Knowledge, germs, and output.

This paper studies the equilibrium and the social optimum in an economy where knowledge diffusion interacts with disease transmission. Knowledge increases productivity and is diffused through learning. A learner chooses the intensities in normal learning, isolated learning and production. Normal learning is more effective than isolated learning but requires a learner to contact a teacher. A higher intensity in normal learning increases a learner's contact rate with a teacher, thereby speeding up both knowledge diffusion and the transmission of an infectious pathogen. An infection reduces productivity and possibly results in death. Calibrating the pathogen to Covid-19, the model shows that the unexpected arrival of the pathogen induces a susceptible learner to adjust the normal learning intensity in a V-shaped pattern over time. Aggregate output also follows V-shaped adjustments. Switching from the equilibrium to the social optimum reduces infections and deaths substantially and increases social welfare. I also examine temporary lockdowns in the equilibrium. Crown

Introduction

Knowledge diffusion and disease transmission have been intertwined throughout human history. As Diamond (1999) argued convincingly, knowledge of agriculture (farmer power) spread from early civilizations together with infectious pathogens in domesticated animals. In the present time, the primary use of knowledge is for industrial production and services instead of agriculture. This may change the sources of knowledge and infectious pathogens, but it does not change two fundamental features of their diffusion/transmission. First, knowledge and infectious pathogens are both non-rivalrous. One is a public good and the other a public bad. Second, individuals ignore the externalities generated by their actions in the transmission of knowledge and diseases. These features put the interactions between knowledge diffusion and disease transmission squarely in the domain of economic analysis. However, the economic literature on epidemiology has largely ignored knowledge diffusion (see a partial review later). I construct a model to focus on the dynamic tradeoff between knowledge acquisition and disease infections.

The model economy has continuous time, a unit measure of risk-neutral individuals, two knowledge levels and one pathogen. The frontier knowledge has higher productivity than the baseline knowledge. An infection of the pathogen reduces the effectiveness in learning and production, and possibly results in death. An individual can divide a unit flow of time (intensity) among normal learning, isolated learning and production. Production does not require a contact between individuals but a contact can occur nevertheless. Isolated learning prevents a contact but is less efficient than normal learning. Normal learning requires a contact, where a learner randomly contacts a teacher who has the frontier knowledge. The contact rate increases in the learner's effective intensity. In such a contact, a learner successfully acquires the frontier knowledge with a positive probability. Independently of the learning outcome, an infected individual can transmit the pathogen to a susceptible individual in the meeting. A recovery from an infection can come with or without immunity. An infection increases the death rate. When an individual dies, a newborn enters the economy with the baseline knowledge and draws the disease status as either susceptible or infected.

The key mechanism of the model is that increasing the intensity of normal learning increases knowledge diffusion and disease transmission simultaneously by increasing the contact rate between learners and teachers. Thus, optimal choices of learning intensities involve a tradeoff between knowledge acquisition and the risk of infections. This tradeoff is dynamic and depends on the endogenous distribution of individuals over the levels of knowledge and disease statuses. A higher intensity in normal learning increases the fraction of infected individuals in the population, thereby increasing the risk of infection for an uninfected and reducing the benefit of normal learning in the future. This forward-looking consideration affects how much and how long a learner wants to postpone normal learning or to substitute it into isolated learning in the time of a pandemic.

To illustrate the dynamic tradeoff concretely, I calibrate the pathogen to Covid-19 and start the economy with a moderate ratio of learners to teachers. After the pathogen arrives unexpectedly, aggregate output drops significantly in a short time and the death toll from infections increases quickly. After being infected, a learner increases the normal learning intensity. In contrast, a susceptible learner adjusts the normal learning intensity in a V-shaped pattern over time. The learner shifts the intensity first from normal learning to production, then from normal learning and production to isolated learning, and finally from isolated learning back to normal learning and production. These adjustments in intensities may slow down the pathogen transmission, but increase the decline in aggregate output. Active cases of infections reach the peak quickly and then decline. Aggregate output also follows V-shaped adjustments over time.

The transmission of the pathogen feeds on knowledge diffusion. To illustrate this interaction, I compare two alternative economies with the baseline. One alternative economy starts in the steady state, where the ratio of learners to teachers is low. This economy does not grow, and learning activities are low. As a result, active cases of infections are much lower than in the baseline, and the pathogen does not affect the economy much. In another alternative, the economy starts with a higher ratio of learners to teachers than in the baseline. In the absence of the pathogen, this economy has faster diffusion of knowledge and faster growth in the short run and the medium run than the baseline economy. After the pathogen arrives in this economy, active cases of infections have a higher peak and the peak is reached earlier than in the baseline economy. The pathogen induces a substantially larger fall in aggregate output than in the baseline.

The baseline model abstracts from the policy controls implemented in the Covid-19 pandemic, such as testing, social distancing, mandatory wearing of masks and enforcing quarantines. A purpose of this abstraction is to show how the pathogen can be transmitted and interact with learning in the absence of such controls. This helps evaluate the welfare gain from the policy controls. I conduct two analyses related to the controls. The first is to characterize the social optimum under the assumption that the planner can costlessly distinguish different types of individuals. The second analysis is to examine the equilibrium with temporary lockdowns that are uniformly enforced in the entire population. In both analyses, I measure social welfare by subtracting from aggregate output the value of each life lost to the infection, which is calibrated in section 5.1.

The equilibrium is socially efficient without the pathogen but inefficient with the pathogen. With the pathogen, the social optimum requires an infected learner to devote all intensity to isolated learning, and none to normal learning or production, until infections become small. Since the infection rate is low, this allocation requires a susceptible learner to devote almost the same intensity to normal learning as without the pathogen. These socially efficient choices reduce infection-induced deaths to a miniscule level, with little reduction in output. Relative to the equilibrium, the welfare gain from the social optimum is 2.1% of permanent output or, equivalently, 44% of the first-year aggregate output.

I analyze two lockdowns each lasting for 19 weeks. The early lockdown starts before the peak of new infections in the baseline equilibrium, and the late lockdown starts after the peak. The early lockdown delays the peak of new infections and significantly reduces the number of deaths from infections. Output falls by less after the pathogen arrives but the reduction in output lasts for a longer time than without the lockdown. The late lockdown does not delay the peak of infections significantly, but reduces cumulative deaths by a similar amount to the early lockdown. Both lockdowns improve social welfare by about a half of the gain from the social optimum.

One might have asked: since knowledge diffusion is relatively slow, how can it play an important role in a fast spreading pathogen like Covid-19? The answer is that the infectivity of a pathogen interacts with knowledge diffusion. When a pathogen is highly infectious, even a small change in learning intensities can have a large effect on how fast the pathogen spreads by affecting the contact rate. This is why the number of infections changes significantly with the initial distribution of knowledge in the population.

It is important to emphasize that knowledge includes all non-physical elements that increase productivity but need time to acquire. Evidently, these elements are not limited to knowledge acquired in formal settings such as schools and universities. Even though I use the terms “teachers” and “learners” for the lack of concise alternative phrases, one should not interpret learning in this paper as activities limited to formal settings. For example, learning on the job is an example of learning in this paper. Section 2.1 will elaborate further on the normal learning process.

The paper is organized as follows. The next subsection reviews the literature. Section 2 constructs the baseline model, and section 3 analyzes the equilibrium. Section 4 presents the results of the calibrated model. Section 5 examines the social optimum and the equilibrium with lockdowns. Section 6 concludes the main text. The appendices provide proofs and sensitivity analyses. The supplementary appendix describes the procedures of calibration and computation, and provides additional sensitivity analyses.

The literature

On knowledge diffusion, this paper follows Lucas and Moll (2014) to focus on the time allocation between learning and production as modeled by Ben-Porath (1967). Lucas and Moll (2014) analyze the balanced growth path by assuming that knowledge is unbounded. In contrast, the current model has only two levels of knowledge and so growth dies down eventually. By analyzing the transition to the steady state, this paper improves the understanding of how the stage of knowledge diffusion interacts with disease transmission. Another difference from Lucas and Moll (2014) is that contacts in the learning process are between learners and teachers. A learner never meets someone with lower or the same knowledge, a possibility existing in Lucas and Moll (2014). Jovanovic and Rob (1989) first use a random meeting model to show that the distribution of knowledge in the population is important for knowledge diffusion. However, they do not model the choice of search/learning intensity. More broadly, the research on the relationship between economic growth and the population goes back at least to Malthus (1798).

In epidemiology, the most commonly used model is the SIR model by Kermack and McKendrick (1927) that describes the dynamics of the population among the susceptible (S), infected (I) and recovered (R) states. An extension is SEIR that adds an exposed (E) state between the susceptible and the infected state (see Linka et al., 2020, for a use of this model). There are also extensions allowing for a recovery to come with temporary immunity that can be lost with a positive probability (e.g., Anderson and May, 1979). Atkeson (2020) uses the SEIR model to construct scenarios of the Covid-19 pandemic. Emphasizing asymptomatic transmission of Covid-19, Berger et al. (2020) advocate testing and case-dependent quarantine to control the rate of exposure. Given the contact rates, the law of motion of the distribution of individuals in the current paper is similar to that in the SEIR model. I allow for the possibility that a recovery comes with no immunity and hence is susceptible to repeat infections, possibly by variants of the pathogen.

Incorporating economic choices is necessary for understanding disease transmission in the human population. Because choices affect the contact rates, the infectivity of a pathogen is endogenous. Also, choices are often based on forward-looking considerations that induce dynamic tradeoffs between gains and losses. Moreover, economic choices are necessary for assessing the economic damage of a pandemic and the welfare effect of policies. In particular, a model needs to take into account how individuals respond to a policy.

The economic literature on epidemics and, especially on Covid-19, has been expanding rapidly. Greenwood et al. (2019) study an equilibrium model of HIV/AIDS infections. Keppo et al. (2020) advocate that the SIR model should incorporate individuals' choices in order to explain the spread of an epidemic, such as the swine flu and Covid-19. However, they model individuals' choices as functions of only the distribution of individuals in the population, which do not capture the forward-looking behavior discussed above. Incorporating the forward-looking behavior, Farboodi et al. (2020) demonstrate how the dependence of individuals' decisions on future values changes the dynamics of a pandemic. Alvarez et al. (2020) analyze the planning problem of a lockdown.

In a macro context, Eichenbaum et al. (2020) show that individuals cut back work and consumption in an epidemic, which deepens a recession. Acemoglu et al. (2020) emphasize heterogeneity among different groups in the risk of infections. They show that an optimal containment policy, such as a lockdown, should target old-age individuals. Also emphasizing the age heterogeneity, Brotherhood et al. (2020) construct a dynamic macro model to incorporate the choices of formal work, telework and social activities. They demonstrate the importance of testing and contact tracing infections. Kapicka and Rupert (2020) focus on heterogeneity in the employment status instead of age to study the consequence of the Covid-19 pandemic on unemployment. Hall et al. (2020) calculate the welfare cost of the Covid-19 pandemic by computing the consumption equivalent of deaths from Covid-19 infections while fixing individuals' choices.

This paper is related to all the papers above, but the emphasis on the interactions between knowledge diffusion and disease transmission is new. The paper captures the dynamic tradeoff between knowledge acquisition and disease infections. Because this tradeoff depends on the endogenous distribution of individuals over the disease statuses and knowledge levels, the equilibrium is not block recursive, in contrast to Shi (2009) and Menzio and Shi (2010). Non-block recursivity makes the equilibrium intractable analytically. As a result, the quantitative analysis is an important component of this paper.

The model

Model environment

Time is continuous. There is a unit measure of individuals in the economy who are risk neutral with a rate of time preference r. An individual is endowed with L units of the flow of time (intensity) that can be allocated to learning and production. L is normalized to 1 in the benchmark model but will be reduced in the analysis of a lockdown later. There are two levels of knowledge: the baseline indexed by 0 and the frontier indexed by 1. The productivity of knowledge is , where . An individual with the baseline knowledge is a learner and an individual with the frontier knowledge is a teacher.

A pathogen can be transmitted in the population. An individual's disease status is denoted as .1 An individual with is currently not infected and is susceptible to an infection. An individual with is infected by the pathogen, who is infectious with a positive probability. An individual with is immune to the pathogen. A susceptible individual gains immunity at a rate ϕ. This is a crude approximation for the possibility of being vaccinated effectively.2 A recovery from an infection can be one of the two types. One is a recovery with immunity, which occurs at a rate μ and makes the individual immune permanently. The other is a recovery without immunity, which occurs at a rate ρ and changes the individual to . Such a recovery leaves the individual susceptible for a repeat infection. Allowing for is a tractable way to capture the clinical finding (e.g., Long et al., 2020) that after recovering from Covid-19, some individuals lose the antibody even in the early convalescent phase. It also captures the possibility that the pathogen can mutate into an infectious variant.

An individual in the state is referred to as an individual az. An individual az dies at the rate , where . When an individual dies, a newborn enters the economy with knowledge and draws z randomly from . The probability of drawing is equal to the fraction of susceptible individuals in the population without immunity.3 Thus, although knowledge cannot be inherited, pathogens can pass from one generation to the next with a positive probability.

The disease status affects an individual's effectiveness in learning and production. Denoted as , the effectiveness iswhere is a constant. The symbol is kept for convenience. The effectiveness is assumed to be the same in production as in learning, and so the productivity of an individual az is .

Only individuals with the baseline knowledge spend time in learning. Learning can be normal or isolated. Normal learning requires a learner to be in close proximity to a teacher, which opens a path for the pathogen to be transmitted between the two. Isolated learning does not require such a contact but is less effective. As in Ben-Porath (1967), learning can be in a formal environment, such as a school, or informal, such as a workplace. It may be helpful to map various labels here to reality. In a formal learning environment, the roles of a teacher and a student are apparent. In an informal learning environment, a teacher can be a supervisor or a senior co-worker and a learner (student) a relatively new worker. Even if the senior co-worker is not assigned to teach a new worker, the latter may need to spend time observing how the senior co-worker completes a task.4 For normal learning, a learner can go to a classroom or a training session, or follow a teacher through a task. For isolated learning, a learner can watch lessons on the Internet or observe in a distance how a task is completed. Because isolated learning has less interaction between a learner and a teacher, it is less effective than normal learning.

Denote the flow of time (intensity) a learner spends in normal learning as ℓ and in isolated learning as u. Let me refer to ℓ as the normal intensity and to u as the isolated intensity. The effective normal intensity is , where ε is the individual's effectiveness. A learner with randomly meets an individual with the frontier knowledge at a rate , where , , and .5 Such a contact is necessary, but not sufficient, for acquiring the frontier knowledge. In fact, knowledge diffusion may be significantly slower than disease transmission. Conditional on meeting a teacher, a learner acquires the frontier knowledge with the probability .6 As in Lucas and Moll (2014), the meeting (contact) rate depends on an individual's own effective learning intensity but not on the teacher-learner ratio in the economy. This assumption captures the important feature that knowledge is non-rivalrous. Someone's learning does not crowd out other individuals' ability to learn. In contrast to Lucas and Moll (2014), a learner only meets a teacher and not another learner. This assumption is reasonable in the current setting where there are only two knowledge levels at any given time.

A contact can transmit pathogens regardless of which side initiates the contact and of whether the learner succeeds in learning. In a meeting between two individuals with the disease statuses I and S, the susceptible individual contracts the pathogen with a probability . The probability captures the possibility that a contact with an infected individual may not result in an infection either because the infected individual has passed the incubation period or because of luck.

In contrast to normal learning, isolated learning does not transmit the pathogen. With an effective isolated intensity , a learner acquires the frontier knowledge with the probability , where , and . Isolated learning is less efficient than normal learning, as will be made precise by Assumption 1 later. If the pathogen did not exist, a learner would allocate all learning intensity to normal learning.

Production can also transmit the pathogen. A learner generates an output flow, , and a teacher generates εLγ. Although production does not require a match between individuals, a producer may still come into contact with another producer. The contact rate is equal to a constant, , times the producer's effective production intensity. The type of the contacted individual is a random draw among producers, where each producer is weighted by the effective the production intensity. As in normal learning, a contact in production transmits the pathogen with the probability ψ between a susceptible and an infected individual. I assume for all z. This captures realism that normal learning requires at least as intensive human interactions as production and so the contact rate per intensity is at least as high in normal learning as in production. However, it is possible that the production intensity can be sufficiently higher than the normal intensity for the actual contact rate to be higher in production than in normal learning.

The modeling of the pathogen transmission has two implicit assumptions: an infection is asymptomatic, and a person cannot prevent others from contacting. Both assumptions can be relaxed at the cost of complicating the model. For example, one can introduce a cost to test for an infection or to reduce contacts by others.7 I will later examine the effects of a lockdown. The modeling of knowledge diffusion, following Lucas and Moll (2014), abstracts from the payment from a learner to a teacher. Such a payment does not affect the joint surplus in a meeting. How the payment affects the choice of the learning intensity depends on the pricing mechanism. If the pathogen is absent in the economy, the pricing mechanism satisfying the Hosios (1990) condition enables the equilibrium to achieve the social optimum. With the meeting technology specified above, this condition requires a learner to take all the surplus in a match (see Proposition 5.1).8 In the presence of the pathogen, the Hosios condition may need modifications. Although it is interesting to explore such modifications, I abstract from them to simplify the analysis.

The aggregate state of the economy at any time t is the distribution of individuals over , denoted as , where is the measure of individuals with knowledge a and the disease status z at time t. Denote the measure of all individuals with knowledge a at t asSimilarly, denote the measure of individuals with the disease status z asThe average death rate in the population at time t is:Because the population is constant, is also the measure of newborns at time t. As assumed earlier, a newborn at time t draws with the probability .

The infection rate of the pathogen depends on the composition of individuals in the economy and on the activity an individual undertakes. Let denote the ratio of individuals az in the population to teachers at time t. Then,To calculate a teacher's infection rate, denote as a learner 0z's contract rate in normal learning at time t under the optimal intensity . The measure of contacts in normal learning made by all such learners is . Thus, the rate at which a teacher is contacted by a learner with z isA susceptible teacher is infected by a learner in normal learning at the rate , and the measure of such infections at t is . Similarly, a susceptible learner is infected in normal learning at the rate , and the measure of such infections at t is . At t, the measure of active cases of infections is and the measure of additional deaths caused by infections is .

In production, the effective intensity is for an individual 0z and for an individual 1z. Denote as the ratio of individuals az's effective production intensities to the total effective the production intensity. That is,The asterisk indicates the optimal learning intensities. Denote . Conditional on a contact in production, the contacted individual is 0z with the probability and 1z with the probability .9 The contacted individual is an infected individual with the probability and a susceptible individual with the probability .

Value functions and optimal decisions

Value functions and aggregate output are normalized by the frontier productivity γ. Consider an individual az at time t. At any time , the individual's knowledge is and the disease status is . The productivity of the individual's knowledge is and the individual's output is , both of which are normalized by γ. Denote the value function as , where the dependence on the aggregate state is abbreviated as t. The value function represents the following expected utility from t onward:The expectations are taken over the shocks and the meeting outcomes during the lifetime, while the death probability is already incorporated in the effective discount rate . For a learner 0z, the Bellman equation for the value function is:The constraints on the maximization problem are and . On the left-hand side of (2.7) is the “permanent income” of the individual. The right-hand side is the sum of capital gains and income flows. The first term is the capital gain caused by the change in the aggregate state. The term with the indicator exists only if the individual is an infected type, i.e., if . This term represents the expected change in an infected learner's value from a recovery. A recovery with immunity occurs at the rate μ, which changes the value to , and a recovery without immunity occurs at the rate ρ, which changes the value to . The term with the indicator is the expected change in a susceptible learner's value from gaining immunity, which occurs at the rate ϕ.

The maximization problem in (2.7) is the flow value generated by the choices of learning intensities. Normal learning generates the expected gain , where the risk of being infected is in specified below. Isolated learning succeeds at the rate and changes the value to . The effective the production intensity is . A learner's output per intensity, normalized by γ, is . In addition, if , the individual is susceptible and will be infected by a producer at the rate , where is the fraction of infected individuals in production weighted by their production intensities. An infection changes the value from to .

In a contact in normal learning, the expected gain to a learner 0z is:In a meeting, the learner succeeds in learning with the probability β. The gain is if the learner does not change the disease status. However, the learner can become infected if he/she is susceptible, i.e., if . The term with the indicator is the reduction in the learner's value resulting from such an infection. In a contact, a susceptible learner is infected with the probability , where is the fraction of teachers who are infected, as defined in (2.4). An infection reduces the gain to the learner by if the learner succeeds in learning, and by if the learner fails to learn.

In the decision problem in (2.7), the learner takes as given. Denote the optimal effective intensity in normal learning as and the implied contact rate as . The first-order condition of is:The direct marginal benefit of normal learning is , which includes the risk of being infected in normal learning. An indirect benefit of learning is that it crowds out the production intensity and, hence, reduces the exposure to infections in production. The term with the indicator in (2.9) captures this indirect benefit. The marginal cost of normal learning is the sacrifice in output, . Note that a susceptible learner's optimal decision depends on the distribution of individuals through and , where the dependence on goes through and .

For a learner 0z, denote the optimal effective intensity in isolated learning as and the implied learning rate as . The first-order condition of is:This is similar to (2.9), except that the direct marginal benefit of isolated learning is . If a learner is either infected or immune, the concern of being infected is moot. In this case, the allocation of intensity between the two modes of learning comes down to comparing the marginal learning efficiencies, versus (see Proposition 3.1, Proposition 3.2 later).

For a teacher () with the disease status z, the Bellman equation for is:In contrast to a learner, a teacher has no intensity in learning and a teacher's productivity relative to γ is 1. Also, there are two groups of terms with the indicator . The first group is the expected loss in a susceptible teacher's value resulting from an infection. For a susceptible teacher, the rate of being infected is by a learner and by a producer. The second group with is the expected change in a susceptible teacher's value from gaining immunity.

An infected learner's choices of affect the extent of the externality of transmitting the pathogen to a susceptible individual. An infected teacher can also transmit the pathogen to a susceptible individual, but the extent of this negative externality is exogenous since a teacher does not make a decision. On the positive side, a teacher generates a positive externality by passing the frontier knowledge to a learner. Both types of externalities related to a teacher would emerge if a teacher could choose the extent to which to participate in the learning process or the effort to fend off contacts by learners.

Distribution of individuals: epidemiology

Given the learning intensities, the distribution of individuals in the economy follows dynamics similar to those in the SEIR model in epidemiology. However, there are two noteworthy details. First, there is heterogeneity in the infection rate between learners and teachers, because only learners spend time in learning, and between different activities. Second, a recovery from an infection can come without immunity, which occurs at the rate ρ.

The group of susceptible learners (): The measure obeysThe first flow into the group is births. The measure of all newborns is , and a newborn draws with the probability . The second flow into the group is infected learners who recover from the infection without immunity, which has the measure . The remaining terms in the equation are outflows from the group. Death takes the individual out of the group at the rate . The individual is infected in production at the rate . Isolated learning enables the individual to acquire the frontier knowledge at the rate . Normal learning takes the individual out of the group at the rate . This is the sum of the rate of acquiring the frontier knowledge, , and the rate of failing to acquire knowledge but being infected in normal learning, . The last term in (2.12) is the outflow from gaining immunity.

The group of infected learners (): The measure obeysOn the right-hand side, the first flow into the group is the measure of births into the group. The second flow into the group comes from learners who fail to learn but contract the pathogen in meetings. The third flow into the group is susceptible producers with who are infected in production. Of the flows out of the group, the first one results from successfully acquiring knowledge through normal and isolated learning. The last outflow is the sum of the outflows from the group caused by the two types of recoveries and death.

Other groups of individuals: Similarly, the measures, , obey the following laws of motion:The explanation for these equations is in the Supplementary Appendix D.

Reproduction numbers

The effective reproduction number is the expected number of infections that an infected individual can cause in the lifetime starting with the current distribution of individuals. The basic reproduction number is the effective reproduction number when the pathogen just arrives. Denote the reproduction number at t as for an infected individual aI and as for an infected individual on average. Examine an infected teacher first. In a small interval of time dt, evolves as follows:The term is the rate at which the infected teacher is contacted by a susceptible learner in normal learning and infects the latter. The term is the rate at which the infected teacher infects another individual in production. Multiplying each rate by dt yields the corresponding probability of transmitting the pathogen in the interval dt. Each transmission increases the infected number by one. If the teacher recovers from the infection, with or without immunity, or if the teacher dies, the future reproduction number becomes zero. With the complementary probability, , the teacher remains infected, and the reproduction number from t+dt onward is . In the limit d, the equation above becomes:

Similarly, for an infected learner, the reproduction number obeys:The main difference from comes from learning activities. The learner succeeds at the rate in normal learning and at the rate in isolated learning. In both cases, the individual's state changes to 1I and the reproduction number changes from to .

Weighting the expected reproduction number of each infected group by the group's relative size, the average reproduction number of an infected individual at t is:The reproduction numbers are forward-looking variables. For example, the derivative in (2.17) is a forward derivative which is the limit of the change in the future number as dt becomes arbitrarily small. The dynamic equations (2.17) and (2.18) should be solved forward, similar to the Bellman equations for the value functions .

In SIR models and its applications in economics, it is common to short-circuit the forward-looking feature of the reproduction number by setting terms like and to zero (e.g., van den Driessche, 2017; Acemoglu et al., 2020; Farboodi et al., 2020). I refer to the solutions obtained in this way as the “effective reproduction numbers” and add the superscript e to them. The actual solutions to (2.17) and (2.18) are referred to as the expected reproduction numbers. Setting in (2.17) and (2.18) yields:These effective reproduction numbers vary over time as the choices and the distribution of individuals change. Yet, the numbers are obtained as if they do not change momentarily, which results in an inconsistency. I will compare with R in the calibrated model.

In epidemiology, another common approach to computing reproduction numbers relies on statistical models. Starting with a prior belief (distribution) of a pathogen's infectivity in the future, this approach uses observed number of infections to update the distribution of infectivity through Bayesian updating and then update the forecast on reproduction numbers (e.g., Cori et al., 2013). By allowing future infectivity to vary over time, this approach does take into account expected variations in future reproduction numbers. However, the updating of the distribution of future infectivity relies entirely on past observations of infections. In this sense, the approach is backward-looking.

The equilibrium

An equilibrium consists of the value functions , the choices and the distribution , where and , that satisfy (i)-(ii): (i) Given the distribution, the choices are optimal for an individual 0z at time t and the value functions obey (2.7) and (2.11), and (ii) The distribution satisfies (2.12)-(2.16), for all az, and .

A critical feature of the equilibrium is that the distribution of individuals interacts with optimal choices. The distribution affects the value functions through the infection rates. In turn, the value functions affect optimal learning intensities through the gains from learning. In the reverse direction, learning intensities affect the evolution of the distribution.

To examine the above dynamic interaction, denote as the gain from successful learning without changing the disease status. Including the possibility of getting infected, the gain from normal learning is in (2.8) which I rewrite asSubtracting (2.7) from (2.11) yields:

Examine immune individuals first, i.e., those with . (3.1) implies , and (3.2) impliesAn immune individual does not have the concern of being infected. Regardless of whether the pathogen exists, an immune learner's choice between normal learning and isolated learning amounts to the comparison between the marginal efficiencies in the two modes of learning, and . I assume that isolated learning is less efficient than normal learning in the sense , so that an immune learner allocates no time to isolated learning. Moreover, I assume that an immune learner allocates positive intensities to both normal learning and production. The return on normal learning for an immune learner is the highest among all learners in the economy. For the normal intensity to be positive in the economy, it is necessary that the net return on normal learning be positive for an immune learner. This requires the marginal meeting efficiency in normal learning to be sufficiently high at ; i.e., should be sufficiently high. Similarly, for an immune learner to spend some intensity in production, the marginal meeting efficiency in normal learning should not be too high at ; i.e., should not be too high. The following assumption specifies the conditions:

, and for , where

The following proposition holds (see Appendix A for a proof):

Maintain Assumption 1 and denote . Then, , , and for all t, where is the unique solution to and . Also, and for all t, where

For an immune learner, the optimal mode of learning is normal learning. For an immune teacher, the knowledge level does not change either. Since the flow value for an immune teacher is equal to the constant output flow, the value function is constant over time. As a result, the expected gain to an immune learner's success in learning is constant. In addition, because knowledge is non-rivalrous, a learner's contact rate depends only on the learner's own learning intensity and not on the teacher/learner ratio. Thus, an immune learner's optimal learning intensity is constant over time. Because the income flow, the gain from learning, and the optimal learning intensity are all constant, an immune learner's value function is constant over time. Note that this result implies that immune individuals' value functions are independent of the distribution of individuals, because the distribution necessarily varies over time.

For the disease status , (3.1) implies , and (3.2) impliesA notable term in this equation is , which appears because an infected individual can recover without immunity. Such a recovery changes z to S, after which the gain from learning without getting infected will be . Because changes over time with the aggregate state, and vary over t. If every recovery comes with immunity, i.e., if , then vanishes from (3.7). The following proposition states the features of the value functions, optimal learning intensity, and the expected gain from learning for infected individuals (see Appendix A for a proof):

Maintain Assumption 1 . Then, regardless of whether or . Consider the case in the remainder of this proposition. For all t, , , and , where is the unique solution to and . Infected individuals' value functions are and for all t, where Moreover, , and . Furthermore, .

Because an infected learner has no concern of being infected, the learner spends no time in isolated learning, provided that the frontier knowledge is valuable in the sense . In the case , infected individuals only transition among themselves or into immune ones. Because these transition rates and immune individuals' value functions are independent of the distribution, all flow values and capital gains to an infected individual are independent of the distribution. Thus, an infected individual's value function is independent of the distribution.

With the same knowledge, an infected individual has a lower value than an immune individual. This is not surprising. Relative to an immune individual, an infected individual has a lower effectiveness in learning and production and has a higher death rate. The expected gain from normal learning is lower for an infected learner, as captured by in Proposition 3.2. In turn, this lower gain from normal learning for an infected learner induces the learner to have a lower effective normal intensity and a lower contact rate. That is, and . However, the learning intensity not adjusted for the effectiveness is not necessarily lower for an infected learner than for an immune learner.

In contrast to an immune learner and an infected learner, a susceptible learner faces the risk of an infection that may drive the expected gain from normal learning to be sufficiently low or even negative. Thus, a susceptible learner's optimal normal intensity can be zero. In addition, a susceptible learner's optimal intensities in the two types of learning vary over time and depend on the distribution of individuals in the economy, since the gains from learning do. These dynamic interactions between susceptible learners' intensities and the distribution are a main driver of the dynamics of infections. The distribution affects susceptible learners' optimal learning intensities by affecting the rate at which they will meet infected teachers. This works primarily through the ratios , where is the fraction of infected individuals among teachers, is the ratio of infected learners to the population of teachers, and is the fraction of infected individuals in production weighted by their production intensities. Because individuals' optimal decisions depend on the distribution, the equilibrium is not block recursive, in contrast to Shi (2009) and Menzio and Shi (2010). To solve such equilibrium dynamics is not feasible analytically. I resort to the quantitative analysis.

Quantitative analysis

Calibration and computation

I calibrate the model at the weekly frequency. The contact/meeting rates in learning have the following functional form:The intensity endowment for market activities is normalized to . Table 1 lists the parameters, their values and the calibration targets. The Supplementary Appendix E describes the calibration procedure in detail. Appendix C conducts sensitivity analyses on and . The relative productivity of the frontier knowledge to the baseline is set to . As a reference, the average wage growth for young male workers in the U.S. in the first 8 years of career was 55% in the NLSY79 (Light, 2005). Such wage growth reflects only a part of the increase in a worker's productivity because it results only from the labor market experience. In addition, an individual's productivity in the model can increase because of formal learning, which is a part of the college premium.

Table 1

Parameters, values and calibration targets.

Parameters	Value	Target
r	9.3827 × 10−4	annual interest rate = 0.05
γ	2	relative wage
b	10.7	health economics
η	0.5	Lucas and Moll (2014)
ℓ_	10−5	baseline value close to 0
u_	3.7 × 105	σ^′(0)=βα^′(ε_0L)
αp	1.1736	α_p=α^′(ε_0L)
A0	2.3472	expected time to get knowledge with full-time normal learning = 5 years
β	0.0016	reproduction number R0 = 3.1
ρ	0.0738	expected time of the two types of recoveries together = 4 weeks
μ	0.1762	prob(immunity|recovery)=0.8
δC = δS	2.9586 × 10−4	expectancy of working = 65 years
δI	2.8114 × 10−3	infection increases death prob. by 0.01
ψ	0.5	incubation period = 2 weeks
ϕ	0.01	expected time of getting vaccinated effectively = 100 weeks

For the productivity loss caused by an infection, there is no direct evidence to use. A patient on the ventilator has negative productivity, but many infected ones continue to work without knowing about the infection. I set so that an infected individual loses 30% of productivity on average; i.e., . As a comparison, Dworsky et al. (2016) find that a permanently disabled worker experiences 30% earnings loss on average in the second year following the injury. Since an infection of Covid-19 may not be a permanent disability, the value b is applied only when an individual is undergoing an infection. It seems reasonable that b should be much higher than 1/0.7 for Covid-19 infections, because an individual tested positive of Covid-19 is often quarantined. I use the value for two reasons. First, there is no reliable evidence on the fraction of Covid-19 infections that are detected. Second, the relatively low value of b leads to a conservative estimate of the output loss caused by the pandemic. Increasing b increases the output loss.

The functional form of is taken from Lucas and Moll (2014), who set and . I use this value of η but set to be a sufficiently small positive number ( to avoid division by 0 in the computation. The value of is such that achieves the upper bound in Assumption 1 (see the related discussion). Similarly, the contact efficiency in production achieves the upper bound given by the assumption . To determine in and , I assume that an uninfected individual who spends full-time in normal learning takes 5 years as the expected time to acquire the frontier knowledge. Such a long time in learning is required because the frontier knowledge doubles productivity. This target depends on and β. To identify both parameters, I add the target on the reproduction number of the pathogen at : . The reproduction number helps identify a learning parameter because it depends on the contact rate in normal learning. To generate , the equilibrium requires (weeks); that is, the expected length of time for a susceptible learner who spends full time in normal learning to get infected is 6 days. Given ψ, this target on and the target on the expected time to learn together determine .

The pathogen is calibrated to Covid-19. First, the average reproduction number at the onset of the pathogen is , the incubation period is two weeks, and the average length of a recovery is four weeks. These characteristics all fall in the range estimated for Covid-19 by Linka et al. (2020).10 Second, the additional death probability from an infection is set to 0.01 to identify . The fatality rate of Covid-19 among confirmed cases is currently near 1.42% worldwide (Worldometers, 2022). The lower number used here is more realistic both because some infections are not detected and because a significant number of deaths have occurred to patients who had pre-medical conditions. Third, the probability of gaining permanent immunity upon a recovery from an infection is set to 0.8, and so 20% of the recoveries do not have immunity. This approximates the clinical evidence in Long et al. (2020).11 Fourth, to determine ϕ, I set the expected length of time for being vaccinated effectively to 100 weeks, i.e., . This length is longer than the time for the first Covid-19 vaccine to arrive. However, the target is reasonable because a susceptible individual who experiences the event ϕ becomes permanently immune in the model, which the vaccines in reality have not been able to achieve.12

To measure economic activities, I use aggregate output (GDP) normalized by γ:

The epidemiology in the model is straightforward to compute: Given the contact rates and the initial distribution , integrating the laws of motion, (2.12)-(2.16), yields n. The economics in the model can take a large time to compute. To solve the optimal learning intensities, one needs to solve the value functions from the forward-looking Bellman equations, (2.7) and (2.11). The value functions are functions of n which contains five independent variables and evolves endogenously.13 To reduce the computing time, I write the value functions as functions of the one-dimensional variable, t, and compute the functions by forward iterations on (2.7) and (2.11). The Supplementary Appendix E describes the computation procedure in detail.

Baseline results

To simulate the model needs an initial distribution of individuals over knowledge.14 Although it is common in macro calibration to set the initial state of the economy to the steady state, doing so undermines the key message that knowledge diffusion can speed up the transmission of the pathogen. In the steady state, the learner/teacher ratio is too low to generate significant learning (see section 4.3), which obscures the main message of the paper that pathogen transmission feeds on knowledge diffusion. On the other hand, setting the initial learner/teacher ratio to be high would lead to large numbers of infections and deaths. To balance the two concerns, I start the population at with 30% learners and 70% teachers, i.e., and . This initial distribution implies that knowledge diffusion generates a growth rate in output per capita equal to 1.8% per year on average in the first 10 years if the pathogen does not arrive. In comparison, the average annual growth rate in real GDP per capita in the U.S. is 2% between 1948 and 2019, and 1.57% between 2010 and 2019 (U.S. Bureau of Economic Analysis, 2021). In the steady state, there will be no growth.

Suppose that the pathogen arrives at unexpectedly, with a measure of learners and a measure of teachers being infected.15 I simulate the equilibrium and depict selected variables in Fig. 1 .16 The top left panel depicts new infections and the decomposition into new infections of learners and teachers. In about 14 weeks, the number of new infections reaches the peak which is 9.6% of the population. The top right panel in Fig. 1 depicts the currently active cases of infections, , and cumulative deaths caused by infections (multiplied by 10). reaches the peak level of 27.3% of the population 17 weeks after the pathogen arrives. Cumulative deaths are the integral of over τ from 0 to t. This cumulative measure increases on an S-shaped path and approaches the maximum asymptotically, which is near 1% of the population.

Fig. 1

The dynamics in the baseline economy (dY is % deviation of Y from Y0). (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)

The middle panels in Fig. 1 depict the reproduction numbers of the pathogen. The reproduction number of an infected learner is higher than that of an infected teacher, because an infected learner actively seeks contacts with teachers in normal learning. The reproduction numbers are the highest at the onset of the pathogen and decline over time as infections reduce the fraction of susceptible individuals in the population. The average reproduction number R, depicted in the middle right panel, has similar features to its two components. The level at , , is set to 3.1 as a calibration target.17 R is above 2 in the first 8 weeks and falls below 1 after week 14. The myopic measure given by (2.20) follows similar dynamics to R, but it over-estimates the reproductive capacity of the pathogen. The discrepancy between the two is the largest precisely when new infections start to increase sharply near week 9, where and .

The bottom left panel in Fig. 1 depicts the three types of learners' optimal normal intensities and a susceptible learner's intensity in isolated learning. As shown in Proposition 3.1, an immune learner's learning intensity is constant over time and independent of whether the pathogen exists. An infected learner's normal intensity is almost constant over time, and it is higher than an immune learner's normal intensity. That is, the pathogen induces a learner to increase the normal intensity once the learner becomes infected. However, adjusted for the effectiveness, an infected learner's effective intensity in normal learning is slightly lower than that of an immune learner's. This is consistent with Proposition 3.2 even though the parameter value is outside the consideration of that proposition. Moreover, infected learners and immune learners do not spend time in isolated learning. That is, for all t (not depicted).

Among all learners, the pathogen induces the largest changes in a susceptible learner's intensities. A susceptible learner's normal intensity falls quickly after week 4, reaches the trough in week 20, and then increases back toward the steady state. Such large swings in are a result of the optimal tradeoff between learning and the risk of an infection. In fact, adjusted for the scale difference, is almost the opposite image of the number of active infections in the top right panel in Fig. 1. When the number of active infections is relatively small at the beginning, the risk of being infected is relatively low, and so a susceptible learner keeps the normal intensity high. As the risk of an infection increases quickly with the number of active cases, a susceptible learner curtails normal learning and increases isolated learning. When the number of active cases starts to decline after week 20, a susceptible learner increases normal learning and reduces isolated learning.

Not all changes in normal learning are substituted into isolated learning. A susceptible learner's isolated intensity is zero except in the time interval between weeks 14 and 30. Before week 14, a susceptible learner's isolated intensity is zero, although the normal intensity has been falling. During this time, the learner shifts intensity from normal learning into production instead of isolated learning. After week 30, returns to zero, and the increase in comes from reducing the production intensity. However, in week 20, a susceptible learner stops normal learning and production altogether, and devotes all intensities to isolated learning.

The bottom right panel in Fig. 1 depicts the percentage change in output from , the level immediately before the pathogen arrives. Before week 7, aggregate output increases as learning intensities are still high, which increase the fraction of the population with the frontier knowledge. Aggregate output reaches a local peak in week 7 and drops into the trough in week 17. The percentage drop in aggregate output between the peak and the trough is 11.1%. After week 17, aggregate output starts to recover and, by week 30, it has exceeded the previous peak. Thus, the arrival of the pathogen causes a large fall in output, followed by a V-shaped recovery.

It is useful to compare the quantitative predictions of the model with the U.S. data. The large fall and the quick reversal in aggregate output resemble those of the U.S. GDP during the Covid-19 pandemic. The U.S. GDP per capita fell by 10.3% from the peak in the fourth quarter of 2019 to the trough in the second quarter of 2020 and, in the first quarter of 2021, it reached a level higher than the previous peak (see U.S. Bureau of Economic Analysis, 2021). However, the number of deaths in the model is much higher than that in the U.S., because the model abstracts from policy controls that have existed in reality, such as social distancing, mandatory wearing of masks, and lockdowns. If such controls were not imposed in the early stage of the pandemic before the vaccines became available, it is conceivable that the number of deaths can be as high as the model predicts. Section 5 will show that a lockdown can significantly reduce the number of deaths. Whether output falls by more or less than in the baseline depends on the timing of the lockdown.

Pathogen transmission feeds on knowledge diffusion

To illustrate a main message that a pathogen is transmitted more quickly if an economy has higher growth fueled by knowledge diffusion, I consider two alternative economies that differ from the baseline only in the initial ratio of learners to teachers, . In the baseline, this ratio is . In one alternative economy, the initial learner/teacher ratio is 1, which leads to higher knowledge diffusion and higher growth on the transition path to the steady state than the baseline. In the other alternative economy, the initial learner/teacher ratio is in the steady state, which is . By construction, this economy does not grow in the absence of the pathogen. Fig. 2 compares equilibrium dynamics in these two economies with the baseline. In all three economies, an infected learner's intensities are similar and almost constant, as shown in the lower left panel.

Fig. 2

Pathogen transmission from three initial states. Each panel depicts a variable in the three economies. dY is % deviation of Y from Y0 in each economy.

In the economy with a higher initial ratio of learners to teachers, active cases of infections have a higher peak that comes earlier than in the baseline economy. Aggregate output falls from the peak to the trough by 21%, as opposed to 11.1% in the baseline. The higher number of infections and the larger fall in aggregate output occur because of the effect in the extensive margin. As there are more learners in this alternative economy, their normal learning activities transmit the pathogen more quickly. In the intensive margin, susceptible learners reduce the normal intensities earlier and in a larger amount than in the baseline, as shown in the lower right panel in Fig. 2. These responses mitigate the transmission of the pathogen but do not overcome the effect in the extensive margin.

If the economy starts in the steady state, all variables respond to the pathogen by much smaller amounts than in the baseline. These responses are barely discernible in comparison with the baseline. Because learners are less than 6% of the population in the steady state, their learning activities do not spread the pathogen quickly. Active cases of infection take a long time to peak, in week 62, and the peak is low (0.006% of the population). Aggregate output barely changes.

Social welfare and lockdowns

This section characterizes the social optimum and analyzes the effects of temporary lockdowns in the equilibrium.

Social optimum

A proper measure of social welfare in the model economy is the discounted sum of aggregate output, , adjusted for the lives lost to infections. At any time t, the flow of deaths caused by infections is . The flow of social welfare is:where is calibrated later. At time t, social welfare is18 :The permanent flow of social welfare associated with is .19 This welfare measure varies with t because the economy is on the transitional path to the steady state.

The planner chooses the allocation for each z and the implied . The constraints are the laws of motion, (2.12)-(2.16), and the feasibility constraints on learning intensities for each z: ; . Denote as the social marginal value of , which is the multiplier of the law of motion of in the planner's problem. Appendix B sets up the Hamiltonian of this problem and characterizes the socially efficient allocation. In a contact in normal learning, let be the social gain per learner 0z, measured in output at t normalized by γ. and have similar expressions to the private gains and in (2.8), except that the social marginal value replaces the private value . However, differs from :The term multiplied by is the negative externality that an infected learner creates on a susceptible teacher through infections. A similar term does not appear in because individuals ignore the externality. Also contrasting to the equilibrium, the social gain from normal learning may be negative for an infected learner, because the social loss from the negative externality of infecting others can be greater than the social benefit of higher productivity.20 If , the choice is irrelevant. If , the social optimum requires to satisfy:Notice the differences between (5.4) and the equilibrium counterpart, (2.9). Besides replacing the private values by the social values , the term multiplied by is absent from individuals' optimal choices. This term captures the externality that learning reduces the exposure to infections in production. Individuals in the equilibrium ignore this externality since they take as given.

Similarly, if , the social optimum requires to satisfy:For an immune learner, isolated learning is zero in the social optimum, as in the equilibrium. In contrast to the equilibrium, the social optimum may require an infected learner to spend a positive amount of time in isolated learning in order to reduce infections. The social benefit of an infected learner's normal intensity, , differs from the private benefit, , by an additional term that incorporates the infection externality (see (5.3)). It may be possible that , in which case the social optimum may require .

The optimality condition of differs from the equilibrium counterpart by taking into account the externalities discussed above and the effects of on births and deaths. For the latter, the planner incorporates the linkage that the measure of births must be equal to the measure of deaths to keep the population constant (see Appendix B for the detail).

If the pathogen does not exist, all of these differences between the equilibrium and the social optimum vanish. In this case, the externalities associated with the pathogen transmission do not arise. Teachers generate a positive externality by passing knowledge to learners but this externality does not affect the allocation because teachers do not choose the intensity in knowledge diffusion. Moreover, in the absence of the pathogen, the flow of births required to keep the population constant is equal to , which is independent of the distribution of individuals. Formally, one can prove the following proposition by comparing equilibrium conditions with the social optimum (the proof is straightforward and omitted):

In the absence of the pathogen, the equilibrium is socially efficient.

To compute the social optimum when the pathogen exists, I set ζ as such that a life lost to the infection is equivalent to 49.88 years of an individual's full-time output in this model. This value comes from the estimate that the average life expectancy of those who die from a Covid-19 infection is 14.5 years and the estimate by the U.S. Environmental Protection Agency (2018) that the value of a lost life is $7.4 million in 2006 in the group age 25 to 55.21 Since the average life expectancy in the group age 25 to 55 is 40 years but the average life expectancy of those who die from Covid-19 is 14.5 years, the effective number of years lost per death is 36.25% of a productive life. The EPA's estimate implies that the value of a life lost to Covid-19 is . Dividing by GDP per capita in the US in 2006, $46300, the value is 58 years of output. In the baseline model, because the average output flow per worker in the first 15 years after the pathogen arrives is 0.86, instead of 1, the value of a life lost to Covid-19 is years of full-time output (with the frontier productivity). Then,

Fig. 3 contrasts the social optimum with the equilibrium allocation. The sharpest contrasts occur in an infected learner's intensities. The social optimum requires an infected learner to spend no intensity in normal learning (see the middle left panel), which contrasts to the high normal intensity in the equilibrium. Before week 15, an infected learner is required to spend no intensity in isolated learning and, instead, use all intensity to produce. After week 17, an infected learner devotes all intensity to isolated learning in the social optimum, which contrasts with zero isolated intensity in the equilibrium (see the top right panel). An infected learner's isolated learning intensity will fall back to zero in the social optimum only after a long time (not depicted).

Fig. 3

Compare the equilibrium with the social optimum. Each panel depicts a variable in the equilibrium and the social optimum. Welfare gain is the % deviation of rU from that in the equilibrium.

By shutting down infected learners' normal intensity, the social optimum succeeds in reducing the transmission of the pathogen. Active cases of infections stay close to zero in the social optimum, as shown in the top left panel in Fig. 3. Deaths from infections are also small (not depicted). Because the risk of being infected is low, a susceptible learner is able to keep the normal intensity high and relatively flat over time, as shown in the middle right panel in Fig. 3. In contrast, in the equilibrium exhibits large declines at the beginning of the pathogen transmission and large increases after week 20.

The bottom panels in Fig. 3 depict (the left panel) and welfare gain relative to the baseline equilibrium (the right panel). The welfare gain is the percentage deviation of from its value in the baseline equilibrium. By construction, this is zero for the equilibrium. For the social optimum, the welfare gain is the percentage of permanent income that the society is willing to give up to switch from the equilibrium to the social optimum. By controlling infections and maintaining stable intensities of susceptible learners, the social optimum yields more stable aggregate output than the equilibrium. The welfare gain from the social optimum is large at the onset of the pathogen, because the social optimum can avoid most of the deaths from infections looming in the equilibrium. As deaths occur in the equilibrium, further gains from the social optimum dissipate over time. The average value of this gain in the first ten weeks is 2.1% of permanent income.22 It is useful to express this average gain relative to the first-year aggregate output. The latter is equal to . Since the average value of rU in the equilibrium in the first ten weeks is equal to 0.7884, the ratio of the average welfare gain to the first-year aggregate output is . That is, the society is willing to pay 44% of the first-year output to switch from the equilibrium to the social optimum.23 Because the U.S. GDP in 2020 is roughly $21 trillion, the welfare gain for the U.S. from the social optimum is $9.24 trillion or $27900 per capita.

There are three remarks on the welfare gain. First, I have assumed that the society can identify infected individuals without a cost. In reality, it is costly to identify and control infections, such as testing, contact tracing, social distancing, requiring mask wearing, and enforcing quarantines. The welfare gain suggests that the society should be willing to spend up to 44% of the first-year GDP to control the transmission of the pathogen. Second, the welfare gain dissipates quickly over time. Third, the welfare gain is larger in an economy with more knowledge diffusion, because the pathogen transmission is faster and causes more deaths in such an economy (see section 4.3).

Effects of lockdowns

A lockdown reduces the contact efficiency and the flow of time an individual can spend in the market. I model a lockdown as a reduction in the intensity endowment from to and a reduction in the contact efficiency in normal learning from to . The efficiency parameter in isolated learning remains at . Precisely,The lockdown starts at and the economy reopens at . I set and . The value of approximates severe social distancing and is only suggestive. The value of approximates the increase in the unemployment rate in the U.S. during the lockdown in 2020.24 Consider two lockdowns:Both lockdowns last for 19 weeks. The early lockdown starts when the number of new infections increases rapidly, and the late lockdown starts when the number of new infections has passed the peak (see Fig. 1). I assume that a lockdown is anticipated at time 0.25

Fig. 4 depicts the effects of the two lockdowns. As shown in the top left panel, the early lockdown delays the peak of new infections and reduces the peak by more than a half. It also slows down cumulative deaths and significantly reduces the maximum of such deaths, as shown in the top right panel. The middle panels in Fig. 4 depict the normal intensity of an infected learner (in the left panel) and a susceptible learner (in the right panel). By reducing the contact efficiency, the early lockdown reduces an infected learner's normal intensity. This reduces a susceptible learner's risk of being infected in normal learning relative to the baseline, resulting in a smaller reduction in a susceptible learner's normal intensity. As all learners reduce the intensity in normal learning, they increase the production intensity. In addition, the effectiveness in production falls by less than in the baseline because the early lockdown delays the peak of infections. Thus, aggregate output falls in the early lockdown by less than in the baseline, as shown in the bottom left panel in Fig. 4. However, once the economy reopens, aggregate output is lower than in the baseline, because the delayed peak of infections arrives albeit at a lower level.

Fig. 4

Effects of an early and a late lockdown. Each panel depicts a variable in the baseline and under the two lockdowns. Welfare gain is the % deviation of rU from that in the baseline.

The early lockdown improves social welfare, as shown in the bottom right panel in Fig. 4. In the first ten weeks, the average welfare gain from the early lockdown is about 1% of permanent income or, equivalently, about 21% of the first-year aggregate output in the baseline equilibrium. The gain falls as the reopening of the economy increases infections and slows down the recovery of aggregate output.

With the late lockdown, the dynamics are a mixture of those in the baseline and the early lockdown. The late lockdown reduces the number of infections but does not delay the peak of infections by much. Deaths caused by infections accumulate more slowly than in the baseline but more quickly than under the early lockdown. However, the late lockdown saves nearly as many lives as the early lockdown, since cumulative deaths from infections approach a similar maximum under the two lockdowns. During the late lockdown, a susceptible learner shifts more intensity into isolated learning than in the baseline, which leads to larger and more prolonged reductions in aggregate output. An infected learner's intensity responds to the late lockdown in a similar pattern to the early lockdown, with an apparent delay of 8 weeks. Moreover, the welfare gain from the late lockdown is similar to that from the early lockdown.

Conclusion

This paper has studied the equilibrium and the social optimum in an economy where knowledge diffusion interacts with disease transmission. Knowledge increases productivity and is diffused through learning. A learner chooses the intensities in normal learning, isolated learning and production. Normal learning requires a learner to make a contact with a teacher, which can transmit an infectious pathogen. An infection reduces productivity and can result in death. By increasing the contact rate, a higher intensity in normal learning increases knowledge diffusion and disease transmission simultaneously. There are dynamic tradeoffs between knowledge acquisition and disease infections. Calibrating the pathogen to Covid-19, the model shows that the unexpected arrival of the pathogen induces a susceptible learner to adjust the normal learning intensity in a V-shaped pattern over time. Aggregate output also follows V-shaped adjustments. Switching from the equilibrium to the social optimum reduces infections and deaths substantially and increases social welfare. I also examine temporary lockdowns in the equilibrium.

Two extensions of the model in this paper are worthwhile pursuing. First, it is useful to model the arrival of an effective vaccine more realistically than the constant probability in this paper. Although there are attempts in the literature to do so, they either do not model individuals' choices or assume that these choices are myopic.26 Since individuals choose how long to wait for a vaccine, their decisions are central to answering the question of how long a society should maintain a lockdown to wait for a vaccine. Second, allowing teachers to choose the intensity can enrich the model. As teachers increase the intensity to reduce the risk of infections, active cases of infections will increase less quickly but aggregate output will fall by more than in the baseline after the pathogen arrives. Also, endogenizing teachers' intensity raises interesting questions about how the two sides of learning should divide the gain in knowledge diffusion and how teachers may change the intensity in response to the pathogen.