#  Impact of COVID-19 type events on the economy and climate under the
stochastic DICE model

Pavel V. Shevchenko1,∗, Daisuke Murakami2, Tomoko Matsui3, Tor A. Myrvoll4

(draft, 1st version, 22 December 2020, revised 9 October 2021)

###### Abstract

The classical DICE model is a widely accepted integrated assessment model for
the joint modeling of economic and climate systems, where all model state
variables evolve over time deterministically. We reformulate and solve the
DICE model as an optimal control dynamic programming problem with six state
variables (related to the carbon concentration, temperature, and economic
capital) evolving over time deterministically and affected by two controls
(carbon emission mitigation rate and consumption). We then extend the model by
adding a discrete stochastic shock variable to model the economy in the
stressed and normal regimes as a jump process caused by events such as the
COVID-19 pandemic. These shocks reduce the world gross output leading to a
reduction in both the world net output and carbon emission. The extended model
is solved under several scenarios as an optimal stochastic control problem,
assuming that the shock events occur randomly on average once every 100 years
and last for 5 years. The results show that, if the world gross output
recovers in full after each event, the impact of the COVID-19 events on the
temperature and carbon concentration will be immaterial even in the case of a
conservative 10% drop in the annual gross output over a 5-year period. The
impact becomes noticeable, although still extremely small (long-term
temperature drops by $0.1^{\circ}\mathrm{C}$), in a presence of persistent
shocks of a 5% output drop propagating to the subsequent time periods through
the recursively reduced productivity. If the deterministic DICE model policy
is applied in a presence of stochastic shocks (i.e. when this policy is
suboptimal), then the drop in temperature is larger (approximately
$0.25^{\circ}\mathrm{C}$), that is, the lower economic activities owing to
shocks imply that more ambitious mitigation targets are now feasible at lower
costs.

_Keywords: Dynamic Integrated Climate-Economy model, climate change, optimal
control, carbon emission, COVID-19, stochastic DICE model_

1 Department of Actuarial Studies and Business Analytics, Macquarie
University, Australia; e-mail<EMAIL_ADDRESS>
2 Institute of Statistical Mathematics, Japan; e-mail<EMAIL_ADDRESS>
3 Institute of Statistical Mathematics, Japan; e-mail<EMAIL_ADDRESS>
4 Norwegian University of Science and Technology, Norway; e-mail:
<EMAIL_ADDRESS>
$\ast$ Corresponding author

##  1 Introduction

The impact of the COVID-19 pandemic on the global economy is more severe than
the impact from the 2008 global financial crisis (see e.g., International
Monetary Fund (2020)), and the projection of the COVID-19 impact on the
economy and climate is a major concern. In this paper, we study the impact of
COVID-19 type events on the economy and climate using the dynamic integrated
climate-economy (DICE) model, extended to include stochastic shocks to the
economy. The DICE model introduced by Nordhaus111Nordhaus was the winner of
the Nobel Prize in Economic Sciences in 2018 “for integrating climate change
into a long-run macroeconomic analysis”. is an extremely popular integrated
assessment model (IAM) for the joint modeling of economic and climate systems.
It has been regularly revised over the last three decades with the first
version dating back to Nordhaus et al. (1992), and the most recent revision
being DICE-2016 Nordhaus (2017)222The most recent version of the DICE model is
available at https://sites.google.com/site/williamdnordhaus/dice-rice.. The
DICE model is one of the three main IAMs (the other two are FUND and PAGE)
used by the United States government to determine the social cost of carbon;
see Interagency Working Group on Social Cost of Greenhouse Gases (2016). It
balances parsimony with realism and is well documented with all published
model equations; in addition, its code is publicly available, which is an
exception rather than the rule for IAMs. At the same time, it is important to
note that IAMs and the DICE model in particular have significant limitations
(in the model structure and model parameters), which have been criticized and
debated in the literature (see the discussions in Ackerman et al. (2009);
Pindyck (2017); Grubb et al. (2021); Weitzman (2011)). Despite the criticism,
the DICE model has become the iconic typical reference point for climate-
economy modeling, and is therefore used in our study.

The DICE model is a deterministic approach that combines a Ramsey-Cass-
Koopmans neoclassical model of economic growth (also known as the Ramsey
growth model) with a simple climate model. It involves six state variables
(atmospheric and upper and lower ocean carbon concentrations; atmospheric and
lower ocean temperatures; and economic capital) evolving in time
deterministically, two control variables (savings and carbon emission
reduction rates) to be determined for each time period of the model, and
several exogenous processes (e.g., population size and technology level).
Uncertainty about the future of the climate and economy is then typically
assessed by treating some model parameters as random variables (because we do
not know the exact true value of the key parameters) using a Monte Carlo
analysis (see Nordhaus (2018); Ackerman et al. (2010)).

Modeling uncertainty owing to the stochastic nature of the state variables
(i.e., owing to the process uncertainty that is present even if we know the
model parameters exactly) requires the development and solution of the DICE
model as a dynamic model of decision-making under uncertainty, where we
calculate the optimal policy response, under the assumption of continuing
uncertainty throughout the time frame of the model. This is a much more
difficult problem that requires more computational and mathematical
sophistication, whereas the deterministic DICE model can be solved using an
Excel spreadsheet or GAMS (a high-level programming language for mathematical
modeling https://www.gams.com/).

Few attempts have been made to extend the DICE model to incorporate
stochasticity in the underlying state variables and solve it as a recursive
dynamic programming problem. For example, Kelly and Kolstad (1999) and Leach
(2007) formulated the DICE model with stochasticity in the temperature-time
evolution and solved this as a recursive dynamic programming problem. These
studies are seminal contributions to the incorporation of uncertainty in the
DICE model (although their numerical solution approach is difficult to extend
to higher dimensional space and time-frequency). Cai et al. (2012) formulates
DICE as a dynamic programming problem with a stochastic shock on the economy
and climate. In addition, Traeger (2014) developed a reduced DICE model with a
smaller number of state variables, whereas Lontzek et al. (2015) studied the
impact of climate tipping points. There are other studies that approached
optimal strategies addressing climate change through a simple minimization of
the damage function to find the optimal timing for an investment, such as in
Conrad (1997); Luo and Shevchenko (2013), which is extremely different from
the DICE modeling approach and is not pursued in our paper.

In our study, we extend the DICE model by adding a discrete stochastic shock
variable, shifting the economy into a stressed regime owing to events such as
COVID-19. This is similar to the model formulation in Cai et al. (2012) but
with different types of jump processes for the shocks. The economy after our
shocks is allowed to recover, whereas the jump shock considered in Lontzek et
al. (2015); Cai et al. (2012) is an irreversible climate tipping point event.
One of the scenarios we consider allows for stochastic shocks affecting
productivity, which leads to a persistent impact on the economy. This is
somewhat similar to the tipping point modeling in Lontzek et al. (2015); Cai
et al. (2012). However, it is important to note that shocks considered in our
paper reduce both the world net output and emission through the shock
reduction of the gross output while tipping point models assume shock on the
net output and no shock on the emission. Thus our shocks lead to a reduction
in a policy stringency while tipping point shocks lead to the opposite effect.
In addition, our base model is the more recent version of DICE-2016, whereas
Lontzek et al. (2015); Cai et al. (2012) use older DICE versions.

COVID-19 has spread across the globe, with over 75 million confirmed cases and
1.6 million deaths from December 30, 2019, to December 20, 2020, according to
the _Weekly Epidemiological Update_ from the World Health Organization on
December 22, 2020333 www.who.int/emergencies/diseases/novel-
coronavirus-2019/situation-reports (with over 4.6 million new cases and 79,000
deaths since the previous weekly updates). Large amounts of emergency loans
are needed around the world to develop therapeutic agents and vaccines, as
well as to implement various interventions to prevent the spread of
infections, such as “stay-at-home” policies, and provide financial support for
them. However, in recent years, the effects of climate change on global
warming have become more serious on a global scale and the “Paris Agreement”
to limit the global temperature increase444The Paris Agreement is an agreement
of over 180 countries to reduce greenhouse gas emissions and limit the global
temperature increase by $2^{\circ}\mathrm{C}$ (above the pre-industrial
levels) by 2100, United Nations Treaty Collection (2015). was adopted in 2015.
This has resulted in more public and private funds being provided for green
projects involving renewable energy and energy conservation, as the world
works to prevent the effects of global climate change. In a pandemic of
infectious diseases such as COVID-19, it is important to consider the economic
impacts of both such pandemics and global warming at the same time.

One of the unique features of the COVID-19 pandemic is extreme and widespread
disruption to the global economy when compared to other global pandemics, such
as the 1918 Influenza Pandemic (“Spanish Flu”) or the Hong Kong Flu of
1968555The real GDP was not significantly affected during previous global
pandemics; see for example GDP data for the United States available from
https://www.measuringworth.com. On the official government website of the
Bureau of Economic Analysis of the United States (see BEA (2020)), it was
reported on September 30, 2020 that the real gross domestic product (GDP)
decreased at an annual rate of 31.4% in the second quarter of 2020. According
to a news report on September 2, 2020 in The Japan Times (2020), Japan’s
April-June GDP is expected to be revised after making an annualized 27.8% drop
on a preliminary basis, which is the largest contraction in the post-World War
II period. Though, at the time of revision of this paper, the reported drop of
the real GDP in the United States in 2020 (compared to 2019) is only 3.4%
(https://www.measuringworth.com accessed on 9 October 2021). The amounts of
the stimulus packages released by governments in many countries to limit the
human and economic impacts of the COVID-19 pandemic have been unprecedented.
The International Monetary Fund policy tracker666www.imf.org/en/Topics/imf-
and-covid19/Policy-Responses-to-COVID-19 presented a summary of the key
economic responses around the world (e.g., the Coronavirus Aid, Relief and
Economy Security Act introduced in United States in March 2020 has been
estimated at 2.3 trillion USD (around 11% of the nation’s GDP)).

Although the impact of COVID-19 on the economy, human capital, and well-being
in the long run is unknown, the historical experience of global pandemics and
global recessions can provide valuable insight. Arthi and Parman (2021)
provides an excellent review of the long-run effects on health, labor, and
human capital from both historical pandemics and historical recessions. It has
been argued that, from a historical perspective, the impact of COVID-19 has
been similar to that of the “Spanish Flu” (1918) in terms of direct effects on
the health and well-being of individuals, and similar to the Great Depression
(1929-1939) in terms of economic disruption.

In this study, we consider the impact of COVID-19 type events on the DICE
model outputs. We reformulate the DICE model as an optimal control problem and
solve it using dynamic programming involving six state variables evolving over
time deterministically and affected by two controls (emission control and
savings rates). We then extend the model by adding a discrete stochastic shock
variable to the gross world output to shift the economy into a stressed regime
owing to events such as COVID-19, assuming that the economy recovers in full
after the stressed period. The extended model is solved as an optimal
stochastic control problem under different scenarios for the world gross
output drop owing to these shock events. With the reference to the Great
Depression (1929-1939) and “Spanish Flu” (1918), for our scenarios, we assume
that shocks occur on average once during a 100-year period and last for 5
years. In addition, the world economic output during a stressed regime
decreases by 5%-10%. We note that during the Great Depression, the real GDP in
the United States dropped for 6 years (1930-1935) comparing to the pre-
depression level in 1929 (averaging to 17% drop per annum over that
period)777See, for example, real GDP data for the United States available from
www.measuringworth.com, thus our assumption for the shock magnitude is a bit
less conservative. Under all considered conservative scenarios, the impact of
COVID-19 type events on the long-term temperature and carbon concentration
appear to be quite small. The results show that if the world’s gross output
recovers in full after each event, the impact of COVID-19 on the temperature
and carbon concentration will be immaterial even in the case of a conservative
10% drop in the annual gross output over a 5-year period. The impact becomes
noticeable, although remaining extremely small (i.e., a long-term temperature
drop by $0.1^{\circ}\mathrm{C}$), if the shocks are persistent 5% drops in
productivity, leading to a 5% drop in output propagating to the subsequent
time periods. If the deterministic DICE model policy is applied to the
stochastic model (i.e., a suboptimal policy is applied in the case of
stochastic shocks), then the drop in temperature will be larger (approximately
$0.25^{\circ}\mathrm{C}$), that is, the lower economic activities owing to the
occurrence of a shock imply that more ambitious mitigation targets are now
feasible at lower costs, which is qualitatively consistent with the results
presented in Meles et al. (2020).

The remainder of this paper proceeds as follows. The model is defined in
Section 2. Section 3 describes the numerical method used to solve the model.
The results are presented in Section 4, and some concluding remarks are given
in Section 5.

##  2 DICE model

The DICE model maximizes the utility of consumption (social welfare) over an
infinite time horizon with a tradeoff between consumption, investment, and
$\mathrm{CO}_{2}$ abatement. Let $t=0,1,\ldots$ be a discrete time measured in
steps of $\Delta$ years (e.g., $t=2$ corresponds to $2\Delta$ years). Using
the DICE-2016 model as the foundation888
www.econ.yale.edu/~nordhaus/homepage/homepage/DICE2016R-091916ap.gms, the
stochastic DICE model can be formulated as

$V_{0}(\bm{X}_{0})=\sup_{\bm{\mu},\bm{c}}{\mathbb{E}}\left[\sum_{t=0}^{\infty}e^{-\widetilde{\rho}\Delta
t}U(c_{t},L_{t})\right],$ (1)

subject to the state vector $\bm{X}_{t}=(K_{t},\bm{M}_{t},\bm{T}_{t},I_{t})$
evolving over time999State variables are affected by controls
$\bm{\mu},\bm{c}$. It is standard in the mathematical literature to indicate
this by corresponding upperscript of the state vector, however, for simplicity
of notation we omit this upperscript. as

$\displaystyle
K_{t+1}=K_{t}(1-\delta_{K})^{\Delta}+\left[\Delta\times(Q_{t}(K_{t},T_{t},\mu_{t})-c_{t})\right]e^{\epsilon^{K}_{t+1}},$
(2) $\displaystyle\bm{M}_{t+1}=\bm{\Phi}^{M}\bm{M}_{t}+\Delta\times(\beta
E_{t}(K_{t},\mu_{t}),0,0)^{\prime}e^{\epsilon^{M}_{t+1}},$ (3)
$\displaystyle\bm{T}_{t+1}=\bm{\Phi}^{T}\bm{T}_{t}+\Delta\times\left(\begin{array}[]{c}\xi_{1}F_{t}(M^{AT}_{t})\\\
0\\\ \end{array}\right)+\bm{\epsilon}^{T}_{t+1},$ (6) $\displaystyle
I_{t+1}=\mathcal{T}^{D}(I_{t},\epsilon_{t+1}^{I}).$ (7)

Here, superscript $\prime$ denotes a transposition, $\widetilde{\rho}$ is the
utility discount rate101010This is the rate at which the social planner
discounts the future utility. In the literature, this is also called the “rate
of pure time preference,” “subjective discount rate,” “pure rate of time
preference,” or “welfare discount rate.” , $\delta_{K}$ is the annual rate of
depreciation of capital, and
($\bm{\Phi}^{M}$,$\bm{\Phi}^{T}$,$\beta$,$\xi_{1}$) are parameters for the
carbon and temperature transition from $t$ to $t+1$. Other variables and
functions are as follows.

  * •

$U(c_{t},L_{t})$ is the utility function defined as

$U(c_{t},L_{t})=\frac{\Delta
L_{t}}{1-\alpha}\left(\left(\frac{c_{t}}{L_{t}}\right)^{1-\alpha}-1\right),$

where $\alpha\geq 0$ is the risk aversion parameter ($\alpha=1$ corresponds to
a logarithmic utility), and $L_{t}$ is the world population in billions at
time $t$.

  * •

$\bm{c}=(c_{0},c_{1},\ldots)$ is the consumption $c_{t}>0$ and
$\bm{\mu}=(\mu_{0},\mu_{1},\ldots)$ is the carbon emission mitigation rate
$\mu_{t}\geq 0$. Their optimal values were determined by solving the
optimization problem (1).

  * •

$K_{t}$ is the world economic capital (in trillions of USD as of 2010).

  * •

$\bm{M}_{t}=(M_{t}^{AT},M_{t}^{UP},M_{t}^{LO})^{\prime}$ is the carbon
concentration (in billions of metric tons) in the atmosphere ($M_{t}^{AT}$),
the upper oceans ($M_{t}^{UP}$), and the lower oceans ($M_{t}^{LO}$).

  * •

$\bm{T}_{t}=(T_{t}^{AT},T_{t}^{LO})^{\prime}$ is the temperature in the
atmosphere ($T_{t}^{AT}$) and the lower oceans ($T_{t}^{LO}$) measured in
degrees Celsius (${}^{\circ}\mathrm{C}$) above the temperature during the year
1900.

  * •

$(I_{t})_{t\geq 0}$ is the random shock process used to model stressed regime
in the economy during events such as COVID-19. Depending on the model setup,
it may have two or more states. For example, one can set $I_{t}=0$
corresponding to the normal regime and $I_{t}=1$ corresponding to the stressed
regime. Additional states can be introduced to model situations where the
stressed regime has to continue for two or more time periods.

  * •

($\epsilon^{K}_{t},\epsilon_{t}^{M},\bm{\epsilon}_{t}^{T},\epsilon_{t}^{I}$)
are independent and identically distributed random disturbances for
$t=1,2,\ldots$. The random disturbance $\epsilon_{t}^{I}$ corresponds to the
transition probability of the shock process $\Pr[I_{t+1}|I_{t}]$. Other random
disturbances correspond to the uncertainties of the world’s net output, carbon
concentration, and temperature, and can be modeled using e.g. a Gaussian
distribution. In this study, all numerical results are presented for the case
in which all random disturbances are set to zero, except for
$\epsilon_{t}^{I}$.

  * •

$Q_{t}(K_{t},T_{t},\mu_{t})$ is the world’s net output (output net of the
damages and abatement) divided between the consumption and investment,
$E_{t}(K_{t},\mu_{t})$ is the carbon emission (in billions of tons per year),
and $F_{t}(M^{AT}_{t})$ is the radiative forcing, which are modelled as

$\displaystyle
Q_{t}(K_{t},T_{t},\mu_{t})=\Omega_{t}(\mu_{t},\sigma_{t},T_{t}^{AT})Y(A_{t},K_{t},L_{t}),$
(8) $\displaystyle
E_{t}(K_{t},\mu_{t})=(1-\mu_{t})\sigma_{t}Y(A_{t},K_{t},L_{t})+E_{t}^{Land},$
(9) $\displaystyle
F_{t}(M^{AT}_{t})=\eta\log_{2}(M_{t}^{AT}/\widetilde{M}^{AT})+F_{t}^{EX},$
(10)

where

$\displaystyle
Y(A_{t},K_{t},L_{t})=(1-\chi(I_{t}))A_{t}K_{t}^{\gamma}L_{t}^{1-\gamma},$ (11)
$\displaystyle\Omega_{t}(\mu_{t},\sigma_{t},T_{t}^{AT})=1-\frac{\sigma_{t}550(1-0.025)^{t}\mu_{t}^{\theta_{2}}}{1000\theta_{2}}-\pi_{2}[T^{AT}_{t}]^{2}.$
(12)

Here, $\chi(I_{t})$ is the impact of COVID-19 type shocks on the Cobb–Douglas
production function
$\widetilde{Y}(A_{t},K_{t},L_{t})=A_{t}K_{t}^{\gamma}L_{t}^{1-\gamma}$,
$A_{t}$ is the total productivity factor, and
$\Omega_{t}(\mu_{t},\sigma_{t},T_{t}^{AT})$ is the damage abatement cost
factor. The damage function as a fraction of the gross output is
$\pi_{2}[T^{AT}_{t}]^{2}$; see (12). The model parameter values and
deterministic functions $F_{t}^{EX}$, $E_{t}^{Land}$, $A_{t}$, and
$\sigma_{t}$ are specified in Table 1.

_Note that, $Y(A_{t},K_{t},L_{t})$ is the annual gross world output (output
before damage and abatement costs) affected by the shocks $\chi(I_{t})$; thus,
the shocks affect both the net output $Q_{t}$ in (8) and the carbon emissions
$E_{t}$ in (9)._

  * •

The carbon price (USD per ton) is calculated as

$P_{t}=\frac{\partial Q_{t}/\partial\mu_{t}}{\partial
E_{t}/\partial\mu_{t}}\times 1000={550(1-0.025)^{t}\mu_{t}^{\theta_{2}-1}}.$

  * •

The typically quoted _savings rate_ output from the DICE model is defined as
$(1-c_{t}/Q_{t})$.

Given that $(\bm{X}_{t})_{t\geq 0}$ is a Markov process, the solution to the
stochastic DICE model (1) is a standard optimal stochastic control problem for
a controlled Markov process (the transition of $\bm{X}_{t}$ to $\bm{X}_{t+1}$
is affected by $\mu_{t},c_{t}$). For a good textbook treatment of such
problems in finance, see Bäuerle and Rieder (2011). This type of problem can
be solved using the dynamic programming performed recursively backward in time
for $t=N-1,\ldots,0$ through the backward induction Bellman equation:

$V_{t}(\bm{X}_{t})=\sup_{\mu_{t},c_{t}}\left(U(c_{t},L_{t})+e^{-\widetilde{\rho}\Delta}\mathbb{E}[V_{t+1}(\bm{X}_{t+1})|\bm{X}_{t}]\right),\;\mbox{s.t.}\;V_{N}(\bm{X}_{N})=0,$
(13)

and the optimal strategy can be found as

$(\mu_{t}^{\ast}(\bm{X}_{t}),c_{t}^{\ast}(\bm{X}_{t}))=\arg\sup_{\mu_{t},c_{t}}\left(U(c_{t},L_{t})+e^{-\widetilde{\rho}\Delta}\mathbb{E}[V_{t+1}(\bm{X}_{t+1})|\bm{X}_{t}]\right).$
(14)

Note that, the optimal strategy (optimal decision for the values to be set for
carbon emission reduction $\mu_{t}$ and consumption $c_{t}$) depends on the
information available at time $t$, that is, depends on the state variable
$\bm{X}_{t}$. In addition, note that to solve the DICE model under the
infinite time horizon numerically, one should use a large enough number of
time steps $N$ (it should be confirmed by the sensitivity of the numerical
solution that $N$ is sufficiently large and thus its impact on the solution
for the period of interest is immaterial).

If the random disturbances and impact from the random shock $\chi(I_{t})$ are
all set to zero, then the above model is reduced to the standard deterministic
DICE-2016. The dynamic programming solution (13) is still valid in this case
and can be used to solve the model. Note that the standard DICE-2016 solution
is a brute force maximization in (1) with respect to
($c_{0},\ldots,c_{N-1},\mu_{0},\ldots,\mu_{N-1}$) and their constraints
simultaneously (a total of 200 parameters plus their constraints when
$N=100$).

In one of the scenarios presented in this paper, to introduce a persistent
shock to the gross output $Y_{t}(A_{t},K_{t},L_{t})$, we consider the total
productivity $A_{t}$ affected by the shock variable $I_{t}$. Then, $A_{t}$
becomes an additional state variable, and the new state vector is
$\bm{X}_{t}=(A_{t},K_{t},\bm{M}_{t},\bm{T}_{t},I_{t})$ with the additional
state transition equation,

$A_{t+1}=A_{t}(1+g_{A}(t))\times(1-\varphi(I_{t})),$ (15)

where $\varphi(I_{t})$ is the impact of shock $I_{t}$ on productivity, which
leads to a persistent shock on the annual economic net output and emission
through the above recursive formula.

###### Remark 2.1

Utility discounting can be interpreted as the relative weighting given to the
well-being of various generations. The choice of an appropriate utility
discount rate $\widetilde{\rho}$ is a controversial subject in the literature
on global warming models. Some economists have argued that a small or zero
utility discount rate $\tilde{\rho}$ should be used to weight different
generations. This topic is discussed in detail in a Stern Review (Stern, 2007,
Chapter 9). In DICE-2016, the utility discount rate $\tilde{\rho}$ is set to
1.5% per year, and the risk-aversion parameter $\alpha$ is 1.45. These
parameters are set to generate consumption rates and real returns on capital,
consistent with observations (see the discussions in Nordhaus (2018)). This
approach to setting the discount rate in the DICE model is called a
“descriptive approach.” Under this approach, the real return on capital $r$ is
not an exogenous but endogenous variable determined through the Ramsey
equation $r=\tilde{\rho}+\alpha g^{\ast}$, where $g^{\ast}$ is the rate of
growth of consumption; see (Nordhaus, 2008, chapter 3). Thus, we assume that
the economy shock events do not affect the utility discounting rate, although
the real return on capital $r$ implied by the affected consumption and risk
aversion can change.

Table 1: The DICE 2016 model parameters, available from
www.econ.yale.edu/~nordhaus/homepage/homepage/DICE2016R-091916ap.gms.
$t=0,\ldots,N,N=99$ with time step $\Delta=5\mbox{ years}$, $t=0$ corresponds
to the year 2015
---
$L_{t}=L_{t-1}\left(\frac{11.500}{L_{t-1}}\right)^{0.134},\;L_{0}=7.403\;(\mbox{in
billions, }10^{9})$
$A_{t}=A_{t-1}(1+g_{A}(t-1)),\;\;g_{A}(t-1)=\frac{0.076\exp(-0.005t\Delta)}{1-0.076\exp(-0.005t\Delta)},\;\;A(0)=5.115$
$\sigma_{t}=\sigma_{t-1}e^{g_{t-1}\Delta},\;\;g_{t}=g_{t-1}(1-0.001)^{\Delta},\;\;\sigma_{0}=\frac{35.85}{105.5(1-0.03)},\;\;g_{0}=-0.0152$
$E_{t}^{Land}=2.6(1-0.115)^{t}$,
$F_{t}^{EX}=\left(0.5+\frac{t}{34}\right)1_{t<17}+1_{t\geq 17}$
$K_{0}=223$, $M_{0}^{AT}=851,\;M_{0}^{UP}=460,\;M_{0}^{LO}=1740$,
$T_{0}^{AT}=0.85,\;T_{0}^{LO}=0.0068$
$\alpha=1.45,\;\gamma=0.3,\;\rho=0.015,\;\widetilde{\rho}=\ln(1+\rho),\;\delta_{K}=0.1$
$\bm{\Phi}^{M}=\left(\begin{array}[]{ccc}\phi_{11}&\phi_{12}&0\\\
\phi_{21}&\phi_{22}&\phi_{23}\\\ 0&\phi_{32}&\phi_{33}\\\
\end{array}\right),\;\;\bm{\Phi}^{T}=\left(\begin{array}[]{cc}1-\xi_{1}\xi_{2}-\xi_{1}\xi_{3}&\xi_{1}\xi_{3}\\\
\xi_{4}&1-\xi_{4}\\\ \end{array}\right)$
$\beta=1/3.666,\;\phi_{21}=0.12,\;\phi_{32}=0.007,\;\phi_{11}=1-\phi_{21},\;\phi_{12}=\phi_{21}588/360$
$\phi_{22}=1-\phi_{12}-\phi_{32},\;\phi_{23}=\phi_{32}360/1720,\;\phi_{33}=1-\phi_{23}$
$\xi_{4}=0.025,\;\xi_{1}=0.1005,\;\xi_{3}=0.088,\;\xi_{2}=3.6813/3.1$
$\eta=3.6813,\;\widetilde{M}^{AT}=588,\;\pi_{2}=0.00236,\;\theta_{2}=2.6$

##  3 Numerical solution

The stochastic DICE model can be solved using the Bellman equation (13) and
the optimal decisions $\mu_{t}^{\ast}(\bm{X}_{t}),c_{t}^{\ast}(\bm{X}_{t})$
can be found using (14) applied backward in time through numerical
deterministic dynamic programming (and then if needed we can simulate forward
in time random trajectories of $\bm{X}_{t}$ based on the calculated optimal
decisions to assess the uncertainty). The logical steps of this numerical
procedure are presented in Algorithm 1. This type of algorithm is often
referred to in the literature as a _value function iteration_. Hereafter,
$\mathcal{T}(\cdot)$ denotes the transition function for the evolution of
state variables

$\bm{X}_{t+1}=\mathcal{T}(\bm{X}_{t},\mu_{t},c_{t},\bm{\epsilon}_{t+1})$

implied by the state processes (2-6), where $\bm{\epsilon}_{t+1}$ is the
vector of random disturbances of the state variables. Algorithm 1 is the
standard approach for solving dynamic programming problems numerically. Its
performance depends on problem-specific details, such as the type of
interpolation across grid points and the type of method used to calculate the
required expectations. For example, Cai et al. (2012) utilizes Chebyshev nodes
for grid points and the Chebyshev polynomial approximation for interpolation.
Cubic spline interpolation is also a possible choice. In our numerical
experiments, we observed that even the simplest linear interpolation works
extremely well for DICE model (it is not the most efficient but is the
simplest and quickest way to implement the algorithm).

Algorithm 1 Dynamic programming with deterministic grid

1:Discretize the state variable space to obtain nodes $x_{j}$, $j=1,\ldots,J$.
This discretization can be different for different time slices, $t$. The state
variable vector may include discrete and continuous variables (in this case,
only continuous variables should be discretized).

2:Initialize $\widehat{V}_{N}(x_{j})=0$ for $j=1,\ldots,J$.

3:for ${t}=N-1\,\,\mathbf{to}\,\,0$ do

4: Interpolate across $\widehat{V}_{t+1}(x_{j})$, $j=1,\ldots,J$ to obtain the
approximation $\widehat{V}_{t+1}(x)$ for any $x$. This step is not necessary
when $t=T-1$ because the maturity condition can be found for any $x$ without
interpolation.

5: for $j=1,\ldots,J$ do

6:
$\widehat{V}_{t}(x_{j})=\sup_{\mu_{t},c_{t}}\left(U(c_{t},L_{t})+e^{-\widetilde{\rho}\Delta}\mathbb{E}[\widehat{V}_{t+1}(\widetilde{x})|\bm{X}_{t}=x_{j}]\right),$

7: where $\widetilde{x}=\mathcal{T}(x_{j},\mu_{t},c_{t},\epsilon_{t+1})$.

8: end for

9:end for

10:Simulation of $M$ optimal trajectories forward in time starting from
$\bm{X}_{0}$.

11:for ${m}=1\,\,\mathbf{to}\,\,M$ do

12: for ${t}=0\,\,\mathbf{to}\,\,N-1$ do

13:
$(\mu_{t}^{\ast},c_{t}^{\ast})=\arg\sup_{\mu_{t},c_{t}}\left(U(c_{t},L_{t})+e^{-\widetilde{\rho}\Delta}\mathbb{E}[\widehat{V}_{t+1}(\widetilde{\bm{X}})|\bm{X}^{(m)}_{t}]\right),$

14: where
$\widetilde{\bm{X}}=\mathcal{T}(\bm{X}^{(m)}_{t},\mu_{t},c_{t},\bm{\epsilon}_{t+1})$
and

15: $\widehat{V}_{t+1}(\widetilde{\bm{X}})$ is found by interpolation across
$\widehat{V}_{t+1}(x_{j})$, $j=1,\ldots,J$.

16: Simulate $\bm{\epsilon}^{(m)}_{t+1}$ from the model specified
distributions.

17: Calculate
$\bm{X}^{(m)}_{t+1}=\mathcal{T}(\bm{X}^{(m)}_{t},\mu^{\ast}_{t},c^{\ast}_{t},\bm{\epsilon}^{(m)}_{t+1})$.

18: end for

19:end for

The calculation of $\mathbb{E}[\widehat{V}_{t+1}(\widetilde{x})|\bm{X}_{t}]$
in Algorithm 1 can be accomplished through a simulation or quadrature
integration methods with respect to the continuous state random variables and
simple summation with respect to the discrete state random variables. In our
study, we consider only one discrete random variable representing the shock of
COVID-19 type events on the world gross output. Thus, the expectation is
simply the sum over the states of the shock variable $I_{t}$. In addition,
note that the interpolation on line 4 in Algorithm 1 is required across grid
points of the continuous state variables only. In the case of many stochastic
state variables (i.e., if we want to account for stochasticity in all state
variables), we can use the least squares Monte Carlo with control
randomization proposed in Kharroubi et al. (2014) with some special
adjustments to handle the expected utility problems introduced in Andréasson
and Shevchenko (2021). This goes beyond the purpose of this study and is the
subject of our ongoing research project.

For numerical calculations, we implemented Algorithm 1 in the statistical
computing programming language R111111https://www.r-project.org/ and then in
Fortan because of the long computational times for some
scenarios121212Depending on the scenario, the computational time was between 2
and 20 h. We used the following settings:

  * •

Each of the six continuous state variables is discretized using equally spaced
points (9 points for $K_{t}$ and 5 points for other variables) and we use a
two-state discrete shock variable $I_{t}$, i.e., in total, there are $2\times
9\times 5^{5}$ points in the deterministic grid in Algorithm 1. We verified
that the increase in the number of discretization points does not have a
material impact on the results. In the case of productivity $A_{t}$ affected
by shocks (see Eq. (15), we discretize $A_{t}$ using 9 points.

  * •

To approximate the infinite time horizon, we use $N=80$ (i.e., 400-year time
horizon) and then report the results for $t=0,1,\ldots,40$ (i.e., up to 200
years). We verified that increasing the time horizon did not materially change
the results.

  * •

The range for $K_{t}$ state variable is selected to be time-varying because
this variable changes from $K_{0}=223$ to approximately $8,000$ at $t=40$. We
denote the solution of the standard DICE-2016 model for capital $K_{t}$ as
$\widetilde{K}_{t}$. The range is then set to
$[0.6\widetilde{K}_{t},\;1.4\widetilde{K}_{t}]$ for $t=0,\ldots,N$. The ranges
for temperatures $T_{t}^{b}$ ($b=AT,LO$) are set to
$[0,\;1.4\widetilde{T}_{\max}^{b}]$, and the ranges for carbon concentrations
$M_{t}^{a}$ ($a=AT,UP,LO$) are set to
$[0.6\widetilde{M}_{\min}^{a},\;1.4\widetilde{M}_{\max}^{a}]$. Here,
$\widetilde{T}_{\max}^{b}$, $\widetilde{M}_{\min}^{a}$, and
$\widetilde{M}_{\max}^{a}$ are the maximum temperature, minimum concentration
and maximum concentration of the standard DICE-2016 solution, respectively. It
was verified that increasing the bounds did not cause any material difference.
In the case of stochastically affected productivity $A_{t}$, the range is set
as $[0.6\widetilde{A}_{t},\;\widetilde{A}_{t}]$, where $\widetilde{A}_{t}$ is
a deterministic function of productivity used in the standard DICE-2016 (see
Table 1).

  * •

The optimal values of the control variables $(\mu_{t},c_{t})$ are not
calculated at $t=0$ but set to the values produced by the standard DICE-2016
because $t=0$ corresponds to the year 2015, which is already in the past. For
other time periods, optimal controls are calculated in Algorithm 1 on lines 6
and 13 using numerical maximization with the same bounds on $\mu_{t}$ as in
the standard DICE-2016.

  * •

We verified that when stochasticity is set to zero, then our numerical dynamic
programming solution leads to virtually the same results as from the original
deterministic DICE model.

  * •

We also set $I_{0}=0$ and $I_{1}=1$ for all trajectories in Algorithm 1 to
reflect the fact that there is a shock at the beginning of 2020 and no shock
in 2015.

To allow for a random change from the normal economy regime to the stressed
regime for each time period owing to COVID-19 like events, one could consider
the shock variable $I_{t}$ with two states such that $I_{t}=0$ corresponds to
the normal regime and $I_{t}=1$ corresponds to the stressed regime. To enforce
the change from the stressed to the normal regime, the matrix of transition
probabilities $\Pr[I_{t+1}|I_{t}]$ can be defined as

$\left(\begin{array}[]{cc}q&1-q\\\ 1&0\\\ \end{array}\right).$ (16)

Here, $q$ is the probability of moving from $I_{t}=0$ to $I_{t+1}=0$, and
$(1-q)$ is the probability of moving from $I_{t}=0$ to $I_{t+1}=1$. If the
annual probability of a COVID-19 type event is $p$, then the transition
probability $q$ over $\Delta$ years can be approximated as $q=(1-p)^{\Delta}$.

Note that modeling of climate tipping point shocks such as in Cai et al.
(2012) can be achieved using the above setup with the important difference;
the probability of shock $(1-q)$ should become a function of temperature
$T_{t}^{AT}$ and the second row of the transition probability matrix (16)
should be changed from $(1\quad 0)$ to $(0\quad 1)$. That is, the shock of the
tipping point event is irreversible, whereas in the case of COVID-19 type
shocks, the stressed regime is forced to be followed by the normal regime.
Scenario B calculated and discussed in the next section assumes persistent
shocks somewhat similar to the tipping point modeling but note that tipping
point modeling assumes shocks to the net output only and no shocks to the
emission.

##  4 Results

To study the impact of the COVID-19 type events on the world economy and
climate under the DICE model, we calculated the following four scenarios. We
set the recovery duration, the decrease in gross world output, and frequency
of such events with reference to the Great Depression (1929-1939) for the
economic impact and to the “Spanish Flu” (1918) for the frequency of the
events.

Scenario A1) Random shocks reduce the gross world output by 5% and it takes 5
years to recover in full. That is, in equation (11) for gross output, we set
$\chi(I_{t})=0.05$ if $I_{t}>0$, and is zero otherwise. These events occur on
average once in a 100-year period (i.e., the annual event probability is
$p=0.01$).

Scenario A2) Random shocks reduce the gross world output by 10% and it takes 5
years to recover in full. That is, in equation (11) for gross output, we set
$\chi(I_{t})=0.1$ if $I_{t}>0$, and is zero otherwise. These events occur on
average once in a 100-year period (i.e., the annual event probability is
$p=0.01$).

Scenario B) Random shocks reduce productivity $A_{t}$ by 5%, i.e., in equation
(15) we set $\varphi(I_{t})=0.05$ if $I_{t}>0$, and is zero otherwise. We also
set $\chi(I_{t})$ in equation (11) to be the same as $\varphi(I_{t})$ so that
gross output persistent drop starts at time $t$. This leads to the persistent
drop in the net output and emission.

Scenario C) The same shock parameters as in Scenario B are used, i.e.,
$\varphi(I_{t})=0.05$ and $\chi(I_{t})=0.05$ for $I_{t}>0$; however, we assume
that the control decisions $\mu_{t}$ and $c_{t}$ undertaken are the same as in
the deterministic DICE model. That is, when simulating trajectories, in
Algorithm 1 on line 13, we do not calculate the optimal stochastic control for
the stochastic DICE model but use controls found by deterministic DICE. This
also means that we undertake suboptimal decisions.

Figures 1, 2, and 3 show the DICE outputs under the four scenarios described
above.

  * •

Figure 1 presents numerical results for the carbon emission mitigation rate
$\mu_{t}$, savings rate $(1-c_{t}/Q_{t})$, economic capital $K_{t}$, and the
net gross world product $Q_{t}$, corresponding to plot titles MIU, S, K, Ynet.

  * •

Figure 2 plots the results for temperatures $T_{t}^{AT}$ and $T_{t}^{LO}$,
carbon price $P_{t}$, and fraction of output lost owing to a temperature
increase $\pi_{2}(T^{AT}_{t})^{2}$, see equation (12), corresponding to plot
titles TATM, TOCEAN, Cprice, DamFct.

  * •

Figure 3 plots the results for carbon concentrations $M^{AT}_{t}$,
$M^{LO}_{t}$, and $M^{LO}_{t}$, corresponding to the plot titles MAT, MU, and
ML.

All plots show results under the standard DICE model (i.e., in the case of no
random shocks) using the dashed line. For the case of stochastic DICE model,
to see the uncertainty/range of outcomes introduced by the shock process
$(I_{t})_{t\geq 0}$, all plots show the 95% probability intervals (indicated
by the gray area in the plots); these intervals are calculated by simulating
1000 random trajectories and calculating 2.5% and 97.5% quantiles over the
trajectories at each $t=1,\ldots,N$ to form the interval. These trajectories
were simulated forward in time using optimal controls
$\mu_{t}^{\ast}(\bm{X_{t}})$ and $c_{t}^{\ast}(\bm{X_{t}})$ obtained by
solving the stochastic DICE model, except for scenario C, where controls are
taken from the deterministic DICE solution.

The results for Scenario A1 show no material change for all DICE outputs from
the deterministic case; only the net output Ynet has a small visible gray area
owing to stochastic shocks. The gray area is below the deterministic DICE
solution for $\tt Ynet$ as expected because shocks reduce net output $Q_{t}$.
The capital state variable $K_{t}$ also has very small gray area below the
deterministic DICE solution consistent with the saving rate S virtually
unaffected by stochastic shocks while output Ynet is reduced by the shocks.

Scenario A2 leads to a more visible (compared to Scenario A1) impact on the
economic variables K and Ynet. The gray area for these variables is below the
deterministic solution and corresponds to approximately 10% variation in 200
years. This is expected because the shock size in Scenario A2 is 10%, larger
than 5% shock under Scenario A1. Saving rate S is also slightly but visibly
affected with most of trajectories below the deterministic case. However, the
impact on climate variables (temperature, carbon concentration, emission
control rate) from shocks in this scenario is immaterial. Though, we still can
note a very small decrease is emission control rate MIU and as a result a
small decrease in carbon price Cprice; also a tiny drop in TATM and in
resulting DamFct. This is also not surprising because shocks reduce not only
the net output but the carbon emission too.

Scenario B clearly leads to larger and material impacts on economic variables
K and Ynet compared to Scenarios A1 and A2 because shocks on the economic
output are persistent (propagate recursively to all subsequent time periods).
Under Scenario B, there is a material impact on the emissions control MIU and
carbon price Cprice, a small but visible impact on the temperature TATM, and a
small but visible impact on the concentrations MAT and MU. Other variables,
such as ML and TOCEAN, are not affected. There is a small impact on the
savings rate S, which is larger than under Scenario A1, but smaller than that
under Scenario A2. The gray area of stochastic DICE trajectories is below
deterministic DICE solution for most of the trajectories across all plotted
variables. More specifically, stochastic trajectories of MIU and Cprice are
always below those under the deterministic DICE solution between now and to
about 100 years (corresponding to approximately 25% range for drop of Cprice
in about 100 years); then, for longer time horizons, there is no difference
between trajectories of these variables and their solutions from the
deterministic DICE. This means that if we account for stochastic persistent
shocks, then the policy for a carbon emission reduction can be less demanding
compared to the deterministic case. This is because persistent shocks reduce
emission (leading to a reduction in concentration and temperature) more than
simple shocks in Scenarios A1 and A2. Trajectories for K and Ynet are also
below those under the deterministic DICE which is explained by persistent
shocks on the net output. In the case of atmospheric temperature TATM, and
concentrations MAT and MU, most of the gray area is below the deterministic
DICE solution, though there are few trajectories going a bit above
deterministic solution after approximately 200 years only.

Finally, the results for Scenario C show the case of applying decisions MIU
and S from the deterministic DICE model to the trajectories of stochastic DICE
model. That is, we apply suboptimal decisions that are optimal under the
deterministic DICE but suboptimal under the stochastic DICE. Thus, one can see
that all trajectories under the stochastic DICE for MIU, Cprice and S are the
same as the deterministic DICE solution. Trajectories for all other variables
are always appear to be below corresponding deterministic DICE solution. Under
this scenario, we see even more material impact on temperature; all
trajectories for TATM in the gray area are below the deterministic DICE
solution. On average, TATM is approximately $0.25^{\circ}\mathrm{C}$ below the
deterministic DICE solution when temperature peaks after about 150 years, with
gray area corresponding to about $0.15^{\circ}\mathrm{C}$ range. The same is
for carbon concentration MAT, where all trajectories are below the
deterministic solution (on average a 10% drop for a concentration at its peak
in about 100 years). In other words, this scenario shows that more ambitious
mitigation targets are now feasible at lower costs, or mitigation targets will
be achieved faster if the policy is unchanged (i.e. not adapted for
environment with stochastic shocks). This is qualitatively the same as the
results of the analysis conducted in Meles et al. (2020). Again, this outcome
is somewhat expected due to persistent shocks reducing not only the net output
Ynet but also carbon emission.

##  5 Conclusion

In this paper, we studied the impact of the COVID-19 type events on the carbon
concentration, temperature, economic capital and other outputs of the DICE
model extended to include corresponding stochastic shocks on the world gross
annual output. We solved the extended model under different scenarios as an
optimal stochastic control problem, assuming that shock events occur randomly
on average once during a 100 period. The results show that if the world gross
output recovers in full after each event, then the impact of the COVID-19
events on the temperature and carbon concentration will be immaterial even in
the case of a conservative 10% decrease in the annual gross output over a
5-year period. The impact becomes noticeable but small (the long-term
temperature in the atmosphere drops on average by $0.1^{\circ}\mathrm{C}$) if
a 5% decrease in the gross output owing to a shock is allowed to propagate
over time (i.e., allowed to be a persistent shock). Finally, if the
deterministic DICE model policy is still applied in the case of stochastic
shocks (i.e., it is a suboptimal policy in this case), then the drop in
temperature will be larger (approximately $0.25^{\circ}\mathrm{C}$). That is,
the lower economic activities owing to the occurrence of a shock imply that
more ambitious mitigation targets are now feasible at lower costs, which is
qualitatively consistent with the results presented in Meles et al. (2020).

Shocks considered in our study reduce both the world net output and emission
through the shock reduction of the gross output while tipping point modeling
studies such as Lontzek et al. (2015); Cai et al. (2012) assume shock on the
net output only and no shock on the emission. Thus our shocks lead to a
reduction in a policy stringency while tipping point shocks lead to the
opposite effect.

In general, incorporation of uncertainty in integrated climate-economy
assessment models, such as the DICE model, is an under-developed research
topic. Typically, the uncertainty is assessed by calculating the models under
the perturbed parameters, and state-of-art stochastic control methods are not
really used. This can be partly explained by the difficulty of implementing
dynamic programming algorithms. This is probably due to the large number of
state variables that call for the use of Monte Carlo simulation methods while
until recently, Monte Carlo techniques have not been used for optimal control
problems that involve controlled processes. A relatively recent development in
this area is the least-squares Monte Carlo approach with the control
randomization technique developed in Kharroubi et al. (2014). However, to
solve stochastic control problems maximizing the expected utility (as in the
DICE model), some special adjustments are required for this technique, as
discussed in Andréasson and Shevchenko (2021). Implementing this approach for
the DICE model incorporating stochasticity in all state variables is the
subject of our research project in progress.

Of course, although the DICE model is a typical reference point for many
climate-economy studies, it is important to remember that IAMs and the DICE
models in particular have significant limitations (in the model structure and
model parameters) and have been criticized and debated in the literature (for
example, see the discussions in Ackerman et al. (2009); Pindyck (2017); Grubb
et al. (2021); Weitzman (2011).

##  6 Acknowledgement

We wish to thank Yoshiki Yamagata for the valuable discussions and comments.
Pavel Shevchenko and Tor Myrvoll acknowledge travel support from the Institute
of Statistical Mathematics, Japan. We also thank the participants of the 11th
conference on the _Transdisciplinary Federation of Science and Technology_ for
their helpful comments on our preliminary results in Matsui et al. (2020).

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Figure 1: Trajectories of optimal solution of DICE model for MU, S, K, YNET
under various scenarios. The horizontal axes represent time (measured in
5-year steps) from 2015 ($t=0$) to 200 years later ($t=40$). The red dashed
lines correspond to the solution of the standard deterministic DICE model
(i.e., in the case of no COVID-19 type events), and the gray area represents
the 95% probability interval calculated over 1,000 random trajectories
simulated. Figure 2: Trajectories of TATM, TOCEAN, Cprice, DamFct. See the
caption of Figure 2 for further details. Figure 3: Trajectories of MAT, MU,
ML. See the caption of Figure 1 for further details.