Why Certainity Eqivalence in PIH only holds for quadratic utilities
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}
&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*}

$$
$$
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} &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}
$$
$$

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.