# Precautionary Savings Two-Period model with a second period quadratic utility function

So preferences are defined as $u(c_0)+Ev(c_1)$

Both utility functions are strictly increasing, concave, and twice differentiable.

I understand in the setting of $u(c_0)+Eu(c_1)$ when $u(c_0)$ is quadratic that it will induce no precautionary savings. (that is the two periods share the same functional form)

However, my problem is that the explicit function of $u(c_0)$ is not given in my case, only that $v(c_1)$ is quadratic.

How do I assess the first period utility function?

The two papers that explored first savings under uncertainty in a two-period setting are

Leland, H. E. (1968). Saving and uncertainty: The precautionary demand for saving. The Quarterly Journal of Economics, 82(3), 465-473.

and

Sandmo, A. (1970). The effect of uncertainty on saving decisions. The Review of Economic Studies, 37(3), 353-360.

They both deal with a general two period utility function $U(c_0,c_1)$ not necessarily additive. But both remark the following related to such a case: The condition for precautionary savings depends on third cross partial derivatives, as well as on the third own derivative with respect to future consumption only.

If the utility function is additive $U=u(c_0)+Ev(c_1)$, then cross-partials are zero, so the result hinges only on the third own derivative for the second-period consumption.

It follows that with an additive over time utility function, it suffices that the second-period utility is quadratic (so third own derivative is zero), in order to not get precautionary savings, irrespective of the form of the first-period utility.

• Wow, thanks a lot. Also, then I'm not sure if the following is correct in showing that optimal consumption only depends on expected income when: $$v^1(c_1)=1-ac_1$$ so $$u^1(c_0)=Ev^1(c_1)$$ then $$1-ac_0=E(1-ac_1)$$ can I assume that? $$c_0=E[c_1]$$ so then $$c_0=E[y_1+(1+r)s]$$? – user12383 Mar 11 '17 at 1:37
• The third line in your previous comment assumes implicitly that utility of the first period is also quadratic. Also "depends only on expected income" contrasted to what? That it does not depend on first-period income and/or the interest rate? – Alecos Papadopoulos Mar 11 '17 at 14:41
• @user12383 Then you don't need to write explicitly first-period utility. Write out the first order condition, (which will be an implicit equation in $c_0$) and only the expected value of income (together with first-period income and interest) will be present there. – Alecos Papadopoulos Mar 11 '17 at 15:05
• @user12383 note that $s$ is expressed in terms of $c_0$. So substitute and re-arrange for completeness. And don't forget the outer $(1+r)$. – Alecos Papadopoulos Mar 11 '17 at 17:40