Why Certainity Eqivalence in PIH only holds for quadratic utilities

Question

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?

Answer 1

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.