**Assumption 7.**

$$\frac{d\theta\_{t+j}}{dx\_{t+j}} = \frac{d\frac{H\_t + \xi\_t}{c\_t}}{dx\_{t+j}} \approx 0.$$

Since climate lowers both income and consumption, the net effect on *ϑ* is likely negligible, justifying an assumption that *ϑ* is not strongly negatively correlated with Λ, *ςk*, *ςe* , *mc*, *nPO*, *mc*, *nPA*, and *np*, all of which are positively correlated with *x*.

Under the assumption that climate shocks are the sole stochastic process *x* driving the economy, the next proposition reflects the fact that variables that are functions of *x* must be correlated with each other. The signs of the covariances listed therein follow from the preceding lemmas.

#### **Lemma 14.** *Signs of key covariances*

*(i) The signs of cov*(Λ, *ϑ*)*, cov*(*ς<sup>e</sup>* , *ϑ*)*, cov*(*ςk*, *ϑ*)*, cov*(*nPO*, Λ)*, cov*(*nPO*, Λ*ϑ*)*, cov*(*nPA*, Λ)*, cov*(*nPA*, Λ*ϑ*)*, cov*(*nPA*, *ϑςe*)*, cov*(*nPA*, *ςk*)*, cov*(*nPA*, *ϑςk*)*, cov*(*mPA*, Λ)*, cov*(*mPA*, *ϑ*Λ)*, cov*(*np*, Λ)*, cov*(*np*, *ςk*)*, cov*(*np*, *mc*)*, cov*(*mp*, *ςk*)*, cov*(*mp*, Λ)*, cov*(*mp*, *ϑ*Λ)*, cov*(*mpςk*, *ϑ*)*, cov*(*mcmp*, Λ)*, cov*(*mc*, Λ)*, and cov*(*mc*, *ϑ*Λ) *are positive.*

$$\text{(ii) The signs of } conv(\boldsymbol{\xi}^{\varepsilon}, \frac{1}{n^{\varepsilon}}), conv(n^{\mathbb{P}}, \frac{1}{n^{\mathbb{P}}}), conv(\boldsymbol{\xi}^{\varepsilon}, \frac{1}{n^{\mathbb{P}}}), and \, conv(\theta \boldsymbol{\xi}^{\varepsilon}, \frac{1}{n^{\mathbb{P}}}) \text{ are negative.}\\(iii) The signs of } conv(\boldsymbol{\xi}^{\varepsilon}, \frac{\mathbb{P}^{\mathbb{P}}}{n^{\mathbb{P}}}), conv(\boldsymbol{\xi}^{\varepsilon}, \frac{\mathbb{P}^{\mathbb{P}}}{n^{\mathbb{P}}}), conv(\boldsymbol{\xi}^{\varepsilon}, \frac{\mathbb{P}^{\mathbb{P}}}{n^{\mathbb{P}}}), and \, conv(\boldsymbol{\xi}^{\varepsilon} \boldsymbol{\theta}, \frac{\mathbb{P}^{\mathbb{P}}}{n^{\mathbb{P}}}) \text{ are indeterminate.}$$

The following propositions distinguish two situations: (i) when the implementability (the marginal-utility-of-consumption value of household wealth) constraint is binding, (Φ > 0), and (ii) when it is not, (Φ = 0). In the latter case, all policies revert to those of an unconstrained social planner.36 To set a baseline for comparison, the first proposition establishes results that obtain with homogeneous beliefs under rational expectations RE.
