Optimal Trade-Off Between Economic Activity and Health During an Epidemic
Tommy Andersson, Albin Erlanson, Daniel Spiro, Robert Östling
OOPTIMAL TRADE-OFF BETWEEN ECONOMICACTIVITY AND HEALTH DURING AN EPIDEMIC ∗ Tommy Andersson † , Albin Erlanson ‡ , Daniel Spiro § and Robert Östling ¶ May 18, 2020
Abstract
This paper considers a simple model where a social planner can influence the spread-intensityof an infection wave, and, consequently, also the economic activity and population health,through a single parameter. Population health is assumed to only be negatively affected whenthe number of simultaneously infected exceeds health care capacity. The main finding is thatif (i) the planner attaches a positive weight on economic activity and (ii) it is more harmfulfor the economy to be locked down for longer than shorter time periods, then the optimalpolicy is to (weakly) exceed health care capacity at some time.
Keywords : Covid-19 pandemic, SI-model, economic activity, health, optimal policy.
JEL Classification : E23, E27, E65, E60.
A central part of many countries’ policies to tackle the Covid-19 pandemic has been to “flattenthe curve.” That is, a more gradual uptick of infected persons prevent health care systems to beoverburdened and save human lives. For example, Greenstone and Nigam (2020) (building onFerguson et al., 2020) estimate that 630,000 lives in the US could be saved by social distancingpolicies assuring that intensive care units are not overwhelmed during the peak of the Covid-19pandemic. At the same time, slowing down disease transmission appear to cause large economiccosts due to a fall in both consumption and production. Fernandes (2020) estimates the costs ∗ We are grateful to Jörgen Weibull for valuable comments and suggestions and, in particular, for his detailednotes on the epidemiological SI-model considered in this paper. † Lund University and Stockholm School of Economics, Departments of Economics. E-mail: [email protected] . ‡ University of Essex, Department of Economics. E-mail: [email protected] . § Uppsala University, Department of Economics. E-mail: [email protected] . ¶ Stockholm School of Economics, Department of Economics. E-mail: [email protected] . a r X i v : . [ ec on . T H ] M a y f the Covid-19 outbreak for 30 countries under different scenarios, and finds a median declinein GDP in 2020 of 2.8 percent, but that GDP can fall by more than 10–15 percent in somescenarios. Many have therefore concluded that the Covid-19 outbreak involves a key trade-off: a higher spread-intensity is advantageous for economic activity, but disadvantageous forpopulation health.This paper analyzes the trade-off between reduced economic activity and population health ina simple and tractable model. In contrast to most previous work, we simplify the epidemiologicalmodel by only allowing two states: individuals are either susceptible or infected. This modellingchoice implies that the whole population is eventually infected and that there is no death orrecovery from the infection. This is overly simplistic for studying disease transmission moregenerally, but we believe it can be an useful simplification when integrating epidemiological andeconomic models. In particular, our model allows the social planner to influence how quickly theinfection spreads, and therefore also the economic activity and population health, by choosinga single parameter. A lower spread-intensity increases economic activity, but harms populationhealth if the number of infected at the peak of the epidemic exceeds health care capacity.We first show that if the social planner only puts weight on population health, health carecapacity will never be exceeded, which is in line with arguments behind “flattening the curve”policies. The same conclusion holds if the social planner is also concerned about upholdingeconomic activity, but production is not affected by how quickly the disease is spreading. Inmore realistic scenarios where the social planner attaches a positive weight on economic activityand it is more harmful for the economy to be locked down for longer than shorter time periods(e.g., because social distancing policies are more harmful for the economy the longer time theyare enforced), the optimal policy is to (weakly) exceed health care capacity during the some timeof the epidemic.Our model is deliberately kept stylized and abstracts from several relevant considerations.The trade-off between economic activity and population health would arise also in more elaboratemodels, but the finding that it is optimal to (weakly) exceed health care capacity is more sensitiveto modelling assumptions. There are several possible reasons why a slower spread of the disease(a flatter curve) below the health care capacity constraint may be optimal in a richer model. Forexample, if patients recover from the disease and develop immunity (as in a SIR-model), a slowerspread of the desease may limit the share of the population that is eventually infected. A slowerspread may also be optimal if population health is negatively affected when more individuals aresimultaneously infected also below the health care capacity constraint. Finally, the possibilitythat a vaccine or better medical treatments becomes available provides additional incentives todelay the epidemic.Although our model implies a sharp trade-off between output and population health, thereare mechanisms that could mitigate that trade-off. In the context of our model, it would be In the SIR-model (Kermack and McKendrick, 1927), the letters S, I and R stand for susceptible, infectious andrecovered, respectively. The considered model is a SI-model since individuals never recover from the infection.
The Model
An infection spreads in the population and a social planner must decide on a policy regarding thespread-intensity. The planner takes both production and population health into consideration andfaces a trade-off: a higher spread-intensity implies that the economy needs to be locked downfor a shorter period of time, but it imposes a higher stress on the health care system. To modelthis trade-off, we first introduce a simple infection model where a social planner can influencethe spread-intensity through a single parameter.
At any time t ≥ , x ( t ) ∈ (0 , represents the share of the population that have been infectedbefore time t . Assume uniform pairwise random matching in the population, and that the infec-tion spreads with probability p when an infected individual meets a susceptible individual (evenif the infection occurred a long time ago). Assume also that the time-rate of pairwise meetings is m > . In this simple infection model, there are only two types of individuals: susceptible andinfected. In particular, there is no death or recovery from the infection, i.e., the infection modelis what epidemiologists refer to as a SI-model (see footnote 1). A social planner can affect both p and m , e.g., by different containment policies, social distancing rules and various hygiene advicecampaigns, but only at time t = 0 . The spread-intensity parameter is given by a = pm .The mean-flow dynamic of the infection over time is then given by the following ordinarydifferential equation: ˙ x = ax (1 − x ) , (1)with initial value x (0) ∈ (0 , . Equation (1) uniquely determines the dynamic evolution of thedisease, and its solution is given by: x ( t ) = e at e ab + e at , (2)where: b = 1 a ln (cid:18) x (0) − (cid:19) . (3) To the best of our knowledge the first time the logistic model was used to describe a population growth, as theone we have above, was by Verhulst (1838) and further developed by the same author in Verhulst (1845). Note that e ab = x (0) − . This way of writing the constant in the ODE simplifies later arguments. ˙ x ( t ) = ae at ( e ab + e at ) − ae at e at ( e ab + e at ) = ae ab e at ( e ab + e at ) . Note that ˙ x ( t ) : R → R + is a probability density function that describes the infection wave(in this case, a hump-shaped function, see the dashed-dotted lines in Figure 1), and its integral x ( t ) : R → [0 , a cumulative density function. Consequently, for a given spread-intensityparameter a , the “size” of the infection wave at time t is given by ˙ x ( t ) , and the “peak” of thewave occurs at the time ˆ t where ¨ x (ˆ t ) = 0 . It can be verified that ˆ t = b where b is given byequation (3) for any value of a . Because the value of b is proportional to /a , the greater a isthe smaller ˆ t = b is. A high spread-intensity rate therefore yields an early peak of the infectiousdisease. A social planner determines the optimal spread-intensity of the pandemic by taking both eco-nomic activities and population health into consideration. To spell out this trade-off formallywe introduce a production function y and a health function h as a measure of how the economicactivity and the health is affected by the pandemic, respectively. We begin by specifying the production function y ( t | b ) . Similar to for example Eichenbaumet al. (2020), we consider a problem in the short run (see also footnote 5), so capital is fixed andwe thus only need to consider labor when specifying the production function. In particular, itis assumed that production at time t depends on the share of non-infected individuals (i.e., theavailable labor force at time t ) together with a continuous and differentiable function g : [ b, T ] → R + . The idea is that g controls how production is affected by the “length” of the period until thepandemic hits its peak at b . It is assumed that g ( b ) ≥ and g (cid:48) ( b ) ≥ for all b ≥ b . Theseassumptions on g captures that the further away in time the peak of the pandemic is, the moreharmful it is for production. The following reverse hump-shaped production function (see thedashed lines in Figure 1) describes this relation between the peak of the pandemic and the outputproduced in the economy y ( t | b ) = 1 − g ( b ) ˙ x ( t ) . (4)To ensure that y ( t | b ) ≥ , it is also assumed that g ( b ) ˙ x ( t ) ≤ for all ( b, t ) ∈ [ b, T ] × [0 , T ] .From the production function (4) it follows that in the normal state of the economy, the entirepopulation is working and produces an output equal to 1 (this is only a normalization, any positivenumber instead of 1 is fine). When the share of the population that has been infected approaches5, the production again approaches the normalized output of 1. In all other time periods t , theproduction level depends both on the share of infected individuals ˙ x ( t ) and the function g ( b ) asspecified in equation (4).Let us now look at the health function h ( t | b ) that determines the impact on health fromthe pandemic. We assume that there is a fixed capacity in the health care system, denoted by c ∈ [0 , , so if the peak is “too high,” not all infected individuals can get proper health care at alltimes t . In fact, it can be shown that if a > c , then the health care capacity is (weakly) exceededin the time interval [ t l , t r ] where (see also the left panel of Figure 1): t l = b + 1 a ln − c − a c − (cid:115)(cid:18) c − a c (cid:19) − , (5) t r = b + 1 a ln − c − a c + (cid:115)(cid:18) c − a c (cid:19) − . (6)From these conditions and equation (3), it follows that the “height” of the peak is exactly equalto the health care capacity at time ˆ t = b ∗ , i.e., ˙ x ( b ∗ ) = c , when: b ∗ = 14 c ln (cid:18) x (0) − (cid:19) . For a < c , the capacity constraint c is never binding and all infected patients can receivetreatment.Infected individuals need medical treatment at the time t when they are infected but notbefore or after. From the social planner’s perspective, this means that the health measure h ( t | b ) in period t is given by the share of the population that has not yet been infected, or has beeninfected before time t , or are infected at time t but receives proper health care: h ( t | b ) = (cid:26) t ∈ [0 , t l ] or t ∈ [ t r , T ] , − ( ˙ x ( t ) − c ) if t ∈ ( t l , t r ) . (7)Note also that t l and t r are functions of b . The results presented in the next section will not qualitatively change if we add the assumption that immunityis reached when a given proportion of the population has been infected, and that the production instead equals 1 assoon as immunity is reached. .2.2 The Social Welfare Function Having specified how the economy and the health in the society is affected by the pandemic wecan now write down the welfare in the society at time t as: w ( t | b ) = λy ( t | b ) + (1 − λ ) h ( t | b ) , (8)where y ( t | b ) is the production function, h ( t | b ) is the health function, and λ ∈ [0 , is a welfareweight reflecting the importance attached to production and health. The total welfare during thepandemic is obtained by integrating the welfare measure from time 0 up to some given time T , W ( b ) = λ (cid:90) T y ( t | b ) dt + (1 − λ ) (cid:90) T h ( t | b ) dt. (9)The (short-run) objective for the planner is to select a spread-intensity that maximizes welfare.For convenience, we shall describe the planner’s problem in terms of deciding on the time ˆ t where the infection wave peaks. Note also that one can equivalently consider the problem ofselecting the optimal spread-intensity parameter a since the exact relationship between a and b isgiven by equation (3). As it is likely that it is practically difficult for the social planner to spreadthe disease “very fast,” we shall assume that the peak cannot occur before some point in time b > . We are, however, agnostic about how close in time b is to t = 0 .The planner’s objective to maximize the social welfare function (9) can be written as max b ∈ [ b,T ] W ( b ) = λ (cid:90) T (1 − g ( b ) ˙ x ( t )) dt +(1 − λ ) (cid:18)(cid:90) t l dt + (cid:90) t r t l (1 − ( ˙ x ( t ) − c )) dt + (cid:90) Tt r dt (cid:19) . Thus, for a given welfare weight λ ∈ [0 , , the objective for the social planner is to decide onthe time where the infection wave peaks to maximize the social welfare given by (9). The first result concerns the two extreme cases where the social planner puts all weight on eitherproduction or health. It is not immediately clear how to choose T . In the remaining part of the paper, it is assumed that T is a“sufficiently large” constant. This is one of three natural choices of T listed by Hansen and Day (2011, p. 428). Theother two are (i) T → ∞ and (ii) some constant I min indicating the first time period when the number of infectedindividuals is sufficiently large to end the pandemic. All results presented in this paper holds qualitatively also forthe two alternative definitions of T . Note also that because we consider a short-run problem, there is no need tointroduce discount factors. In case the capacity not is exceeded for any t ≥ , i.e., when there is no solution to equations (5)–(6), thisexpression can be simplified. See equation (12) in Section 4. roposition 1. Suppose that b ≥ b and g (cid:48) ( b ) > for all b ≥ b . If (i) λ = 0 , then any b ≥ b ∗ maximizes the welfare function (9), and if (ii) λ = 1 , then b = b maximizes the welfare function(9).The first part of the proposition states that if the social planner only is concerned about health,the optimal policy is to never exceed the health care capacity at any time. The second part ofthe proposition states that if the social planner only is concerned about production, the optimalpolicy is to make the infection wave peak as soon as it is feasible.We next state another special case, namely the case when g (cid:48) ( b ) = 0 for all b ≥ b , i.e.,when production is equally affected independently of the spread-intensity and when in time thepandemic peaks. In this case, the optimal policy is again to never exceed the health care capacityat any time. Proposition 2.
Suppose that λ ∈ [0 , , and that g (cid:48) ( b ) = 0 for all b ≥ b . Then b ≥ b ∗ maximizesthe welfare function (9).The above two propositions states that if the social planner only values health or if the economyis equally affected independently of when the infection wave peaks, the optimal policy is to neverexceed health care capacity. However, the assumption that g (cid:48) ( b ) = 0 is rather unrealistic sincethis means that it is not more harmful for the economy to be locked down for longer than shortertime periods and that the social planner cannot affect the function g by any policy measures.If these assumptions are dropped and if, in addition, the planner attaches a positive weight onproduction, Proposition 3 and Example 1 show that the optimal policy is to (weakly) exceedhealth care capacity. Proposition 3.
Let λ ∈ (0 , , and suppose that g (cid:48) ( b ) > for all b ≥ b . If b maximizes the socialwelfare function (9), then it cannot be the case that b > b ∗ . Example 1.
Suppose that x (0) = 0 . , T = 15 , b = 3 . , and c = 0 . . If g ( b ) = 1 + bT forall b ≥ b and λ = 0 . , the welfare maximizing peak of the infection wave occurs at b = 6 . implying that the health care capacity is exceeded in the interval [4 . , . . This is illustratedin the left panel of Figure 1 where the infection waves (dashed-dotted lines) are illustrated in thebottom of the figure, and the production functions (dashed lines) and the health functions (solidlines) are illustrated in the top of the figure for 11 different values of b between 3.06 and 7.66.The corresponding functions for the optimal b = 6 . are marked in red color.The right panel of Figure 1, illustrates the situation for λ = 0 . . In this case, the welfaremaximizing policy is to set the peak of the infection wave at b = 7 . , i.e., at the time where the“height” of the infection wave equals the health care capacity ( c = 0 . ). (cid:3) Note that the results in the example will not change if b < . . The only thing that will change in Figure1 is that additional curves has to be added to the left of t = 3 . but they will not be a solution to the planner’soptimization problem for the given parameter values. λ or health care capac-ity c , the optimal value of b weakly decreases. Thus, if the social planner attaches more weighton production or if health care capacity increases, the optimal policy is to select an infection peakcloser in time.Figure 1: In the left panel, the health care capacity is exceeded at the welfare maximizing solution( b = 6 . ). In the right panel, the “height” of the infection wave at the welfare maximizing solution( b = 7 . ) equals health care capacity (0.15). Because all proofs are based on the same ideas, we start by stating some general remarks thatwill be useful in all of the proofs.Note first that t r > b > t l > b for all b ∈ ( b, b ∗ ) since capacity c is exceeded for all b ∈ [ b, b ∗ ) ,and t l = t r = b > b for b = b ∗ since the “height” of the peak equals c for b = b ∗ . Hence, thewelfare function (9) is for any b ∈ [ b, b ∗ ] given by: W ( b ) = λ (cid:90) T y ( t | b ) dt + (1 − λ ) (cid:90) T h ( t | b ) dt, = λ (cid:90) T (1 − g ( b ) ˙ x ( t )) dt + (1 − λ ) (cid:18)(cid:90) t l dt + (cid:90) t r t l (1 − ( ˙ x ( t ) − c )) dt + (cid:90) Tt r dt (cid:19) , = λ [ t − g ( b ) x ( t )] T + (1 − λ ) (cid:16) [ t ] t l + [ t + ct − x ( t )] t r t l + [ t ] Tt r (cid:17) , = T − λg ( b )( x ( T ) − x (0)) + (1 − λ )( c ( t r − t l ) − ( x ( t r ) − x ( t l ))) . (10)Note also that if b ∈ [ b, b ∗ ] , it follows that: c ( t r − t l ) ≤ x ( t r ) − x ( t l ) , (11)9ith strict inequality for b ∈ [ b, b ∗ ) . Finally, if b > b ∗ the capacity c is never exceeded so thewelfare function (10) can be simplified to: T − λg ( b )( x ( T ) − x (0)) . (12) Proof of Proposition 1.
Consider first part (i), and suppose that b ∈ [ b, b ∗ ] . By the aboveconclusions, it follows that t r > b > t l > b for all b ∈ [ b, b ∗ ) , and t l = t r = b > b for b = b ∗ .Furthermore, since λ = 0 the welfare function (10) can be written as: W ( b ) = T − ( c ( t r − t l ) − ( x ( t r ) − x ( t l ))) . From equation (11), it follows that W ( b ) < T for any b ∈ [ b, b ∗ ) and W ( b ) = T for b = b ∗ . If,on the other hand, b > b ∗ and λ = 0 , the capacity is never exceeded so W ( b ) = T by equation(12). This proves part (i) of the proposition.To prove part (ii), note that since λ = 1 , the welfare function (10) reduces to: W ( b ) = T − g ( b )( x ( T ) − x (0)) . (13)Since g ( b ) ≥ and g (cid:48) ( b ) > for all b ≥ b , and x ( T ) − x (0) > , it follows that equation (13) ismaximized when b is minimized. Hence, b = b maximizes equation (13). (cid:3) Proof of Proposition 2.
Consider first the case when b ∈ [ b, b ∗ ] . Because g ( b ) = 1 and g (cid:48) ( b ) = 0 for all b ≥ b by assumption, equation (10) reduces to: W ( b ) = T − λ ( x ( T ) − x (0)) + (1 − λ )( c ( t r − t l ) − ( x ( t r ) − x ( t l ))) . (14)Because T − λ ( x ( T ) − x (0)) is a constant, equation (14) is, by condition (11), maximized when b = b ∗ , i.e., when W ( b ∗ ) = T − λ ( x ( T ) − x (0)) . To complete the proof, we need only todemonstrate that the welfare equals T − λ ( x ( T ) − x (0)) for any b > b ∗ . But if b > b ∗ , thecapacity is never exceeded so the welfare function is given by equation (12) for g ( b ) = 1 and g (cid:48) ( b ) = 0 , i.e., W ( b ) = T − λ ( x ( T ) − x (0)) for all b > b ∗ , which concludes the proof. (cid:3) Proof of Proposition 3.
Consider first the case when b ∈ [ b, b ∗ ] . Then the welfare function isgiven by equation (10), W ( b ) = T − λg ( b )( x ( T ) − x (0)) + (1 − λ )( c ( t r − t l ) − ( x ( t r ) − x ( t l ))) . For b ≥ b ∗ , this equation reduces to W ( b ) = T − λg ( b )( x ( T ) − x (0)) . Because x ( T ) − x (0) is aconstant, g ( b ) ≥ and g (cid:48) ( b ) > for all b ≥ b , it then follows that W ( b ∗ ) > W ( b ) for any b > b ∗ .Hence, the welfare cannot be maximized for any b > b ∗ . (cid:3) eferences Abakuks, A. (1973). An Optimal Isolation Policy for an Epidemic.
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