*6.2. The Ramsey Planner's Implementability Constraints*

In solving for optimal taxation in the case of a Ramsey planner, I use a so-called primal approach due to Chamley (1986) that searches directly for allocations by solving the government's problem subject to an *implementability* constraint. A starting point is the household's dynamic budget constraint (4), which, when solved forward for *bt*, utilizing the no-arbitrage condition (14) and a no-Ponzi game condition, yields the intertemporal budget constraint<sup>23</sup>

$$b\_t \quad \ge \quad \mathbb{E}\_t \lim\_{T \to \infty} \sum\_{j=0}^{T-1} \mathfrak{p}\_{t+j} [c\_{t+j} - H\_{t+j} - g\_{t+j}].\tag{35}$$

Let W*<sup>t</sup>* denote household wealth in period t, composed of government bonds, the after-tax equity value of physical capital, and fossil fuels still in the ground, valued at current after-tax energy prices:

$$\mathcal{W}\_t \equiv \ \ b\_t + R\_t^k k\_t + (p\_t^\varepsilon - \tau\_t^\varepsilon) Q\_t. \tag{36}$$

Appendix B shows that

$$\mathcal{W}t \ge \left. c\_t - H\_t - \mathbf{g}\_t + \beta \mathbb{E}\_t m\_{t+1} \frac{\mathbf{u}\_{\mathcal{E}\_{t+1}}}{\mathbf{u}\_{\mathcal{E}\_t}} \mathcal{W}\_{t+1} \right. $$
 
$$= \left. c\_t - H\_t - \mathbf{g}\_t + \mathbb{E}\_t \boldsymbol{\mathcal{P}}\_{t+1} \mathcal{W}\_{t+1} \right. \tag{37}$$

where *p*ˆ*t*+*j*,*<sup>t</sup>* is the market's previously defined distorted *t* + *j* equilibrium price of an Arrow–Debreu security in terms of consumption at history *x<sup>t</sup>* . Solved forward, household wealth is

$$\mathcal{W}\_t \quad \ge \quad \mathbb{E}\_t \sum\_{j=0}^{\infty} \mathfrak{p}\_{t+j,t} [\mathfrak{c}\_{t+j} - H\_{t+j} - \mathfrak{g}\_{t+j}].\tag{38}$$

For later use, it is convenient to define the marginal-utility-of-consumption-scaled market value of wealth Υ*<sup>t</sup>* = *uct*W*t*, so that the Ramsey planner's implementability constraint (37) becomes, equivalently,

$$
\mathcal{Y}\_t \quad \ge \quad \Omega\_t + \mathbb{E}\_t m\_{t+1} \mathcal{Y}\_{t+1} \,. \tag{39}
$$

where

$$
\Omega\_t = \|\mathbf{u}\_{\mathbf{c}\_t}[\mathbf{c}\_t - H\_t - \mathbf{g}\_t]\_\prime \tag{40}
$$

with derivative,

$$\begin{array}{ll} \Omega\_{\mathfrak{c}\_{l}} &=& \frac{\mathsf{u}\_{\mathfrak{c}\_{l}}}{\mathsf{u}\_{\mathfrak{c}\_{l}}} (\mathsf{c}\_{l} - H\_{l} - \mathsf{g}\_{t}) \mathsf{u}\_{\mathfrak{c}\_{l}} + \mathsf{u}\_{\mathfrak{c}\_{l}} \\\\ &=& [1 - \gamma \mathsf{c}\_{t}^{-1} (\mathsf{c}\_{t} - H\_{t} - \mathsf{g}\_{t})] \mathsf{u}\_{\mathfrak{c}\_{l}} = [1 - \gamma + \gamma \frac{H\_{t} + \mathsf{g}\_{t}}{\mathsf{c}\_{t}}] \mathsf{u}\_{\mathfrak{c}\_{l}} \geq 0,\end{array} \tag{41}$$

since, from Section 3.1, *ucct uct* = −*γ*/*ct*. Note that if the constraint (39) is nonbinding, the government is the social planner widely treated in the literature.

Solved forward,

$$\mathbf{Y}\_t = \mathbb{E}\_t \boldsymbol{u}\_{\mathbf{c},t} \sum\_{j=0}^{\infty} \boldsymbol{\mathcal{p}}\_{t+j} (\mathbf{c}\_{t+j} - \mathbf{H}\_{t+j} - \mathbf{g}\_{t+j}) = \boldsymbol{u}\_{\mathbf{c}t} \mathbf{b}\_{t\prime} \tag{42}$$

is the government's surplus valued in terms of the marginal utility of consumption.

Finally, by its very definition, a Ramsey planner heeds all equilibrium constraints imposed by competitive markets. In particular, when consumers are pessimistic with belief distortions defined in terms of their continuation values, where *mt* = *m<sup>c</sup> <sup>t</sup>*+<sup>1</sup> <sup>=</sup> *<sup>e</sup> <sup>σ</sup>c*U*t*+<sup>1</sup> E*te <sup>σ</sup>c*U*t*+<sup>1</sup> , the planner faces two additional implementability constraints that come from (6) and (7):

$$\mathcal{M}\_{t+1}^{\mathfrak{c}} = \begin{array}{c} \mathfrak{e}^{\sigma^{\varepsilon} \mathcal{U}\_{t+1}} \\ \mathbb{E}\_{\mathfrak{t}} \mathfrak{e}^{\sigma^{\varepsilon} \mathcal{U}\_{t+1}} \end{array} \mathcal{M}\_{\mathfrak{t}}^{\mathfrak{c}} \tag{43}$$

$$\mathcal{U}\_{\rm tr} = \left. u(\alpha\_{\rm t}) + \frac{\beta}{\sigma^c} \log \mathbb{E}\_{\rm t} e^{\sigma^c \mathcal{U}\_{\rm t+1}} \right. \tag{44}$$

#### **7. Ramsey Planning in Four Belief Regimes**

Subject to the fossil resource constraint (2) and (3), the national income identity (33), and the implementability constraint (39), the taxing Ramsey authority chooses {*c*, *E*, *k*, *Q*, Υ} to maximize society's expected welfare, and {*N*, *n*} to minimize discounted *relative entropy* defined in (1),

$$\max\_{\mathbf{x}, \mathbf{E}, k, \mathbf{Q}, \mathbf{Y}} \min\_{\mathbf{N}, \mathbf{n}\_r} \mathbb{E}\_0 \sum\_{t=0}^{\infty} \beta^t \mathcal{N}\_t(\mathbf{x}^t) \left[ \mu(\mathbf{c}\_t) - \frac{\beta}{\sigma} (\mathbb{E}\_t n\_{t+1} \log n\_{t+1}) \right],\tag{45}$$

subject to *Nt*+<sup>1</sup> <sup>=</sup> *nt*+1*Nt*, *<sup>N</sup>*<sup>0</sup> <sup>=</sup> 1, <sup>E</sup>*tnt*+<sup>1</sup> <sup>=</sup> 1, and 0 <sup>&</sup>gt; *<sup>σ</sup>* <sup>&</sup>gt; <sup>−</sup>∞, where *Nt* is the government's martingale multiplier, equivalent to *Mt* defined earlier for the consumer, and *σ* < 0 is the planner's *parameter of ambiguity aversion*.

Exploiting a linear homogeneity property such that *V*(Υ*t*, *kt*, *Qt*, *Nt*) = *Nt*V*t*(Υ*t*, *kt*, *Qt*), the problem may be cast as the recursive (Bellman) Lagrangian

$$\begin{split} \mathcal{W}\_{t}(\mathbf{Y}\_{t},k\_{t},\mathbf{Q}\_{t}) &= \max\_{\boldsymbol{c}\_{t},\boldsymbol{E}\_{t},\mathbf{k}\_{t+1},\mathbf{Q}\_{t+1}\mathbf{Y}\_{t+1}} \min\_{\boldsymbol{n}\_{t+1}} u(\mathbf{c}\_{t}) - \frac{\beta}{\sigma} \mathbb{E}\_{t} n\_{t+1} \log n\_{t+1} \\ &+ \quad \beta \mathbb{E}\_{t} n\_{t+1} \mathcal{V}\_{t+1}(\mathbf{Y}\_{t+1},k\_{t+1},\mathbf{Q}\_{t+1}) \\ &+ \quad \Phi\_{t}[\Omega\_{t} + \mathbb{E}\_{t} m\_{t+1} \mathbf{Y}\_{t+1} - \mathbf{Y}\_{t}] + \mathcal{M}\_{t} \end{split} \tag{46}$$

where

$$\mathcal{M}\_{t} = \begin{pmatrix} \lambda\_{t} [(1 - \varepsilon^{\mathbf{x}\_{l}} (\mathbf{Q}\_{0} - \mathbf{Q}\_{l})) F(k\_{t}, H\_{t}, E\_{t}) + (1 - \delta) k\_{t} - \mathfrak{c}\_{t} - \mathfrak{g}\_{t} - k\_{t+1}] \\\ \mathfrak{q}\_{t} [\mathbf{Q}\_{t} - E\_{t} - \mathbf{Q}\_{t+1}] \\\ \upsilon [\mathbf{Q}\_{0} - \sum\_{i=0}^{\infty} E\_{i}] \end{pmatrix},$$

is a group of Lagrangian constraints and *λt*, Φ*t*, *ϕ<sup>t</sup>* are non-negative Lagrangian co-state variables, and *υ* is constant.24

The preceding problem constitutes a Stackelberg game between the government as the leader and the private sector as the follower. Embedded in this game is another game with a minimizing opponent to represent worst-case outcomes. This latter game, framed as *inner minimization*, may be played by either government or consumers, as in Section 3.4, or even both, depending on respective attitudes toward extreme risk. The solution to this subgame implies a plan for *outer maximization* of an indirect value function, typically a *risk sensitivity recursion*, such as the one introduced in Section 3.4.2.
