7.1.1. Inner Game with Nature

Ambiguity for this planner produces an *ex post* worst-case probability model with distorted *ex post homogeneity* in beliefs between the government and the private sector.

Minimization of (47) with respect to *nt*+<sup>1</sup> leads to

$$n\_{t+1} = \frac{e^{\sigma \left[ \mathcal{V}\_{t+1} + \Phi\_t \mathcal{Y}\_{t+1} \right]}}{\mathbb{E}\_t e^{\sigma \left[ \mathcal{V}\_{t+1} + \Phi\_t \mathcal{Y}\_{t+1} \right]}} \equiv n\_{t+1}^{PO} \tag{48}$$

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

7.1.2. Outer Maximization with Implied Risk-Sensitive Recursion

The continuation value V*t*+<sup>1</sup> in the exponent of the formula for *<sup>n</sup>PO* comes from the planner's distrust of the reference model itself, while continuation wealth Υ*t*+<sup>1</sup> points to the planner's ignorance of private-sector beliefs. Formula (48), indicates that, given *σ* < 0, ambiguity leads this robust planner to put more probability weight on histories with low continuation values V*t*+<sup>1</sup> and Υ*t*+1, and lower probabilities on histories with high continuation values.

Substituting (48) for both *nt*+<sup>1</sup> and *mt*+<sup>1</sup> in (47) produces a variation on Hansen and Sargent's (1995) discounted risk-sensitive recursion, one that is augmented with household wealth Υ, a forward-looking entity valued at the shadow price Φ, in the exponent,27

$$\mathcal{V}(\mathbf{Y}\_{l}, k\_{l}, Q\_{l}) \quad = \max\_{\mathbf{c}\_{l}, \mathbf{E}\_{l}, k\_{l+1}, Q\_{l+1}, \mathbf{Y}\_{l+1}} u(\mathbf{c}\_{l}) + \Phi\_{l}[\Omega\_{l} - \mathbf{Y}\_{l}] + \mathcal{M}\_{l}$$

$$+ \quad \frac{\beta}{\sigma} \log \mathbb{E}\_{l} e^{\sigma(\mathbf{V}\_{l+1} + \Phi\_{l}\mathbf{Y}\_{l+1})}.\tag{49}$$

The distortion *nPO* that attains the minimum of the right side of (49) tilts the *xt*+<sup>1</sup> distribution exponentially toward lower continuation values via multiplication of *π*(*xt*+1) by *nPO <sup>t</sup>*+<sup>1</sup> in (48).

The first-order conditions (FONCs) for {*ct*, *Et*} for an interior maximum are: ∀*t* ≥ 1, **Flows**

$$
\alpha\_t: \quad \lambda\_t = \mu\_\mathcal{E} + \Omega\_{\mathcal{E}\_t} \Phi\_{t,\prime} \tag{50}
$$

$$E\_t: \quad \mathcal{Y}\_{E\_t} \lambda\_t = \mathfrak{p}\_t + \mathfrak{v}\_t \tag{51}$$

where *YEt* = *<sup>ν</sup> Yt Et* , is the marginal product of fossil energy based on the Cobb-Douglas assumption.

**Envelope conditions**

$$k\_{\rm t}: \quad \mathcal{V}\_{\rm k\_{\rm t}} = (1 - \delta + \mathcal{Y}\_{\rm k\_{\rm t}}) \lambda\_{\rm t} \tag{52}$$

$$Q\_t: \quad \mathcal{V}\_{Q\_t} = \varphi\_t + \lambda\_t \mathbf{x}\_t \mathbf{D}\_t \mathbf{F}\_{\mathbf{t}\_t} \tag{53}$$

$$\mathcal{Y}\_t: \quad \mathcal{V}\_{\mathcal{Y}\_t} = -\Phi\_t. \tag{54}$$

Note that *λ* ≥ 0 and *ϕ* ≥ 0 imply *Vk* ≥ 0 and V*Qt* ≥ 0, respectively. Further, Φ ≥ 0 implies *V*<sup>Υ</sup> ≤ 0.

**Stocks**

$$\begin{array}{rcl} \mathbb{K}\_{t+1} : & & \\ 1 & = & \frac{\beta}{\lambda\_t} \mathbb{E}\_t n\_{t+1}^{\mathrm{PO}} \mathcal{V}\_{k\_{t+1}} = \beta \mathbb{E}\_t n\_{t+1}^{\mathrm{PO}} \frac{\lambda\_{t+1}}{\lambda\_t} (1 - \delta + \mathcal{Y}\_{k\_{t+1}}) \end{array} \tag{55}$$

$$\begin{array}{rcl} Q\_{t+1}: & & \\ \varphi\_t & = & \beta \mathbb{E}\_t n\_{t+1}^{PO} \mathcal{V}\_{Q\_{t+1}} = \beta \mathbb{E}\_t n\_{t+1}^{PO} (\varphi\_{t+1} + \lambda\_{t+1} \mathbf{x}\_{t+1} D\_{t+1} F\_{t+1}), & \end{array} \tag{56}$$

$$\begin{array}{rcl} \mathsf{Y}\_{t+1}: & & \\ \Phi\_t & = & -\mathsf{V}\_{\mathsf{Y}\_{t+1}} = \Phi\_{t+1} = \Phi\_t \end{array} \tag{57}$$

where the second equality in each equation follows from the envelope conditions (52)–(54), and (56), giving a marginal utility valuation of the benefit of fossil energy net of climate damages. Given (57), the Lagrangian multiplier Φ*<sup>t</sup>* is a constant and, by the Kuhn-Tucker condition, is zero if the wealth constraint is nonbinding, indicating that the planner is a social and not a Ramsey planner.

Note that because both the private sector's and the planner's probabilities are twisted by a martingale multiplier, the ambiguity for this planner effectively delivers *ex post* a model of endogenously distorted *homogeneous* beliefs. Hansen and Sargent (2012) emphasize that *nPO* is not intended to "solve" an impossible inference problem, and being the planner's *cautious* inference about unknown private beliefs, should be viewed as merely a device to construct a robust Ramsey policy. If the planner were to solve the private sector's Euler equations using the minimizing *nPO* in order to derive its *ex post* decision rules for consumption, labor, and energy, it would not necessarily end up reproducing their observed values, meaning that private beliefs cannot be reverse-engineered from such observations.

#### 7.1.3. Paternalistic Planner

Woodford (2010) originally introduced a monetary authority facing a type of ambiguity described here: while trusting the reference model *π*, the government is ignorant about the private sector's distorted beliefs *m<sup>s</sup>* . It expresses its ambiguity by setting *σ* < 0. Trust in its own model means that the planner sets *nt*+<sup>1</sup> multiplying continuation value V*t*+<sup>1</sup> equal to unity.28
