*3.4. Pessimistic Consumer*

The general framework for the consumer's problem is a game against some malevolent force representing extreme uncertainty about the model. Given the resource constraint in (2) and the budget constraint (4), the representative consumer solves the Lagrangian

$$\max\_{\{\varepsilon\_{l},\mathbb{E}\_{l},b\_{l+1},k\_{l+1},Q\_{l+1}\}} \min\_{\{M\_{l},m\_{l+1}\}} \qquad \mathbb{E}\_{0} \sum\_{t=0}^{\infty} \beta^{t} M\_{l} \left(\mu(c\_{l}) + \lambda\_{l}^{c} \mathcal{L}\_{l} - \frac{\beta}{\sigma^{c}} m\_{l+1} \log m\_{l+1}\right),$$

subject to *Mt*+<sup>1</sup> = *mt*+1*Mt*, *<sup>M</sup>*<sup>0</sup> = 1, E*tmt*+<sup>1</sup> = 1, where *<sup>λ</sup><sup>c</sup> <sup>t</sup>* is a time-varying Lagrangian multiplier associated with the household budget constraint (4), and 0 ≥ − <sup>1</sup> *<sup>σ</sup><sup>c</sup>* < ∞ denotes a Lagrangian shadow cost for the penalty of deviating from rational expectations represented by the approximating distribution *π*, also known as the Kullback-Leibler distance (see Kullback and Leibler 1951) between the two probability measures *π* and *π*ˆ; so 0 > *σ<sup>c</sup>* > −∞ may be considered the consumer's *parameter of ambiguity aversion*. <sup>16</sup> As in all of this paper, expectations E*<sup>t</sup>* are taken over the measure *π*.

The preceding criterion has the Bellman recursion

$$\begin{aligned} \mathcal{M}\_t(k\_t, b\_t, Q\_t) &= \max\_{c\_t, E\_t, k\_{t+1}, b\_{t+1}, Q\_{t+1}} \min\_{m\_{t+1}} u(c\_t) + \lambda\_t^c \mathcal{L}\_t(\mathbf{x}^t) \\ &+ \beta \mathbb{E}\_t \left( m\_{t+1} \mathcal{U}\_{t+1}(k\_{t+1}, b\_{t+1}, Q\_{t+1}) - \frac{\beta}{\sigma^c} m\_{t+1} \log m\_{t+1} \right) \\ &+ \quad q\_t^c (Q\_t - E\_t - Q\_{t+1})\_t \end{aligned} \tag{5}$$

where *ϕ<sup>c</sup> <sup>t</sup>* is the Lagrangian shadow price associated with the resource constrains (2).
