Program: MSc Economics
Track: Monetary policy and banking Date of version: 15-07-2021
Supervisor: Dr Isabelle L. Salle
The Losers of the Lockdown: an Age Group Analysis in an Agent-
Based COVID Model Nicolas Schrama
11881437
Statement of originality
This document is written by Student Nicolas Schrama who declares to take full responsibility for the contents of this document.
I declare that the text and the work presented in this document are original and that no sources other than those mentioned in the text and its references have been used in creating it.
The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
Abstract
Long COVID is a new medical phenomenon with a relatively high prevalence among young generations.
Therefore, it sheds new light on the balancing of government containment policy. This study aims to find out if the young economically benefit from a lockdown. To that end, I use a calibrated agent-based SIR model, motivated by Eichenbaum et al. (2020), with rich heterogeneity and long COVID. I simulate the COVID pandemic in the Netherlands with and without government intervention. In the model, the government institutes a lockdown by imposing a homogeneous cap on agents’ weekly consumption. In the simulations, the lockdown forces the young to cut back consumption more than the old. Furthermore, the number of deceased agents decreases in all age groups but the youngest. The difference is largest in the oldest group. Moreover, during the pandemic, the young lose utility because of the lockdown while the old gain utility. The utility loss of the young is slim and temporary, but the utility gain of the old is slim and permanent.
I. Introduction
The COVID pandemic has brought the lives versus livelihoods tradeoff to the fore. Governments must strike a balance between the loss of lives and the loss of economic activity. The youngest generations in Dutch society complain that the balance struck by the government favors the elderly. Since they face few of the coronavirus’s risks, young people refuse to abide by the social distancing rules.
Long COVID sheds new light on this balance. Long COVID encompasses a range of symptoms, such as fatigue and brain fog, that persist after the acute phase of infection (British Office for National Statistics, 2021). 61.6% of individuals with long COVID symptoms experience limitations in their day- to-day activities. 17.9% reports severe limitations in their day-to-day activities. Crucially, symptoms are independent of symptoms during the acute phase of infection. Therefore, long COVID symptoms are prevalent in the 25-34 age group, even though this group experiences few symptoms during the acute phase of the virus. Indeed, the media report many twenty-year-olds who are exhausted after a thirty- minute walk and cannot work, e.g., Kirby (2021).
These reports prompt the question of whether young people benefit from a lockdown economically. To answer that question, I use a calibrated agent-based SIR model with rich heterogeneity and long COVID, motivated by Eichenbaum et al. (2020). The authors enrich the canonical SIR model by Kermack and McKendrick (1927), such that consumption and labor decisions influence the spread of the virus. Agents’ decisions to cut back economic activity reduce the severity of the pandemic, as measured by its death toll, but exacerbate the size of the recession caused by the pandemic.
The simple model set out by Eichenbaum et al. (2020) makes the intuition behind economic and epidemiological dynamics transparent. It is a great starting point for an agent-based model since extra layers of complexity cloud this intuition.
Agent-based modeling gained traction after the Great Recession revealed the inadequacy of the predominant theoretical framework grounded on DSGE models (Dosi & Roventini, 2019)1. First, the DSGE framework’s predictions of stable equilibrium paths under rational expectations are inconsistent with crisis periods. This finding suggests that individuals do not forecast future economic dynamics using full information (Jang & Sacht, 2021). Instead, they process past information through backward- looking rule-of-thumb behavior, i.e., using forecast heuristics (Akerlof & Shiller, 2009). Therefore, rational expectations became a primary target for criticism.
Second, as opposed to DSGE models, agent-based models make it possible to consider the economy as a complex evolving system, i.e., as an ecology populated by heterogeneous agents, whose far-from-equilibrium interactions continuously change the system’s structure.
I adapt to these criticisms by introducing a heterogeneous nonrational model that alters the model by Eichenbaum et al. (2020) in four ways. First, it incorporates nonrational expectations. Second,
it uses agent-based modeling to replace the homogeneous population used by Eichenbaum et al. with a heterogeneous population that mirrors the Dutch populations’ age structure and COVID mortality. As a result, susceptible agents behave heterogeneously depending on their age. The behavior of infected, long COVID, and recovered agents does not depend on age. Third, the model is calibrated using recent Dutch data to study economic effects on the Dutch youth. Eichenbaum et al., however, were limited to early US and Chinese data. Finally, I incorporate long COVID in the model. As far as I know, this is the first paper to include long COVID in a macroeconomic model.
Because of four market failures, the government can improve social outcomes with a containment policy. Individuals’ atomistic behavior causes the first: infected agents take the economywide infection rate as given and do not internalize how it is affected by their economic decisions (Eichenbaum et al., 2020). Second, if susceptible agents decrease economic activity, hence slowing the virus’ propagation today, infection rates increase tomorrow because of the relative increase in the number of susceptible individuals (Rachel, 2020). Third, susceptible agents do not consider that they will infect others if they contract the virus (Garibaldi et al., 2020). Finally, susceptible agents are unaware of long COVID and do not consider its effects when making economic decisions, leading to an internality.
To deal with these market failures, the government institutes a lockdown by imposing a homogeneous maximum on agents’ weekly consumption, called the consumption cap. Each week, the government assesses the state of the economy and the pandemic and sets a new consumption cap. It sets the consumption cap as if it persists for the rest of the pandemic, though it reneges each week.
To answer the research question, I distinguish three models: (i) the perfect foresight model by Eichenbaum et al. (2020)2; (ii) the homogeneous nonrational model, which differs from the perfect foresight model by its agent-based nature and by incorporating nonrational expectations; (iii) the heterogeneous nonrational model, which differs from the perfect foresight model by its agent-based nature and by incorporating nonrational expectations, long COVID, heterogeneity, and recent Dutch data for the calibration. Comparing the perfect foresight model to the homogeneous nonrational model, I find that the recession is less severe under nonrational expectations since agents underestimate the value of remaining susceptible. Comparing the homogeneous nonrational model to the heterogeneous nonrational model, I find that long COVID increases the severity of the recession, that heterogeneity increases the death toll in the model under nonrational expectations, and that a smaller share of the population is infected under the calibration based on recent data.
Then, I compare aggregate consumption, lives taken, and aggregate utility in the heterogeneous nonrational model between the age groups 16-30, 31-50, 51-66, and 67+ with and without a containment policy. I find that a lockdown forces the youngest group to cut back consumption more than the oldest group. Furthermore, the number of deceased agents decreases in all age groups but the youngest. The
difference is largest in the oldest group. Moreover, during the pandemic, the young lose utility because of the lockdown while the old gain utility. The utility loss of the young is slim and temporary, but the utility gain of the old is slim and permanent.
In the next section, I summarize the literature on the macroeconomic consequences of COVID.
Section three elaborates on the theoretical model. Section four details the calibration of the theoretical model. Section five assesses the empirical validity of the model. Section six discusses the model’s simulation results, and section seven addresses its robustness to various calibrations. Finally, section eight concludes.
II. Literature review
The literature on the interaction between economics and epidemiology, as reviewed in Perrings et al.
(2014), has long pointed out the negative externality of too little effort towards preventing the spread of infectious disease by self-interested agents (Brotherhood et al., 2020), giving rise to optimal policy problems. Seminal works vary from studies on the effects of private vaccination incentives on epidemic dynamics and optimal public health policy (Philipson, 2000) to heterogeneous-agent choice-theoretic equilibrium models for HIV/AIDS (Greenwood et al., 2019). As the coronavirus took over the world, so did it take over this literature. Because of the prevalence of COVID economics research, this section fixates on the macroeconomic consequences of COVID.
The debate has revolved around two questions (Delli Gatti & Reisl, 2020). First, what kind of shock hit the macroeconomy following the epidemic outbreak, and what are its transmission channels?
Second, is there a trade-off between lives and livelihoods, and how can policy balance it?
A consensus has emerged regarding the first question: the epidemic generated both a demand and a supply shock with profound contractionary consequences (Baldwin & Weder di Mauro, 2020).
For example, Guerrieri et al. (2020) characterize the COVID shock as a Keynesian supply shock, i.e., a supply shock capable of causing movements in aggregate demand of even greater magnitude. Faria-e- Castro (2021) studies the effect of an epidemic modeled as a negative shock to the utility of consumption of contact-intensive services in a model with borrowers and savers. Buera et al. (2020) analyze the effects of a temporary shock made persistent by firms’ deteriorating balance sheets and labor market frictions. Finally, Baqaee and Farhi (2020) study supply and demand shocks in a disaggregated model with multiple sectors.
Answering the second question required the development of theoretical macroeconomic models in which agents’ health states change. Most research projects use compartmental models except for a few, such as Inoue and Todo (2020). Many of the former build upon the canonical equation-based SIR
link to economic activity. Containment policies typically reduce production or consumption, hence inducing an economic shock. The interplay between containment measures and economic costs is then studied as an optimization problem from a social planner’s perspective or embedded in a macroeconomic framework where agents individually optimize their decisions.
In this vein, Atkeson (2020) provides an overview of SIR models and explores their implications for the COVID pandemic. Other examples include Eichenbaum et al. (2020), who modify the canonical SIR framework such that economic activity affects the propagation of the virus. They implement a proportional tax on consumption that reduces the severity of the pandemic at the cost of a deeper recession. Krueger et al. (2020), however, argue that more than two-thirds of this decline in aggregate consumption is avoided if agents can endogenously adjust in which sectors they consume. Alvarez et al.
(2020) study optimal lockdown policy in a version of the canonical SIR model in which the case-fatality rate increases with the number of infected people. Finally, Toxvaerd (2020) analyzes the equilibrium amount of social distancing and argues it is not socially optimal.
The lack of heterogeneity along dimensions as age and implied COVID mortality in the first wave of SIR models received criticism. Therefore, newer papers have tended to incorporate them. For example, Glover et al. (2020) study optimal mitigation policies in a model with retired and non-retired households. They find that relative to predictions from models without heterogeneity, the utilitarian optimal shutdown is relatively mild. Brotherhood et al. (2020) conclude that short and mild lockdowns have minor benefits or even adverse consequences. Harsher and longer lockdowns that last until the arrival of a vaccine save lives and increase utility for the old, while for the young, they save lives at the cost of utility. Acemoglu et al. (2020) study targeted lockdowns in a model with young, middle-aged, and old agents. They find that optimal policies differentially targeting age groups outperform optimal uniform policies and that having stricter lockdown policies on the oldest groups reaps most of the gains.
Other examples of papers that include age heterogeneity are Favero et al. (2020) and Gollier (2020).
Though, not all SIR-type models build upon deterministic equations to describe how agents’
health states change. Taking from Epstein (2009) and Gourinchas (2020), Delli Gatti and Reissl (2020) criticize these models for not capturing the local and complex social interactions associated with economic activities, which play an essential role in the transmission of the coronavirus. They propose an agent-based SIR model with three age groups and find that liquidity support for firms, a short-time working scheme with compensation, and direct transfer payments to households are helpful policy tools to alleviate the economic impact of the epidemic and the lockdown.
Many economists share this view on epidemiological dynamics. For example, Silva et al. (2020) employ an agent-based SIR model that simulates the pandemic using a society of agents emulating people, businesses, and government to study the effects of social distancing interventions. Finally, Basurto et al. (2020) use an agent-based SIR model with a multisectoral economy and two age groups to analyze the impact of various lockdown policies. They find that any efficient policy should impose a
low threshold of newly infected for moving from the lockdown to the opening-up stage and that in the opening-up stage, all restrictions on economic activity should cease.
III. Model description
This section sketches the heterogeneous nonrational model, a macroeconomic agent-based SIR model motivated by Eichenbaum et al. (2020). The first subsection establishes economic behavior in the pre- epidemic economy. Subsection B adds all features necessary for an epidemiological model. For a description of the perfect foresight model, refer to Eichenbaum et al. (2020). The theoretical description of the homogeneous nonrational is identical to the description of the heterogeneous nonrational model, ignoring heterogeneity and long COVID.
A: the pre-epidemic economy
The model functions in discrete time, and each period symbolizes one week in the real world. 𝑁 agents populate the economy and maximize the objective function
𝑈 = $ 𝛽!
"
!#$
𝑢'𝑐%,!, 𝑙%,!+. (1)
In Equation 1, 𝛽 ∈ (0,1) denotes the discount factor, 𝑐%,! the consumption of agent 𝑖 in period 𝑡, and 𝑙%,! the number of hours worked by individual 𝑖 in period 𝑡. Momentary utility takes the form
𝑢'𝑐%,!, 𝑙%,!+ = ln'𝑐%,!+ −𝜃
2𝑙%,!' . (2)
Agents face the budget constraint
𝑐%,! = 𝐴𝑙%,!. (3)
In Equation 3, 𝐴𝑙%,! equals the wage. Consequently, all agents homogeneously consume 𝑐%,! = 𝐴 √𝜃⁄ and work 𝑙%,! = 1 √𝜃⁄ hours in the pre-epidemic economy.
Agents are employed at a continuum of competitive firms that produce consumption goods using hours worked according to a linear production technology:
$ 𝑐%,!
(
%#)
= 𝐴 $ 𝑙%,!.
(
%#)
(4)
B: the epidemic economy
After the epidemic outbreak, agents pertain to one of five health states: susceptible, infected, long COVID, recovered, or deceased. Susceptible individuals have not yet fallen ill from the virus. Infected individuals are in the acute phase of the disease and can infect others. Agents with long COVID have survived the acute phase of the virus and can no longer infect others but suffer from symptoms such as fatigue. Recovered individuals have survived the virus, are free of all symptoms, and can no longer contract the virus. Deceased agents have not survived the acute phase of the virus.
Infected and long COVID agents face diminished productivity. As a result, the firm production technology in the epidemiological economy is
$ 𝑐%,!
(
%#)
= 𝐴 $'1 − ℎ%,!) − ℎ%,!'+𝑙%,!+ 𝜑ℎ%,!) 𝑙%,!+ 𝜌ℎ%,!' 𝑙%,!.
(
%#)
(5) In Equation 5, ℎ%,!) equals one for infected individuals, ℎ%,!' equals one for individuals with long COVID, 𝜑 measures the relative productivity of infected individuals, and 𝜌 measures the relative productivity of individuals with long COVID. 𝜑 and 𝜌 lie between zero and one.
Figure 1 illustrates how agents’ health states change.
Figure 1: Health state dynamics in the epidemiological economy
Susceptible agents may contract the virus through three channels: during consumption, at work, or in a random encounter. The probability of being infected during consumption in period 𝑡 is
𝜋)𝑐%,!∑(%#)ℎ%,!) 𝑐%,!
𝑁 .
The summation gives the aggregate consumption of all infected individuals and is divided by the number of agents alive 𝑁. 𝜋) is thus the probability of being infected per unit of consumption if all infected individuals together procure 𝑁 units of consumption. The model abstracts from heterogeneity in types of consumption activities.
Similarly, the probability of being infected at work in period 𝑡 is 𝜋'𝑙%,!∑(%#)ℎ%,!) 𝑙%,!
𝑁 .
𝜋' is the probability of being infected per hour of labor if all infected agents work 𝑁 hours on aggregate. The model also abstracts from heterogeneity in types of labor activities.
Third, susceptible and infected agents meet in ways unrelated to work or consumption, i.e., random encounters. The probability that a susceptible individual contracts the virus during a random encounter in period 𝑡 is
𝜋+∑(%#)ℎ%,!) 𝑁 .
Here, 𝜋+ is approximately the probability of being infected in a random encounter if all other agents are infected. For simplification, I exclude the probability that agents are infected through more than one of these channels.
Infected individuals move on to one of three health states. First, they may die with probability 𝜋%,. Second, they may recover from the virus and suffer from no long COVID symptoms with probability 𝜋%-∙ 𝜋.. Finally, they may recover from the virus but suffer from long COVID with probability 𝜋%-(1 − 𝜋.). Then, they have a homogeneous probability 𝜋. of recovering from their long COVID symptoms each period.
The epidemic starts through zoonotic exposure, i.e., in period 𝑡 = 1, the virus is transmitted from animals to a fraction 𝜀 of agents. All agents are aware of the initial infection and understand the health dynamics. Though, agents are unaware of long COVID and expect that the infected pass away or recover completely. Furthermore, agents take as given aggregate variables, such as
∑(%#)ℎ%,!) 𝑙%,!
𝑁 and
∑(%#)ℎ%,!) 𝑐%,!
𝑁 .
Recovered agents optimize the value function
𝑚𝑎𝑥/!,# 0!,# 𝑈%,!1 = 𝑢'𝑐%,!, 𝑙%,!+ + 𝛽𝑈%,!2)1 (6) subject to the budget constraint in Equation 3.
𝑈%,!1 is the lifetime utility of recovered agent 𝑖 in period 𝑡. In all health states, lifetime utility in the final period 𝑇 equals momentary utility. Solving the Lagrangian gives
𝑢/3'𝑐%,!, 𝑙%,!+ = −𝑢03'𝑐%,!, 𝑙%,!+
𝐴 . (7)
According to Equation 7, the marginal benefit of consumption must equal its marginal costs in the optimum. Solving the optimization problem gives that 𝑐%,! = 𝐴 √𝜃⁄ and 𝑙%,! = 1 √𝜃⁄ . Because the
heterogeneous mortality rates are irrelevant to the recovered, they behave homogeneously. The heterogeneous mortality rates matter only to susceptible agents.
Agents with long COVID unexpectedly find themselves in this health state. They do not know how long it takes to recover from long COVID but correctly assume that the symptoms cease eventually.
Thus, they do not know the value of 𝜋., but this is inconsequential for optimization. Agents with long COVID optimize
𝑚𝑎𝑥/!,# 0!,# 𝑈%,!. = 𝑢'𝑐%,!, 𝑙%,!+ + 𝛽(1 − 𝜋.)𝑈%,!2). + 𝛽𝜋.𝑈%,!2)1 (8) subject to
𝑐%,! = 𝜌𝐴𝑙%,!. (9)
Solving the optimization problem gives that 𝑐%,! = 𝜌𝐴 √𝜃⁄ and 𝑙%,! = 1 √𝜃⁄ . In words, agents with long COVID (i) work as many hours as recovered agents but (ii) consume less owing to their diminished productivity.
First, is the assumption that less productive agents work as many hours realistic, or should they work fewer hours? In the real world, most long-COVID patients maintain their labor hours (British Office for National Statistics, 2021). Some, e.g., twenty percent of long-COVID patients, are physically too impaired to work, however. Then, aggregate consumption and labor fall by twenty percent on aggregate in the real world. In the model, aggregate labor does not fall, but aggregate consumption falls by twenty percent. Thus, the model tracks aggregate consumption well but not aggregate labor. Under a utility specification in which less productive agents cut back labor, agents’ consumption would fall further. As a result, labor hours in the model become closer to those in the real world. Though, aggregate consumption would fall by more than twenty percent, creating a new problem. Thus, the utility specification determines where the mistake is. I choose the utility specification in Equation 2 to accurately track consumption, which is the key variable I use to analyze the model dynamics.
Second, compared to the real world, Dutch employees are covered by social security when ill, but the model abstracts from social security.
Infected agents optimize
𝑚𝑎𝑥/!,# 0!,# 𝑈%,!4 = 𝑢'𝑐%,!, 𝑙%,!+ + 𝛽'1 − 𝜋-− 𝜋%,+𝑈%,!2)4 + 𝛽𝜋-𝑈%,!2)1 (10) subject to
𝑐%,! = 𝜑𝐴𝑙%,!. (11)
Solving the optimization problem gives that 𝑐%,! = 𝜑𝐴 √𝜃⁄ and 𝑙%,! = 1 √𝜃⁄ . Like individuals with long COVID, infected individuals work as many hours as recovered agents but consume less.
Tracking the aggregate hours of infected agents runs into the same issue as tracking the aggregate hours of long COVID patients. Similar to how only some long COVID patients are physically too impaired to work, only some infected agents suffer from symptoms and do not work. Furthermore, Equation 10 assumes that the cost of death is the foregone utility of life.
Susceptible agents optimize
𝑚𝑎𝑥/!,# 0!,# 5!,# 𝑈%,!6 = 𝑢'𝑐%,!, 𝑙%,!+ + 𝛽'1 − 𝜏%,!+𝑈%,!2)6 + 𝛽𝜏%,!𝑈%,!2)4 (12) subject to the budget constraint in Equation 3 and
𝜏%,! = 𝜋)𝑐%,!∑(%#)ℎ%,!) 𝑐%,!
𝑁 + 𝜋'𝑙%,!∑(%#)ℎ%,!) 𝑙%,!
𝑁 + 𝜋+∑(%#)ℎ%,!)
𝑁 . (13)
𝜏%,! is the probability that agent 𝑖 is infected in period 𝑡. Agents must optimize 𝜏%,! because the virus implies a tradeoff between health and economic activity. Susceptible agents consume and work less to decrease 𝜏%,!.
Solving the Lagrangian gives two equations:
𝑘%,! = 𝛽'𝑈%,!2)6 − 𝑈%,!2)4 + (14)
𝑢/3'𝑐%,!, 𝑙%,!+ =−𝑢03'𝑐%,!, 𝑙%,!+ + 𝑘%,!𝜋'∑(%#)ℎ%,!) 𝑙%,!
𝑁
𝐴 + 𝑘%,!𝜋)∑(%#)ℎ%,!) 𝑐%,!
𝑁 . (15)
𝑘%,! is the Lagrangian multiplier of the constraint in Equation 13. It is the decrease in lifetime utility if agent 𝑖 is infected in period 𝑡. Hence, the left-hand side of Equation 14 gives the marginal cost of reducing 𝜏%,!, while the right-hand side gives the marginal benefit of reducing 𝜏%,!. According to Equation 15, the marginal benefit of consumption must equal its marginal cost in the optimum.
Susceptible agents must form expectations about future momentary utility and future infection probabilities since these determine 𝑈%,!2)6 . Consequently, they must form expectations about the number of infected, the number of deceased, and their own labor and consumption in every future period. The determinants of 𝑈%,!2)4 , on the other hand, are homogeneous across infected individuals and time.
Contrary to the perfect foresight model, agents do not form expectations rationally but use rules of thumb. For the ratio of infected agents to the number of agents alive, they expect a bell-shaped curve.
Precisely, the resulting ratios in the perfect foresight model form these expectations. Furthermore, all susceptible agents assume that their future consumption and labor will equal their consumption and labor in period 𝑡.
At the start of the real-world pandemic, many images of bell-shaped infection curves circulated in the media, e.g. (Anderson et al., 2020; RIVM, 2020b). The government intended to make people aware of the necessity of policy to flatten the curve so that healthcare systems could deal with the pressure of extra patients. Therefore, it is reasonable to assume individuals in the real world also expected a bell-shaped infection curve. Furthermore, naive expectations, such as agents’ consumption and labor expectations, are common in the literature. For example, Dosi et al. (2013) assume that firms in their model form naive expectations about future demand and find that increasing firms’
computational capabilities does not influence the dynamics of the economy.
Government intervention
Three externalities and an internality encourage social welfare-improving government policy in the
10. Therefore, infected agents do not consider that they may infect susceptible agents. Second, if susceptible agents decrease economic activity, hence slowing the virus’ propagation today, infection rates increase tomorrow because of the relative increase in the number of susceptible individuals.
Agents’ value functions do not consider this effect, thus neglecting herd immunity. Third, because the economywide infection rate is absent in infected agents’ value function, it also does not enter susceptible agents’ value function through 𝑈%,!2)4 . Therefore, susceptible agents do not consider that they will infect others if they contract the virus. Finally, long COVID does not enter susceptible agents’ value functions, leading to an internality.
Because of these market failures, the government can improve social outcomes with a containment policy. At the start of every period, when the number of individuals in each health state is known, the government sets a homogeneous maximum on agents’ weekly consumption, called the consumption cap. It calibrates the consumption cap such that it is optimal for the remaining 𝑇 − 𝑡 periods. Still, each period, the government reconsiders and adjusts its policy.
The government knows how agents come to their decisions and how they form expectations.
Furthermore, the government is aware of long COVID and incorporates it into the consumption cap.
The government aims to maximize aggregate discounted lifetime utility. The cap is given by
𝑐!789= 𝑐̅(1 − 𝛼!). (16)
In Equation 16, 𝑐̅ denotes individual consumption in the pre-epidemic economy. The parameter 𝛼! governs the strictness of the containment policy. When it takes a value of, for example, 0.1, agents can consume up to ninety percent of pre-epidemic consumption.
To set 𝛼!, the government must form expectations on every agent’s future labor and consumption to determine their future momentary utility. The government assumes that agents maintain their consumption and labor of period 𝑡. Because the government knows how agents make economic decisions, it can determine agents’ behavior in period 𝑡 based on the number of agents in each health state.
For the recovered, this assumption holds since recovered individuals consume either the pre- epidemic amount or the consumption cap if there is one. This assumption does not hold for infected and long COVID agents since they will recover or die in the coming weeks. For the susceptible, it is more complicated. Because the government sets 𝛼! as if it persists for the remainder of the pandemic, susceptible agents that consume the cap in period 𝑡 likely will in the future too. Section six discusses the effects of this assumption in detail.
As a result of this assumption, policy decisions are not truly optimal. Though, perfectly calculating the number of infected and deceased individuals in all future periods and determining how every individual will respond is too complex, both in the model and the real world.
Furthermore, the government must form expectations on the future infection probabilities of susceptible agents. For two reasons, this is a complex task. First, future infection probabilities are not
linearly related to future consumption and labor because of their dependency on the future number of infected and deceased agents. Second, heterogeneous mortality rates mean every agent responds differently. As a rule of thumb, the government assumes that all infection probabilities in periods after 𝑡 equal 𝜏%,!(1 − 𝛼!), i.e., 𝜏%,!2: with 𝑣 > 0 equal 𝜏%,!(1 − 𝛼!). At the start of the pandemic, infection probabilities are rising, such that this assumption does not hold. If the government, however, does not believe that its policy lowers future infection probabilities, the best policy in its eyes is no intervention.
The sequence of events that occur every period summarizes the model:
1. The number of susceptible, infected, long COVID, recovered, and deceased agents are announced. Thus, the probability of being infected during work and consumption per unit of labor and consumption are known.
2. The government announces the consumption cap.
3. Agents choose labor and consumption, given the consumption cap and the spread of the virus.
4. Agents work, receive their wage, and spend it on the produced consumption goods.
5. After these activities and random encounters have taken place, agents’ health states may change:
susceptible agents may become infected, infected agents may die, recover, or move on to the long COVID phase, and agents with long COVID may fully recover.
IV. Data and calibration
This section reports the values of the parameters used in the simulations. I use recent Dutch data for the heterogeneous nonrational model instead of early pandemic US and Chinese data – as Eichenbaum et al. (2020) use for the perfect foresight model – since the study aims to investigate economic effects on the Dutch youth. The homogeneous nonrational model uses the calibration of the perfect foresight model. Even after a year of COVID research, the true values of these parameters are still uncertain.
Section seven investigates the robustness of the model to different configurations of these values.
𝜃 governs the number of hours worked. Eichenbaum et al. (2020) set it at 0.001275 so that the pre-epidemic representative agent works 28 hours per week, in line with data from the Bureau of Labor Statistics’ 2018 ATUS. Furthermore, they set 𝐴 at 39.835 such that the representative agent earns a weekly income of $58 000 52⁄ , in line with the 2019 estimate for per capita income of the US Bureau of Economic Analysis. Finally, 𝛽 equals 0.96)/<' in their model to have a value of life consistent with economic values used by government agencies.
I do not adjust these values. First, the value of 𝐴 is irrelevant in the model, as its sole task is to scale consumption. Second, Dutch National Accounts data, e.g., for the first quarter of 2019 (CBS, 2021a), indicate that Dutch workers work around 28 hours per week. Third, changing 𝛽 leads to different
results because agents have different preferences and, therefore, to a comparison between apples and oranges. Keeping annual 𝛽 at 0.96 allows for a better comparison with the literature.
To choose the mortality and the recovery rate, Eichenbaum et al. (2020) use mortality by age data from the South Korean Ministry of Health and Welfare from March 2020 and apply them to the US age distribution to obtain a weighted average of the US mortality rate. They exclude people above seventy since the elderly have a low labor force participation rate. They obtain a mortality rate of 0.5%.
Furthermore, Eichenbaum et al. assume that it takes 18 days to either recover or die from the virus on average. They base this estimate on two results. First, Yang et al. (2020) find that the median time from disease onset to diagnosis among confirmed patients is five days. Second, Chen et al. (2020) find that the average hospitalization period is 12.39 days. 12.39 + 5 yields an upper bound of 18 days.
These estimates use early Wuhan data. Since the Eichenbaum model is weekly, they set the weekly mortality rate 𝜋, to 7/18 ∙ 0.005 and the weekly recovery rate 𝜋- to 7/18 ∙ 0.995.
The Dutch National Institute for Public Health and the Environment (henceforth RIVM) estimates the number of days until disease onset 5 to 6 days (RIVM, 2021b). Furthermore, the RIVM reports that the average hospitalization period was 8.2 days outside the ICU and 18.2 in the ICU (RIVM, 2020c). From January 2021 to April 2021, 29.1% of Dutch COVID-related hospitalization concerned the ICU (LCPS, 2021). Thus, the average hospitalization period is 11.3 days. Using 11.3 days as the average hospitalization period and 5 to 6 days as the time to disease onset yields an upper bound of 18 days to recover or die from the virus on average.
To set a heterogeneous mortality rate per age, I use the 2019 CBS Population Pyramid (CBS, 2021b) for the number of people per age in the Netherlands before the pandemic. In 2019, the youngest person was 0 and the oldest person 100. Furthermore, Brazeau et al. (2020) find in a meta-study that COVID mortality doubles with every eight years of age and that that the average COVID mortality in the Dutch sample is 0.62%. Although COVID mortality in the Dutch sample is based on a sample of blood donors and thus possibly not representative, they find that average mortality in developed countries is similar at 0.68%.
Two arguments explain the increase in average mortality from 0.5% in Eichenabum et al.
(2020). First, according to Brazeau et al. (2020), mortality was typically biased downward in the early days of the pandemic. Second, Eichenbaum et al. (2020) exclude the elderly. I keep the elderly since they are a crucial age group in the study.
Applying average mortality of 0.62% and doubling mortality with every eight years of age on the weighted Dutch age distribution yields that the mortality of someone aged zero equals 0.003%, while someone aged one hundred has a mortality of 18.2%. Since the model is weekly, these rates are multiplied by 7/18. An individual’s periodic recovery rate equals 7/18 minus their mortality. After this procedure, I exclude individuals younger than 16 from the model since they are too young to make economic decisions.
For the calibration of 𝜋), 𝜋', and 𝜋+, Eichenbaum et al. (2020) assume that one in six infections relates to consumption, one in six to labor, and the rest to random encounters. They base this on several American data sources, the argument by Ferguson et al. (2006) on influenza transmission, and the common assumption in epidemiology that the relative importance of these three transmission channels is equal across respiratory diseases. Furthermore, they assume that sixty percent of the population either recovers or dies from the virus in the limit of the canonical SIR model, based on Bennhold & Eddy (2020). Eichenbaum et al. (2020) obtain the values 7.8408 ∙ 10=>, 1.2442 ∙ 10=?, and 0.3901 for 𝜋), 𝜋', and 𝜋+.
No new estimates of the percentage of people that will have had COVID by the end of the pandemic are available. Currently, the RIVM is working on the PIENTER Corona study (RIVM, 2021a), in which they collect data on antibodies in the blood of Dutch citizens. This study should give a better estimate.
Moreover, little data on the relative importance of each transmission channel are available.
Contact tracing research by the Dutch GGD (RIVM, 2020a) provides estimates. In contact tracing research, positively tested individuals indicate in which setting they suspect they contracted the virus.
These data have two flaws. First, people are uncertain where they were infected. Second, people choose from a list of potential infection locations, but the only consumption-related activity on the list is the hospitality sector.
From May fourth to August eighteenth, when government regulation was negligible, the workplace accounted for 15.8% of infections in the survey, and the hospitality sector for 1.9% of infections. The former indicates that the estimate of the relative importance of the workplace in viral transmission is accurate. If consumption is approximately as important as the workplace, Eichenbaum et al. (2020) estimate 𝜋) and 𝜋' accurately. To obtain a better picture of the relative importance of the transmission channels, I contacted the RIVM, but they had no time. Therefore, the values for 𝜋), 𝜋', and 𝜋+ used in the simulations are equal to those reported in Eichenbaum et al. (2020).
Eichenbaum et al. base the calibration of 𝜑 on the observation that only symptomatic people stop working. Since the China Center for Disease Control and Prevention finds that eighty percent of infected individuals are asymptomatic, they set 𝜑 equal to 0.8. Byambasuren et al. (2020), however, find in a recent meta-analysis that only seventeen percent of infected individuals are asymptomatic.
Moreover, Buitrago-Garcia et al. (2020) find an estimate of twenty percent. Therefore, 𝜑 equals 0.2, the more conservative of the two estimates.
Eichenbaum et al. (2020) appear to set 𝜀 equal to 0.001 arbitrarily. I keep this value for comparability.
Finally, two parameters are new in this study, 𝜌 and 𝜋.. These parameters determine the productivity of individuals with long COVID and the longevity of the symptoms. The British Office for National Statistics (2021) finds that 82% of individuals with long COVID are slightly limited in their
day-to-day activities. The other 18% are severely limited and cannot work. Thus, I set 𝜌 equal to 0.82.
Furthermore, they find that 21% of long COVID patients suffer from long COVID symptoms after five weeks and 13.7% after twelve weeks. These estimates are independent of symptoms experienced during the acute phase of the infection. Periodic recovery 𝜋. = 0.2 fits these data reasonably, assuming that recovery in week zero is possible. It yields that 33% of individuals suffer from long COVID symptoms after five weeks and 7% after twelve weeks. Increasing 𝜋. leads to a better approximation after twelve weeks but a worse approximation after five weeks, and vice versa.
Table 1 summarizes the calibration of the models.
Parameter Description Calibration
heterogeneous nonrational model
Calibration perfect foresight model and
homogeneous nonrational model
𝜃 Disutility of labor 0.001275 0.001275
𝐴 Labor productivity 39.835 39.835
𝛽 Discount rate 0.96)/<' 0.96)/<'
𝜋, Periodic mortality rate 0.0062 ∙ 7/18
(average)
0.005 ∙ 7/18
𝜋- Periodic recovery rate 0.9938 ∙ 7/18
(average)
0.995 ∙ 7/18 𝜋) Viral transmission during
consumption
7.8408 ∙ 10=> 7.8408 ∙ 10=>
𝜋' Viral transmission at work 1.2442 ∙ 10=? 1.2442 ∙ 10=?
𝜋+ Viral transmission in random encounters
0.3901 0.3901
𝜑 Infected productivity 0.2 0.8
𝜀 Fraction of initial infections 0.001 0.001
𝜌 Long COVID productivity 0.82 −
𝜋. Long COVID periodic recovery rate
0.2 −
Table 1: Calibration of the models
V. Empirical validation
This section assesses the empirical relevance of the perfect foresight model and the heterogeneous nonrational model by comparing the models’ predictions with and without a lockdown to the US and Dutch data. This section does not include the homogeneous nonrational model since that model exists only to isolate the effect of nonrational expectations. First, I compare the perfect foresight model’s predictions to the US data. Then, I compare the heterogeneous nonrational model’s predictions to the
Dutch data. The lockdown imposed by the US and Dutch governments was likely less restrictive than the ones in the models. Therefore, the numbers indicated by the US and Dutch data should lie in between the models’ predictions with and without a lockdown.
Table 2 presents the models’ predictions and the US and Dutch data for aggregate consumption, aggregate hours, output, the number of infected individuals, and the number of deceased individuals.
Consumption % a Hours % b GDP % c Infected rate % d Mortality e Perfect foresight model – no lockdown
−4.7% −4.7% −4.7% 0.322% 890 000
Perfect foresight model – with lockdown
−17.0% −17.0% −17.0% 0.169% 690 000
US data
−2.9% f 1.0% g −3.5% h 0.091% i j 604 069 i
Heterogeneous nonrational model – no lockdown
−2.2% −0.7% −2.2% 0.155% 46 625
Heterogeneous nonrational model – with lockdown
−10.2% −5.9% −10.2% 0.108% 42 625
Dutch data
−4.9% f −2.5% k −4.3% l 0.075% m 17 745 n
Table 2: the perfect foresight model’s and the heterogeneous nonrational model’s predictions and the US and Dutch data for aggregate consumption, aggregate hours, output, the number of infected individuals, and the number of deceased individuals.
a - Decrease in aggregate consumption in the first year of the pandemic (2020Q2-2021Q1) relative to the previous year (2019-Q2-2020Q1)
b - Decrease in aggregate hours in the first year of the pandemic relative to the previous year
c - Decrease in output in the first year of the pandemic relative to the previous year. In the models, output equals consumption.
d - Most new cases in a single day as a percentage of the pre-epidemic population. The US and Dutch data use confirmed cases up to June 30th, 2021, and likely underestimate the true number of cases. I use most new cases in a day instead of the peak number of infected since there is no US or Dutch data on the total number of infected individuals.
e - Total number of deaths at the end of the pandemic. The US and Dutch data use confirmed cases on June 30th, 2021, and likely underestimate the true number of deceased individuals.
f - OECD (2021a)
g - Bureau of Labor Statistics (2021) h - US Bureau of Economic Analysis (2021) i - The New York Times (2021)
j - assuming 331 million American citizens before the pandemic k - Centraal Bureau voor de Statistiek (2021a)
l - OECD (2021b). Dutch GDP is based on a nominal GDP series. Given inflation, this figure underestimates the fall of real GDP.
m - Rijksoverheid (2021a) n - Rijksoverheid (2021b)
First, in the perfect foresight model, the aggregate consumption drop exceeds the US consumption drop even without a lockdown. The output drop predicted by the model also exceeds the US real GDP drop. Thus, the model predicts too large a fall in consumption.
Second, the perfect foresight model’s results indicate that aggregate labor dynamics are identical to aggregate consumption dynamics, even though the model should underestimate the fall of aggregate labor - see Appendix B. In any case, the model predicts a decline in aggregate hours in the first year of the pandemic. The US data, on the contrary, indicate an increase in aggregate hours, leading to a qualitative error in the model’s predictions.
Third, the largest increase in the number of cases indicated by the US data is below the range predicted by the model. Even though the data underestimate the true number of cases, the range indicated by the model starts two times higher than the data indicate. Thus, the model likely overestimates the true number of cases.
Finally, the US data indicate that the number of deceased individuals lies closely below the range indicated by the model. Given that (i) the pandemic is not yet over, (ii) the US data underestimate the true number of deceased individuals, and that (iii) the calibration of the perfect foresight model does not include the elderly, the model’s prediction for the number of deceased individuals appears accurate.
In the heterogeneous nonrational model, both the aggregate consumption fall and the GDP fall indicated by the Dutch data lie in the range predicted by the model. The real GDP drop is larger than the nominal GDP drop but likely still falls within the model’s predicted range.
Moreover, the fall in aggregate labor according to the Dutch data lies within the range predicted by the model. Even though the model underestimates aggregate hours by assumption, the number indicated by the Dutch data would still fall into the range predicted by the model if the model’s prediction for aggregate hours equaled its prediction for aggregate consumption.
Furthermore, the largest increase in the number of cases indicated by the Dutch data lies below the range predicted by the model. Since the data underestimate the true number of cases, the model’s prediction may be accurate.
Finally, the number of deceased individuals indicated by the Dutch data lies far below the range predicted by the model. The Dutch data likely underestimate the true number of individuals that passed away because of COVID, however. Statistics Netherlands (2021c) suspects that by February 2021, 28 000 people had died of COVID in the Netherlands. Even then, the range predicted by the model is too high.
In sum, the heterogeneous nonrational model’s predictions appear to replicate the Dutch data more closely than the perfect foresight model’s predictions replicate the US data, except for the number of deceased agents. Specifically, nonrational expectations lead to better predictions for aggregate consumption and aggregate hours, while the recalibration of 𝜑 leads to better predictions for the number of infected.
VI. Results
This section analyzes the simulation results of the perfect foresight model, the homogeneous nonrational model, and the heterogeneous nonrational model. Subsection A compares the simulation results of the perfect foresight model with those of the homogeneous nonrational model when the government institutes no lockdown. Subsection B compares the simulation results of the homogeneous nonrational model with those of the heterogeneous nonrational model when the government institutes no lockdown.
Finally, subsection C discusses the simulation results of the heterogeneous nonrational model with a lockdown.
Appendix A contains technical details.
A: the perfect foresight model and the homogeneous nonrational model
Nonrational expectations create new economic dynamics and form the first step in addressing the criticisms on the perfect foresight model. To isolate their effects, this subsection compares the perfect foresight model’s simulation results to the homogeneous nonrational model’s simulation results.
Figure 2 presents the simulation results of the perfect foresight model and the homogeneous nonrational model.
Figure 2: Simulation results of the perfect foresight model (blue curves) and the homogeneous nonrational model (red curves)
The blue curves show the results of the perfect foresight model, as presented in Eichenbaum et al. (2020). The red curves show the results of the homogeneous nonrational model.
Epidemiological dynamics are similar across models. In both models, the number of infected peaks at 5.24% of the initial population around week 33. Also, the number of recovered stabilizes at 53.31% of the initial population in the perfect foresight model and at 55.5% in the homogeneous nonrational model. Finally, the number of deceased agents stabilizes at 0.27% of the initial population in the perfect foresight model and at 0.29% in the homogeneous nonrational model.
Furthermore, the bottom-middle and bottom-right panels show that the pandemic induces a recession in both models. The recession in the homogeneous nonrational model is less severe than in the perfect foresight model, as consumption falls by 3.73% relative to the pre-epidemic steady state at the pandemic’s peak instead of 9.8%. As a result, aggregate consumption decreases by 1.87% in the first year of the pandemic in the homogeneous nonrational model, compared with 4.7% in the perfect foresight model. The recession results from lower productivity among the infected, the death toll of the pandemic, and the shifting behavior of susceptible agents to prevent infection.
Susceptible agents reduce consumption since infection could result in death and certainly results in temporarily decreased productivity. As a result, the lifetime utility of being susceptible in period 𝑡 + 1 is higher than the lifetime utility of being infected in period 𝑡 + 1. Therefore, susceptible agents optimally cut back economic activity to decrease 𝜏%,! according to their value function. Because of this mechanism, there is a reaction in susceptible agents’ behavior even without a government-imposed lockdown.
Moreover, of the three mechanisms, the shifting behavior of susceptible agents is the principal determinant of the fall in aggregate consumption. It is responsible for 68.3% of the decline in aggregate consumption at the pandemic’s peak, whereas the decreased productivity of infected agents is responsible for 28.3% of the decline and the death toll for 3.4% of the decline. To determine these shares, I calculate how much aggregate consumption would have fallen if the agents in a health state, e.g., all susceptible agents, consumed the pre-epidemic level.
Aggregate hours worked fall by 2.70% in the homogeneous nonrational model at the pandemic’s peak and by 9.8% in the perfect foresight model. Aggregate hours exhibit a smaller decline than consumption because the infected consume less than in the pre-epidemic steady state but work as many hours. This effect inexplicably - see Appendix B - appears absent in the perfect foresight model.
The two differences in the models’ expectations of susceptible agents may explain the discrepancies in economic dynamics. First, in the perfect foresight model, agents form expectations rationally, hence expectations by definition equal the dynamics that materialize. In the homogeneous nonrational model, the materialized dynamics from the perfect foresight model are the basis for agents’
expectations. The resulting dynamics in the nonrational homogeneous model are similar to the resulting dynamics in the perfect foresight model. Thus, expectations in the homogeneous nonrational model are
similar to the resulting dynamics. Therefore, epidemiological expectations unlikely causes discrepancies between the perfect foresight model and the homogeneous nonrational model. Second, agents expect that they will maintain their current consumption and labor in all future periods. How does this assumption influence their behavior? Figure 3 presents a graphical illustration of susceptible agents’
expectations relative to what materializes.
Figure 3: Graphical illustration of susceptible agents' consumption expectations in period 𝑡.
The solid red curve in Figure 3 represents the consumption of agent 𝑖 that materializes during the pandemic. The dashed blue line represents consumption expectations formed in period 𝑡. Period 𝑡 is an arbitrary period soon before the pandemic’s peak. Because the pandemic’s peak is in the future, agent 𝑖 will choose to further lower consumption in upcoming periods. In period 𝑡, however, they expect to maintain their current consumption. Thus, consumption expectations for upcoming periods are too high.
The red-shaded area represents consumption overestimation. After the pandemic’s peak, agent 𝑖 will choose to increase their consumption so that realized consumption will exceed consumption expectations formed in period 𝑡. From this point on, consumption expectations are too low. The blue- shaded area represents consumption underestimation. Because the effect of the pandemic on consumption has faded chiefly after 50 weeks and agents optimize for the first 120 weeks after the start of the pandemic, the blue-shaded area is larger than the red-shaded area. Thus, agent 𝑖 net underestimates their future consumption in period 𝑡. Because mistakes further in the future are discounted away more, the net mistake is smaller than it appears.
Generalizing, the growth of agent 𝑖’s consumption is nonnegative after the pandemic’s peak. As a result, they underestimate future consumption in all periods after the pandemic’s peak: there is no red- shaded area, while the blue-shaded area is large. In the earliest days of the pandemic, however, consumption expectations are close to pre-epidemic levels such that agent 𝑖 net overestimates future
consumption. The location of the tipping point, i.e., the first period in which agent 𝑖 net underestimates future consumption, depends on the discount factor, the number of periods, and the propagation of the virus. Thus, agent 𝑖 net underestimates future consumption and labor, except before the tipping point.
Furthermore, after the tipping point but before the pandemic’s peak, net consumption expectations become less accurate every period, as the red-shaded area becomes increasingly smaller while the blue-shaded area becomes increasingly larger. After the pandemic’s peak, there is no red- shaded area, but the blue-shaded area becomes increasingly smaller. Thus, expectations first become less accurate, then more accurate. Expectations are the least accurate at the pandemic’s peak.
Accordingly, Figure 2 indicates the largest discrepancy between the perfect foresight and the homogeneous nonrational model in that period.
To confirm these conjectures, Figure 4 presents the net mistake in susceptible agents’
consumption expectations.
The red line in Figure 4 shows the discounted lifetime difference between the expected and materialized consumption of agent 𝑖. When the line is in positive territory, agent 𝑖 net overestimates their future consumption, whereas agent 𝑖 net underestimates future consumption when the line is in negative territory. Figure 4 confirms that agents net underestimate future consumption while the pandemic rages. Furthermore, it shows that the tipping point occurs in week 15 of the pandemic and confirms that, after the tipping point, consumption expectations first become less accurate and then more accurate. Finally, Figure 4 reveals that consumption expectations have near-perfect accuracy after week 70.
Figure 4: The net mistake in susceptible agents’ consumption expectations in the homogeneous nonrational model.
The net underestimation of future consumption and labor has two effects. First, agent 𝑖 net underestimates future momentary utility. As a result, they underestimate 𝑈%,!2)6 and insufficiently drop consumption and labor in period 𝑡. Second, they underestimate 𝜏%,!2: for all 𝑣 > 0. Since susceptible lifetime utility exceeds infected lifetime utility in every period, underestimation of future infection probabilities causes agent 𝑖 to overestimate 𝑈%,!2)6 and to lower consumption and labor in period 𝑡 too much. The results in Figure 2 show that aggregate consumption and labor exhibit a weaker response to the pandemic, thus indicating that the first channel dominates the second.
For the infected, recovered, and deceased, expectations do not influence the consumption and labor decision. Therefore, flawed expectations do not cause decisional mistakes. As the pandemic progresses, the number of susceptible agents dwindles, meaning that aggregate behavior in the economy tends towards optimal behavior.
B: the heterogeneous nonrational model without a lockdown
This subsection incorporates heterogeneity, long COVID, and recent Dutch data for the calibration on top of nonrational expectations. I compare the simulation results of the homogeneous nonrational model to those of the heterogeneous nonrational model without a lockdown to analyze how these features affect economic and epidemiological dynamics.
Figure 5 presents the simulation results of the heterogeneous nonrational model.
Figure 5: Simulation results of the heterogeneous nonrational model
The red curves show the simulation results for the heterogeneous nonrational model without a containment policy. Subsection C discusses the blue curves, which show the results with a containment policy. The number of infected reaches 2.4% of the population at its peak, which occurs in week 40 of the pandemic. The number of susceptible agents stabilizes at 56.0% of the initial population, the number of recovered at 43.7% of the initial population, and the number of deceased at 0.32% of the initial population. The number of agents with long COVID reaches 3.52% at its peak. Furthermore, aggregate consumption is 3.98% lower at the pandemics’ peak than in the pre-epidemic steady state and aggregate labor 1.3%. As a result, aggregate consumption decreases by 2.2% in the first year of the pandemic.
Compared with the homogeneous nonrational model, the number of infected and aggregate consumption at the pandemic’s peak are lower in the heterogeneous nonrational model, while aggregate labor is higher. By the end of the pandemic, the number of deceased and the number of susceptible agents are higher, but the number of recovered agents is lower. Thus, there are three main differences with the homogeneous nonrational model to explain. First, what is the mechanism underlying the decrease in the number of infected? In turn, this decrease explains the increase in the number of susceptible agents and the decrease in the number of recovered agents at the end of the pandemic.
Second, by which mechanisms has the number of deceased agents increased? Third, which mechanisms underlie the deepening of the aggregate consumption trough?
The decrease in the number of infected agents
The decrease in the number of infected agents may have two causes. First, the increase in average mortality 𝜋, and the decrease in infected productivity 𝜑 amplify the adverse effects of the virus, inducing more effort to decrease 𝜏%,! and resulting in less economically active susceptible agents.
Second, because a decrease in 𝜑 lowers the consumption of infected agents, it decreases the probability of being infected per unit of consumption for susceptible agents, resulting in fewer infections.
With respect to the first reason, running the heterogeneous nonrational model with average mortality 𝜋,= 0.005 reveals that average mortality has little influence on epidemiological dynamics.
The effect of 𝜑 on epidemiological dynamics through amplifying the virus’s adverse effects is more complex since it introduces reverse causality. Susceptible agents cut back economic activity more if the adverse effects of the virus are amplified, resulting in fewer infections. In turn, susceptible agents cut back economic activity less since the number of infected agents decreases. Comparing the consumption of susceptible agents between the homogeneous nonrational model and the heterogeneous nonrational model reveals that susceptible agents cut back consumption by 3.5% at the pandemic’s peak in the homogeneous nonrational model and by 1.6% in the heterogeneous nonrational model. This decrease in consumption across models is proportional to the decrease in the number of infected agents at the pandemic’s peak. Therefore, it is unlikely that a shift in the behavior of susceptible agents owing to 𝜑 caused the decrease in the number of infected agents.
With respect to the second reason, running the heterogeneous nonrational model with 𝜑 = 0.8 reverts the number of infected at the pandemic’s peak to 5.2%. Thus, the decrease in the probability of being infected per unit of consumption resulting from the decrease in 𝜑 causes the decrease in the number of infected agents.
The increase in the number of deceased agents
Second, I turn to which mechanisms affect the number of deceased agents in the heterogeneous nonrational model relative to the homogeneous nonrational model. As explained above, the decrease in 𝜑 lowers the number of infected agents, resulting in fewer deceased agents. The increase in average mortality, however, more than offsets this decrease, leading to an increase in the number of deceased agents.
The deepening of the aggregate consumption trough
Third, which mechanisms underlie the deepening of the aggregate consumption trough? Relative to the homogeneous nonrational model, the aggregate consumption trough depends on the aggregate
