household's maximization problem is </p>

<p><span class="math-container">\begin{equation*}
\begin{aligned}
&amp; \underset{\{c_t, a_{t+1}\}}{\text{max}}
&amp; &amp; E_0\sum_{t=0}^{\infty} \beta^{t}u(c_t), \text{ where } \beta = \frac{1}{1 + \rho} \text{ , } \rho &gt; 0 \\
&amp; \text{s.t.}
&amp; &amp; c_t + a_{t+1} = (1+r)a_t + y_t &amp; \forall t.\\
\end{aligned}
\end{equation*}</span></p>

<p><span class="math-container">$$
$$</span>
Therefore, the Lagrangian for this maximization problem is</p>

<p><span class="math-container">\begin{equation*}
\mathcal{L} = \sum_{t=0}^{\infty} \beta^{t} \{E_t[u(c_t) + \lambda_t((1+r)a_t+y_t-c_t-a_{t+1})]\}.
\end{equation*}</span>
<span class="math-container">$$
$$</span></p>

<p>The first-order conditions of the Lagrangian are
<span class="math-container">\begin{align}
\frac{\partial \mathcal{L}}{\partial c_t} &amp;= \beta^{t} u'(c_t) - \beta^t\lambda_t = 0 &amp;&amp;(1)\\
\frac{\partial \mathcal{L}}{\partial a_{t+1}} &amp;= -\beta^{t}\lambda_t + \beta^{t+1} (1+r)E_t[\lambda_{t+1}] = 0 &amp;&amp;(2).
\end{align}</span>
<span class="math-container">$$
$$</span></p>

<p>Solving for <span class="math-container">$\lambda_t$</span> using (1) and substituting into (2) gives
<span class="math-container">\begin{align*}
u'(c_t) = \beta (1+r)E_t[u'(c_{t+1})].
\end{align*}</span>
<span class="math-container">$$
$$</span></p>

<p>Take <span class="math-container">$\rho = r$</span> in this economy for simplicity. Thus <span class="math-container">$\beta(1+r) = 1$</span>:
<span class="math-container">\begin{align*}
u'(c_t) = E_t[u'(c_{t+1})].
\end{align*}</span>
<span class="math-container">$$
$$</span></p>

<p>Since utility is quadratic, marginal utility is linear which implies that the expectation of marginal utility is the marginal utility of the expectation:
<span class="math-container">\begin{align*}
u'(c_t) = u'(E_t[c_{t+1}])
\end{align*}</span>
which implies that
<span class="math-container">\begin{align*}
c_t = E_t[c_{t+1}]
\end{align*}</span>
since <span class="math-container">$u'(\cdot)$</span> is injective. Thus we have certainty equivalence.</p>

CC BY-SA 4.0
2018-10-18T18:47:05.217Z
Fantastic @Kenneth Rios, that makes so much more sense now, thank you! I would just like to ask for a clarification regarding the expectation signs when we form the Lagrangian. If I understand it correctly $E_t[ ]$ is the expectation at times $t$. So is it then trivial that $E_0[ \sum^\infty_0 \beta^t u(c_t)]  = \sum^\infty_0 \beta^t E_t[u(c_t)]$? Indeed why don't we have to use expected consumption and assets?
Jhonny
CC BY-SA 4.0
2018-10-18T19:29:48.390Z
Yes they are notationally equivalent. $E_0$ is more correct as it emphasizes that the consumption profile $\{c_t\}$ is decided at $t=0$ and that there is uncertainty over future consumption in periods $t$. Compacting notation by using $E_t$ makes taking the FOCs easier because it emphasizes that consumption takes place at each time $t$. Also, in my answer I should have noted that the Euler equation for consumption implies that consumption follows a random walk.
Kenneth Rios
CC BY-SA 4.0
2018-10-18T19:44:35.913Z
I fixed a bracket typo in the Lagrangian that may have caused confusion.
Kenneth Rios
CC BY-SA 4.0
2018-10-18T15:59:56.613Z
2018-10-18T19:33:45.677Z
2018-10-18T19:33:45.677Z
Kenneth Rios
<p>In my current macro economics course, it has been stated that there is <em>certainty equivalence</em> in the random walk permanent income hypothesis ("which implies individuals act as if future consumption was at its conditional mean value and ignore its variation") only holds if the utility is assumed to be on form <span class="math-container">$$ u(c) = c - ac^2/2 $$</span>
I have consulted the coursebook but could not find any rigorous proof or intuition of this. Could anybody help please?</p>

CC BY-SA 4.0
2018-10-18T12:58:23.170Z
Please let us see a reference for your claim.
Ali
CC BY-SA 4.0
2018-10-18T13:46:09.630Z
@Ali please see for instance *Intertemporal Macroeconomics*, Gernot Doppelhofer, 2009, page 14
Cambridge Essays in Applied Economics. (I believe pdf copies can be found on google)
Jhonny
CC BY-SA 4.0
2018-10-18T14:31:12.030Z
Hi: I think it has to do with the conditional mean of x ( which is an expectation )  being equivalent to the projection of x when x is gaussian. If x is not gaussian, then there are non-linear estimators x that have better performance than the conditional mean. In the latter case, taking the expectation is not  necessarily optimal.
mark leeds
CC BY-SA 4.0
2018-10-18T16:00:53.797Z
It has to do with the nature of expectations and how the Euler equation simplifies at $\rho = r$. See if my answer is helpful.
Kenneth Rios
<macroeconomics><utility><uncertainty><friedman>
Why Certainity Eqivalence in PIH only holds for quadratic utilities
Jhonny
# Why Certainity Eqivalence in PIH only holds for quadratic utilities

(Asked by: username_0 on 2018-10-18)

In my current macro economics course, it has been stated that there is *certainty equivalence* in the random walk permanent income hypothesis ("which implies individuals act as if future consumption was at its conditional mean value and ignore its variation") only holds if the utility is assumed to be on form $$ u(c) = c - ac^2/2 $$
I have consulted the coursebook but could not find any rigorous proof or intuition of this. Could anybody help please?

**Tags:** macroeconomics, utility, uncertainty, friedman

> **Comments:**
> **username_1**: Please let us see a reference for your claim.  
> **username_0**: @username_1 please see for instance *Intertemporal Macroeconomics*, Gernot Doppelhofer, 2009, page 14 Cambridge Essays in Applied Economics. (I believe pdf copies can be found on google)  
> **username_2**: Hi: I think it has to do with the conditional mean of x ( which is an expectation )  being equivalent to the projection of x when x is gaussian. If x is not gaussian, then there are non-linear estimators x that have better performance than the conditional mean. In the latter case, taking the expectation is not  necessarily optimal.  
> **username_3**: It has to do with the nature of expectations and how the Euler equation simplifies at $\rho = r$. See if my answer is helpful.  

---

## 1 answers

### Answer by username_3 on 2018-10-18. Score: 4 | Accepted answer

Certainty equivalence in the context of the Permanent Income Hypothesis implies that $u'(E\_t[c\_{t+1}]) = E\_t[u'(c\_{t+1})]$, which only holds if marginal utility $u'(\cdot)$ is linear (by linearity of expectations) and thus only if actual utility $u(\cdot)$ is quadratic.

To see where this comes from, consider the utility maximization problem of a representative household over an infinite horizon where consumption stream $\{c\_t\}$ and asset holdings stream $\{a\_{t+1}\}$ are chosen, taken as given the real interest rate, fixed at $r$, initial wealth $a\_0$ and a stream of income $\{y\_t\}$. Lastly, take utility to be quadratic over consumption: $u(c\_t) = \alpha c\_t - \frac{1}{2}c^2\_t$ with sufficiently large $\alpha$.

So the household's maximization problem is

\begin{equation\*}
\begin{aligned}
& \underset{\{c\_t, a\_{t+1}\}}{\text{max}}
& & E\_0\sum\_{t=0}^{\infty} \beta^{t}u(c\_t), \text{ where } \beta = \frac{1}{1 + \rho} \text{ , } \rho > 0 \\
& \text{s.t.}
& & c\_t + a\_{t+1} = (1+r)a\_t + y\_t & \forall t.\\
\end{aligned}
\end{equation\*}

$$
$$
Therefore, the Lagrangian for this maximization problem is

\begin{equation\*}
\mathcal{L} = \sum\_{t=0}^{\infty} \beta^{t} \{E\_t[u(c\_t) + \lambda\_t((1+r)a\_t+y\_t-c\_t-a\_{t+1})]\}.
\end{equation\*}
$$
$$

The first-order conditions of the Lagrangian are
\begin{align}
\frac{\partial \mathcal{L}}{\partial c\_t} &= \beta^{t} u'(c\_t) - \beta^t\lambda\_t = 0 &&(1)\\
\frac{\partial \mathcal{L}}{\partial a\_{t+1}} &= -\beta^{t}\lambda\_t + \beta^{t+1} (1+r)E\_t[\lambda\_{t+1}] = 0 &&(2).
\end{align}
$$
$$

Solving for $\lambda\_t$ using (1) and substituting into (2) gives
\begin{align\*}
u'(c\_t) = \beta (1+r)E\_t[u'(c\_{t+1})].
\end{align\*}
$$
$$

Take $\rho = r$ in this economy for simplicity. Thus $\beta(1+r) = 1$:
\begin{align\*}
u'(c\_t) = E\_t[u'(c\_{t+1})].
\end{align\*}
$$
$$

Since utility is quadratic, marginal utility is linear which implies that the expectation of marginal utility is the marginal utility of the expectation:
\begin{align\*}
u'(c\_t) = u'(E\_t[c\_{t+1}])
\end{align\*}
which implies that
\begin{align\*}
c\_t = E\_t[c\_{t+1}]
\end{align\*}
since $u'(\cdot)$ is injective. Thus we have certainty equivalence.

> **Comments:**
> **username_0**: Fantastic @username_3, that makes so much more sense now, thank you! I would just like to ask for a clarification regarding the expectation signs when we form the Lagrangian. If I understand it correctly $E_t[ ]$ is the expectation at times $t$. So is it then trivial that $E_0[ \sum^\infty_0 \beta^t u(c_t)]  = \sum^\infty_0 \beta^t E_t[u(c_t)]$? Indeed why don't we have to use expected consumption and assets?  
> **username_3**: Yes they are notationally equivalent. $E_0$ is more correct as it emphasizes that the consumption profile $\{c_t\}$ is decided at $t=0$ and that there is uncertainty over future consumption in periods $t$. Compacting notation by using $E_t$ makes taking the FOCs easier because it emphasizes that consumption takes place at each time $t$. Also, in my answer I should have noted that the Euler equation for consumption implies that consumption follows a random walk.  
> **username_3**: I fixed a bracket typo in the Lagrangian that may have caused confusion.  

---

