The typical dynamic consumption-saving under certainty model can be written as:
$$ \max V(c)=\sum_{t=1}^{T} \beta^{t-1}\; u(c_t) $$ Subject to the intertemporal budget constraint $$ \sum_{t=1}^{T}\frac{y_t}{1+r}=\sum_{t=1}^{T}\frac{c_t}{1+r} $$
Where $T\leq \infty$ (finite or infinite horizon), $\beta \in (0,1)$ is the subjective discount factor, $y_t$ is the individual's income and $c_t$ is the individual's consumption in every time period. Finally, $u(\cdot)$ is the agent's subutility function, with $u^{\prime}>0$ whose concavity is usually related to the agent's preference for consumption smoothing. The argument is equivalent to the one justifying that an individual will reveal risk aversion iff his utility function over wealth is strictly concave. The argument that leads to consumption smoothing is the following: The first order condition for the problem is the well known Euler equation
$$ \frac{u^{\prime}(c_t)}{u^{\prime}(c_{t+1})}=\beta \; (1+r) $$ It is assumed that $\beta^{-1}=(1+r)$ so that $$ u^{\prime}(c_t)=u^{\prime}(c_{t+1}) \quad \Rightarrow \text{by strict concavity of} \; u \quad c_t=c_{t+1} $$
I have two questions regarding this argument which are somehow related:
- The result of consumption smoothing seems to be independent of the preference for such smoothing: one needs the condition $\beta =(1+r)$ to hold in orther to obtain the result.
- It seems to me that the implication $$ u^{\prime}(c_t)=u^{\prime}(c_{t+1}) \quad \Rightarrow c_t=c_{t+1} $$
Would still hold for a strictly convex utility function, as one needs a much general condition to the implication to hold, namely that the derivatives of the utility function will be unique for any increasing utility function.. it seems to be that the uniqueness of $u^{\prime}$ (i.e $u^{\prime}(a)=u^{\prime}(b) \quad \Rightarrow a=b$) will hold irrespective of the sign of $u^{\prime \prime}$ provided it is not equal to zero.
If my thinking is correct this would imply that the desire for smoothing doesn't play any role on what the individual does at the end, as this possibility is open only when $\beta =(1+r)$ holds, and furthermore, it would be optimal to smooth out consumption even to an agent with a strictly convex utility function.