A Bayesian model of Knightian uncertainty

Abstract

A long tradition suggests a fundamental distinction between situations of risk, where true objective probabilities are known, and unmeasurable uncertainties where no such probabilities are given. This distinction can be captured in a Bayesian model where uncertainty is represented by the agent’s subjective belief over the parameter governing future income streams. Whether uncertainty reduces to ordinary risk depends on the agent’s ability to smooth consumption. Uncertainty can have a major behavioral and economic impact, including precautionary behavior that may appear overly conservative to an outside observer. We argue that one of the main characteristics of uncertain beliefs is that they are not empirical, in the sense that they cannot be objectively tested to determine whether they are right or wrong. This can confound empirical methods that assume rational expectations.

Appendix

Proof of Proposition 2.2

$$\begin{aligned} V(\mu )&= E_\mu E_\theta u\left( {\frac{1}{n}}\sum \limits _{i=1}^n f_i\right) \end{aligned}$$

(10)

$$\begin{aligned}&\simeq E_\mu u\left( {\frac{1}{n}}E_\theta \sum \limits _{i=1}^n f_i\right) \\&\simeq E_\mu u\left( {\frac{1}{n}}E_\theta f\right) \nonumber \\&\simeq E_\mu V\left( \theta ,\,{\frac{1}{n}}f\right) \nonumber \\&< V\left( \bar{\theta }^\mu \right) , \nonumber \end{aligned}$$

(11)

where the last inequality follows from the strict concavity of u.

Proof of Proposition 3.1

We first note that since we are optimizing a concave function, it suffices to show that the Euler and transversality conditions are satisfied. Applying $u^{-1}$ to both sides of $u(c_t^*) = E_t[u(c_{t+k}^*)]$ and substituting in (5), we can express the evolution of wealth as:

$$\begin{aligned} w_{t+1}&= (1+r)\left[ w_t + f_{t} - {\frac{r}{1+r}}w - r^{-1} \mathrm{CE}({\textit{rf}}\,| \theta )\right] \\&= w_t + (1+r) f_{t} - (1+r)r^{-1} \mathrm{CE}({\textit{rf}}\,| \theta ), \end{aligned}$$

implying

$$\begin{aligned} c_{t+1}&= {\frac{r}{1+r}}\left[ w_t + (1+r) f_{t} - (1+r)r^{-1} \mathrm{CE}({\textit{rf}}\,|\theta )\right] + r^{-1} \mathrm{CE}({\textit{rf}}\,| \theta )\\&= {\frac{r}{1+r}}w_t + r f_{t} - \mathrm{CE}({\textit{rf}}\,| \theta )+ r^{-1} \mathrm{CE}({\textit{rf}}\,|\theta )\\&= c_t + r f_{t} - \mathrm{CE}({\textit{rf}}\,| \theta ). \end{aligned}$$

Under CARA, certainty equivalents are independent of wealth, so the last equation implies that $c_{t+1}$ has a time t certainty equivalent of $c_t,$ which we argued earlier is equivalent to the Euler equation.

To get the formula for the evolution of consumption, note that $\theta$ does not change in time (this is the key property of lack of uncertainty) so we can deduce from Eq. (6) that

$$\begin{aligned} c_{t+1} - c_t = {\frac{r}{1+r}}\left( w_{t+1} - w_t\right) , \end{aligned}$$

after which substituting for $w_{t+1}$ according to Eq. (5) and $c_t$ from Eq. (6) gives (7). It shows that consumption, which is linear in wealth, can grow only linearly so long as f is bounded, so wealth also grows at most linearly also and the transversality condition $\lim _{t \rightarrow \infty } (1+r)^{-t}w_t$ is satisfied with probability 1. $\square$

Next, we establish the concavity of the value function under risk. Note that for any lottery over two outcomes, we have:

$$\begin{aligned} u^{-1}\left( \lambda u\left( r f_1\right) + (1-\lambda ) u\left( r f_2\right) \right) \ge r u^{-1}\left( \lambda u\left( f_1\right) + (1-\lambda ) u\left( f_2\right) \right) . \end{aligned}$$

(12)

This observation, along with the previous proposition, allows us to show:

Proposition 5.1

For known $\theta ,$

• (1) Optimal consumption is convex in $\theta .$

• (2) Value is concave in $\theta$ for any $r<1.$

To illustrate the meaning of this proposition, assume that under distribution $\theta _0$ an asset returns 0 for sure, under $\theta _1$ it returns 1 for sure, and thus under $\theta _2 = 0.5 \theta _0 + 0.5 \theta _1$ the asset returns 0 or 1 each period according to a series of i.i.d. coin flips. With zero wealth, optimal consumption under $\theta _0$ is simply 0, with value u(0), and under $\theta _1$ is 1, with value u(1). The first point then states that consumption under $\theta _2$ is less than 0.5; this is because there is a small precautionary saving to mitigate against the risk of the next coin flip. However, value under $\theta _2$ is greater than the average of u(0) and u(1); this is driven by consumption smoothing, as $\theta _2$ ensures many independent coin flips and entails much less long-run risk than a lottery between $\theta _0$ and $\theta _1.$ There are two countervailing forces on the value, as taking a convex combination of $\theta$s increases the need for precautionary savings but mitigates against long-run risk. The CARA utility allows us to say that the second force dominates for $r < 1.$ When $r \ll 1$ as usual, the second force is much stronger and it is reasonable to think that the CARA form is not essential to this claim. It is interesting to note that if we allow $r>1,$ the precautionary-savings effect dominates, as discounting is fast and long-run risk is unimportant (mostly the current period matters).

Proof of Proposition 5.1

Applying Proposition 3.1, the first item reduces to the statement that the certainty equivalent of a compound lottery is less than the average of the certainty equivalents of the nodes. That is,

$$\begin{aligned} \mathrm{CE}\left( {\textit{rf}}\,| \lambda \theta _1 + (1- \lambda ) \theta _2\right)&= u^{-1} E\left[ u({\textit{rf}}\,\,) | \lambda \theta _1 + (1- \lambda ) \theta _2 \right] \\&= u^{-1} \left( \lambda E\left[ u({\textit{rf}}\,\,)\,| \theta _1\right] + (1- \lambda ) E\left[ u({\textit{rf}}\,\,)| \theta _2\right] \right) \\&= u^{-1} \left( \lambda u\left( \mathrm{CE} \left( {\textit{rf}}\,| \theta _1\right) \right) + (1- \lambda ) u\left( \mathrm{CE}\left( {\textit{rf}}\,| \theta _2\right) \right) \right) \\&\le \lambda \mathrm{CE} \left( {\textit{rf}}\,| \theta _1\right) + (1- \lambda ) \mathrm{CE} \left( {\textit{rf}}\,| \theta _2\right) , \end{aligned}$$

where the last step follows from u being increasing and concave (take u of both sides to get the definition of concavity). Along with Proposition 3.1 this yields the first statement.

For the second statement, use $V(\theta )$ as shorthand for $V(\theta ,\,0).$ By earlier results,

$$\begin{aligned} V(\theta ) = u\left( r^{-1} \mathrm{CE}({\textit{rf}}\,|\theta )\right) = u\left( r^{-1} u^{-1} E(u({\textit{rf}}\,\,) |\theta )\right) . \end{aligned}$$

Now observe that

$$\begin{aligned} V\left( \lambda \theta _1 + (1- \lambda ) \theta _2\right)&= u\left( r^{-1} \mathrm{CE}\left( {\textit{rf}}\,|\lambda \theta _1 + (1 - \lambda ) \theta _2\right) \right) \\&= u\left( r^{-1} u^{-1}\left( E\left[ u({\textit{rf}}\,\,) | \lambda \theta _1 + (1- \lambda ) \theta _2\right] \right) \right) \\&= u\left( r^{-1} u^{-1}\left( \lambda E\left[ u({\textit{rf}}\,\,) | \theta _1\right] + (1- \lambda )E\left[ u({\textit{rf}}\,\,) |\theta _2\right] \right) \right) \\&= u\left( r^{-1} u^{-1}\left( \lambda u\left( r u^{-1}V\left( \theta _1\right) \right) + (1- \lambda ) u\left( r u^{-1}V\left( \theta _2\right) \right) \right) \right) \\&\ge \lambda V\left( \theta _1\right) + (1-\lambda ) V\left( \theta _2\right) , \end{aligned}$$

where the last step uses (12). This shows concavity of V in $\theta$ for w = 0. For general w, Eqs. (3.1) and (6) show that

$$\begin{aligned} V(\theta ,\,w) = \mathrm{e}^{-a {\frac{r}{1+r}}w} V(\theta ), \end{aligned}$$

which implies the result. $\square$

Proof of Lemma 3.2

The Bellman equation tells us

$$\begin{aligned} V(\mu ,\,w) = (1-\delta ) u(c^*(\mu ,\,w)) + \delta E[V(\mu |f,\,(1+r)(w - c^*(\mu ,\,w) + f) ) ], \end{aligned}$$

and we showed earlier that $V(\mu ,\,w) = u( c^*(\mu ,\,w)),$ so the Bellman equation reduces to

$$\begin{aligned} u(c^*(\mu ,\,w)) = E[u(c^*(\mu |f,\,(1+r)(w-c^*(\mu ,\,w)+f)))], \end{aligned}$$

and when we substitute the form in the lemma, this reduces algebraically to the Bellman equation for w = 0. This proves the result. $\square$

Proof of Proposition 3.3

The first inequality (in its weak form) is simply the statement that information about $\theta$ has a non-negative value. Since the utility being optimized is strictly concave, optimal plans are unique and it is strict whenever plans differ under the two beliefs.

The second inequality follows from the first, given our earlier result that $V(\mu ,\,w) = u( c^*(\mu ,\,w))$ and the fact that u is increasing and concave. (Assume it is false, apply u to both sides, then apply concavity to contradict the first inequality.)

The final statement follows from expressing $\mu$ as a convex combination of point masses $\theta _i,$ applying convexity of V with respect to $\mu ,$ then applying concavity of V with respect to $\theta$ (Proposition 5.1). $\square$