# Why Certainity Eqivalence in PIH only holds for quadratic utilities

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?

• Please let us see a reference for your claim. – Ali Oct 18 '18 at 12:58
• @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 Oct 18 '18 at 13:46
• 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 Oct 18 '18 at 14:31
• 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 Oct 18 '18 at 16:00

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.

• 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 Oct 18 '18 at 18:47
• 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 Oct 18 '18 at 19:29
• I fixed a bracket typo in the Lagrangian that may have caused confusion. – Kenneth Rios Oct 18 '18 at 19:44