7.1.4. Inner Game with Nature

The minimizing value of *nt*+<sup>1</sup> is

$$m\_{t+1} \quad = \frac{e^{\sigma[\Phi\_t \mathbf{Y}\_{t+1}]}}{\mathbb{E}\_t e^{\sigma[\Phi\_t \mathbf{Y}\_{t+1}]}} \equiv n\_{t+1'}^{PA} \tag{58}$$

with limits lim*σ*↓−<sup>∞</sup> *<sup>n</sup>PA <sup>t</sup>*+<sup>1</sup> <sup>→</sup> 0, and lim*σ*↑<sup>0</sup> *<sup>n</sup>PA <sup>t</sup>*+<sup>1</sup> → 1, indicating that the conditional likelihood ratio *nPA* is inversely related to the intensity of the planner's doubt about private beliefs, approaching 1 as doubt ceases.

An important distinction between the political planner and the paternalistic planner with ambiguity is that here, the planner's worst-case distortion of beliefs *nPA* is solely determined by continuation values of wealth, since the only sources of ambiguity are private-sector beliefs that distort the consumer's expectation of future household wealth. Formula (58) instructs us that, via *nPA*, a robust paternalistic planner assigns greater probability weights to histories with low continuation values of wealth, weighted with marginal utility of consumption in (39).

#### 7.1.5. Outer Maximization with Implied Risk-Sensitive Recursion

Substitution of the formula for *nPA <sup>t</sup>*+<sup>1</sup> in (46) implies another variation on Hansen and Sargent's (1995) discounted risk-sensitive recursion,

$$\begin{aligned} \mathcal{V}(\mathbf{Y}\_{l},\mathbf{k}\_{l},\mathbf{Q}\_{l}) &= \max\_{\mathbf{c}\_{l},\mathbf{E}\_{l},\mathbf{k}\_{l+1},\mathbf{Q}\_{l+1},\mathbf{Y}\_{l+1}} \{\boldsymbol{u}(\mathbf{c}\_{l}) + \boldsymbol{\Phi}\_{l}[\boldsymbol{\Omega}\_{l} - \mathbf{Y}\_{l}] + \frac{\beta}{\sigma} \mathbb{E}\_{l} \log e^{\sigma \boldsymbol{\Phi}\_{l} \mathbf{Y}\_{l+1}} + \mathcal{M}\_{l} \\ &+ \quad \beta \mathbb{E}\_{l} \mathcal{V}\_{t+1}(\mathbf{Y}\_{t+1},\mathbf{k}\_{t+1},\mathbf{Q}\_{t+1}) \}, \end{aligned}$$

which differs from the recursion (49) for a political planner in that the exponent does not include the continuation value V*t*<sup>+</sup>1, since, here, risk sensitivity does not apply to the planner's own trusted model.

The FONCs for {*c*, *E*} are previously given by (50) and (51). The envelope conditions are also the same as before. However, for choosing {*kt*+1, *Qt*+1, Υ*t*+1}, the government must solve

$$\mathbb{E}\left[1\right] = \mathbb{E}\_l \beta \frac{\lambda\_{l+1}}{\lambda\_l} (1 - \delta + \mathbb{Y}\_{k\_{l+1}}),\tag{59}$$

$$
\varphi\_t \quad = \ \beta \mathbb{E}\_t \mathcal{V}\_{\mathbb{Q}\_{t+1}} = \beta \mathbb{E}\_t (\varphi\_{t+1} + \lambda\_{t+1} \mathbf{x}\_{t+1} D\_{t+1} F\_{t+1}), \tag{60}
$$

$$
\Phi\_t = -\mathcal{V}\_{\mathbf{Y}\_{t+1}} = \Phi\_{t+1} = n\_{t+1}^{PA} \Phi\_t. \tag{61}
$$

From (61) follows that <sup>Φ</sup>*<sup>t</sup>* is a martingale: E*t*Φ*t*+<sup>1</sup> = E*tnPA <sup>t</sup>*+1Φ*<sup>t</sup>* = Φ*t*, unless the wealth constraint 0 <sup>≤</sup> [Ω*<sup>t</sup>* <sup>+</sup> <sup>E</sup>*tnPA <sup>t</sup>*+1Υ*t*+<sup>1</sup> − Υ*t*] is non-binding, in which case Φ*<sup>t</sup>* = 0, and the government reverts to a social planner.

#### *7.2. Pessimistic Planner, Skeptical Consumer*

This section treats a variation of the paternalistic planner in Section 7.1.3, where, instead of trusting the approximating model *π*, the authority has pessimistic doubts about it, meaning it now faces two kinds of ambiguity: one that derives from its ignorance of private beliefs, indexed by *M*, and the other stemming from its own doubts about the model, indexed by *N*. Accordingly, the planner minimizes with respect to both *m* and *n*, given two Lagrange penalty functions. For the sake of simplicity, I will assume that in the following recursion, a single risk sensitivity *σ* applies to both kinds of ambiguities:

$$\begin{split} \mathcal{V}\_{t}(\mathbf{Y}\_{t},\mathbf{k}\_{t},\mathbf{Q}\_{t}) &= \max\_{\mathbf{c}\_{t},\mathbf{E}\_{t},\mathbf{k}\_{t+1},\mathbf{Q}\_{t+1}\mathbf{Y}\_{t+1}} \min\_{n\_{t+1},m\_{t+1}} \boldsymbol{\mu}(\mathbf{c}\_{t}) \\ &- \frac{\beta}{\sigma} \mathbb{E}\_{t}(n\_{t+1}\log n\_{t+1} + m\_{t+1}\log m\_{t+1}) \\ &+ \quad \beta \mathbb{E}\_{t} n\_{t+1} \mathcal{V}\_{t+1}(\mathbf{Y}\_{t+1},\mathbf{k}\_{t+1},\mathbf{Q}\_{t+1}) \\ &+ \quad \Phi\_{t}[\boldsymbol{\Omega}\_{t} + \mathbb{E}\_{t} m\_{t+1} \mathbf{Y}\_{t+1} - \mathbf{Y}\_{t}] + \mathcal{M}\_{t}. \end{split}$$

#### 7.2.1. Inner Minimization

Minimization with respect to *n* and *m* produces the following worst-case multipliers

$$m\_{t+1}^p = \begin{array}{c} \underline{\mathcal{e}^{\sigma \mathcal{V}\_{t+1}}}\\ \underline{\mathbb{E}\_t \mathcal{e}^{\sigma \mathcal{V}\_{t+1}}} \end{array} \tag{62}$$

$$m\_{t+1}^p = \frac{e^{\sigma \Phi\_t \mathbf{Y}\_{t+1}}}{\mathbb{E}\_t e^{\sigma \Phi\_t \mathbf{Y}\_{t+1}}}.\tag{63}$$

#### 7.2.2. Outer Maximization with Implied Risk-Sensitive Recursion

Substitution as before yields the risk-sensitive recursion

$$\mathcal{V}(\mathbf{Y}\_{l},k\_{l},\mathbf{Q}\_{l}) \quad \begin{aligned} \mathcal{V}(\mathbf{Y}\_{l},k\_{l},\mathbf{Q}\_{l}) \quad & = \max\_{\mathbf{c}\_{l},H\_{l},\mathbf{E}\_{l},\mathbf{E}\_{l+1},\mathbf{Q}\_{l+1},\mathbf{Y}\_{l+1}} \boldsymbol{\mu}(\mathbf{c}\_{l},H\_{l}) + \boldsymbol{\Phi}\_{l}[\boldsymbol{\Omega}\_{l} - \mathbf{Y}\_{l}] + \boldsymbol{\mathcal{M}}\_{l} \\ & + \frac{\boldsymbol{\mathcal{B}}}{\boldsymbol{\sigma}}(\log \mathbb{E}\_{l}e^{\sigma \mathcal{V}\_{l+1}} + \log \mathbb{E}\_{l}e^{\sigma \boldsymbol{\Phi}\_{l}\mathbf{Y}\_{l+1}}). \end{aligned}$$

The first-order conditions for {*kt*+1, *Qt*+1, Υ*t*+1} are

$$\mathbb{1}\_{t} = \ \frac{\beta}{\lambda\_{t}} \mathbb{E}\_{t} n\_{t+1}^{p} \mathcal{V}\_{k\_{t+1}} = \beta \mathbb{E}\_{t} n\_{t+1}^{p} \frac{\lambda\_{t+1}}{\lambda\_{t}} (1 - \delta + \mathcal{Y}\_{k\_{t+1}}),\tag{65}$$

$$\mathcal{J}\_t = -\beta \mathbb{E}\_t n\_{t+1}^p \mathcal{V}\_{Q\_{t+1}} = \beta \mathbb{E}\_t n\_{t+1}^p (\boldsymbol{\varphi}\_{t+1} + \boldsymbol{\lambda}\_{t+1} \mathbf{x}\_{t+1} D\_{t+1} F\_{t+1}) \tag{66}$$

$$
\Phi\_l \quad = \quad -\mathcal{V}\_{\mathbf{Y}\_{l+1}} = \Phi\_{l+1} = \frac{m\_{l+1}^p}{n\_{l+1}^p} \Phi\_l. \tag{67}
$$

Those for {*c*, *H*, *E*} remain the same as before. A special case:

1. The consumer has rational expectations (*m<sup>p</sup>* = 1). As noted earlier, this case has been widely treated in papers on robust climate policy.

#### *7.3. Pessimistic Planner, Pessimistic Consumer*

This section treats a variation on the preceding belief regime by replacing its skeptical consumers with the pessimistic consumers from Section 3.4. Doing so will require adding two more implementability constraints: (1) the law of motion for the households' worstcase beliefs *M<sup>c</sup> <sup>t</sup>* in (43), because the authority needs to keep track of its evolution, and (2) the consumer's risk-sensitive utility recursion (44), because increments to the worst-case

likelihood ratio *M<sup>c</sup> <sup>t</sup>* are determined by that household's utility U*t*. <sup>29</sup> For this policy maker, the Bellman recursion is,

$$\begin{split} \mathcal{V}\_{t}(\mathbf{Y}\_{t},k\_{t},\mathbf{Q}\_{t}) &= \max\_{\boldsymbol{\varepsilon}\_{t},\mathbf{E}\_{t},k\_{t+1},\mathbf{Q}\_{t+1},\mathbf{Y}\_{t+1},M\_{t}^{c},\mathcal{U}\_{t+1}} \min\_{\boldsymbol{\varepsilon}\_{t}} u(\boldsymbol{c}\_{t}) - \frac{\beta}{\sigma} \mathbb{E}\_{t}(n\_{t+1}\log n\_{t+1}) \\ &+ \quad \beta \mathbb{E}\_{t}n\_{t+1}\mathcal{V}\_{t+1}(\mathbf{Y}\_{t+1},k\_{t+1},\mathbf{Q}\_{t+1}) \\ &+ \quad \Phi\_{t}[\boldsymbol{\Omega}\_{t} + \beta \mathbb{E}\_{t}\frac{\mathcal{M}\_{t+1}^{c}}{\mathcal{M}\_{t}^{c}}\mathbf{Y}\_{t+1} - \mathbf{Y}\_{t}] + \mathcal{M}\_{t} \\ &+ \quad \beta \mathbb{E}\_{t}\mu\_{t+1} \left[\frac{\boldsymbol{\varepsilon}^{\sigma \prime \mathcal{U}\_{t+1}}}{\mathbb{E}\_{t}\boldsymbol{\varepsilon}^{\sigma \prime \mathcal{U}\_{t+1}}}M\_{t}^{c} - \mathcal{M}\_{t+1}^{c}\right] \\ &+ \quad \varepsilon\_{t}\left[u(\boldsymbol{c}\_{t},H\_{t}) + \frac{\beta}{\sigma^{c}}\log \mathbb{E}\_{t}\boldsymbol{\varepsilon}^{\sigma \prime \mathcal{U}\_{t+1}} - \mathcal{U}\_{t}\right] + \mathcal{M}\_{t}, \end{split}$$

where *μt*+<sup>1</sup> and *ε<sup>t</sup>* are the Lagrangian shadow prices for the law of motion for *M<sup>c</sup> <sup>t</sup>*+<sup>1</sup> and the consumer's worst-case utility U*t*, respectively.

Outer Maximization with Implied Risk-Sensitive Recursion

With *m<sup>c</sup>* computed by the consumer in Section 3.4,

$$m\_{t+1}^{\varepsilon} = \frac{M\_{t+1}^{\varepsilon}}{M\_t^{\varepsilon}} = \frac{e^{\sigma\_n \mathcal{U}\_{t+1}}}{\mathbb{E}\_t e^{\sigma\_n \mathcal{U}\_{t+1}}} \cdot \mathbb{I}$$

and *n<sup>p</sup>* the worst-case multiplier chosen by this planner,

$$m\_{t+1}^p = \frac{e^{\sigma\_n \mathcal{V}\_{t+1}}}{\mathbb{E}\_t e^{\sigma\_n \mathcal{V}\_{t+1}}}$$

,

the risk-sensitive recursion to be solved under dual ambiguities is,

$$\begin{split} \mathcal{V}(\mathbf{Y}\_{t},k\_{t},\mathcal{Q}\_{t}) &= \max\_{\boldsymbol{\varepsilon}\_{t},\boldsymbol{\varepsilon}\_{t},\boldsymbol{M}\_{t}^{c},\mathcal{U}\_{t}\neq\boldsymbol{0},\boldsymbol{\varepsilon}\_{t},\boldsymbol{Y}\_{t+1}} u(\boldsymbol{\varepsilon}\_{t}) + \boldsymbol{\Phi}\_{t}[\boldsymbol{\Omega}\_{t} + \boldsymbol{\beta}\mathbb{E}\_{t}\frac{\boldsymbol{M}\_{t+1}^{c}}{\boldsymbol{M}\_{t}^{c}}\mathbf{Y}\_{t+1} - \mathbf{Y}\_{t}] \\ &+ \quad \frac{\beta}{\sigma}\log\mathbb{E}\_{t}e^{\sigma\boldsymbol{\mathscr{V}}\_{t+1}} + \beta\mathbb{E}\_{t}\mu\_{t+1}\left[\frac{e^{\sigma^{c}\boldsymbol{\mathscr{U}}\_{t+1}}}{\mathbb{E}\_{t}e^{\sigma^{c}\boldsymbol{\mathscr{U}}\_{t+1}}}M\_{t}^{c} - \boldsymbol{M}\_{t+1}^{c}\right] \\ &+ \quad \varepsilon\_{t}\left[u(\boldsymbol{\varepsilon}\_{t},\boldsymbol{H}\_{t}) + \frac{\beta}{\sigma^{c}}\log\mathbb{E}\_{t}e^{\sigma^{c}\boldsymbol{\mathscr{U}}\_{t+1}} - \boldsymbol{\mathcal{U}}\_{t}\right] + \boldsymbol{\mathcal{M}}\_{t}. \end{split}$$

The first-order condition for *c* changes a little from before and becomes

$$\boldsymbol{c}\_{t}: \quad \boldsymbol{\lambda}\_{t} = \left[1 + \boldsymbol{\varepsilon}\_{t}\right] \boldsymbol{u}\_{\mathrm{c}} + \boldsymbol{\Omega}\_{\mathrm{c}\_{t}} \boldsymbol{\Phi}\_{t},\tag{68}$$

while the condition for *E* remains (51). The first-order conditions for {*kt*+1, *Qt*+1, Υ*t*+1} are

$$\begin{array}{rcl}1 &=& \beta \mathbb{E}\_{t} n\_{t+1}^{p} \frac{\lambda\_{t+1}}{\lambda\_{t}} (1 - \delta + \mathbb{Y}\_{k\_{t+1}}), \end{array} \tag{69}$$

$$
\rho\_t = \beta \mathbb{E}\_t n\_{t+1}^p \mathbb{V}\_{Q\_{t+1}} = \beta \mathbb{E}\_t n\_{t+1}^p (\varphi\_{t+1} + \lambda\_{t+1} \mathbf{x}\_{t+1} D\_{t+1} F\_{t+1}),
\tag{70}
$$

$$
\Phi\_{t+1} = \frac{m\_{t+1}^c}{n\_{t+1}^p} \Phi\_{t\prime} \tag{71}
$$

$$0 \le \Phi\_t[\Omega\_t + \mathbb{E}\_t m\_{t+1}^c \mathbb{Y}\_{t+1} - \mathbb{Y}\_t].\tag{72}$$

From (71), Φ*<sup>t</sup>* is the submartingale:

$$\mathbb{E}\_t \Phi\_{t+1} \ge \frac{\mathbb{E}\_t m\_{t+1}^c}{\mathbb{E}\_t n\_{t+1}^p} \Phi\_t = \Phi\_{t\_\prime} \tag{73}$$

unless the wealth constraint is nonbinding, when, by the Kuhn-Tucker condition (72), Φ*<sup>t</sup>* = 0. Finally, given endogenous belief distortions *M<sup>c</sup>* in the private sector, the first-order conditions with respect to *M<sup>c</sup> <sup>t</sup>* and U*<sup>t</sup>* are

$$\mathcal{M}\_t^{\mathbb{C}}: \quad \mu\_t - \frac{\Phi\_{t-1}}{\mathcal{M}\_{t-1}^{\mathbb{C}}} \mathbf{Y}\_t = \beta \mathbb{E}\_t m\_{t+1}^{\mathbb{C}} \left(\mu\_{t+1} - \frac{\Phi\_t}{\mathcal{M}\_t^{\mathbb{C}}} \mathbf{Y}\_{t+1}\right), \tag{74}$$

$$\mathcal{M}\_t: \quad \varepsilon\_t = \sigma^c m\_t^c M\_{t-1}^c \left(\mu\_t - \mathbb{E}\_{t-1} m\_t^c \mu\_t\right) + m\_t^c \varepsilon\_{t-1}. \tag{75}$$

The forward solution of (74) implies that the shadow value of increasing *m<sup>c</sup> <sup>t</sup>* is proportional to the value of debt (to the consumer) in units of the marginal utility of consumption,

$$
\mu\_t \quad = \begin{array}{c} \Phi\_{t-1} \\ \overline{M\_{t-1}^c} \end{array} \mathbf{Y}\_t = \frac{\Phi\_{t-1}}{\overline{M\_{t-1}^c}} \mu\_{\varepsilon,t} b\_{t\_\prime} \tag{76}
$$

obtained by using (42). Substituting this in (75) yields

$$
\varepsilon\_t = -\sigma^\varepsilon \Phi\_{t-1} m\_t^\varepsilon u\_{\varepsilon t} (b\_t - \mathbb{E}\_{t-1} m\_t^\varepsilon b\_t) + m\_t^\varepsilon \varepsilon\_{t-1} \tag{77}
$$

where the term in parentheses is the innovation in government debt, with positive surprises producing a negative shock to the pessimistic likelihood ratio.

### **8. The Equilibrium Price of Capital**

The consumption Euler condition (68) implies the discount factor,

$$\begin{array}{rcl} \mathfrak{e}\_{t+j,t}^{\*} & \equiv & \beta \frac{\lambda\_{t+j}}{\lambda\_{t}}\\ &=& \beta \frac{(1+\mathfrak{e}\_{t+j})u\_{\mathfrak{e}\_{t+j}} + \mathfrak{1}\_{\mathfrak{e}\_{t+j}}\Phi\_{t+j}}{(1+\mathfrak{e}\_{t})u\_{\mathfrak{e}\_{t}} + \mathfrak{1}\_{\mathfrak{e}\_{t}}\Phi\_{t}}\\ &=& \beta \frac{u\_{\mathfrak{e}\_{t+j}}}{u\_{\mathfrak{e}\_{t}}} \frac{1+\mathfrak{e}\_{t+j} + [1-\gamma+\gamma\vartheta\_{t+1}]\Phi\_{t+j}}{1+\mathfrak{e}\_{t} + [1-\gamma+\gamma\vartheta\_{t}]\Phi\_{t}}\\ &=& \Psi(\mathfrak{e}\_{t+j}, \mathfrak{n}\_{t+j}^{\*})\mathfrak{e}\_{t+j,t}. \end{array} \tag{78}$$

where

$$\Psi(\varepsilon\_{t+j}, n\_{t+j}^\*) = \frac{1 + \varepsilon\_{t+j} + [1 - \gamma + \gamma \theta\_{t+j}] n\_{t+j}^\* \Phi\_{t+j-1}}{1 + \varepsilon\_t + [1 - \gamma + \gamma \theta\_t] \Phi\_t},\tag{79}$$

*<sup>ϑ</sup><sup>t</sup>* = *Ht*+*gt ct* is the inverse of the average propensity to consume, namely the ratio of wage income plus the lump-sum rebate to consumption, and *n*∗ *t*+*j* , associated with the planner's implementability constraint on household wealth, varies according to policy regime as shown in Table 1. The shadow prices *μ* and *ε* are zero, except when the private sector has pessimistic beliefs *m* = *mc*.

For the Ramsey plans derived previously, the conditions for capital *k* imply the distorted discount factor,

$$
\varrho\_{t+j,t}^{\*\*} \equiv n\_{t+j}^{\*\*} \beta \frac{\lambda\_{t+j}}{\lambda\_t} = n\_{t+j}^{\*\*} \Psi(\varepsilon\_{t+j}, n\_{t+j}^{\*}) \varrho\_{t+j,t} \tag{80}
$$

where, like *n*∗ *t*+*j* , *n*∗∗ *<sup>t</sup>*+*<sup>j</sup>* varies by belief regime as shown in Table 1. <sup>30</sup> Note that from (78) and (80), the discount factor ∗∗ *<sup>t</sup>*+*j*,*<sup>t</sup>* is an *n*∗∗-distorted version of the previous consumption discount factor ∗ *t*+*j*,*t* . The corresponding *t* + *j* distorted equilibrium price of capital is

$$
\mathfrak{d}^{\ast \ast}\_{t+j,t} = n^{\ast \ast}\_{t+j} \mathfrak{d}^{\ast}\_{t+j,t}.\tag{81}
$$


**Table 1.** Equilibrium Arrow–Debreu prices in alternative belief regimes.
