key: cord-0743519-96644do8
authors: Camera, Gabriele; Gioffré, Alessandro
title: The economic impact of lockdowns: A theoretical assessment()
date: 2021-07-18
journal: J Math Econ
DOI: 10.1016/j.jmateco.2021.102552
sha: 5ec71be9c889e551f4496f087dc1013500475cbd
doc_id: 743519
cord_uid: 96644do8

The sudden appearance of the SARS-CoV-2 virus and the onset of the COVID-19 pandemic triggered extreme and open-ended “lockdowns” to manage the disease. Should these drastic interventions be the blueprint for future epidemics? We construct an analytical framework, based on the theory of random matching, which makes explicit how epidemics spread through economic activity. Imposing lockdowns by assumption prevents contagion and reduces healthcare costs, but also disrupts income-generation processes. We characterize how lockdowns impact the contagion process and social welfare. Numerical analysis suggests that protracted, open-ended lockdowns are generally suboptimal, bringing into question the policy responses seen in many countries.

The emergence of the SARS-CoV-2 virus and the onset of the COVID-19 pandemic motivated many governments to bring to a stand-still all human activity, social and economic, for many months. The stated objective is to slow down contagion and prevent healthcare systems from being overwhelmed. Many countries have gone to the extreme of imposing long-lasting and drastic "lockdowns" (e.g., China, Italy, Spain, UK), i.e., mandatory stay-at-home orders, business closures, and sweeping limitations to the freedom of movement. These lockdowns (also known as NPI's for nonpharmaceutical interventions) have been implemented on an openended basis, with a severity and duration primarily tied to the growth rate in infections. Apart from a few notable exceptions (e.g., Sweden), most countries have sought to minimize a single risk, that of contagion from the SARS-CoV-2 virus, without fully accounting for the economic and social consequences of doing so. In the aftermath of these interventions, a public debate emerged questioning the optimality of these policies. Are these drastic interventions optimal from a social welfare perspective? Should we keep them in place to address future epidemics? Should their implementation be open-ended?

The answer to these questions partly depends on how one models the relevant economic tradeoffs. This paper develops an analytical framework that makes explicit the process of contagion, and ties it to the frequency of economic activity.

The model economy has a constant population composed of individuals who can earn income only in periods in which they meet a trade partner. Meetings occur on a market where a matching process pairs individuals at random-all pairs generate a deterministic flow of income, and dissolve at the end of the period. The model assumes transmissibility via asymptomatic individuals, a central feature of the COVID-19 epidemic, as well as no cost from trading while asymptomatic.

This implies that individuals who are unaware of being infected, have no incentive to stay out of the market and, hence, can spread the disease by meeting healthy trade partners. Repeating this random matching process period after period is processes are country-specific, the model indicates that the tradeoffs associated with lockdown policies are also country-specific. In other words, there is no "onesize-fits-all" kind of policy, which seems opposite to the adoption patterns seen so far, where many governments simply followed similar policies.

There is a voluminous literature on infectious diseases, an extensive review of which is beyond the scope of this paper. Due to space limitations, we refer the reader to the recent survey in Avery et al. (2020) and here we explain what our study adds to the existing literature. The novelty of our contribution lies in the approach to studying the diffusion of epidemics, which relies on the theory of random matching. This technique allows us to offer a framework that makes explicit the transmission of a disease in the population-in contrast to the standard epidemiological model, which uses a reduced-form approach. This framework is then used to assess the economic optimality of policy interventions based on the lockdowns imposed in the recent past. 1

To elaborate on this, start by noting that the typical model used in the epidemiology literature -known as the SIR model -is based on three possible states for an individual: S for susceptible (to infection), I for Infected, and R for Recovered. The evolution of these three mutually exclusive states is governed by laws of motion that underlie a reduced-form process of contagion. We retain the 1 Our objective to study the welfare impact of health policies is shared by other recent works, all of which modify the standard SIR framework while maintaining its basic reduced-form approach. For example, Eichenbaum et al. (2020) assumes that economic decisions affect the path of the disease because consuming and working less reduces the probability of becoming infected (hence, transition probabilities between health states). This gives rise to an externality that can be partially internalized by imposing limits to consumption and work activity (such as it happens in a lockdown). The model in Goenka et al. (2014) also considers feedback effects from disease to economic decisions. They embed the SIR model into a neo-classical growth model where investment in health capital alters the incidence of the disease, and the latter affects labor supply. They solve an optimal control problem, showing that both a diseasefree steady state and a disease-endemic steady state may exist. The study in Alvarez et al. (2020) also follows the typical approach in the epidemiology literature where the evolution of the epidemic is a function of exogenous parameters, and extends it by embedding an optimal control problem, whereby a social planner chooses the diffusion parameter to maximize social welfare. An optimal control problem is also at the heart of Acemoglu et al. (2020) , which extends the canonical single-group SIR model to a multi-group version with group-specific parameters. In particular, contact rates are governed by a matching function that is group-specific. In this context, policies that apply differential lockdowns across groups are superior to uniform policies that identically affect everyone in society. The main difference between these frameworks and ours is that we do not use a reduced-form approach; we construct an explicit meeting process that determines the spread of the infection.

three-state representation typical in the literature, and innovate by constructing an explicit model of contagion, which is based on a pairwise random meeting process. In this manner, the model allows us to track the evolution of the disease when the markets are open or close-thanks to the explicit matching process-and thus study the optimality of lockdown policies under alternative scenarios for the initial state of the infection and the probabilistic nature of reaching immunity.

Given the persistent nature of the current epidemic, we also allow recovered individuals to be potential candidates for re-infection-something that is atypical in epidemiological models, where the recovered cannot be re-infected. To explain, in the standard SIR framework the number of susceptible individuals decreases over time due to the recovery process. In this case, threshold parameters exist such that an absorbing state is eventually reached where the infection vanishes as enough individuals contract the disease and herd immunity is naturally achieved. 2

By contrast, we work with a model where everyone in the population is generally susceptible to the disease until a point where the disease can be fully eradicated. It is assumed that in each period there is a probability that contagion stops, and until that happens everyone remains susceptible to infection, even recovered individuals.

In this manner, a state of immunity is reached probabilistically and simultaneously by everyone in the population. This is a mathematically convenient way to capture a prominent aspect of current thinking behind lockdown policies: the COVID-19 disease is so dangerous and hard to contain that the epidemic must be stopped with a mass-vaccination campaign. This set-up allows us to trace a most favorable scenario for imposing lockdowns, which is when they delay the progression of the disease while medical and pharmacological interventions are being developed to address the problem. 3

The paper proceeds as follows. Section 2 presents the model economy. Section 3 characterizes the dynamic process of contagion. Section 4 studies the impact of lockdowns on the spread of the epidemic in a baseline, worst-scenario model when there is no herd immunity threshold. In section 4.4, we extend the analysis to a richer model where herd immunity can be reached by naturally acquired immunity and medical discovery. In Section 5, we apply this machinery to determine how interventions of varying severity impact social welfare; this is done by studying income losses and healthcare cost savings associated with lockdowns via numerical experiments. Broader policy implications of our analysis are discussed in Section 6, which concludes the study. Here, we note that if the market is open, then it is possible (but not certain) for an individual to meet a random trade partner. We interpret a meeting as a trading situation that is advantageous to both individuals. Considering a generic pair (i, j), individual i obtains payoffȳ > 0 from being in the meeting. Symmetrically, we have a payoffȳ for individual j. The payoff corresponds to the instantaneous utility assumed in matching models of the labor market (Mortensen and Pissarides, 1994) . We normalize y = 0 to underline that economic activity is beneficial, as it is necessary to create economic surplus. Market inactivity harms economic welfare because it does not allow surplus to be generated.

Individuals discount future payoffs with a common discount factor δ ∈ (0, 1).

Letting y t ∈ {y,ȳ} denote the generic payoff to an individual at date t, the ex-6 J o u r n a l P r e -p r o o f Journal Pre-proof pected payoff to any individual at the start of the economy is therefore

It is assumed that an infectious disease exists in the population, which can be transmitted only when individuals meet to trade. That is to say, in the model contagion occurs solely via business activity and not social interactions. This process is described in what follows.

The health status of individuals. Partition the population of workers N into three sets denoted healthy, symptomatic and asymptomatic. These last two sets are collectively called infected individuals. It is assumed that an asymptomatic individual becomes symptomatic at the start of a period with constant probability s ∈ (0, 1). An infected individual can fully recover from a period to the next with probability a ∈ (0, 1), thus regaining a healthy state and becoming noninfectious, a standard assumption in the epidemiology literature. 4 This probability of recovery is independent of symptoms. As a result, the healthy set includes both individuals who were never infected or those who were and recovered. We thus have three possible states for an individual: infected and symptomatic, infected and asymptomatic, and healthy (never infected and recovered); see Fig. 1 .

We make two assumptions that match the empirical characteristics of the COVID-19 disease and partly depart from the standard SIR model discussed in the epidemiology literature.

This assumption sets our model apart from the standard epidemiology literature, which typically assumes that recovered individuals cannot be re-infected and cannot infect others (e.g., see Avery et al., 2020) .

An individual who is infected and asymptomatic cannot be distinguished from a healthy individual. healthy individuals meet to trade. It is assumed that in such a meeting contagion occurs with probability η ∈ (0, 1], independent of whether or not the healthy individual is someone who recovered from the disease in the past. In other words, it is assumed there is no permanent immunity to the disease and that it cannot be rooted out by naturally acquired immunity or medical means. That is, we make the infection very difficult to handle, so we give best shot at a lockdown policy to be welfare enhancing. 5 Since not all infected individuals are present in the market-symptomatic individuals are excluded-to calculate the transmission rate of the disease we must first discuss how individuals are matched into potential trade meetings. This is done in the following subsection. 5 We later relax this assumption by introducing the possibility of medical progress or naturally acquired immunity as a way to root out the disease.

Trade Meetings. Consider a period when the market is open. At the start of the period a matching process determines a proposed partition of the entire population of workers, into pairwise trade meetings. This means that every worker is assigned to a proposed pair, for the period. Pairs are selected using a uniform random matching process, e.g., as in Diamond (1982) . Therefore, in each period t, the probability that individual i is assigned to meet any other individual is 1 N −1 . No meeting can last for more than one period, meaning that rematching takes place in each new trading period.

The population partition is only "proposed" because not all meetings necessarily take place: we assume that symptomatic workers are prevented from joining a meeting (e.g., they are sick, so cannot work). As a result, only healthy and asymptomatic workers join their proposed meeting. To clarify, consider Fig. 2 , which displays the timeline of events in a period. First, meetings are proposed (a partition is proposed), then all workers join their assigned meeting but for symptomatic individuals. This is an analytically convenient way to maintain tractability, because we can run the matching process on a stationary population of size N . We now explain the remaining aspects of the timing of events in a period.

If there are k infected individuals in the economy after meetings take place in a period t, then at the start of the following period they may decline tok ≤ k because some of them might recover before a new set of meetings takes place. At 9 J o u r n a l P r e -p r o o f that point a set of N/2 trades is proposed by assigning everyone in the population to a trade pair, using a uniform random matching process. After this assignment, some infected individuals may show symptoms and are prevented from meeting their assigned trade partner. As a result, not all proposed trades take place. In the meetings that do take place, some will involve asymptomatic individuals who may end up transmitting the infection to their healthy partner. Hence, in t + 1 after meetings take place we have k ≥k infected individuals. This process repeats itself indefinitely and governs the progression of the infection in the economy. Summing up, in our model the infection is transmitted by asymptomatic individuals, who cannot be recognized and isolated, in contrast with those who show symptoms-who can be excluded from all trading activity. Hence, the asymptomatic are the main channel of contagion, which reflects the empirical observation that the SARS-CoV-2 virus is infectious even without symptoms. Two other empirically relevant features of the model are that the disease is not necessarily endemic in the population (recovery is possible), and past exposure to the infection is not a guarantee of permanent immunity. Finally, the empirical observation that the disease has a very low fatality rate in the working-age population motivated us to consider a zero fatality rate since we only consider contagion occurring through economic interactions. 7 This is also analytical convenient as doing so makes the population size stationary. We now proceed by showing how to trace the dynamics of the contagion in our model.

As seen above, the evolution of the disease in the economy involves two separate processes. The contagion process that occurs through trading activity, and the recovery process that takes place in-between separate rounds of trading. This section discusses these two processes, starting with that operating in the market.

Consider the start of a period when the market is open. For notational convenience letk = k denote the number of infected individuals. That is to say, there are k = 1, . . . , N infected individuals who might interact in the market. We start by deriving the probability Q kk (N ) that k ≥ k individuals are infected by the end of a period of market activity. Two sources of randomness affect this probability: the matching process, which determines how many meetings occur between infected and healthy individuals, and the biological process, which determines if the infection is transmitted in these meetings.

Here, we derive the probability λ k (N ) that, if we have k infected individuals and N − k healthy individuals, then there will be = 0, 1, . . . , min(k, N − k) pairs composed of one infected and one healthy individual. We call these pairs "mixed matches," i.e., the only meetings were contagion can occur in the model. We have:

where 8

To derive (1), notice that if k is even (odd), then the number of mixed matches cannot be odd (even), which explains why λ k (N ) = 0 in (1) Repeating the process until everyone is matched, there are · ( − 1) · ( − 2) · · · 3 · 2 · 1 = ! possible mixed matches between infected and healthy individuals.

Hence, the number of pairings that give rise to at least mixed matches is

Since we are interested in finding the number of pairings that generate exactly Repeating this procedure until all k − infected individuals have been matched among themselves gives Therefore, the number of pairings that generate exactly mixed matches is 

This functional transformation of λ k (N ) simplifies the calculations in Matlab, where we can use the gammaln() function to work with log linearized expressions.

J o u r n a l P r e -p r o o f of the disease between a healthy and an infected individual occurs with probability p < 1. This has two implications. First, the number of mixed matches that can

i.e., the probability that we have at least k − k mixed matches. By construction, the sum of probabilities

λ k (N ) = 1. To derive the conditional probability that transmission occurs in exactly k − k of these ≥ k − k matches start by considering the case = k − k. Here, the probability that the infection is transmitted in all mixed matches is

Putting together all this information leads to the following:

Lemma 1. The probability that k ≥ k individuals are infected by the end of a period of market activity is:

In deriving Q kk (N ) we have only considered states in which someone is infected, k, k > 0. For the remaining cases k = 0 and k = 0 we define the transition probabilities Q 00 = 1, Q 0k = 0 for all k > 0, and Q k0 = 0 for all k > 0. The first two definitions imply that the zero-infection state is absorbing (if the disease is non-existent we cannot have new infections) and the third simply follows from the observation that market interaction can only generate new infections.

Note that the random process determining the number of infected individuals is a finite Markov chain because the transition probability Q kk depends only on the state k = 0, 1, . . . , N at the start of a trading period and not on the entire history of infections. 11 11 The proof of Lemma 1 immediately implies that all Q kk (N ) ≤ 1, as by construction they are These considerations lead to the following.

Because the number of infections is a finite Markov chain the square matrix Q is a Markov matrix. An important property of Markov matrices is that the transition probabilities taking place after j ≥ 1 steps is determined by the product Q j . Matrix Q is a central element to calculate the evolution of the disease in the economy, but not the only one. The reason is that Q kk (N ) only tells us the probability that k − k ≥ 0 new infections occur as a consequence of market activity. It does not account for the possibility of recovery from the disease, in which case the infection might decline or even be completely eradicated. This additional component is discussed in what follows. Fig. 2 indicates that the number of infected individuals present at the start of a trading period depends on the recovery process that occurs between the end of a period and the beginning of the next. This process is next described.

Assume that an individual who results infected by the end of a period, recovers with probability a by the beginning of the following period. This implies that if we start with k infected individuals, and some recover, then at the beginning of next period we may havek ≤ k infected individuals.

probabilities. Although the analytical calculation N k =0 Q kk (N ) = 1 is not straightforward for a generic N , we note that this holds because Q kk (N ) = 0 for k < k and k > min(2k, N ), and the states with positive transition probability k = k, . . . , min(2k, N ) are mutually exclusive. By means of example, if N = 4, p = 1/2, k = 2 we have λ 20 = 1/3, λ 21 = 0, λ 22 = 2/3, and Q 22 (4) = λ 20 + λ 22

We use R kk to denote the probability that, given k = 0, 1, . . . , N individuals being infected at the end of a period, k −k have recovered by the start of the following period, where

It should be clear that if k = 0, then the contagious process stops and the disease is permanently eradicated. So, given k = 0, 1, . . . , N infected individuals, the disease is eradicated with probability R k0 = a k . With the complementary probability, it continues.

Again, the random process determining the number of infected individuals between the end of a period t at the beginning of t + 1 is a finite Markov chain because the transition probability R kk depends only on the state k = 0, 1, . . . , N ,

i.e., the number of infected individuals at the end of the period (not on the entire history of infections). 12

This discussion immediately implies the following.

Let there be k = 0, 1, . . . , N infected individuals at the end of a period. The number of infected individuals at the start of the following period is fully described by the (N + 1) × (N + 1) lower-triangular Markov matrix

In the next section, we show how to use matrices R and Q to calculate the evolution of the number of infected individuals over time. 

Market interventions are imposed by an external authority (a government), which can select to close the market for any desired extended length of time, without restrictions and without consultation with the population. This is what we call a "policy intervention."

The policy is completely described by the two parameters (T, j), with T, j ≥ 0.

The policy specifies an initial lockdown phase composed of T consecutive rounds of market inactivity, followed by a reopening phase consisting of j consecutive trading rounds. This means that the policy's horizon is T + j periods, after which the policy expires and a new policy can be considered. Fixing the policy horizon T + j to some arbitrary value, the parameter T defines the severity of the lockdown;

T = 0 corresponds to no lockdown. As T increases, the lockdown is stricter. 13 As an illustration, consider a policy (T, j) that is repeated until the infection is eradicated. In this case we have what we call a lockdown cycle (T, j). This kind of open-ended policy intervention is illustrated in Figure 3 . The policy comes into effect in period τ , i.e., the market is open up until period τ − 1. All trading stops for T periods at regular intervals τ then τ + T + j, and so on. The market is also (re)opened for j periods at regular intervals τ + T , then τ + 2T + j and so on. As we assumed that infection can only occur in trade meetings, the contagion spreads at random across the population only when markets are open. Whenever a lockdown comes into effect, trade stops for T consecutive periods, preventing further contagion and allowing some recoveries to occur. Therefore, the two Markov matrices Q and R allow us to characterize the evolution of the infection in the economy for a any policy intervention.

With this machinery, we can calculate the expected number of infections for any given lockdown policy (T, j) and, its welfare consequences.

It is convenient to first consider the case T = 0, i.e., when there is no lockdown.

Here, the contagion process in a period is governed by the matrix product QR.

This product accounts for the trading and recovery processes described in Fig. 2 .

The product QR is a stochastic matrix. It represents the Markov process that governs the law of motion of the number of infected individuals in the economy in any period t in which the market is open. For any given number k = 0, 1, . . . , N of individuals that result infected at the start of period t, it gives us the probabilities that we will have k = 0, 1, . . . , N infected individuals at the start of period t + 1. This product accounts for two components affecting the transition probabilities:

Q determines the new infections that can emerge as a consequence of trading activity in period t; this is an intermediate state reached in period t. The final J o u r n a l P r e -p r o o f Journal Pre-proof state is determined by R, which gives us the probabilities that-after trading takes place-some infected individuals will recover by the start of t + 1.

Because the product QR is a stochastic matrix of the Markov type, we can iterate on it to calculate the number of new infections expected after any given number j ≥ 1 of consecutive trading periods. In this case (QR) j governs the law of motion of the number of infected individuals in the economy between the start of period t and the start of period t + j.

Based on the above, let us now consider an economy of size N where there are k = 0, 1, . . . , N infected individuals at the start of a period. The expected number of infected individuals after j ≥ 1 consecutive trading periods is

Here, κ = (0, 1, . . . , N ) is a column vector that contains all possible numbers of infected individuals in the economy (including the 0 absorbing state, when the disease is eradicated). The vector e k is the (N + 1)-dimensional column vector with 1 in the (1 + k) th position and 0 everywhere else. The transpose of this vector, denoted e k , selects the (1 + k) th row of matrix (QR) j , i.e., the state of the economy corresponding to k ≥ 0 infected individuals. The non-zero elements of that row are the probabilities to transition from k to k = 0, 1, . . . , min(2k, N ) infected people by the end of j consecutive trading periods. Now consider the case when T ≥ 1, i.e., when trading activity restarts after T periods of complete isolation of all traders. During the lockdown phase there cannot be further contagion and there can be some recoveries. Hence, R T determines the decline in the number of infections that we can expect from T rounds of lockdown, which is simply calculated by iterating T times transition matrix R. For each initial infection rate, the three curves are drawn for "low," "medium" and "high" transmissibility of the disease corresponding to η = 1/8, 3/8, 2/3.

Consider a lockdown policy (T, j) that has two phases: a lockdown phase lasting T periods, when markets are closed and there is no contagion (only recoveries); and a reopening phase lasting j periods, when markets reopen and contagion restarts.

We juxtapose a moderate to severe policy, which are differentiated by the length of the lockdown phase T = 10, 20, respectively. As T increases, the recovery process lasts longer, causing a greater reduction in the number of infected individuals before markets reopen. For each of these T values, we consider two alternatives durations j of the reopening phase: predetermined and state-dependent. In the predetermined case, after the lockdown ends markets remain open for j periods.

In the state-dependent case, when markets reopen a lockdown is immediately reimposed if the infection rate climbs above a pre-specified trigger level-hence j varies depending on the progression of the infection.

In the numerical experiments, we consider the case of a highly transmissible disease. The economy starts in period 0 with open markets and two initial infection rates, 1% and 10% (k = 10, 100 respectively, given N = 1000). Two different kinds of responses are considered: quick and delayed. Under a quick response, there is a short, 4-period interval before the start of the lockdown phase. Under a delayed response, the delay grows to 12 periods. We report the expected number of infections for moderate and severe lockdowns, in the top and bottom panels of Fig.   5 , respectively. The dots pin down the period when lockdowns are imposed and the squares the period when markets are reopened, after which the reopening phase continues uninterrupted. The horizontal lines correspond to the initial infection rate; they also allow us to illustrate state-dependent policies according to which a new lockdown is imposed based on, alternatively, a high and a low infection trigger (10% and 1%). There are three main observations about these economies where herd immunity is assumed impossible.

Journal Pre-proof ; these policies correspond to "moderate" and "severe" interventions. Each panel considers two possible initial infection rates, 1% and 10%, identified by the horizontal lines. The solid curves correspond to a "quick response" (the lockdown policy is implemented after 4 periods), while the dashed curves identify a "delayed response" (12 periods delay). Circles identify periods when the lockdown starts, the squares identify the start of the reopening phase.

Consider the top panel. For any initial infection rate, high or low, a moderate lockdown lowers the number of infections in the short run, but not in the long run.

The expected share of infected individuals rapidly falls as soon as the lockdown is imposed, due to natural recoveries and the absence of additional contagion.

This beneficial effect of the lockdown intervention is of short-duration. Infections quickly climb back up as soon as markets reopen, quickly surpassing the initial infection levels. By contrast, in the bottom panel the lockdown lasts twice as long.

This lowers the long-run expected infection rate well below the initial levels, even for the high 10% initial rate. Intuitively, the longer duration of the intervention greatly reduces the number of infected individuals so the progression of the contagion is slow when markets reopen. Seen this way, more severe lockdowns more effectively reduce the expected number of infections in the long-run. However, even in this case infection rates eventually climb back up. This last consideration leads to our second observation.

To illustrate this, consider the top panel of Fig. 5 . Infections climb back up very quickly when markets reopen, and rapidly exceed the initial infection rate. In this sense, a one-time moderate lockdown is ineffective at containing the epidemic in the long-run. To ensure low infection rates, the lockdown phase must be repeated. To illustrate this suppose a new round of lockdowns is triggered whenever the infection rate climbs up to a pre-determine threshold, either high (10%) or low (1%). For a low threshold, this occurs soon after markets reopen. 

In each panel of Fig. 5 , infections quickly decline when lockdowns are implemented. The faster markets are shut down the sooner infection rates drop.

However, implementation delays may also have an impact in the long run as the infections expected after a quick and a delayed response take two different paths in the long run. The speed of policy response is clearly relevant for short-run management of the disease, which may be critical if there are tight constraints on healthcare resources in the short run. In addition, the illustration suggests that the speed of the response may also be relevant in the long-run, especially for determining if lockdowns must be repeatedly applied to contain the disease.

Overall, these numerical illustrations suggest that-presuming that herd immunity is impossible-imposing a one-time lockdown cannot adequately curb the infection, even if the duration of the lockdown is quite long. This brings into question the emphasis on imposing one-time extreme lockdowns to contain the epidemic-the "flattening the curve" notion made popular in the media at the onset of the COVID-19 epidemic. The reason is that, since immunity is assumed impossible, as soon as markets reopen the diseases starts to spread again. In this case, markets must be shut down as long as necessary for the process of natural recoveries to bring down the number of infections to zero. At that point, markets can be reopened. The alternative to such and extreme intervention is to repeat the lockdown to maintain infection rates at a manageable level. The open question is whether these conclusions change when herd immunity is possible, which is what we study next.

In this section we extend the model to include the possibility of achieving herd immunity. For convenience, assume that in each period there is a time-invariant probability h that contagion stops. When this occurs, the number of new infections 26 J o u r n a l P r e -p r o o f becomes zero forever. This can be thought of as the consequence of the massvaccination campaigns currently implemented in many countries (White House, 2021), or scientific discovery leading to the emergence of effective pharmacological interventions (e.g., the repurposing of ivermectin as recently reported in Kory et al., 2021) . It can be also seen as a way of modeling naturally acquired immunity in reduced-form, i.e., without making explicit a process of gradual pathogenic exposure that subsequently gives rise to natural immunity.

The probabilistic herd-immunity assumption implies that we expect the disease will be eradicated 1/h periods in the future, since

Hence, 1 − (1 − h) n is the probability that herd immunity is achieved after n periods. Since this probability is independent of other model parameters, we can calculate an expression equivalent to µ k (T, j), i.e., the number of infected individuals expected when a lockdown phase of T periods is followed by a reopening phase of j periods. 

k (0, j) = (1 − h) j µ k (0, j).

To prove the Lemma, consider a policy consisting of T rounds of lockdown followed by j trading rounds. At that point the policy expires and a (possibly) new policy comes into effect. The probability that herd immunity is achieved at any point during these T + j periods is thus:

With the complementary probability (1 − h) T +j , herd immunity is not achieved. Given that this probability is independent of other factors in the model, the Lemma is easily obtained by construction, using the result in Lemma 3. For low and medium transmission rates, the expected path of the infection is similar to the case illustrated in Fig. 4 . Intuitively, in those cases the natural recovery process is enough to curb contagion so adding the possibility of herd immunity does not change the overall picture. Instead, the possibility of attaining herd immunity is of primary importance when the disease is highly transmissible.

In that case the progression of the disease is much more contained as compared to the case illustrated in Fig. 4 . Compared to the no-herd-immunity case, the infection is expected to peak in the short run, and to reach a lower rate (about 10 percentage points less in the illustration). The problem is that the infection rate declines slowly after it reaches the peak. This suggests that one-time lockdowns may be helpful to manage the disease in the high-transmissibility scenario, because now they can complement the process underlying the development of herd immunity. This is illustrated in Fig. 7 , which considers the same interventions studied shows that this can be accomplished by imposing a more severe lockdown. Even for high initial rates, the one-time severe lockdown is expected to significantly reduce long-run infection rates, and to eventually bring them to zero.

Note that after reaching its peak, the expected share of the infected population eventually declines to zero in all cases considered. This decline occurs even if the lockdown is not repeated, i.e., if markets remain indefinitely open after period 100 in the figure. In this case, the policymaker can select a lockdown length T to contain the epidemic below a target rate, or to accelerate the convergence to a no-infection state. A new lockdown can further speed up the convergence to a no-infection state, and can do so even if this subsequent intervention is more moderate than the first one (shorter duration).

In this sense, our analysis suggests that the repeated, extreme lockdowns that we have been experiencing do not appear to be generally necessary to contain the disease within manageable levels if herd immunity is possible. In this case, a one-time intervention can be sufficient to "flatten the curve" and contain the infection rate below a desirable target. We can make one more observation.

This observation emerges from considering the top panel in Fig. 7 for a starting infection rate of 1%. Suppose that we want to maintain the infection rate below 10% so that new lockdowns would be imposed only if this level is reached when markets reopen. A moderate lockdown implemented with a short delay (solid line) quickly reduces the infection rate and once markets reopen the expected infection rate remains below 10% and eventually declines. However, implementing the lockdown with considerable delay (dashed line) fails to keep the infection rate J o u r n a l P r e -p r o o f Journal Pre-proof below 10% when markets reopen. As the delay affects the long-run path of the infection, this requires the lockdown to be repeated in order to achieve the desired target. The top panel also shows that if the infection is not in the initial stages, say we are at 10% and not 1%, then the moderate lockdown would not help managing the diseases in the long run, and the 10% target is rapidly overcome even if the intervention is implemented relatively quickly. In this case, we need a severe lockdown to bring the disease under control, maintaining the rate of infections below 10%(bottom panel).

We emphasize that these observations are simply illustrative, given the nature of the exercise conducted and the constraints imposed on the model. Yet, the numerical experiments provide some useful insights. The greater the severity of the lockdown intervention, the lower the share of the population expected to contract the disease. The impact is non-linear; a severe intervention is helpful to contain the epidemic, especially when the disease is already quite widespread. In this case, repetition of the intervention might be unnecessary to maintain control over the infection. This suggests that severe lockdowns can be useful to quickly bring a highly transmissible disease under control, but at the same time repeating the lockdowns does not seem to be generally necessary. The fundamental question is whether or not society benefits from this.

In order to look into this question, consider that lockdowns destroy income by precluding business activities. On the other hand, lockdowns prevent the healthcare system form being overwhelmed in the initial stages of an epidemic. They can also stave-off contagion long enough to ensure that-as the sick recoverhealthcare capacity constraints do not bind in the long-run. These two opposite economic considerations give rise to a tradeoff that we study in what follows.

In this section, we offer a measure of social welfare corresponding to per-capita expected payoffs under a lockdown policy (T, j). The social welfare measure discussed in this section only considers the trade-off between healthcare costs and lost J o u r n a l P r e -p r o o f Journal Pre-proof incomes during the periods that encompass the policy duration, which is a limitation introduced to maintain the analysis tractable. Another assumption made for tractability is that the effects of lockdowns do not linger in the long-run; once lockdowns are lifted, markets immediately resume their normal operations.

Expected per-capita income from trading. We start by determining the maximum and minimum per capita income. In a representative period, no income is produced by those who cannot trade andȳ > 0 for those who trade. Hence, an individual that trades in every period has lifetime payoff These vectors define expected payoffs v k given the number k of infected individuals at the start of the period, for the extreme case when the lockdown is only lifted when the disease is eradicated. Given h ≥ 0 we have

It follows that we can express the vector of expected payoffs as

Now we define expected per capita payoffs for general (T, j) policies, i.e., T periods of lockdown followed by j periods of market reopening. Again, let k be the number of individuals infected the period before the lockdown comes into effect.

The present discounted value of the average income generated until the policy

For computational convenience, the calculation does not include incomes past period T + j. The first summation calculates the payoff (in present discounted terms) if herd immunity is attained at any point in time t = 1, . . . , T + j. This can happen in the lockdown phase, or during the reopening phase, comprising periods = T + 1, . . . , T + j. The probability that herd immunity is achieved in period t is

(1 − h) t−1 h. At that point, which we discount by δ t−1 , markets reopen. This gives the representative individual a payoffȳ in each subsequent period t + 1, . . . , T + j;

in period t + 1, the present value of this stream is 1−δ T +j−t+1 1−δ ×ȳ. The second summation focuses on the reopening phase, i.e., periods = T + 1, . . . , T + j, which is why it is discounted by δ T −1 . The summation calculates the payoff if herd immunity is not achieved. In that case, µ k (T, )/N defines the share of infected individuals in the population in a period . Since a fraction s of these individuals is symptomatic and cannot trade, the upper bound on the income generated in that period is 1 − µ k (T, ) N s ȳ. 16 We now proceed by modeling the costs of the disease, current and expected.

Healthcare costs of the epidemic. Our working assumption is that the healthcare cost of the infection depends quadratically on its severity. To measure the severity of the infection we use the share of the population infected in a period.

For concreteness, think of the cost generated from the necessary healthcare equipment and resources in place to fight the infection (e.g., setting up and staffing additional medical facilities).

To calculate this expected number, we use Lemma 4. Given that at the end of last period we have k = 0, 1, . . . , N infected individuals, then the expected number of infected individuals at the end of the first period of lockdown isμ k (1, 0), it iŝ µ k (2, 0), at the end of the second period of lockdown and it isμ k (T, t), t periods after the economy has reopened. We divided this number by N to find the expected share of the infected population.

Given k, the per-capita healthcare expenditure is the present discounted value of the cost associated with the policy (T, j) is

To understand this expression, fix the number k of individuals who are infected before the lockdown policy comes into effect. For computational convenience, the expenditure is only calculated up to period T + j, and does not include cost calculations past period T + j. In the first phase (T periods) there is no market activity so there is no additional contagion -only recoveries are possible. In each period the disease can be eradicated with probability h, if herd immunity is achieved. In this case, the healthcare costs drop to zero. Hence, in the expression above we only see the terms multiplied by (1 − h) t , i.e., the periods t = 1, . . . , T in which herd immunity is not yet attained. In this lockdown phase the expected number of infected individuals is µ k (t, 0). This explains the first summation on was assigned to another symptomatic individuals).

the RHS of the expression above.

The second summation refers to the reopening phase, i.e., periods t = 1, . . . , j post-lockdown. By the time the lockdown is lifted, there is probability (1 − h) T that herd immunity has not been yet achieved. In that case, µ k (T, t) defines the expected number of infected individuals. Since the reopening phase starts T periods after the beginning of the lockdown, we discount the second summation by δ T −1 .

Social welfare. Now that we have both the expected cost and the expected income during the T +j periods of the lockdown policy horizon, we offer a measure of average ex-ante welfare:

It is simply the present value of the per-capita income expected in periods 1, . . . , T + j by the average individual, minus the per-capita cost. We use this expression to study the dynamics of social welfare based on different policies (T, j). The results are reported in the following section. We emphasize that these calculations are meant to offer a numerical illustration, not a carefully calibrated assessment of social welfare. In particular, we emphasize that the welfare measure only considers the trade-off between healthcare costs and lost incomes, and focuses only on the T + j periods that encompass the policy duration. This is done to offer an insight into the tradeoff, while reducing the complexity of the numerical exercise.

We study the welfare impact of lockdowns by means of numerical simulations.

For computational convenience, we work with N = 100 individuals. All other parameters are as in the earlier experiments. We consider an economy where the infection has not spread out yet and is confined to a small group in the population (initial infection rate 1%). We also consider economies where the infection has already spread out more widely, to 3%

and 10% of the population. 3% 10%

1%, 3% 10% Notes: Horizontal axis: duration T of the lockdown phase. Vertical axis: welfare gain relative to a no-lockdown policy. Each point is calculated for a 100 period horizon, with the reopening phase lasting 100 − T periods. The circle markers identify the intervention that maximizes welfare. For the medium transmissibility case, imposing no lockdown maximizes the welfare gain for all initial infection rates considered. For the high transmissibility case, imposing some lockdown maximizes the welfare gain with the maximum reached for T = 1, 3, 6 as the initial infection rate increases.

The welfare gain reported on the vertical axis is the change in average ex-ante welfare for any given intervention T relative to no intervention, w k (T, j) w k (0, T + j) − 1.

The length (or, severity) T of the lockdown intervention, from 0 (no intervention)

to 15, is reported on the horizontal axis. The circle markers identify the intervention that maximizes welfare, for each initial infection rate scenario. The results of our analysis can be organized into two main observations. Observation 6. The welfare impact of lockdowns depends on the diffusion of the epidemic. Immediately imposing a lockdown might lower social welfare.

The numerical illustration suggests that lockdowns do not necessarily increase welfare. Welfare declines in the figure when the infection is not easily transmitted, which is when the size of lost incomes exceeds the reduction in healthcare costs from the lockdown. Moreover, lockdowns do not necessarily increase social welfare even when the disease is highly transmissible. In fact, this depends on the duration of the lockdown. Imposing a six-period lockdown is optimal when the infection rate is 10%. The optimal duration declines to 3 and then 1 periods as the initial infection rate decreases to 3% and 1%. This suggests a threshold infection rate might exists below which imposing a lockdown is not welfare-improving. In other words, starting a lockdown immediately upon detection of the infection is not necessarily the best strategy even if the diseases is highly transmissible. 17

In this illustration, welfare gains are non-monotone in the initial infection rate.

Consider that the maximum welfare gain from lockdowns is about 0.28% when the infection rate is 1%, it climbs to about 0.83% when the infection rate is 3% and then decreases to about 0.01% for an infection rate of 10%. This non-monotonicity is interesting because although we see that longer lockdowns are optimal as the starting infection rate increases, they are not necessarily more socially beneficial.

We can make one more observation.

Observation 7. Prolonged lockdowns can be harmful to social welfare, even for widespread epidemics. Extreme lock-downs are suboptimal.

Two considerations support this observation. First, Fig. 8 shows that overshooting the optimal duration T of a lockdown can be quite harmful, especially if the disease is highly transmissible but not widespread. In the illustration, welfare 17 We thank an anonymous Referee for raising this point.

J o u r n a l P r e -p r o o f Journal Pre-proof gains are rather small and rapidly turn to losses as we move past the optimal duration. For instance, with 1% infection rate, the optimal duration of the lockdown is T = 1 periods. This welfare gain turns to a loss as soon as the lockdown duration exceeds three periods. For higher infection rates welfare gains also eventually turns to losses. All curves monotonically decline beyond T = 15. Consequently, extreme lockdowns are suboptimal in all cases.

Second, if lockdowns fall short of the optimal lockdown duration this does not dramatically reduce the welfare gains, because of their nonlinearity when the infection is no initially widespread. In this case, a large share of the maximum welfare gain can be attained even if the lockdown falls short of the optimal duration. We also observe that shorter than optimal durations can reduce social welfare, when the infection is already widespread. This is because in that case the lockdown does little to rapidly contain the disease and simply reduces current incomes. This creates a welfare loss. However, notice that the welfare reduce from a shorter-than-optimal lockdown is also quite smaller than the welfare loss from a longer-than-optimal lockdown, which again suggests that extreme lockdowns are not optimal. Seen this way, the insight from these numerical illustrations is that it may be best to err on the side of caution, implementing a shorter rather than longer lockdown. On the one hand the risk of overshooting the optimal duration carries a risk of lowering welfare. On the other, falling short of the optimal target does not dissipate too much of potential welfare gains.

An additional aspect suggests a further reason to avoid prolonged lockdowns, or entirely avoid them. The model assumes that the income decline associated with a lockdown is only temporary and as soon as the lockdown is lifted, market activities fully and immediately resume. However, recent experience suggests that this assumption might be empirically unreasonable: some economic activities may be permanently damaged and may not recover when the lockdown is lifted.

Below, we enrich the model by introducing the possibility that lockdowns may permanently damage markets to some extent.

J o u r n a l P r e -p r o o f Journal Pre-proof Permanent Damage to Economic Activity: In this section we augment the model, by assuming that a fraction α(T ) ∈ [0, 1] of per-capita incomeȳ is permanently lost as a consequence of a lockdown. This fraction is an increasing function of the duration of the lockdown, T .

In the numerical illustrations we use α(T ) = (T − 1)/(10T ). Consider a lockdown lasting 5 periods. This induces a permanent loss of a fraction α(T ) = 0.1 of per-capita income, which in present-value terms amounts to 1−δ 100 1−δ ×ȳ. 

Notes: Welfare is calculated using expression 5. The parameters are the same as in Fig. 9 . Fig. 9 shows that when lockdowns have permanent adverse consequences on economic activity, their desirability further declines, from a social welfare perspective. This can be ascertained by comparing Fig. 9 to 8. First, the optimal length of lockdowns decreases even if the epidemic is widespread and highly transmissible. While in Fig. 8 the optimal lockdown lengths increased in the infection rate, 39 J o u r n a l P r e -p r o o f in Fig. 9 the optimal length does not exceed one period and, in fact, is zero for 10% rate. Second, the welfare gain now rapidly turns into a welfare loss if the lockdown duration overshoots the optimal target. While in Fig. 8 the welfare gain turned to a loss after five periods lockdown for an initial infection rate of 3%, now welfare declines immediately, as we get into the second period of the lockdown.

Overall, this is a further indication supporting Observation 7, i.e., the numerical illustrations suggest care in implementing strict lockdown policies because prolonged lockdowns may end up lowering social welfare, even when the disease is widespread.

This study contributes to the debate on how to best address the challenges stemming from contagious diseases, such as COVID-19. It offers a mathematical framework-based on the theory of random matching-which makes explicit how economic activity can contribute to the contagion process. The model is used to assess the welfare consequences of non-pharmaceutical interventions that limit economic and social activities-the so-called "lockdowns." In our model, shutting down all business activity is assumed to be the only way stop the progress of contagion. Lockdowns can thus reduce the burden on the healthcare system, but do so by shutting down all income streams as well. The optimal policy intervention must therefore balance these two aspects.

Three insights emerge from the analysis, that seem relevant for policy. First, there is the question of whether imposing drastic stay-at-home mandates at the onset of the disease is the best course of action. For example, Sweden was criticized by the popular press for having kept businesses and schools open. The analysis suggests that there can be gains from imposing some limits to economic activity at the onset of the epidemic because this can be helpful to bring the epidemic under control (Observations 1-4) . However, such a policy becomes counterproductive as restrictions on business activity are prolonged, because long-lasting lockdowns are unnecessary to contain the epidemic (Observation 5 and 7).

Second, there is the question of whether all countries should adopt the same response pattern. Our simple model indicates that lockdowns are not a "one-sizefits-all" type of policy. The severity of the intervention should depend on the economic structure in terms of costs imposed on the healthcare system, and the anticipated evolution of the disease based on the characteristics of the susceptible population (e.g., the age structure of a country). All else equal, while stricter interventions might be suitable for countries with more fragile healthcare systems, this policy is not globally optimal, especially in the initial stages of an epidemic if the disease is not highly transmissible (Observation 6). In other words, the model does not lend support to the view that interventions should be necessarily identical across countries. By means of example, supposing that the extreme intervention adopted by China was locally optimal, the model does not imply that such an intervention is optimal for the rest of the world.

In fact, and this is a third insight, extreme and open-ended lockdowns are not generally socially optimal (Observation 7). The numerical illustrations show that a nonlinear social welfare response to policy intervention. Tighter restrictions on business activity based on the number of cases does not necessarily constitute an optimal intervention. As policymakers tighten the noose on the economy, they attain progressively smaller healthcare benefits while generating progressively larger income losses. This result clearly depends on the dynamics of healthcare costs, which are assumed quadratic in the share of the infected population, in the numerical experiments presented here.

The numerical experiments reported in this paper should be taken for what they are-illustrative of possible outcomes. They should not be taken a statements having general validity. Indeed, we do not think the model can be practically useful for policy implementation at this stage of its development, as it is still rather rudimentary in more than one aspect. We consider it a first, albeit imperfect, attempt at integrating the contagion process into a richer economic model of the One the one hand, the model assumes that business shutdowns, no matter how long, can only induce temporary reductions in income flows. It does not account for possible negative externalities associated with prolonged economic inactivity, such as human capital decline, economic inequality, disruption of business and financial networks, or declines in firm survival rates-phenomena that can depress economic activity for many years to come. This is an especially important consideration for countries whose economic systems are fragile (e.g., consider Greece as compared to Germany). When we enrich the model with the possibility that protracted lockdowns permanently harm economic activity, then even lockdowns of short duration can reduce welfare. Furthermore, in the model business activity is assumed to spread the infection, and lockdowns prevent further transmission.

Yet, there is neither conclusive evidence nor consensus on the empirical reduction in COVID-19 transmission and deaths that can be attributed to lockdowns, or their overall efficacy in reducing case growth (e.g., Allen, 2021; Atkeson et al., 2020; Bendavid et al., 2021) . Additionally, the model assumes that asymptomatic individuals are undetectable infection vectors who have no incentive to avoid market interaction when in fact this problem is greatly mitigated by wide access to quick testing procedures. These assumptions suggest a bias in favor of restrictive lockdowns.

On the other hand, our social welfare calculations consider only a basic tradeoff: that between health costs directly associated with the care needed by infected individuals, and incomes lost due to business closures. There are additional aspects of the COVID-19 disease that are omitted from the model and, hence, bias welfare results in opposite ways. Consider, for instance, the loss of life directly caused by the disease, and the long-term negative health consequences suffered by some COVID-19 patients. Furthermore, we did not account for the beneficial income-smoothing effect of fiscal policy-many countries swiftly addressed the lockdown-induced income decline with significant government transfers financed 42 J o u r n a l P r e -p r o o f Journal Pre-proof by an increase in public debt. These omissions bias the social welfare calculations against adopting early and possibly longer lockdowns. This being said, we also ruled out indirect harmful health effects of lockdowns, e.g., increase in mortality from lack of care for other diseases, mental health problems and increased suicides (e.g., Bendavid et al., 2021) , and the possibility that outpatient treatments exist that could be effective at reducing death from and COVID-19 hospitalizations (McCullough et al., 2021) . Omitting these aspects biases our welfare calculation in favor of lockdowns.

Overall, these considerations suggest that our analysis, albeit rudimentary, is more likely to be biased in favor of finding a beneficial role for lockdowns than a negative one. Future refinements of the model should relax the assumptions discussed above to enhance the empirical applicability of the model. Other features missing from the current layout should also be included. For instance, our current formulation of the severity of the intervention is only governed by the duration parameter T . One could consider an additional dimension that accounts for the proportion of business activity affected by stay-at-home mandates. Another extension is to consider contagious processes that are affected by the population density, something that can be implemented by assuming the contagion parameter p depends on the population size N . Here, we have chosen to keep the framework nimble in order to lay out as clearly as possible an explicit process of contagionour pairwise random matching process, that is-and show how it can be integrated into economic models of epidemics.

Gabriele Camera declares that he has no relevant or material financial interests that relate to the research described in this paper.

Alessandro Gioffré declares that he has no relevant or material financial interests that relate to the research described in this paper.

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