iso}$-${\mathfrak{m}}/{\mathfrak{s}}$-relations).

Let $k\geq 1$, consider a multiset $\bm{A}_{J_{k}}$ of traceless matrices as
above and fix $\ell>0$. Then, the mean sizes $\mathfrak{m}^{\rm iso}_{k}$ and
$\mathfrak{m}_{k}$, and the standard deviation sizes ${\mathfrak{s}}^{\rm
iso}_{k}$ and ${\mathfrak{s}}_{k}$, satisfy the following inequalities.

  * (i)

The _average sizes_ ${\mathfrak{m}}/{\mathfrak{s}}$ are bounded by the
isotropic sizes ${\mathfrak{m}}^{\rm iso}/{\mathfrak{s}}^{\rm iso}$:

$\begin{split}{\mathfrak{m}}_{k}(\ell;J_{k})\lesssim{\mathfrak{m}}_{k}^{\rm
iso}(\ell;J_{k})\qquad\text{and}\qquad{\mathfrak{s}}_{k}(\ell;J_{k})\lesssim\ell^{1/2}\,{\mathfrak{s}}_{k}^{\rm
iso}(\ell;J_{k})\,.\end{split}$ (A.11)

  * (ii)

The isotropic standard deviation size can be upper bounded by the average
standard deviation size

${\mathfrak{s}}_{k}^{\rm
iso}(\ell;J_{k})\lesssim\,N\,\big{[}1+(N\ell)^{-1/2}\big{]}\,{\mathfrak{s}}_{k}(\ell;J_{k})\,.$
(A.12)

Stochastic control quantities. Consider deterministic vectors
$\bm{x},\bm{y}\in{\mathbb{C}}^{N}$ with $\|\bm{x}\|,\|\bm{y}\|\lesssim 1$,
traceless matrices $A_{1},\dots,A_{k}$, and for $k\geq 1$ define, analogously
to (5.12)–(5.13),

$\begin{split}\Psi_{k}^{\rm iso}(t)&=\Psi_{k}^{\rm
iso}(t;\bm{x},\bm{y};\bm{z}_{[1,k+1]};\bm{A}_{[1,k]}):=\frac{N^{1/2}}{{\mathfrak{s}}_{k}^{\rm
iso}(\ell_{t};\bm{A}_{[1,k]})(N\ell_{t})^{1/4}}\big{|}(G_{[1,k+1],t}-M_{[1,k+1],t})_{{\bm{x}}{\bm{y}}}\big{|}\,,\\\
\Phi_{k}^{\rm iso}(t)&=\Phi_{k}^{\rm
iso}(t;\bm{x},\bm{y};\bm{z}_{[1,k+1]};\bm{A}_{[1,k]}):=\frac{N^{1/2}}{{\mathfrak{s}}_{k}^{\rm
iso}(\ell_{t};\bm{A}_{[1,k]})}\big{|}(G_{[1,k+1],t}-M_{[1,k+1],t})_{{\bm{x}}{\bm{y}}}\big{|}\,,\end{split}$
(A.13)

where we denoted the multisets of deterministic matrices by $\bm{A}_{[1,k]}$
and $\bm{z}_{[1,k+1]}=(z_{1},...,z_{k+1})$ with $z_{i}=z_{i,0}$ (initial
condition), respectively. Note that by (1.4), using the convention
${\mathfrak{s}}_{0}^{\rm iso}(t):=\ell_{t}^{-1/2}$, we have $\Phi_{0}(t)\prec
1$, and that by (6.7) it follows

$\Phi_{k}^{\rm iso}(0)\prec 1,$ (A.14)

for any $k\geq 1$. Similarly to the averaged case (see the two paragraphs
around (5.12)–(5.13)), also in the isotropic case we introduced the two
quantities in (A.13) as we will first prove $\Psi_{k}^{\rm
iso}(t)=\Phi_{k}^{\rm iso}(t)/(N\ell_{t})^{1/4}\prec 1$ using the master
inequalities in Proposition A.4, and then use this as an input to prove
$\Phi_{k}^{\rm iso}(t)\prec 1$ in Proposition A.5.

######  Proposition A.4 (Isotropic master inequalities).

Fix $k\in{\mathbb{N}}$, assume that $\Psi_{l}^{\rm iso}(t)\prec\psi_{l}^{\rm
iso}$, with $\psi_{0}^{\rm iso}:=1$ and $\Phi_{l}(t)\prec 1$ (with
$\Phi_{l}(t)$ from (5.13)), for any $0\leq l\leq
k+\mathbf{1}(k\,\mathrm{odd})$ uniformly in $t\in[0,T]$. Then

$\Psi_{k}^{\rm iso}(t)\prec 1+\frac{\psi_{k}^{\rm
iso}+\psi_{k+\bm{1}(k\,\mathrm{odd})}^{\rm
iso}}{(N\ell_{T})^{1/8}}+\sum_{l=1}^{k-1}\psi_{l}^{\rm iso}$ (A.15)

uniformly in $t\in[0,T]$.

Analogously to Section 5, using Proposition A.4 we show that $\Psi_{k}^{\rm
iso}(t)\prec 1$ and then use this information as input to conclude
$\Phi_{k}^{\rm iso}(t)\prec 1$:

######  Proposition A.5.

Fix $k\in{\mathbb{N}}$, assume that $\Phi_{l}^{\rm iso}(t)\prec 1$, for $0\leq
l\leq k-2$, $\Phi_{l}(t)\prec 1$ for $0\leq l\leq k$ (with $\Phi_{l}(t)$ from
(5.12)), and $\Phi_{k}(t)\prec(N\ell_{t})^{1/4}$ for $l\leq 2k$, uniformly in
$t\in[0,T]$. Then

$\Phi_{k}^{\rm iso}(t)\prec 1$ (A.16)

uniformly in $t\in[0,T]$.

Closing the hierarchy. We start showing that (A.15), using _iteration_ as in
Lemma 5.4, in fact implies $\Psi_{k}(t)\prec 1$. We start with $k=1,2$. By
(A.15), we have

$\Psi_{1}^{\rm iso}(t)\prec 1+\frac{\psi_{1}^{\rm iso}+\psi_{2}^{\rm
iso}}{(N\ell_{T})^{1/8}},\qquad\quad\Psi_{2}^{\rm iso}(t)\prec
1+\frac{\psi_{2}^{\rm iso}}{(N\ell_{T})^{1/8}}+\psi_{1}^{\rm iso}.$ (A.17)

Then, by iteration, we obtain

$\Psi_{1}^{\rm iso}(t)\prec 1+\frac{\psi_{2}^{\rm
iso}}{(N\ell_{T})^{1/8}},\qquad\quad\Psi_{2}^{\rm iso}(t)\prec 1+\psi_{1}^{\rm
iso}.$ (A.18)

Finally, plugging the first inequality into the second one and using iteration
once again we obtain $\Psi_{1}^{\rm iso}(t)+\Psi_{2}^{\rm iso}(t)\prec 1$.

Next, we proceed by induction. Fix an even $k\in{\mathbb{N}}$, and assume that
$\Psi_{l}^{\rm iso}(t)\prec 1$ for $l\leq k-2$, then by (A.15) we have

$\Psi_{k-1}^{\rm iso}(t)\prec 1+\frac{\psi_{k-1}^{\rm iso}+\psi_{k}^{\rm
iso}}{(N\ell_{T})^{1/8}},\qquad\quad\Psi_{k}^{\rm iso}(t)\prec
1+\frac{\psi_{k}^{\rm iso}}{(N\ell_{T})^{1/8}}+\psi_{k-1}^{\rm iso}.$ (A.19)

Proceeding exactly as in (A.17)–(A.18), by (A.19), we conclude that
$\Psi_{l}^{\rm iso}(t)\prec 1$ for any $l\leq k$. Finally, using that as a
consequence of $(N\ell_{t})^{1/4}\Phi_{k}^{\rm iso}(t)=\Psi_{l}^{\rm
iso}(t)\prec 1$ the hypothesis of Proposition A.5 are satisfied, we conclude
that $\Phi_{k}^{\rm iso}(t)\prec 1$ and so the proof of Theorem 6.2. ∎

####  A.2.2. Isotropic master inequalities: Proofs of Propositions A.4–A.5

###### Proof of Proposition A.4.

Note that the second term in the first line of (A.4) can be incorporated into
the lhs. by differentiating
$e^{-kt/2}(G_{[1,k+1,t]}-M_{[1,k+1,t]})_{{\bm{x}}{\bm{y}}}$. The exponential
factor $e^{kt/2}\sim 1$ is irrelevant, we thus neglect this term from the
analysis. In the following we will often use the simple bound (5.2) even if
not stated explicitly. Additionally, every time that two resolvents get next
to each other in a chain we use the integral representation (5.24) to reduce
their number by one at the price of an additional $1/\eta$–factor (see e.g.
(5.23)).

We now start with the computation of the quadratic variation of the martingale
term in (A.4):

$\frac{1}{N}\sum_{i=1}^{k+1}\big{[}G_{[1,i],t}(G_{[1,i],t})^{*}\big{]}_{{\bm{x}}{\bm{x}}}\big{[}(G_{[i,k+1],t})^{*}G_{[i,k+1],t}\big{]}_{{\bm{y}}{\bm{y}}}\,\mathrm{d}t.$
(A.20)

Similarly to the averaged case (see (5.27)–(5.31)), to estimate the quadratic
variation of the martingale term in (A.4) we rely on the following _reduction
inequality_ for $k\in{\mathbb{N}}$ even and $j\leq k$ (here for simplicity we
drop the indices of $G$’s and $A$’s):

$\big{|}(G_{[1,k+j+1]})_{{\bm{x}}{\bm{y}}}\big{|}\lesssim\sqrt{N}\langle|G|A(GA)^{j-1}|G|(AG^{*})^{j-1}A\rangle^{1/2}\prod_{{\bm{v}}\in\\{{\bm{x}},{\bm{y}}\\}}[(GA)^{k/2}|G|(AG^{*})^{k/2}]_{{\bm{v}}{\bm{v}}}^{1/2}.$
(A.21)

The proof of (A.21) is postponed to the end of this section. Recall the
conventions $\psi_{0}^{\rm iso}=1$ and $\mathfrak{s}_{0}^{\rm
iso}(t)=\ell_{t}^{-1/2}$, then using (A.21) for $j=k-2i+2$, $i\leq k/2$ for
even $k$ (and $j=k-2i+1$, $i\leq(k+1)/2$ for odd $k$) and then (A.8), (5.8),
to bound (recall (5.28) in Footnote 16)

$\begin{split}\big{[}(G_{[i,k+1],s})^{*}G_{[i,k+1],s}\big{]}_{{\bm{y}}{\bm{y}}}&\prec\sqrt{N}\left(\mathfrak{m}_{k}^{\rm
iso}(s)+\frac{\mathfrak{s}_{k}^{\rm
iso}(s)(N\ell_{s})^{1/4}}{\sqrt{N}}\right)\left(\ell_{s}\mathfrak{m}_{2(k-2i+2)}(s)+\frac{\mathfrak{s}_{2(k-2i+2)}(s)}{N}\right)^{1/2}\\\
&\lesssim\mathfrak{s}_{k}^{\rm
iso}(s)\sqrt{\mathfrak{s}_{2(k-2i+2)}(s)}\ell_{s}^{1/2}\big{(}\sqrt{N\ell_{t}}+(N\ell_{t})^{1/4}\psi_{k}^{\rm
iso}\big{)},\end{split}$ (A.22)

we estimate each term of (A.20) by

$\begin{split}&\frac{\sqrt{N}}{\mathfrak{s}_{k}^{\rm
iso}(t)(N\ell_{t})^{1/4}}\left(\frac{1}{N}\sum_{i=1}^{k+1}\int_{0}^{t}\big{[}G_{[1,i],s}(G_{[1,i],s})^{*}\big{]}_{{\bm{x}}{\bm{x}}}\big{[}(G_{[i,k+1],s})^{*}G_{[i,k+1],s}\big{]}_{{\bm{y}}{\bm{y}}}\,\mathrm{d}s\right)^{1/2}\\\
&\prec\frac{\sqrt{N}}{\mathfrak{s}_{k}^{\rm
iso}(t)(N\ell_{t})^{1/4}}\Bigg{(}\sum_{i=1}^{\lceil
k/2\rceil}\int_{0}^{t}\frac{\mathfrak{s}_{k}^{\rm
iso}(s)}{N\eta_{s}^{2}}\left(\mathfrak{m}_{2(i-1)}^{\rm
iso}(s)+\frac{\psi_{2(i-1)}^{\rm iso}\mathfrak{s}_{2(i-1)}^{\rm
iso}(s)(N\ell_{s})^{1/4}}{\sqrt{N}}\right)\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\big{(}\sqrt{N\ell_{s}}+\psi_{k+\bm{1}(k\,\mathrm{odd})}^{\rm
iso}(N\ell_{s})^{1/4}\big{)}\sqrt{\ell_{s}\mathfrak{s}_{2(k-i+1)}(s)}\,\mathrm{d}s\Bigg{)}^{1/2}\\\
&\lesssim\frac{\sqrt{N}}{\mathfrak{s}_{k}^{\rm
iso}(t)(N\ell_{t})^{1/4}}\left(\sum_{i=1}^{\lceil
k/2\rceil}\int_{0}^{t}\frac{\mathfrak{s}_{k}^{\rm
iso}(s)\mathfrak{s}_{k-i+1}^{\rm iso}(s)\mathfrak{s}_{i-1}^{\rm
iso}(s)}{N\eta_{s}^{2}}\ell_{s}^{3/2}\left(1+\frac{\psi_{2(i-1)}^{\rm
iso}}{(N\ell_{s})^{1/4}}\right)\big{(}\sqrt{N\ell_{s}}+\psi_{k+\bm{1}(k\,\mathrm{odd})}^{\rm
iso}(N\ell_{s})^{1/4}\big{)}\,\mathrm{d}s\right)^{1/2}\\\ &\lesssim
1+\sum_{i=1}^{\lceil k/2\rceil}\frac{\sqrt{\psi_{2(i-1)}^{\rm
iso}}}{(N\ell_{s})^{1/8}}+\frac{\sqrt{\psi_{k+\bm{1}(k\,\mathrm{odd})}^{\rm
iso}}}{(N\ell_{t})^{1/8}}+\sum_{i=1}^{\lceil
k/2\rceil}\frac{\sqrt{\psi_{2(i-1)}^{\rm
iso}\psi_{k+\bm{1}(k\,\mathrm{odd})}^{\rm
iso}}}{(N\ell_{t})^{1/4}}.\end{split}$ (A.23)

Here in the second inequality (A.8)–(A.9), (A.11), and in the last inequality
we used (A.7), (A.10).

In the following computations, when two $G$’s, with spectral parameters having
imaginary parts of the same sign appear next to each other (i.e. without a $A$
in between), we use the integral representation (5.24) to reduce the number of
$G$’s by one at the price of an additional $1/\eta$. If the imaginary parts of
the spectral parameters have different signs, we use resolvent identity (see
e.g. around (5.23)–(5.24) in the averaged case). For the terms in the second
line of (A.4) we estimate

$\frac{\sqrt{N}}{\mathfrak{s}_{k}^{\rm
iso}(t)(N\ell_{t})^{1/4}}\int_{0}^{t}\langle{G}_{[i,j],s}-M_{[i,j],s}\rangle({M}_{[1,i]\cup[j,k+1],s})_{{\bm{x}}{\bm{y}}}\,\mathrm{d}s\prec\frac{\sqrt{N}}{\mathfrak{s}_{k}^{\rm
iso}(t)(N\ell_{t})^{1/4}}\int_{0}^{t}\frac{\mathfrak{s}_{j-i}(s)}{N\eta_{s}}\frac{\mathfrak{m}_{k-j+i}^{\rm
iso}(s)}{\eta_{s}}\,\mathrm{d}s\lesssim\frac{1}{(N\ell_{t})^{3/4}},$ (A.24)

where we used (A.6)–(A.8), the second inequality in (A.11), and (A.10).

For the terms in the third line of (A.4) we estimate

$\begin{split}&\frac{\sqrt{N}}{\mathfrak{s}_{k}^{\rm
iso}(t)(N\ell_{t})^{1/4}}\int_{0}^{t}\langle{M}_{[i,j],s}\rangle({G}_{[1,i]\cup[j,k+1],s}-M_{[1,i]\cup[j,k+1],s})_{{\bm{x}}{\bm{y}}}\,\mathrm{d}s\\\
&\qquad\qquad\quad\prec\frac{\sqrt{N}}{\mathfrak{s}_{k}^{\rm
iso}(t)(N\ell_{t})^{1/4}}\int_{0}^{t}\frac{\mathfrak{m}_{j-i}(s)}{\eta_{s}}\frac{\mathfrak{s}_{k-j+i}^{\rm
iso}(s)\psi_{k-j+i}^{\rm
iso}(N\ell_{s})^{1/4}}{\sqrt{N}\eta_{s}}\,\mathrm{d}s\lesssim\psi_{k-j+i}^{\rm
iso},\end{split}$ (A.25)

where in the last inequality we used the first inequality of (A.11),
(A.7)–(A.8).

For the terms in the fourth line of (A.4) we estimate

$\begin{split}&\frac{\sqrt{N}}{\mathfrak{s}_{k}^{\rm
iso}(t)(N\ell_{t})^{1/4}}\int_{0}^{t}\langle{G}_{[i,j],s}-M_{[i,j],s}\rangle({G}_{[1,i]\cup[j,k+1],s}-M_{[1,i]\cup[j,k+1],s})_{{\bm{x}}{\bm{y}}}\,\mathrm{d}s\\\
&\qquad\qquad\quad\prec\frac{\sqrt{N}}{\mathfrak{s}_{k}^{\rm
iso}(t)(N\ell_{t})^{1/4}}\int_{0}^{t}\frac{\mathfrak{s}_{j-i}(s)}{N\eta_{s}}\frac{\mathfrak{s}_{k-j+i}^{\rm
iso}(s)\psi_{k-j+i}^{\rm
iso}(N\ell_{s})^{1/4}}{\sqrt{N}\eta_{s}}\,\mathrm{d}s\lesssim\frac{1}{N\ell_{t}}\psi_{k-j+i}^{\rm
iso},\end{split}$ (A.26)

where in the last step we used the second inequality of (A.11) and the second
inequality of (A.7).

Finally, for the term in the last line of (A.4) we estimate

$\begin{split}\frac{\sqrt{N}}{\mathfrak{s}_{k}^{\rm
iso}(t)(N\ell_{t})^{1/4}}\int_{0}^{t}\langle
G_{i,s}-m_{i,s}\rangle({G}^{(i)}_{[1,k+1],s})_{{\bm{x}}{\bm{y}}}\,\mathrm{d}s&\prec\frac{\sqrt{N}}{\mathfrak{s}_{k}^{\rm
iso}(t)(N\ell_{t})^{1/4}}\int_{0}^{t}\frac{1}{N\eta_{s}}\left(\frac{\mathfrak{m}_{k}^{\rm
iso}(s)}{\eta_{s}}+\frac{\mathfrak{s}_{k}^{\rm iso}(s)\psi_{k}^{\rm
iso}(N\ell_{s})^{1/4}}{\sqrt{N}\eta_{s}}\right)\,\mathrm{d}s\\\
&\lesssim\frac{1}{(N\ell_{t})^{3/4}}+\frac{\psi_{k}^{\rm
iso}}{N\ell_{t}},\end{split}$ (A.27)

where we used (A.6) and (A.8) together with (A.8). Combining
(LABEL:eq:estquadvariso)–(A.27), recalling that $\langle
M_{[i,i+1],s}\rangle=0$ in (A.25), using (A.14) to bound $\Psi_{k}^{\rm
iso}(0)\prec 1$ and $N\ell_{T}\geq 1$, $\psi_{l}^{\rm iso}\geq 1$,
$\ell_{t}\gtrsim\ell_{T}$ , we conclude (A.15). ∎

###### Proof of Proposition A.5.

The proof of this proposition is analogous to the proof of Proposition A.5.
All the terms except for the martingale one are estimated as in the proof of
Proposition A.5 after multiplying each line by $(N\ell_{t})^{1/4}$ and setting
$\psi_{l}^{\rm iso}=1$. We conclude the proof pointing out that the only
difference is in the estimate of the quadratic variation of the martingale
term, i.e. (LABEL:eq:estquadvariso) has to be replaced by

$\begin{split}&\frac{\sqrt{N}}{\mathfrak{s}_{k}^{\rm
iso}(t)}\left(\frac{1}{N}\sum_{i=1}^{k+1}\int_{0}^{t}\big{[}G_{[1,i],s}(G_{[1,i],s})^{*}\big{]}_{{\bm{x}}{\bm{x}}}\big{[}(G_{[i,k+1],s})^{*}G_{[i,k+1],s}\big{]}_{{\bm{y}}{\bm{y}}}\,\mathrm{d}s\right)^{1/2}\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\prec\frac{\sqrt{N}}{\mathfrak{s}_{k}^{\rm
iso}(t)}\left(\int_{0}^{t}\frac{\ell_{s}\mathfrak{s}_{2(i-1)}^{\rm
iso}(s)\mathfrak{s}_{2k-2(i-1)}^{\rm
iso}(s)}{N\eta_{s}^{2}}\,\mathrm{d}s\right)^{1/2}\lesssim 1,\end{split}$
(A.28)

where in the first inequality we used the definition (A.13) together with
(A.6), (A.8), and in the second inequality we used (A.7), (A.9)–(A.10). ∎

###### Proof of (A.21).

By spectral decomposition we estimate

$\begin{split}\big{|}(Q_{1}G(z)Q_{2}G(w)Q_{3})_{{\bm{x}}{\bm{y}}}\big{|}&=\left|\sum_{ij}\frac{\langle{\bm{x}},Q_{1}{\bm{u}}_{i}\rangle\langle{\bm{u}}_{i},Q_{2}{\bm{u}}_{j}\rangle\langle{\bm{u}}_{j},Q_{3}{\bm{u}}_{j}\rangle}{(\lambda_{i}-z)(\lambda_{j}-w)}\right|\\\
&\lesssim\sqrt{N}(Q_{1}|G(z)|Q_{1}^{*})_{{\bm{x}}{\bm{x}}}(Q_{3}^{*}|G(w)|Q_{3})_{{\bm{y}}{\bm{y}}}\langle|G(z)|Q_{2}|G(w)|Q_{2}^{*}\rangle^{1/2},\end{split}$
(A.29)

where in the last inequality we used Schwarz inequality to separate $Q_{2}$
from $Q_{1},Q_{3}$. Choosing $z=z_{k/2+1}$, $w=z_{k/2+j+1}$, and

$Q_{1}=G_{1}A_{1}\dots A_{k/2},\qquad Q_{2}=A_{k/2+1}G_{k/2+2}\dots
A_{k/2+j},\qquad Q_{3}=A_{k/2+j+1}G_{k/2+j+2}\dots G_{k+1},$

this concludes the proof. ∎

## References

  * [1] A. Adhikari, J. Huang. Dyson Brownian motion for general $\beta$ and potential at the edge. _Probability Theory and Related Fields_ 178, 893–950 (2020).
  * [2] A. Adhikari, B. Landon. Local law and rigidity for unitary Brownian motion. _Probability Theory and Related Fields_ 187.3, 753–815 (2023).
  * [3] A. Aggarwal, J. Huang. Edge Rigidity of Dyson Brownian Motion with General Initial Data. arXiv: 2308.04236 (2023).
  * [4] A. Bloemendal, L. Erdős, A. Knowles, H.-T. Yau, J. Yin. Isotropic local laws for sample covariance and generalized Wigner matrices. _Electronic Journal of Probability_ 19, 33, 1–53 (2014).
  * [5] O. Bohigas, M.-J. Giannoni, C. Schmit. Characterization of chaotic quantum spectra and universality of level fluctuation laws. _Physical Review Letters_ 52, 1–4 (1984).
  * [6] P. Bourgade. Extreme gaps between eigenvalues of Wigner matrices. _Journal of the European Mathematical Society_ 24, 2823–2873 (2021).
  * [7] G. Cipolloni, L. Erdős, D. Schröder. Eigenstate Thermalisation Hypothesis for Wigner matrices. _Communications in Mathematical Physics_ 388, 1005–1048 (2021).
  * [8] G. Cipolloni, L. Erdős, D. Schröder. Thermalisation for Wigner matrices. _Journal of Functional Analysis_ 282, 109394 (2022).
  * [9] G. Cipolloni, L. Erdős, D. Schröder. On the Spectral Form Factor for Random Matrices. _Communications in Mathematical Physics_ 401, 1665–1700 (2023).
  * [10] G. Cipolloni, L. Erdős, D. Schröder. Optimal multi-resolvent local laws for Wigner matrices. _Electronic Journal of Probability_ 27, 1–38 (2022).
  * [11] G. Cipolloni, L. Erdős, D. Schröder. Rank-uniform local law for Wigner matrices. _Forum of Mathematics, Sigma_ 10, E96 (2022).
  * [12] G. Cipolloni, L. Erdős, D. Schröder. Mesoscopic central limit theorem for non-Hermitian random matrices. Accepted to _Probability Theory and Related Fields_ , arXiv: 2210.12060 (2022).
  * [13] G. Cipolloni, L. Erdős, J. Henheik, D. Schröder. Optimal Lower Bound on Eigenvector Overlaps for non-Hermitian Random Matrices. arXiv: 2301.03549 (2023).
  * [14] G. Cipolloni, L. Erdős, J. Henheik. Eigenstate thermalisation at the edge for Wigner matrices. arXiv: 2309.05488 (2023).
  * [15] G. Cipolloni, L. Erdős, Y. Xu. Universality of extremal eigenvalues of large random matrices. arXiv: 2312.08325 (2023).
  * [16] B. Collins, S. Matsumoto, J. Novak. The Weingarten Calculus. _Notices of the American Mathematical Society_ 69, 5 (2022).
  * [17] J. Cotler, N. Hunter-Jones, J. Liu, B. Yoshida. Chaos, complexity, and random matrices. _Journal of High Energy Physics_ 2017, 48, 1–60 (2017).
  * [18] J. M. Deutsch. Quantum statistical mechanics in a closed system. _Physical Review A_ 43, 2046–2049 (1991).
  * [19] L. Erdős, H.-T. Yau, J. Yin. Rigidity of eigenvalues of generalized Wigner matrices. _Advances in Mathematics_ 229, 1435–1515 (2012).
  * [20] L. Erdős, A. Knowles, H.-T. Yau, J. Yin. The local semicircle law for a general class of random matrices. Electronic Journal of Probability 18, No. 59, 1–58 (2013).
  * [21] L. Erdős, H.-T. Yau. _A dynamical approach to random matrix theory_. American Mathematical Society, Vol. 28 (2017).
  * [22] L. Erdős, T. Krüger, D. Schröder. Random matrices with slow correlation decay. _Forum of Mathematics, Sigma_ 7, E8 (2019).
  * [23] P. J. Forrester. Differential identities for the structure function of some random matrix ensembles. _Journal of Statistical Physics_ 183, 2, 33 (2021).
  * [24] P. J. Forrester. Quantifying dip–ramp–plateau for the Laguerre unitary ensemble structure function. _Communications in Mathematical Physics_ 387, 1, 215–235 (2021).
  * [25] P. J. Forrester, M. Kieburg, S. H. Li, J. Zhang. Dip-ramp-plateau for Dyson Brownian motion from the identity on $U(N)$. Accepted to _Probability and Mathematical Physics_ , arXiv: 2206.14950 (2022).
  * [26] E. M. Fortes, R. A. Jalabert, D. A. Wisniacki. Gauging classical and quantum integrability through out-of-time-ordered correlators. _Physical Review E_ 100, 4, 042201 (2019).
  * [27] I. García-Mata, R. A. Jalabert, D. A. Wisniacki. Out-of-time-order correlations and quantum chaos. _Scholarpedia_ 18, 4, 55237 (2023).
  * [28] P. Hayden, J. Preskill. Black holes as mirrors: quantum information in random subsystems. _Journal of High Energy Physics_ 2007, 09, 120 (2007).
  * [29] J. Huang, B. Landon. Rigidity and a mesoscopic central limit theorem for Dyson Brownian motion for general $\beta$ and potentials. _Probability Theory and Related Fields_ 175, 209–253 (2019).
  * [30] A. Kitaev. Hidden correlations in the Hawking radiation and thermal noise. _Talk at Breakthrough Prize Symposium_ YouTube (2014).
  * [31] A. Knowles, J. Yin. The isotropic semicircle law and deformation of Wigner matrices. _Communications on Pure and Applied Mathematics_ 66, 1663–1749 (2013).
  * [32] G. Kreweras. Sur les partitions non croisées d’un cycle. _Discrete mathematics_ 1, 333–350 (1972).
  * [33] B. Landon, P. Sosoe. Almost-optimal bulk regularity conditions in the CLT for Wigner matrices. arXiv: 2204.03419 (2022).
  * [34] M. Lemm, S. Rademacher. Out-of-time-ordered correlators of mean-field bosons via Bogoliubov theory. arXiv: 2312.01736 (2023).
  * [35] L. Leviandier M. Lombardi, R. Jost, J. Pique. Fourier transform: A tool to measure statistical level properties in very complex spectra. _Physical Review Letters_ 56, 2449–2452 (1986).
  * [36] D. Marković, M. Čubrović. Detecting few-body quantum chaos: out-of-time ordered correlators at saturation. _Journal of High Energy Physics_ 2022, 5, 1–28 (2022).
  * [37] S. Pappalardi, F. Fritzsch, T. Prosen. General Eigenstate Thermalization via Free Cumulants in Quantum Lattice Systems. arXiv: 2303.00713 (2023).
  * [38] J. Reker. Multi-Point Functional Central Limit Theorem for Wigner Matrices. arXiv: 2307.11028 (2023).
  * [39] J. Reker. Fluctuation Moments for Regular Functions of Wigner Matrices. arXiv: 2307.11029 (2023).
  * [40] Y. Sekino, L. Susskind. Fast scramblers. _Journal of High Energy Physics_ 2008,10, 065 (2008).
  * [41] S. Xu, B. Swingle. Scrambling Dynamics and Out-of-Time-Ordered Correlators in Quantum Many-Body Systems. _PRX Quantum_ 5, 1, 010201 (2024).


# Climate uncertainty, financial frictions and constrained efficient carbon
taxation††thanks: I thank William Brock, Harold Cole and Herakles
Polemarchakis for very useful conversations and comments. This work was
supported by the Swiss National Science Foundation (SNF), under project ID
“Can Economic Policy Mitigate Climate-Change?”.

Felix Kübler Department for Banking and Finance, University of Zürich; Swiss
Finance Institute (SFI); Email<EMAIL_ADDRESS>
###### Abstract

In this paper, I consider a simple heterogeneous agents model of a production
economy with uncertain climate change and examine constrained efficient carbon
taxation. If there are frictionless, complete financial markets, the simple
model predicts a unique Pareto-optimal level of carbon taxes and abatement. In
the presence of financial frictions, however, the optimal level of abatement
cannot be defined without taking a stand on how abatement costs are
distributed among individuals. I propose a simple linear cost-sharing scheme
that has several desirable normative properties. I use calibrated examples of
economies with incomplete financial markets and/or limited market
participation to demonstrate that different schemes to share abatement costs
can have large effects on optimal abatement levels and that the presence of
financial frictions can increase optimal abatement by a factor of three
relative to the case of frictionless financial market.

##  1 Introduction

The negative effects of anthropogenic climate change will differ dramatically
across regions and across different individuals within regions (see, e.g.,
Nordhaus and Yang [1996], Dennig et al. [2015], Cruz and Rossi-Hansberg
[2021], Kotlikoff et al. [2021b], Krusell and Smith Jr [2022]). Moreover, the
extent of future damages is highly uncertain (see, e.g., Hassler et al. [2018]
or Barnett et al. [2022a]). Many agents do not have access to financial
markets, and existing markets are incomplete in that climate uncertainty
cannot be hedged. These financial frictions, together with agents’
heterogeneity, obviously exacerbate welfare losses from climate change. Carbon
pricing can reduce emissions and the level as well as the uncertainty about
future climate-related damages. However, there is obviously a trade-off
between the costs of abatement and the future costs of climate change.
Normative statements about the optimal level of abatement are only possible if
one takes a stand on how the abatement costs should be shared across agents.
Financial frictions can prevent those agents who suffer the most from future
damages from using resources from those future states with little damage to
combat climate change today. Depending on the abatement cost-sharing scheme,
carbon policy might only help little to combat welfare losses from climate
change.

In this paper, I consider a simple stochastic growth model with heterogeneous
agents and climate change to formalize these ideas. Financial frictions take
the form of incomplete financial markets and restricted participation in the
existing markets. The primary goal is to find a constrained Pareto-efficient
path of carbon-risk taxes and abatement cost sharing and to show how they
differ from the ones in an economy without financial frictions. I also examine
the welfare losses that are caused by a combination of financial frictions and
climate damages.

This is hardly the first quantitative analysis of uncertainty’s importance to
optimal carbon policy. Prior major studies include Jensen and Traeger [2014],
Cai et al. [2013], and Traeger [2019] who all focus on uncertainty arising
from tipping points. More closely related to this paper’s treatment of
uncertainty are Hassler et al. [2018], Van den Bremer and Van der Ploeg
[2021], Kotlikoff et al. [2021a], Barnett et al. [2020], Barnett et al.
[2022a] and Barnett et al. [2022b]111See, e.g., Gillingham et al. [2015] and
Cai [2020] for thorough reviews of the literature.. This said, this paper
appears to be the first study of constrained optimal carbon-risk tax policy in
a heterogeneous agents model with financial frictions. Given the importance of
the abatement cost-sharing scheme, this paper is also closely related to the
literature on optimal carbon taxation and redistribution in economies with
heterogeneous agents (see, e.g., Barrage [2020] and Douenne et al. [2022]).
This literature focuses largely on who bears the burden of carbon taxes, while
in my paper, the problem is modeled in an overly simplified fashion. There is
also literature on the role of financial frictions in the amplification of
transition risk (see, e.g., Carattini et al. [2021]). While financial
frictions also play a role in my paper, there is no close connection.

In the model, higher temperature negatively affects labor productivity. These
effects differ in magnitude across different agents. Future temperatures are
uncertain because of uncertainty about the climate sensitivity parameter.
Markets are incomplete in that agents can only trade risky capital. In
addition, some agents might have no access to asset markets whatsoever.

The supply side of the model is a slightly simplified version of the model in
Nordhaus [2017]. Final goods are produced using capital, labor, clean energy,
and dirty energy, emitting CO2 into the atmosphere. Importantly, there is a
single capital stock. This makes an interpretation of the heterogenous agents
in the economy as agents in different regions somewhat problematic. In a model
with region-specific production functions, one would need to take a stand on
capital mobility and international financial markets, which is a bit beyond
the scope of this study.

Abatement can reduce CO2 emissions at some reduced form cost function. I adopt
the carbon cycle and temperature equation from Golosov et al. [2014], which
simplifies Nordhaus [2017]’s treatment of these elements.222 As Folini et al.
[2021] point out, this model misses various important features of climate
change, but for the purposes of this study, it seems an acceptable
simplification. In particular, the length of a period is taken to be 10 years
which alleviates some of the issues raised in Folini et al. [2021].

The uncertainty about the climate-sensitivity parameter (ECS) is the key
ingredient of the model. While the climate-sensitivity coefficient plays a
crucial role in understanding the effects of CO2 emissions on global warming,
there is, unfortunately, little scientific consensus on its value. As Knutti
et al. [2017] put it, “Equilibrium climate sensitivity characterizes the
Earth’s long-term global temperature response to increased atmospheric CO2
concentration. The temperature response has reached almost iconic status as
the single number that describes how severe climate change will be.
Interestingly, the consensus on the ‘likely’ range of temperature increase –
1.5 to 4.5 degrees Celsius – is the same today as the projection made by Jule
Charney in 1979.”A large part of the uncertainty about the ECS is typically
attributed to model uncertainty. In other words, researchers believe that the
time variation in the true parameter is much smaller than this range but that
our current climate models cannot determine this true parameter. However,
there is some evidence that even beyond model uncertainty, the equilibrium
climate sensitivity might vary over time (see, e.g., Roe and Baker [2007] but
also Zaliapin and Ghil [2011]). Based on my reading of the climate-science
literature, I model this parameter as a Markov process whose persistence
increases over time so that in 300 years, the model is essentially
deterministic.

The policy instruments are i) a tax on dirty energy’s use that depends on time
and the state of the economy and ii) a sharing rule (also varying with time
and the state) of each period’s abatement costs. Critically, I do not allow
for arbitrary transfers but only for those that compensate agents for their
losses caused by abatement. Because of financial frictions, there is no
Pareto-efficient policy using only these two instruments. The focus of this
paper is on “constrained interim efficient” policies. In each period, current
carbon-taxes are Pareto-efficient in the sense that there are no other taxes
and transfers in the period that make everybody better off. On one hand, ex-
interim efficiency seems to be a reasonable requirement since each period,
taxes and transfers are optimal in that period. On the other hand, in the
presence of financial frictions, one can increase everybody’s ex-ante welfare
by dropping the interim efficiency requirement, even if transfers are
restricted to only compensate for abatement costs. If agents pay more in
states where they are relatively rich and less in states where there are
relatively poor, since marginal rates of substitution differ across agents,
everybody’s ex-ante welfare can increase. In this paper, possible ex-ante
Pareto-improvements are ignored, and the focus lies on ex-interim efficiency.
The concept is referred to as interim efficiency since it can be interpreted
as a generalization of ex-post efficiency to a multi-period model. Starr
[1973] already notes the tension between ex-ante and ex-post efficiency in a
different context. This problem is also somewhat similar to issues arising in
time-consistent public policy models (e.g., Klein et al. [2008]). I do not
explicitly model a policy game, but one reason for the constraints on policies
to be ex-interim efficient can lie in a lack of commitment.

Of course, there are infinitely many policies that satisfy the condition of
constrained interim Pareto-efficiency, and they all differ with respect to
abatement levels and transfers. I focus on a particular one, where each
period, all agents are compensated exactly for the losses due to abatement,
and the transfers necessary for this are shared proportionally to agents’
marginal willingness to pay for the reduction in CO2 emissions. In other
words, prices are held fixed at their pre-abatement levels, and abatement
costs are shared according to a linear rule. The concept can be viewed as a
variation of a Balanced Linear Cost Share Equilibrium as in Mas-Colell and
Silvestre [1989] or a Ratio Equilibrium in the sense of Kaneko [1977]. As Mas-
Colell and Silvestre [1989] point out, “no justification in terms of fairness
should be attempted for this distribution scheme” (see also Moulin [1987] for
cost-sharing schemes that can be justified by normative principles), but there
are a variety of practical arguments in its favor. It is easy to prove that
this policy satisfies the definition of constrained efficiency and that, given
the transfer rule, there are no other taxes that lead to an ex-ante Pareto-
improvement. By construction, the policy constitutes a Pareto-improvement
relative to the laissez-faire outcome. Most importantly, it is tractable in
that optimal abatement and transfers can be calculated relatively easily. In a
slight abuse of language, this policy is sometimes referred to as the
constrained optimal, or constrained efficient policy.

I compare the policy to a policy where taxes are set to be interim efficient,
but there are no transfers. It is easy to see that in this setup, these taxes
will generally not be ex-ante efficient. The resulting allocation might not
constitute a Pareto-improvement over the laissez-faire outcome. These taxes
are generally higher than the constrained optimal taxes, and the hand-to-mouth
consumers that are hurt most by climate change are better off, while the
consumers that are hurt least by climate change are worse off. However, the
effects on welfare are quantitatively small in our calibration.

I also compare the constrained efficient policy to the optimal policy in a
world without financial frictions. Optimal taxes are significantly lower by a
factor of 2-3, (and as a result, future temperatures and damages are higher)
in the absence of financial frictions. This might seem counter-intuitive at
first, but is caused by the precautionary savings motive resulting from
uncertainty.

In the calibrations, the welfare properties of the constrained efficient
policy depend crucially on abatement costs. For low abatement costs, the
welfare losses due to financial frictions are very small, about two percent of
certainty equivalent lifetime consumption for the agents hit most by climate
change. If abatement costs become very high, overall welfare losses obviously
increase but, more importantly, welfare losses from financial frictions
increase as much as welfare costs from emissions.

The remainder of the paper is organized as follows: Section 2 presents the
basic model. In Section 3 I introduce constrained efficient carbon policy
Section 4 discusses the calibration strategy and Section 5 presents the
results. Section 6 concludes. An appendix discusses some computational
details.

##  2 Model

In this section, the economic model is introduced. The climate change
externality is described and economic policy is defined.

###  2.1 The physical economy

This paper considers a production economy with long-lived agents. Time is
indexed by $\displaystyle t=1,\ldots,T\leq\infty$. Exogenous shocks
$\displaystyle z_{t}$ realize in a finite set
$\displaystyle{\mathbf{Z}}=\\{1,\ldots,Z\\}$, and follow a first-order Markov
process with transition probability $\displaystyle\mathbb{P}_{t}(.|z)$ at each
date $\displaystyle t$ . A history of shocks up to some date $\displaystyle t$
is denoted by $\displaystyle z^{t}=(z_{0},z_{1},\ldots,z_{t})$ and called a
date event. Whenever convenient, I simply use $\displaystyle t$ instead of
$\displaystyle z^{t}$.

At each date-event there is a single good available for consumption. Capital
can be stored and agents are endowed with labor.

###  2.2 Firms

A representative firm produces the consumption good using capital and labor.
The production function is Cobb-Douglas and given by

$f(K_{t},l_{t},\mu_{t})=A_{t}(\mu_{t})K_{t}^{\alpha}L_{t}^{(1-\alpha)}+(1-\delta)K_{t},$
(1)

where $\displaystyle A_{t}$, $\displaystyle K_{t}$, $\displaystyle L_{t}$
refer to total factor productivity, capital and labor respectively. The
parameter $\displaystyle\alpha\in(0,1)$ represents the capital share,
$\displaystyle\mu_{t}\in[0,1]$ denotes abatement and $\displaystyle\delta$ the
depreciation rate. Throughout it is abstracted from TFP growth and I take
$\displaystyle A_{t}(0)=1$ for all $\displaystyle t$. It is useful to define
$\displaystyle Y_{t}=K_{t}^{\alpha}L_{t}^{(1-\alpha)}$ as potential output
without any abatement. I assume throughout that for all $\displaystyle t$,
$\displaystyle A_{t}$ is decreasing, concave and differentiable and that
$\displaystyle A_{t}(0)=1$.

Production emits CO2 into the atmosphere. I take as exogenous a sequence
$\displaystyle(e_{t})_{t=0}^{T}$ and assume that total emissions in period
$\displaystyle t$ are given by $\displaystyle E_{t}=(1-\mu_{t})e_{t}Y_{t}$,
where $\displaystyle\mu_{t}$ is the level of abatement that reduces production
via $\displaystyle A_{t}(\mu)$.

Firms rent capital and buy labor on spot markets, facing a wage $\displaystyle
w_{t}$ and a price of capital $\displaystyle r_{t}$. Since they make no
intertemporal decisions, there is no issue about the appropriate objective
function in a model with financial frictions. This is an important
simplification that is crucial for the analysis; in a more realistic setup
that models the use of oil, gas, and coal explicitly, this is unlikely to be
the case, and without taking a stand on the theory of the firm in incomplete
markets, one cannot talk about optimal policy (see, e.g., Dierker et al.
[2002], for some of the complications that arise).

###  2.3 Households

There are $\displaystyle H$ types of $\displaystyle T$-period lived agents,
$\displaystyle h\in{\mathbf{H}}=\\{1,\ldots,H\\}$, that maximize lifetime
expected utility, given by

$U(C)={\mathbb{E}}_{0}\sum_{t=0}^{T}\beta^{t}\frac{c_{t}^{1-\sigma}-1}{1-\sigma},$
(2)

subject to

$c_{t}+a_{t+1}=(1+r_{t})a_{t}+w_{t}l_{h,t}-\theta_{t,h},\
a_{t}\in{\mathbf{Y}}^{h}$ (3)

where $\displaystyle\beta\in(0,1)$ is the time preference factor,
$\displaystyle\sigma>0$ denotes the coefficient of relative risk aversion,
$\displaystyle c_{t}$, $\displaystyle a_{t}$, $\displaystyle r_{t}$,
$\displaystyle w_{t}$ correspond to consumption, capital holdings, returns to
capital and wages at time $\displaystyle t$, respectively, and labor supply at
$\displaystyle t$ is $\displaystyle l_{h,t}$. Capital holdings of agent
$\displaystyle h$ are restricted to lie in a set
$\displaystyle{\mathbf{Y}}^{h}$ – some agents might not be able to participate
in capital markets at all. In the analysis below, I also consider the case
where, in addition to risky capital, there is a complete set of Arrow
securities available for trade and where no agents face restrictions on their
trades. I refer to this case as complete markets and contrast it to the
economy with financial frictions described here.

The variable $\displaystyle\theta_{t,h}$ denotes the possibly state-specific
net tax paid by the agent $\displaystyle h$ at time $\displaystyle t$. In our
specification of transfers below this can depend on the agents’ beginning of
period capital holdings, a fact the agent takes into account when making
savings decisions. It is assumed throughout that
$\displaystyle\sum_{h}\theta_{t,h}=0$ for all $\displaystyle t$.

I assume that an agent’s labor productivity is impacted negatively by climate
change. For each agent $\displaystyle h\in{\mathbf{H}}$ there is a agent-
specific damage function $\displaystyle D_{h}(T_{t})$, where $\displaystyle
T_{t}$ denotes average surface temperature (in degrees Kelvin) relative to
pre-industrial levels. An agent’s labor supply at time $\displaystyle t$ is
denoted by $\displaystyle l_{t,h}=D_{h}(T_{t})\bar{l}_{h}$, where
$\displaystyle\bar{l}_{h}>0$ can be thought of as the agents labor supply
without climate change. Aggregate damages to labor supply are denoted by

$\displaystyle D(T_{t})=\sum_{h\in{\mathbf{H}}}D_{h}(T_{t})\bar{l}_{h},$

and given the assumption on Cobb-Douglas production, they translate to an
effect of total output that can be written as $\displaystyle
D(T_{t})^{1-\alpha}$.

###  2.4 Modeling climate change

The carbon cycle is modeled as in Golosov et al. [2014]. The temperature
$\displaystyle T_{t}$ in period $\displaystyle t$ is determined by the stock
of carbon in the atmosphere, $\displaystyle S_{t}$,

$T_{t}=\lambda_{t}\frac{\log(S_{t}/S)}{\log(2)},$ (4)

where $\displaystyle S$ is the pre-industrial carbon stock. We model
$\displaystyle\lambda_{t}$ in a stochastic manner, assuming that
$\displaystyle\lambda_{t}$ only depends on the current shock $\displaystyle
z_{t}$.

Following Golosov et al. [2014], it is assumed that the CO2 stock in the
atmosphere has two components—that is,

$S_{t}=S_{1t}+S_{2t},$ (5)

where

$S_{1t}=\xi_{1}\xi_{2}\cdot E_{t}+\delta_{S1}\cdot S_{1,t-1},$ (6)

and where

$S_{2t}=\xi_{1}(1-\xi_{2})\cdot\xi\cdot E_{t}+\delta_{S2}\cdot S_{2,t-1}.$ (7)

The depreciation parameters satisfy $\displaystyle\delta_{S2}<\delta_{S1}\leq
1$. I calibrate the former at a low value and the latter at a high value,
again following Golosov et al. [2014]. Hence, $\displaystyle S_{1t}$ is a
slowly depreciating stock of carbon, whereas $\displaystyle S_{2t}$ is a
rapidly depreciating stock. The parameters $\displaystyle\xi_{1}$ and
$\displaystyle\xi_{2}$ control the fraction of CO2 emissions entering the
atmosphere. We take $\displaystyle\xi_{1},\xi_{2}$, and the depreciation
parameters as fixed. As Golosov et al. [2014] point out, there is no consensus
in the literature concerning the values of these parameters. Folini et al.
[2021] discuss better models for the carbon cycle in detail. In any case, the
main results in this paper are robust to moderate differences in these
parameters as well as to time-varying shocks they may experience.

Climate change reduces output productivity by destroying agents’ labor
endowments, as described above.

###  2.5 Abatement

Firms can reduce CO2 emissions via abatement, and a carbon tax will lead to a
reduction in emissions that depends on the abatement technology. Instead of
modeling dirty energy supply explicitly, I use the reduced form formulation
from Nordhaus [2017]. As mentioned above, it is assumed that emissions
$\displaystyle E_{t}$ are given by the level of production, an exogenous time-
varying parameter, and abatement $\displaystyle\mu$, and therefore the
abatement necessary to achieve emissions $\displaystyle E_{t}$ in period
$\displaystyle t$ can be written as

$\displaystyle\mu_{t}=1-\frac{E_{t}}{e_{t}Y_{t}}.$

Given a tax $\displaystyle\tau$ per unit of CO2 emissions firms choose
$\displaystyle\mu$ so that profits are maximized. The first order condition
for optimal abatement is therefore given by

$\displaystyle A_{t}^{\prime}(\mu_{t})\frac{1}{e_{t}Y_{t}}Y_{t}=\tau_{t},$

which give a unique $\displaystyle\mu_{t}(\tau)$ for each $\displaystyle\tau$.
Instead of searching for a tax policy, one can, therefore, directly search for
optimal abatement, $\displaystyle\mu_{t}$. The revenue from the carbon tax,
$\displaystyle\tau E_{t}$ is assumed to be redistributed lump-sum to the firms
so that transfers only have to compensate for the abatement costs,
$\displaystyle(1-A_{t}(\mu_{t}))Y_{t}$.

###  2.6 Policy

A climate policy taxes emissions and institutes transfers between agents to
compensate them for the losses incurred by taxation. As explained above, I
will formally write this policy as abatement levels and transfers but will
often refer to it as taxes and transfers.

In the setup of this paper, it is natural to assume that taxes can depend on
time and the state of the economy (i.e., the exogenous shock as well as CO2
concentrations and endogenous economic variables). Since the period length
amounts to 10 years, it would be unreasonable to assume that with the arrival
of new information regarding the ECS and CO2 concentrations, taxes and
transfers cannot be adjusted. Agents take as given a tax- and transfer
function that depends on time, the shock, and the state of the economy. The
endogenous state, $\displaystyle s$, consists of beginning-of-period capital
holdings across agents, $\displaystyle(a_{1},\ldots,a_{H})$ and beginning-of-
period CO2 stocks $\displaystyle(S_{1},S_{2})$. I write
$\displaystyle\mu_{t}(z,s)$ and
$\displaystyle(\theta_{h,t}(z,s,\mu))_{h\in{\mathbf{H}}}$ to denote the
policy. Crucially, taxes are assumed not to depend on individual variables,
i.e. a single household’s choices at $\displaystyle t-1$ do not affect taxes.
Transfers, on the other hand, will compensate for lower returns due to
abatement and, therefore, will depend on agents’ choices at $\displaystyle
t-1$. As mentioned above, I directly work with abatement $\displaystyle\mu$
instead of taxes $\displaystyle\tau$ since this simplifies the notation. In a
slight abuse of language, $\displaystyle\mu$ is sometimes referred to as the
tax.

###  2.7 Equilibrium

Given taxes and transfers, a competitive equilibrium consists of a stochastic
process for choices and prices so that asset markets and labor markets clear
and all agents maximize utility given the tax and transfer policies. In this
specification this implies that $\displaystyle K_{t}=\sum_{h}a_{t}^{h}$,
$\displaystyle L_{t}=\sum_{h}l_{ht}$ and wages and the rental rate of capital
are given by the marginal product of labor and capital in the firm’s
production function (given optimal abatement). I sometimes write
$\displaystyle r(\mu)$ and $\displaystyle w(\mu)$ to make explicit the
dependence on abatement.

Throughout the analysis, it is assumed that taxes and transfers are a function
of time, the current shock, and the endogenous state and that there exists a
recursive equilibrium all optimal choices are functions of the current shock,
the endogenous state, and time, $\displaystyle t$, alone. The endogenous
state, $\displaystyle s$, consists of beginning-of-period capital holdings
across agents, $\displaystyle(a_{1},\ldots,a_{H})$ and beginning-of-period CO2
stocks $\displaystyle(S_{1},S_{2})$. An agent’s value function, $\displaystyle
V^{h}_{t}(z,s,a)$ gives his equilibrium utility as a function of time, the
aggregate state, i.e, the endogenous state, and the current shock,
$\displaystyle z_{t}$, and the agent’s beginning of period asset holding,
$\displaystyle a$. It is useful to define the Q-function as the value also
depending on current policy choices, i.e. for every agent $\displaystyle h$,

$\displaystyle Q^{h}(z,s,a;\mu,\theta)=\max_{a^{\prime}}u(c)+\beta
E_{t}V^{h}(z^{\prime},s^{\prime},a^{\prime})\mbox{ s.t. }$ $\displaystyle
c=a(1+r(\mu))+l_{h}(T(s))w(\mu)-a^{\prime}-\theta_{h}$ $\displaystyle
S_{1}^{\prime}=\xi_{2}\cdot\xi_{1}\cdot(1-\mu)Y_{t}e_{t}+\delta_{S1}\cdot
S_{1},\
S_{2}^{\prime}=(1-\xi_{2})\cdot\xi_{1}\cdot(1-\mu)Y_{t}e_{t}+\delta_{S2}\cdot
S_{2}.$

It is assumed throughout that a recursive equilibrium exists – even in simple
models without climate change this might be a strong assumption (see, e.g.
Brumm et al. [2017]). However, clearly, one needs to avoid the possibility of
multiple equilibria in the comparative statics exercises.

##  3 Constrained efficient policy

In standard economic models, climate change is treated as an externality, and,
going back all the way to the work of Pigou [1920], it is well known that
Pareto efficiency can be restored by adding the marginal market value of the
externality to the market price of carbon. In settings where the welfare
theorems hold, this can be shown in some generality, although there is, of
course, an issue with uniqueness. In this setting, this value cannot be
deduced from market prices and has to be obtained by adding up each agent’s
marginal valuation. Since the welfare theorems fail due to financial
frictions, a notion of constrained efficiency has to be employed in order to
examine carbon taxes that have desirable normative properties.

As explained above, taxes are functions of time and the exogenous and
endogenous state of the economy. In the first best, transfers would also
depend on the state and lead to a Pareto-optimal allocation. However, since it
is assumed that there are no financial assets that pay contingent on the
realization of the climate shock, it seems reasonable to assume that transfers
cannot compensate for the lack of asset markets. I have in mind a model where
at time $\displaystyle t$, the value of $\displaystyle\lambda_{t}$ is
observable but where it is not ex-ante (i.e. at any time before $\displaystyle
t$) contractible. At time $\displaystyle t$, once $\displaystyle\lambda_{t}$
is observed taxes can be adjusted accordingly, but transfers are restricted to
compensate only for losses that are caused by taxes in the current period.
Pareto efficiency can generally not be obtained with such a scheme, and, in
general, it is difficult to characterize constrained efficient taxes and
transfers. Since markets are incomplete, the tax policy will have an insurance
aspect that depends on taxes across states in the future. The key assumption
in this paper is that policy is restricted to be ex-interim efficient.
Formally, we say that a tax- and transfer policy is interim constrained
Pareto-efficient if at any node $\displaystyle z^{t}$, there are no taxes and
transfer $\displaystyle(\widetilde{\mu},\widetilde{\theta})$ such that for all
$\displaystyle h\in{\mathbf{H}}$

$\displaystyle
Q^{h}_{t}(z_{t},s(z^{t}),\widetilde{\mu},\widetilde{\theta})\geq
V^{h}_{t}(z_{t},s(z^{t})),$

where the inequality holds strict for at least one agent.

The requirement of interim efficiency puts obvious restrictions on taxes. An
interim-efficient tax solves

$\displaystyle\max\sum_{h}\lambda_{h}Q^{h}(z,s,\mu,\theta)\mbox{ s.t.
}\sum_{h}\theta_{h}=0$

for some weights $\displaystyle\lambda\in\mathbb{R}^{H}_{++}$. It is easy to
see that a necessary condition for a constrained interim efficient
$\displaystyle\mu$ is as follows.

$\sum_{h\in{\mathbf{H}}}\frac{\frac{\partial
Q^{h}}{\partial\mu}}{u^{\prime}(c^{h}_{t})}=0.$ (8)

The sum of the marginal benefits of an infinitesimal increase in abatement
must equal the marginal cost. In the general equilibrium setting of this
paper, the condition will generally not be sufficient to determine the optimal
level of abatement uniquely. In the following, I make the strong assumption
that there exists a unique solution to this equation that is denoted by
$\displaystyle\mu^{*}_{t}(z,s)$.

While constrained optimal taxes must satisfy (8), they obviously depend
crucially on the welfare weights $\displaystyle\lambda\in\mathbb{R}^{H}_{+}$,
which have been substituted out of the expression and are replaced by a
transfer scheme. Without specifying the welfare weights, there are obviously
infinitely many possible transfers, even if one focuses on a system of
transfers that is interim Pareto-improving relative to the laissez-faire
outcome. Moreover, the levels of constrained optimal taxes and of abatement
depend critically on the transfer scheme, and it is impossible to characterize
optimal taxes without, at the same time, characterizing transfers.

In general, this is to be expected in a model with heterogeneous agents – the
distribution of wealth affects asset prices and hence the evaluation of future
damages. However, in this paper, it is assumed that agents have identical CRRA
utility functions, and in this setup, optimal abatement is independent of
transfers when markets are complete (see, e.g. Gollier [2001]). With financial
frictions (incomplete financial markets and/or limited participation) optimal
abatement always depends on transfers. The agents’ first-order conditions with
respect to asset holdings are (generically) linearly independent of Equation
(8) and the solution of $\displaystyle(\ref{optax})$ depends on the
distribution of income in the current period. A calibrated example in Section
5 below demonstrates that different cost-sharing rules can imply significant
differences in optimal abatement.

The joint determination of optimal abatement and its cost-sharing is formally
equivalent to a model of the optimal supply of a public good (at least in the
simple model considered in this paper, see Chichilnisky and Heal [1994] for a
discussion of the problem in a more sophisticated setup with heterogeneous
produces and heterogeneous consumers). The classical formalization of a
Lindahl equilibrium in Foley [1970] takes ownership shares (in the abatement
technology) as given and is therefore not suitable for our setting. Kaneko
[1977] and Mas-Colell and Silvestre [1989] do not consider ownership shares
and derive a linear cost-sharing rule which I adopt in this study. The scheme
fits well with the idea that individual contributions to the cost reflect
personal valuations and is very tractable compared to other approaches (e.g.
Moulin [1987]).

To formulate the scheme, define future benefits from abatement today as

$\displaystyle\tilde{Q}^{h}_{t}(\mu)=\frac{\partial
Q_{t}^{h}}{\partial\mu}-u^{\prime}(c^{h}_{t})A_{t}^{\prime}(\mu)\left(l^{h}_{t}(1-\alpha)K_{t}^{\alpha}L_{t}^{-\alpha}+k^{h}_{t}\alpha
K_{t}^{\alpha-1}L_{t}^{1-\alpha}\right)$

and set transfers to be

$\displaystyle\displaystyle\theta_{th}(z,s)$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{\tilde{Q}^{h}_{t}(\mu^{*}_{t}(z,s))}{\sum_{i}\tilde{Q}^{h}_{t}(\mu^{*}_{t}(z,s))}(A(\mu^{*}_{t}(z,s))-A(0))K_{t}^{\alpha}L_{t}^{1-\alpha}-$
$\displaystyle\displaystyle(A(\mu_{t})-A(0))\left(l^{h}_{t}(1-\alpha)K_{t}^{\alpha}L_{t}^{-\alpha}+k^{h}_{t}\alpha
K_{t}^{\alpha-1}L_{t}^{1-\alpha}\right).$

In the scheme, capital holders are compensated for the lower returns, and
workers are compensated for the lower wages caused by abatement. The capital
holders take this into account when making investment decisions at
$\displaystyle t-1$. The costs of abatement are shared according who gains
most from the abatement irrespective of the initial endowments in capital and
labor. In a model with technical change, where agents have an option to invest
in dirty versus clean energy technologies, this compensation causes an obvious
inefficiency. In this paper, this is not the case, and, as mentioned above, a
more realistic supply side would entail taking a stand on the problem of the
policy of the firm with financial frictions. Something that needs to be
avoided. It seems that it is important to leave asset returns unchanged
through the transfers. Changes in prices lead to changes in savings, and in
the incomplete markets setup considered here, that alone generally has welfare
consequences (see Davila et al. [2012]). As mentioned above, a collection of
taxes and cost-sharing across nodes that is ex-ante efficient would take all
of this into account and might result in abatement levels that are very
different than the ones reported below. However, it is important to emphasize
that the concept here is purely forward-looking.

As it is shown in Theorem 1 below, this scheme of taxes and transfers has
several desirable properties. It is an interim improvement relative to the
laissez-faire, that is, at all $\displaystyle z^{t}$, for all $\displaystyle
h\in{\mathbf{H}}$,

$\displaystyle V^{h}(z_{t},s(z^{t}))\geq Q^{h}_{t}(z_{t},s(z^{t}),0,0).$

Moreover, given transfer functions $\displaystyle\theta(s,z,\mu)$, the tax
policy is ex ante efficient, i.e. there is no other policy,
$\displaystyle\tilde{\mu}$ that satisfies for all $\displaystyle
h\in{\mathbf{H}}$

$\displaystyle Q^{h}_{0}(z_{0},s_{0},|(\widetilde{\mu}))\geq
V^{h}_{0}(z_{0},s_{0}),$

where the inequality is strict for at least one agent.

The following theorem summarizes the main results of this section.

######  Theorem 1.

Suppose there exists a recursive equilibrium and suppose Equation (8) always
has a unique solution. Then

  * •

The policy given by (8) and (3) is constrained interim efficient.

  * •

It constitutes an interim Pareto-improvement relative to the laissez-faire
equilibrium.

  * •

There is no other tax policy $\displaystyle(\tau(z^{t}))_{z^{t}}$ with
transfer given by (3) that makes all agents ex ante better off.

The first two properties follow directly by the construction of the policy.
Ex-ante efficiency is guaranteed because each agent’s welfare at each
$\displaystyle t$ reaches its maximum at $\displaystyle\mu^{*}$. Any change in
taxes must make at least one agent worse off. By construction, the linear
cost-sharing scheme implies that at each $\displaystyle t$ all agents agree on
the optimal level of abatement.

As mentioned in the introduction, the policy defined by (8) and (3) is
sometimes referred to as the constrained optimal policy in this paper.

The assumption that the $\displaystyle Q$ function is convex in emissions is
implicit in the assumption that there is a unique solution to Equation (8) .
It is the analog to a standard convex cost assumption. In the case of climate
change, this assumption is more problematic, however, even in a representative
agent model. There are two competing supply-side non-linearities in climate
models. First, damages are assumed to be a strictly convex function of global
average surface temperature. Second, the average surface temperature is
modeled as a logarithmic and, therefore, a concave function of atmospheric
CO2. Often the convexity outweighs its supply-side concavity in determining
carbon-risk damages. Moreover, when it comes to the welfare effects, risk
aversion is likely to outweigh the concavity in the forcing equation.

I compare this transfer scheme with constrained efficient taxation without
transfers below. Note that without transfers, taxes will be ex-interim
constrained efficient if they satisfy 8. In the presence of financial
frictions, they will generally not be ex-ante efficient, and they might not
lead to an allocation that Pareto dominates the laissez-faire.

I also compare the constrained optimal policy to the optimal policy in an
economy without financial frictions. In this case, the optimal tax is simply
set to the discounted (at the stochastic discount factor) sum of future
damages caused by one extra infinitesimal unit of emissions in the current
period. Since agents are assumed to have identical CRRA utility, this so-
called social cost of carbon does not depend on who bears the abatement cost.
To determine optimal taxes, one can simply use a representative agent economy.
Transfers can then be chosen independently to trace out the Pareto-frontier of
the economy.

##  4 Calibration

I take a model period to be 10 years and assume $\displaystyle T=30$ – a
horizon of 300 years is convenient for calibrating emissions, and a model
period of 10 years is the same as in Golosov et al. [2014]. A longer, even
infinite time horizon has only small effects on optimal abatement and welfare
in the first periods.

Technology and preferences are standard. It is assumed that the capital share
in the aggregate production function is $\displaystyle\alpha=0.33$. The
capital depreciation rate, $\displaystyle\delta$ is set to 0.57 (corresponding
to 8 percent depreciation annually). For simplicity, there is no TFP growth,
and absent climate change the economy would be in a steady state. Adding trend
TFP growth is unlikely to change any of the results. Of course, it is an
important assumption that climate damages do not affect TFP growth.

There are $\displaystyle H=3$ agents with identical CRRA utility functions.
The coefficient of relative risk aversion, $\displaystyle\sigma$, is set to
$\displaystyle 5$, and the time preference factor, $\displaystyle\beta$, is
set to 0.74 (corresponding to an annual $\displaystyle\beta$ of 0.97).
Different examples of labor endowments $\displaystyle\bar{l}_{h}$,
$\displaystyle h\in{\mathbf{H}}$ and market participation are considered
below.

The climate part of the model follows Golosov et al. [2014], and it is assumed
that $\displaystyle\xi_{2}=0.5$ , and $\displaystyle\xi_{1}=0.4$. The slow and
fast depreciation rates, $\displaystyle\delta_{S1}$ and
$\displaystyle\delta_{S2}$, are set at 1.0 and 0.97, respectively. Initial
conditions for $\displaystyle t=0$ are $\displaystyle S_{0}=(118,684)$ as the
carbon in the two reservoirs at $\displaystyle t=0$ and $\displaystyle S=581$
as the pre-industrial level of carbon in the atmosphere.

As explained in the introduction, there is a large disagreement in the climate
science literature about the value of the equilibrium climate sensitivity
(ECS). In this context, it is important to note that the ECS is not a free
parameter in Earth System models but that it emerges as one characteristic of
simulated climate change describing the very long-run behavior of temperature.
Determining a suitable stochastic process for this parameter that can be used
in macro-economic modeling is challenging, and there is no consensus on this
issue. In this study, the ECS-parameter, $\displaystyle\lambda$, is assumed to
take one of 6 possible values each period,

$\displaystyle\lambda\in\\{1.1,2.6,3.1,3.6,4.1,5.6\\}$

where the range from 2.6 to 4.1 is consistent the range of reasonable values
from climate science. I add extremes to capture additional uncertainty in the
damage function. As explained in the introduction, it is assumed that the
uncertainty about $\displaystyle\lambda$ reduces over time. I take all six
shocks to be equiprobable for the first two periods, $\displaystyle t=0,1$ and
then set

$\displaystyle\mathbb{P}_{t}(z|z)=1-0.5^{\frac{t+1}{3}}$

and

$\displaystyle\mathbb{P}_{t}(z^{\prime}|z)=\frac{1-\mathbb{P}_{t}(z|z)}{5}\mbox{
for }z^{\prime}\neq z.$

At $\displaystyle t=30$, the probability of switching climate states is
therefore only about 1/1000. There seems to be a consensus in the literature
that, in the long run, the uncertainty about the ECS will be very small (see,
e.g., Sherwood et al. [2020]).

Figure 1: Global CO2 emissions (in GtCO2) in DICE 2016, the BAU-scenario in
DICE 2016 and in Kotlikoff et al. [2021b] and the different RCP scenarios
(year 0 is 2017).

The emissions to output ratio, $\displaystyle e_{t}$, is calibrated to match
RCP 4.5 emissions over the next 300 years at steady state output. The
representative concentration pathways (RCPs) describe the results of different
socioeconomic narratives that produce particular concentration profiles of
greenhouse gases, aerosols, and other climatically relevant forcing agents
over the 21st century, see [Meinshausen et al., 2011, http://www.pik-
potsdam.de/~mmalte/rcps/]. The RCP85 scenario, for instance, reflects a “no
policy” narrative, in which total anthropogenic forcing reaches approximately
$\displaystyle 8.5W/m^{2}$ in the year 2100. Conversely, the RCP2.6 scenario
involves aggressive decarbonization, causing radiative forcing to peak at
approximately $\displaystyle 3W/m^{2}$ around 2050 and to decline to
approximately $\displaystyle 2.6W/m^{2}$ at the end of the 21st century. As
Hausfather and Peters [2020] point out, RCP8.5 was intended to explore an
unlikely high-risk future, entails unrealistically high coal usage. In this
light, the RCP6.0 or the RCP4.5 scenario seem more relevant for forecasting
emissions absent additional abatement. The BAU scenario in Nordhaus [2017]
leads to emissions that lie somewhere between RCP6.0 and RCP 8.5, but closer
to RCP6.0 (see e.g. Folini et al. [2021]). In this paper, RCP4.5 emissions are
assumed to describe the path of future emissions without abatement. The RCP4.5
scenario is a stabilization scenario, which means the radiative forcing level
stabilizes before 2100 by employment of a range of technologies and strategies
for reducing greenhouse gas emissions. In the standard DICE2016 calibration
there is little room for additional abatement given the parameter values
assumed (see Folini et al. [2021]). As will be shown, uncertainty together
with financial frictions lead to substantial abatements on top of the ones
already present in the RCP4.5 scenario. Figure 1 (from Kotlikoff et al.
[2021b]) shows the emissions path for the different scenarios. Clearly, this
paper starts with a very optimistic one.

The abatement technology is taken from Nordhaus [2017] and it is assumed that

$\displaystyle A_{t}(\mu_{t})=1-\phi_{1t}\mu_{t}^{\phi_{2}}.$

To calibrate the parameters $\displaystyle\phi_{1t}$ and
$\displaystyle\phi_{2}$ one needs to take a stand on the elasticities of
substitution between dirty energy, capital, and labor, as well as on future
possibilities of carbon removal. To simplify the analysis, it is assumed in
our calibration that these coefficients are constant over time. In Nordhaus
[2017] abatement becomes more cost-effective over time, but, as pointed out
above, BAU emissions are assumed to stay relatively constant. The RCP4.5
scenario used in this paper builds in a steep decrease of emissions starting
at about 2050, and hence it feels natural to keep the abatement technology
constant. It is assumed that $\displaystyle\phi_{2}=2$ and
$\displaystyle\phi_{1}=0.25$. This implies that reducing emissions by 20
percent reduces output by 1 percent, and reducing emissions by 50 percent
reduces output by 10 percent. As pointed out several times, this is in
addition to the large reductions already assumed in the RCP 4.5 scenario and
therefore seems rather optimistic.

As Hassler et al. [2012] point out, assuming a low elasticity of substitution
between capital, labor, and dirty energy is unrealistic when the time period
is short. In fact, they find that a Leontief specification provides a good fit
for annual data. However, Hassler et al. [2018] make the assumption that the
production function is Cobb-Douglas in capital, labor, and energy in a model,
which, like the model in this paper, has periods lasting for ten years.

To calibrate damages, I use the functional form in Nordhaus [2017] who assumes
that damages to output are given by

$\displaystyle D(T_{t})=1-\zeta T_{t}^{2}.$

There is an active debate on the specification of the damage function. Hänsel
et al. [2020] strongly criticize the function posited by Nordhaus in his DICE
model. They follow Howard and Sterner [2017] and use Nordhaus’ functional
form, simply changing parameters so that damages reach $\displaystyle 6.7$
percent of output for a 3-degree temperature increase, as opposed to only 2.1
percent in Nordhaus’ calibration333It is also not clear what a good functional
form for the damage function would look like, see Weitzman [2009]. For
simplicity, this paper focuses on quadratic damage functions. As Folini et al.
[2021] point out the functional form plays an important role in the
determination of optimal carbon policy..

As mentioned above, in this paper, climate damages are modeled as a reduction
in agents’ labor endowments. I calibrate losses to the labor efficiency units
to be quadratic (hence losses to output are not quadratic but instead
determined by the term $\displaystyle(1-\zeta T^{2})^{1-\alpha}$) and take the
coefficient to be $\displaystyle\zeta=0.007$. Together with the specification
of the labor share, one then obtains a 4.2 percent aggregate output loss for a
3-degree temperature increase. This value lies right in the middle between
Nordhaus’ and more pessimistic calibration from Howard and Sterner [2017].

Labor endowments and damages vary across agents. In the benchmark calibration,
it is assumed that labor endowments are given by
$\displaystyle\bar{l}_{1}=\frac{3}{4},\ \bar{l}_{2}=\bar{l}_{3}=\frac{1}{8}$.
Agents $\displaystyle h=1,2$ trade risky capital, while agent $\displaystyle
h=3$ is assumed to be a hand-to-mouth consumer who consumes his labor income
every period. Agent-specific damages are

$\displaystyle D_{1}(T)=1-\frac{1}{2}\zeta T^{2},\quad D_{2}(T)=1-\zeta
T^{2},\quad D_{3}(T)=1-4\zeta T^{2}.$

Given the individual labor endowments across agents, the resulting individual
damages always add up to aggregate damages, but, of course, damages can become
very large for the third agent. This benchmark calibration is referred to as
BM below.

The fact that the hand-to-mouth agents are hardest hit by climate change plays
a crucial role in the results. One could alternatively imagine a situation
where, as in the main calibration, some agents suffer much more from climate
change than others, but where these agents can save in risky capital. In this
alternative calibration, the roles of agents 2 and 3 are reversed, i.e., it is
assumed that

$\displaystyle D_{2}(T)=1-4\zeta T^{2},\quad D_{3}(T)=1-\zeta T^{2}.$

Agent 2 now suffers most from climate change, but he can save in risky capital
to insure against some of the damages. This benchmark calibration is referred
to as AC below.

I also consider a third calibration where the only friction consists of market
incompleteness. In that calibration, agents 1 and 2 are identical and
$\displaystyle\bar{1}_{1}=\bar{l}_{2}=\frac{2}{5}$ while
$\displaystyle\bar{l}_{3}=\frac{1}{5}$. Damages are

$\displaystyle D_{1}(T)=D_{2}(T)=1-\frac{1}{4}\zeta T^{2},\quad
D_{3}(T)=1-4\zeta T^{2}$

The calibration is referred to as IM below.

A more careful calibration of relative damages across different individuals
would require a larger number of individuals and several regions. This is
beyond the scope of this paper.

##  5 Results

As explained in the introduction, the primary focus is to explore constrained
efficient taxes and transfers. Welfare and optimal carbon taxes for an economy
where all three agents can trade in a complete set of Arrow securities are
compared to those in the economy with financial frictions. Without any climate
change, there would be no uncertainty, and welfare across agents would be
identical for the two cases (one agent is not allowed to invest in capital,
however, absent any uncertainty, the agent has no reason to do so). However, I
consider a situation where abatement is costly and where there will always be
climate change and the associated uncertainty about temperature and damages.
It is then a quantitative question of how the equilibria will differ and how
optimal taxes depend on financial markets.

All results are obtained numerically. A short description of the numerical
method and an error analysis can be found in the appendix.

###  5.1 Constrained efficient abatement

Table 1 reports optimal abatement levels for three different cases: A
frictionless economy (CM), optimal taxes (BM), and constrained efficient taxes
without transfers (NT) in the benchmark economy. Obviously, these abatement
levels vary across time and states, and they are reported for nine specific
date events. It is assumed that in the first two periods, the realized shock
is $\displaystyle z=3$, in the third period shock $\displaystyle z=6$
realizes, and in the fourth period abatement is reported for all six shocks.
For different histories of shocks the results are comparable, in particular in
the first two periods where shocks are i.i.d. differences between shocks are
somewhat small.

\ | $\displaystyle z_{0}=3$ | $\displaystyle z_{1}=3$ | $\displaystyle z_{2}=6$ | $\displaystyle z_{3}=1$ | $\displaystyle z_{3}=2$ | $\displaystyle z_{3}=3$ | $\displaystyle z_{3}=4$ | $\displaystyle z_{3}=5$ | $\displaystyle z_{3}=6$
---|---|---|---|---|---|---|---|---|---
CM | 4.49 | 5.17 | 6.69 | 4.04 | 4.43 | 4.63 | 4.88 | 5.18 | 6.96
OT | 11.12 | 12.97 | 8.70 | 14.81 | 12.71 | 11.81 | 10.84 | 9.92 | 8.40
NT | 11.53 | 14.33 | 9.44 | 17.71 | 14.48 | 13.21 | 11.91 | 10.68 | 8.85
Table 1: Constrained optimal abatement

Optimal abatement with financial frictions is almost always about 2.5 times
higher in an economy with financial frictions than in the Pareto-optimal
outcome in the frictionless economy (with the exception of $\displaystyle
t=2,\ z_{t}=6$ and $\displaystyle t=3,z_{t}=6$). This seems surprising: in an
optimal world, emissions, temperature increases, and the resulting damages are
significantly higher than in the sub-optimal equilibrium with financial
frictions. The economic reasoning is clear; in an economy with financial
friction, high carbon taxes substitute for precautionary savings, which are
impossible for the hand-to-mouth agents and only possible only in a risky
asset for other agents.

In the frictionless case, optimal abatement after the fourth period (i.e.
$\displaystyle t=3$) is higher if the economy is in a worse climate state.
This makes sense since the marginal benefits of abatement are high precisely
in those states. Financial frictions reverse this pattern. Optimal abatement
at $\displaystyle t=3$ is highest in the low ECS states and decreases as the
ECS increases. This is caused by the fact that in high ECS states at
$\displaystyle t=3$, there are already substantial damages, and while the
marginal benefits of abatement are large, most of the costs have to be borne
by the hand-to-mouth agents who are already hard hit. If these agents could
insure against this bad outcome they would be able to spend more on abatement.

A comparison between the constrained optimal case (BT) and the case without
transfers (NT) shows that this is not only caused by the fact that in the
optimal case, the hand-to-mouth agents have to pay a large fraction of the
abatement costs. Without transfers, all agents pay proportionally to their
capital holdings and their labor endowments; in this case, the hand-to-mouth
agents actually only bear a small fraction of total costs. Optimal abatement
is uniformly larger. In some instances, e.g. in $\displaystyle z_{1}=3$ or
$\displaystyle z_{2}=6$ the difference is quite substantial and larger than 10
percent.

The issue becomes clearer if one looks at transfers and total costs of
abatement as detailed in Table 2

\ | $\displaystyle z_{3}=1$ | $\displaystyle z_{3}=2$ | $\displaystyle z_{3}=3$ | $\displaystyle z_{3}=4$ | $\displaystyle z_{3}=5$ | $\displaystyle z_{3}=6$
---|---|---|---|---|---|---
costs | 1 | 0.73 | 0.63 | 0.52 | 0.43 | 0.30
share | (8, 4, 88) | (11, 5, 84) | (13,5,82) | (15,6,79) | (17, 7, 75) | (29, 12, 59)
Table 2: Optimal abatement cost and cost-sharing

The table shows relative costs (relative to costs at $\displaystyle z_{3}=1$
which are normalized to one) as well as the share of these costs each agent
bears in the transfer scheme. For good climate shocks with small damages,
agent 3 bears almost the entire cost, and the total expenditure for abatement
is high. For a bad climate shock, damages to agent 3’s labor endowments are
already very high (about 30 percent loss in shock six) and the agent simply
does not have the resources to finance a large abatement.

\ | $\displaystyle z_{0}=3$ | $\displaystyle z_{1}=3$ | $\displaystyle z_{2}=6$ | $\displaystyle z_{3}=1$ | $\displaystyle z_{3}=2$ | $\displaystyle z_{3}=3$ | $\displaystyle z_{3}=4$ | $\displaystyle z_{3}=5$ | $\displaystyle z_{3}=6$
---|---|---|---|---|---|---|---|---|---
BM | 11.12 | 12.97 | 8.70 | 14.81 | 12.71 | 11.81 | 10.84 | 9.92 | 8.40
IM | 6.14 | 7.62 | 9.53 | 7.62 | 7.59 | 7.58 | 7.59 | 7.64 | 9.94
AC | 5.01 | 6.01 | 7.70 | 5.88 | 8.13 | 8.44 | 8.36 | 8.31 | 8.27
Table 3: Optimal abatement, alternative calibration

Of course, the magnitude of the differences between the complete markets
abatement and the constrained optimal abatement with financial frictions
depends crucially on the presence of hand-to-mouth consumers that are hit
hardest by future damages from climate change. While this is arguably a
realistic assumption, it is useful to also examine other calibrations. Table 3
compares the optimal abatement in the main calibration (BM) to the optimal
abatement in the alternative calibration (AC) and to optimal abatement in the
calibration where the only financial friction consists of market
incompleteness (IM). In the (AC) calibration, participation in financial
markets, even if they are incomplete, leads to much lower optimal abatement
levels, certainly in cases where the damages of climate change are not yet too
high (in $\displaystyle z_{3}=6$ optimal abatement is very similar in the two
cases). Note that abatement levels are still significantly higher than in the
frictionless case.

Interestingly, in the calibration without any hand-to-mouth consumers, optimal
abatement levels are again significantly higher than in (AC). This is caused
by the fact that the agents that are hardest hit by climate change now
constitute a significant fraction of the total population (1/5 instead of only
1/8 in the BM calibration). The constrained optimal abatement is much larger
than in the CM markets case (for this case, it is the same as in Table 1,
since aggregate damages remain the same). The example illustrates that even
when the only friction is the incompleteness of financial markets, optimal
abatement levels can differ by a lot. It should be emphasized that in this
case of incomplete markets, like in the case of the complete market, abatement
in the 4th period is higher in worse climate states. This is in stark contrast
to the calibration with hand-to-mouth consumers, where the opposite is true.
In fact, at $\displaystyle z_{2}=6$ and at $\displaystyle z_{3}=6$,
constrained optimal abatement in the CM calibration is significantly higher
than in the BM calibration. Even with incomplete financial markets, agents
that suffer the most from climate change have enough liquidity in bad climate
shocks to pay for higher abatement.

Finally, I consider the effects of delayed implementation on optimal abatement
levels. Table 4 shows optimal abatement for the cases where abatement is
delayed for one period and or two periods (i.e., for 10 or 20 years). Delayed
implementation slightly increases optimal taxes (also, compared to the results
in Table 1). This is also true in the frictionless case (here only reported
for a delay by one period) but certainly much less so. In the low ECS states
at $\displaystyle t=3$ optimal taxes in the calibration with frictions are
more than three times higher than optimal taxes in the frictionless case.

\ | $\displaystyle z_{1}=3$ | $\displaystyle z_{2}=6$ | $\displaystyle z_{3}=1$ | $\displaystyle z_{3}=2$ | $\displaystyle z_{3}=3$ | $\displaystyle z_{3}=4$ | $\displaystyle z_{3}=5$ | $\displaystyle z_{3}=6$
---|---|---|---|---|---|---|---|---
1, OT | 13.08 | 8.24 | 15.51 | 13.28 | 12.33 | 11.29 | 10.22 | 8.48
2, OT | 0 | 8.99 | 16.71 | 14.19 | 13.13 | 11.97 | 10.76 | 8.60
1, CM | 5.20 | 6.72 | 4.06 | 4.46 | 4.65 | 4.90 | 5.20 | 6.98
Table 4: Optimal abatement, delayed implementation

To summarize the findings so far: Financial frictions have very large effects
on optimal carbon taxes and optimal abatement. The less sophisticated possible
transfer schemes, the higher the optimal taxes (and the lower the resulting
climate damages).

###  5.2 Welfare

A natural next step is to investigate the welfare consequences of these
findings. We report all welfare losses in lifetime consumption equivalent,
i.e., we ask by what percentage consumption at all times and date events has
to change to compensate for welfare changes caused by climate change or
financial frictions.

Since the model starts off with the RCP 4.5 emissions scenario, welfare losses
from climate change are generally small444In size somewhat comparable to
welfare losses in DICE2016.. Clearly, welfare losses from climate change are
much larger in a very pessimistic no-policy scenario such as RCP8.5 or the BAU
scenario from DICE2016. Note, however, that with discounting and moderate risk
aversion welfare losses from climate change and imperfect insurance against
climate change tend to be small. Potentially large welfare gains hinge
critically on Inada-conditions on the utility functions.

For the three agents, I find that the welfare gains, in percentage, of
constrained efficient carbon policy (relative to no further mitigation in RCP
4.5) are given by

$\displaystyle(0.066,0.486,3.68).$

These are obviously smaller than total welfare losses from climate change
since abatement is assumed to be costly.

Relative to these numbers, the potential welfare gains from complete markets
and optimal taxes, relative to the situation with financial frictions and
optimal taxes, seem relatively large. With appropriate transfers that leave
agent 1 at his status-quo welfare level, the welfare of agents 2 and 3 can be
increased by 2.82 percent by erasing all financial frictions. The constrained
optimal abatement in the economy with financial frictions cannot nearly
compensate for the climate risks that individuals face and cannot insure.
Despite the fact that in the frictionless outcome, aggregate climate damages
are significantly higher than in the benchmark, each agent’s welfare is
significantly higher because the individual risks can be diversified away.

If abatement is impossible, the welfare losses from financial frictions are
naturally significantly larger. Aggregate damages now are very similar, and
the larger individual risks in the economy with frictions have relatively
large effects on welfare. Welfare gains from completing the markets, together
with transfers that keep agent 1 at his status-quo welfare and ensure uniform
gains for agents 2 and 3, lead to welfare gains for these agents of 4.01
percent.

Lastly, welfare for the optimal abatement case is compared to that in the case
with efficient taxes but without any transfers. Moving to such a scheme
implies welfare changes (in percent) for the three agents of

$\displaystyle(-0.331,-0.290,2.149).$

Not surprisingly, Agent 3, who are most affected by climate damages and has no
access to financial markets, gains a lot if he no longer has to pay a large
fraction of the abatement. However, agent 2 and agent 1 incur significant
losses. As before, since abatement is higher in the scenario than in the
constrained optimal policy (and much higher than in the optimal policy with
complete markets), climate change damages are lowest in this case.
Nevertheless, welfare losses with respect to the complete markets optimum are
substantial, in fact, slightly (about 0.01 percent) higher than in the
constrained optimal case.

##  6 Conclusion

Climate change poses large aggregate risks that will likely have very
different effects on different households. Unfortunately, these risks cannot
be shared in existing financial markets. I give some quantitative examples to
show that the inability to share these risks leads to much higher abatement
and lower emissions at a constrained optimum.

In the calibrated examples, I focus on uncertainty about future temperature
and aggregate damages. Of course, there might be many other risks caused by
anthropogenic climate change which cannot be shared in markets and have
significant effects on agents’ well beings. Risks associated with tipping
points have played a prominent role in the literature (e.g., Lemoine and
Traeger [2014], Cai et al. [2013]), but it might be as important to model the
effects of climate change on idiosyncratic income risk. It is subject to
further research to understand how these uncertainties interact and how large
the quantitative impact on welfare and optimal taxes are.

This paper focuses on heterogeneity across consumers. Another important factor
in optimal carbon taxation lies in heterogeneity across producers,
particularly across regions.

## Bibliographies

  * Barnett et al. [2020] Michael Barnett, William Brock, and Lars Peter Hansen.  Pricing uncertainty induced by climate change.  _The Review of Financial Studies_ , 33(3):1024–1066, 2020.
  * Barnett et al. [2022a] Michael Barnett, William Brock, and Lars Peter Hansen.  Confronting uncertainty in the climate change dynamics.  _working paper_ , 2022a.
  * Barnett et al. [2022b] Michael Barnett, William Brock, and Lars Peter Hansen.  Climate change uncertainty spillover in the macroeconomy.  _NBER Macroeconomics Annual_ , 36(1):253–320, 2022b.
  * Barrage [2020] Lint Barrage.  Optimal dynamic carbon taxes in a climate–economy model with distortionary fiscal policy.  _The Review of Economic Studies_ , 87(1):1–39, 2020.
  * Brumm et al. [2017] Johannes Brumm, Dominika Kryczka, and Felix Kubler.  Recursive equilibria in dynamic economies with stochastic production.  _Econometrica_ , 85(5):1467–1499, 2017.
  * Cai [2020] Yongyang Cai.  The role of uncertainty in controlling climate change.  _arXiv preprint arXiv:2003.01615_ , 2020.
  * Cai et al. [2013] Yongyang Cai, Kenneth L Judd, and Thomas S Lontzek.  The social cost of stochastic and irreversible climate change.  Technical report, National Bureau of Economic Research, 2013.
  * Carattini et al. [2021] Stefano Carattini, Garth Heutel, and Givi Melkadze.  Climate policy, financial frictions, and transition risk.  Technical report, National Bureau of Economic Research, 2021.
  * Chichilnisky and Heal [1994] Graciela Chichilnisky and Geoffrey Heal.  Who should abate carbon emissions?: An international viewpoint.  _Economics Letters_ , 44(4):443–449, 1994.
  * Cruz and Rossi-Hansberg [2021] Jose-Luis Cruz and Esteban Rossi-Hansberg.  The Economic Geography of Global Warming.  2021\.
  * Davila et al. [2012] Julio Davila, Jay H Hong, Per Krusell, and José-Víctor Ríos-Rull.  Constrained efficiency in the neoclassical growth model with uninsurable idiosyncratic shocks.  _Econometrica_ , 80(6):2431–2467, 2012.
  * Dennig et al. [2015] Francis Dennig, Mark B Budolfson, Marc Fleurbaey, Asher Siebert, and Robert H Socolow.  Inequality, climate impacts on the future poor, and carbon prices.  _Proceedings of the National Academy of Sciences_ , 112(52):15827–15832, 2015.
  * Dierker et al. [2002] Egbert Dierker, Hildegard Dierker, and Birgit Grodal.  Nonexistence of constrained efficient equilibria when markets are incomplete.  _Econometrica_ , 70(3):1245–1251, 2002.
  * Douenne et al. [2022] Thomas Douenne, Albert Jan Hummel, and Marcelo Pedroni.  Optimal fiscal policy in a climate-economy model with heterogeneous households.  2022\.
  * Foley [1970] Duncan K Foley.  Lindahl’s solution and the core of an economy with public goods.  _Econometrica: Journal of the Econometric Society_ , pages 66–72, 1970.
  * Folini et al. [2021] Doris Folini, Felix Kubler, Aleksandra Malova, and Simon Scheidegger.  The climate in climate economics.  _Available at SSRN 3885021_ , 2021.
  * Gillingham et al. [2015] Kenneth Gillingham, William D Nordhaus, David Anthoff, Geoffrey Blanford, Valentina Bosetti, Peter Christensen, Haewon McJeon, John Reilly, and Paul Sztorc.  Modeling uncertainty in climate change: A multi-model comparison.  Technical report, National Bureau of Economic Research, 2015.
  * Gollier [2001] Christian Gollier.  Wealth inequality and asset pricing.  _The Review of Economic Studies_ , 68(1):181–203, 2001.
  * Golosov et al. [2014] Mikhail Golosov, John Hassler, Per Krusell, and Aleh Tsyvinski.  Optimal taxes on fossil fuel in general equilibrium.  _Econometrica_ , 82(1):41–88, 2014.
  * Hänsel et al. [2020] Martin C Hänsel, Moritz A Drupp, Daniel JA Johansson, Frikk Nesje, Christian Azar, Mark C Freeman, Ben Groom, and Thomas Sterner.  Climate economics support for the un climate targets.  _Nature Climate Change_ , pages 1–9, 2020.
  * Hassler et al. [2012] John Hassler, Per Krusell, and Conny Olovsson.  Energy-saving technical change.  Technical report, National Bureau of Economic Research, 2012.
  * Hassler et al. [2018] John Hassler, Per Krusell, and Conny Olovsson.  The consequences of uncertainty: climate sensitivity and economic sensitivity to the climate.  _Annual Review of Economics_ , 10:189–205, 2018.
  * Hausfather and Peters [2020] Zeke Hausfather and Glen P Peters.  Emissions–the ‘business as usual’story is misleading, 2020.
  * Howard and Sterner [2017] Peter H Howard and Thomas Sterner.  Few and not so far between: a meta-analysis of climate damage estimates.  _Environmental and Resource Economics_ , 68(1):197–225, 2017.
  * Jensen and Traeger [2014] Svenn Jensen and Christian P Traeger.  Optimal climate change mitigation under long-term growth uncertainty: Stochastic integrated assessment and analytic findings.  _European Economic Review_ , 69:104–125, 2014.
  * Kaneko [1977] Mamoru Kaneko.  The ratio equilibrium and a voting game in a public goods economy.  _Journal of Economic Theory_ , 16(2):123–136, 1977.
  * Klein et al. [2008] Paul Klein, Per Krusell, and Jose-Victor Rios-Rull.  Time-consistent public policy.  _The Review of Economic Studies_ , 75(3):789–808, 2008.
  * Knutti et al. [2017] Reto Knutti, Maria AA Rugenstein, and Gabriele C Hegerl.  Beyond equilibrium climate sensitivity.  _Nature Geoscience_ , 10(10):727–736, 2017.
  * Kotlikoff et al. [2021a] Laurence Kotlikoff, Felix Kubler, Andrey Polbin, and Simon Scheidegger.  Pareto-Improving Carbon-Risk Taxation.  _Economic Policy_ , 02 2021a.  ISSN 0266-4658.  doi: 10.1093/epolic/eiab008.  URL https://doi.org/10.1093/epolic/eiab008.  eiab008.
  * Kotlikoff et al. [2021b] Laurence J Kotlikoff, Felix Kubler, Andrey Polbin, and Simon Scheidegger.  Can today’s and tomorrow’s world uniformly gain from carbon taxation?  Working Paper 29224, National Bureau of Economic Research, September 2021b.  URL http://www.nber.org/papers/w29224.
  * Krusell and Smith Jr [2022] Per Krusell and Anthony A Smith Jr.  Climate change around the world.  Technical report, National Bureau of Economic Research, 2022.
  * Lemoine and Traeger [2014] Derek Lemoine and Christian Traeger.  Watch your step: optimal policy in a tipping climate.  _American Economic Journal: Economic Policy_ , 6(1):137–66, 2014.
  * Malin et al. [2011] Benjamin A Malin, Dirk Krueger, and Felix Kubler.  Solving the multi-country real business cycle model using a smolyak-collocation method.  _Journal of Economic Dynamics and Control_ , 35(2):229–239, 2011.
  * Mas-Colell and Silvestre [1989] Andreu Mas-Colell and Joaquim Silvestre.  Cost share equilibria: A lindahlian approach.  _Journal of Economic Theory_ , 47(2):239–256, 1989.
  * Meinshausen et al. [2011] M. Meinshausen, S. C. B. Raper, and T. M. L. Wigley.  Emulating coupled atmosphere-ocean and carbon cycle models with a simpler model, MAGICC6 - Part 1: Model description and calibration.  _Atmospheric Chemistry & Physics_, 11:1417–1456, February 2011.  doi: 10.5194/acp-11-1417-2011.
  * Moré and Wild [2014] Jorge J Moré and Stefan M Wild.  Do you trust derivatives or differences?  _Journal of Computational Physics_ , 273:268–277, 2014\.
  * Moulin [1987] Herve Moulin.  Egalitarian-equivalent cost sharing of a public good.  _Econometrica: Journal of the Econometric Society_ , pages 963–976, 1987.
  * Nordhaus [2017] William D Nordhaus.  Revisiting the social cost of carbon.  _Proceedings of the National Academy of Sciences_ , page 201609244, 2017.
  * Nordhaus and Yang [1996] William D Nordhaus and Zili Yang.  A regional dynamic general-equilibrium model of alternative climate-change strategies.  _The American Economic Review_ , pages 741–765, 1996.
  * Pigou [1920] Arthur C Pigou.  Wealth and welfare, london, 1912.  _Later editions, The Eco $\displaystyle\neg$ nomics of Welfare_, 1920.
  * Roe and Baker [2007] Gerard H Roe and Marcia B Baker.  Why is climate sensitivity so unpredictable?  _Science_ , 318(5850):629–632, 2007.
  * Sherwood et al. [2020] SC Sherwood, Mark J Webb, James D Annan, Kyle C Armour, Piers M Forster, Julia C Hargreaves, Gabriele Hegerl, Stephen A Klein, Kate D Marvel, Eelco J Rohling, et al.  An assessment of earth’s climate sensitivity using multiple lines of evidence.  _Reviews of Geophysics_ , 58(4):e2019RG000678, 2020.
  * Starr [1973] Ross M Starr.  Optimal production and allocation under uncertainty.  _The Quarterly Journal of Economics_ , 87(1):81–95, 1973.
  * Traeger [2019] Christian P Traeger.  Ace–analytic climate economy (with temperature and uncertainty).  _Working paper_ , 2019.
  * Van den Bremer and Van der Ploeg [2021] Ton S Van den Bremer and Frederick Van der Ploeg.  The risk-adjusted carbon price.  _American Economic Review_ , 111(9):2782–2810, 2021.
  * Weitzman [2009] Martin L Weitzman.  On modeling and interpreting the economics of catastrophic climate change.  _The Review of Economics and Statistics_ , 91(1):1–19, 2009.
  * Zaliapin and Ghil [2011] Ilya Zaliapin and Michael Ghil.  Reply to g.h. roe’s and m.b. baker’s comment on ”another look at climate sensitivity”.  _Nonlinear Processes in Geophysics_ , 18(1):129–131, feb 2011.  doi: 10.5194/npg-18-129-2011.

##  Appendix A Computational details

It is assumed that recursive equilibrium with constrained optimal carbon
policy can be characterized by a system of functional equations involving
unknown policy functions but also value functions. To solve this system
numerically, I use standard projection methods (see, e.g., Malin et al.
[2011]). The endogenous state space consists of the capital holdings of agents
$\displaystyle h=1,2$ and of the two CO2 reservoirs $\displaystyle
S_{1},S_{2}$ and is, therefore, four-dimensional. Since there are no
occasionally binding constraints, Smolyak-polynomials, as in Malin et al.
[2011], are ideally suited for approximating the unknown functions. The
maximal errors in agents’ Euler equations are below $\displaystyle 10^{-3}$ in
all computations. Unfortunately, another potential source of numerical error
is the numerical computation of the optimal tax. For this, one needs to
estimate the derivative of the agents’ value function with respect to CO2
concentrations in the atmosphere. The error in the computation of the level of
agents’ value functions can be controlled by verifying the quality of the
numerical approximation in each iteration. The derivative needs to be computed
using finite differences. The method from Moré and Wild [2014] is used to
estimate the noise in the value function and to determine the optimal step
length for the finite-difference approximation. Obtaining an estimate for the
error in the level of optimal abatement appears difficult.