EPIDEMICS IN THE NEW KEYNESIAN MODEL

This paper documents the behavior of key macro aggregates in the wake of the Covid epidemic. We show that a unique feature of the Covid recession is that the peak-to-trough decline is roughly the same for consumption, investment, and output. In contrast to the 2008 recession, there was only a short-lived rise in financial stress that quickly subsided. Finally, there was mild deflation between the peak and the trough of the Covid recession. We argue that a New Keynesian model that explicitly incorporates epidemic dynamics captures these qualitative features of the Covid recession. A key feature of the model is that Covid acts like a negative shock to the demand for consumption and the supply of labor.


Introduction
This paper documents the behavior of key macro aggregates in the wake of the Covid epidemic, using data for six industrialized countries. The Covid epidemic is associated with a large recession in which consumption, investment, and output comove positively. In contrast to the 2008 recession, there was only a short-lived rise in …nancial stress that quickly subsided. In addition, there was mild de ‡ation between the peak and the trough of the Covid recession. A unique feature of the Covid recession is that the peak-to-trough decline is roughly the same for consumption, investment, and output.
What class of models can account for these stylized facts? In Eichenbaum, Rebelo, and Trabandt (2021) we argue that the Covid epidemic acts like a simultaneous negative shock to the demand for consumption goods and the supply of labor. We extend that framework to incorporate investment as well as nominal price rigidities standard in New Keynesian (NK) models. We show that the model captures the main qualitative features of the Covid recession, generating sizable declines of similar magnitude in consumption, investment, and output, along with moderate de ‡ation.
The intuition for our results is as follows. Suppose that people can become infected through consumption activities but not by working. Then, an epidemic leads to a large drop in consumption and a boom in investment. The latter boom re ‡ects two forces: people want to consume more once the infection wanes and they want to smooth hours worked over time.
By building up the capital stock, they can accomplish both objectives. Investment rises and consumption falls, so this version of the model cannot account for the severity and the comovement properties of the Covid recession. Now suppose that people can become infected by working but not through consumption activities. Then, an epidemic leads to a small decline in consumption but a large fall in hours worked and output. There is also a large fall in investment because people smooth consumption in the face of a transitory fall in income. This version of the model does produce a large recession. However, it cannot account for consumption falling by roughly as much as investment during the Covid recession.
In the calibrated version of the model, consistent with the empirical evidence (see e.g., Barbieri et al. (2020) and Cai et al. (2020)), people can become infected through both consumption and work-related activities. We …nd that the shift in labor supply dominates the shift in consumption demand. So, an epidemic generates a steep recession along with sharp declines in both consumption and investment. The model is also consistent with the mild de ‡ation observed in the data.
We …nd that sticky prices increase the depth of the recession relative to a model with ‡exible prices. But this e¤ect is relatively small. The intuition for this result is as follows. It is well known that nominal price rigidities exacerbate the impact of negative demand shifts.
But they alleviate the impact of negative supply shifts. 1 Since both shifts are operative during an epidemic, sticky prices do not, on net, have a strong e¤ect on the response of output to an epidemic. Nevertheless, sticky prices are important for the overall performance of the model. With ‡exible prices, the model overstates the de ‡ation associated with the Covid recession.
Starting with Barro and King (1984), there is an extensive literature that emphasizes the role of di¤erent shocks and frictions in generating comovement between consumption, invest- To articulate in a transparent way the impact of the epidemic on aggregate demand and supply, we focus on the …rst wave of Covid infections. For the same reason, we abstract from government interventions such as containment measures and transfers to the households.
We view our work as a …rst step towards integrating epidemic models and DSGE models.
Our e¤orts complement research that mimics the impact of an epidemic in DSGE models through a sequence of demand and supply shocks (see e.g., Faria-e-Castro (2021) and Chen et al. (2021)).
Our analysis relates to the work of Guerrieri et al. (2021). In their model, an epidemic is equivalent to a supply shock. These authors show that, in multi-sector models with nominal rigidities, a supply shock can trigger a demand shortage that leads to an aggregate contraction. In contrast, in our model, an epidemic necessarily leads to both a contraction in supply and demand.
The remainder of this paper is organized as follows. In Section 2, we summarize some key facts about the Covid recession. Section 3 describes an NK model with endogenous epidemic dynamics. Section 4 uses this model to study the role of demand and supply shocks induced by the epidemic. Section 5 concludes.

The Covid recession
In this section, we summarize the behavior of key economic aggregates during the Covid recession and compare it to their behavior in other recessions. Our primary focus is the U.S., but we show that our key …ndings hold for other developed economies.

Data
In this subsection, we discuss the data used in our analysis. We use quarterly data for the U.S., Canada, France, Italy, Germany, and the UK.

U.S. data sources
For the U.S., we obtain seasonally-adjusted data from the Bureau of Economic Analysis on real GDP (A191RX), real consumption of nondurable goods (DNDGRA), real expenditures on services (DSERRA), real expenditures on durable goods (DDURRA), real …xed residential investment (A011RA), and real non-residential investment (A008RA). 2 We compute real expenditures on nondurables and services by taking a weighted average of real consumption of nondurable goods and real expenditures on services. The weights are given by the beginning-of-sample share of nominal expenditures on nondurables goods (DNDGRC) and services (DSERRC) in total expenditures on nondurable goods and services. We construct a series for real investment by taking a weighted average of expenditures on real durable goods, …xed residential investment, and non-residential investment at the beginning of the sample of interest. The data on the headline Consumer Price Index (CPI) and the core CPI (CPIL-FESL and CPIAUCSL, respectively) is from the Bureau of Economic Analysis, obtained from the Federal Reserve Bank of St. Louis database (FRED). We obtain data on the FRB Chicago National Financial Conditions Index along with its sub-indices from the Federal Reserve Bank of Chicago. 3 We also obtain data on the Federal Reserve Bank of St. Louis Financial Stress Index. 4 We use the dates for the beginning of the recessions determined by the NBER except for the Covid recession which we time as starting in 2019.Q4.

Canadian data sources
For Canada, we obtain seasonally-adjusted data from Statistics Canada on real GDP (v62305752), real consumption of nondurable goods (v62305728), real consumption of semidurable goods (v62305727), real expenditures on services (v62305729), and real expenditures on durable goods (v62305726). 5 We compute real expenditures on nondurable goods and services using the beginningof-sample weighted average of the share of nominal expenditures on nondurable goods (v62305759), semi-durable goods (v62305758), and services (v62305760) in total nominal expenditures on nondurable goods, semi-durable goods and services. We compute total real investment as the weighted sum of real expenditures on residential Investment (v62305734), real investments of non-pro…t institutions serving households (v62305739), machinery and equipment (v62305735), intellectual property products (v62305738), and durable goods (v62305726). The weights are given by the beginning-of-sample share of nominal expenditures on residential investment ( v62305765), non-pro…t institutions (v62305770), machinery and equipment (v62305766) plus intellectual property products (v62305769), and durable goods(v62305757) in total expenditures on these categories. We obtain data on the headline CPI (v41690973) and core CPI (v41755376) from the Bank of Canada. 6 We were unable to obtain a …nancial stress index for Canada. We use the C. D. Howe institute dates for the beginning of recessions in Canada.

UK Data Sources
For the UK, we obtain seasonally-adjusted data from the O¢ ce of National Statistics on real GDP (YBIL), real consumption of nondurable goods (UTIL), plus semi durables (UTIT), services (UTIP), and durable goods (UTID). 7 We compute aggregate real nondurable and services using the beginning-of-sample weighted average of the share in total nominal expenditures on nondurable goods (UTIJ), semi-durables (UTIR), and services (UTIN). 3 https://www.chicagofed.org/publications/nfci/index 4 https://fred.stlouisfed.org/series/STLFSI2 5 https://www150.statcan.gc.ca/n1/pub/13-607-x/2016001/64-eng.htm 6 https://www.bankofcanada.ca/rates/price-indexes/cpi 7 https://www.ons.gov.uk/ We compute real total investment plus durable goods as the weighted sum of total …xed capital formation (NPQT) and real durable goods (UTID). The weights are given by the share of nominal expenditures in total …xed capital formation (NPQS) and durable goods (UTIB) in total nominal expenditures on total …xed capital formation and durable goods. We obtain data on the headline CPI (CPIH) and core CPI (DKC6) from the O¢ ce of National Statistics. We use the Economic Cycle Research Institute for the beginning of recessions in the UK.

Data for France, Germany, and Italy
For France, Germany, and Italy we obtain data on nominal and real GDP as well as consumption, all investment categories, and headline and core CPI from Eurostat. 8 . We compute aggregate real categories as weighted averages of the underlying nominal categories using the same method as for the U.S., Canada, and the UK. Our …nancial stress indices for the UK, Germany, France, and Italy are from the European Central Bank. 9 We use the Economic Cycle Research Institute dates for the beginning of recessions.

Empirical results
We begin by discussing our results for the U.S. Figures 1 and 2 display the behavior of key macro aggregates during the Covid recession, the Great Recession that started in the fall of 2008, and the average of the recessions that occurred between 1947 Q1 and 2008 Q3. 8 We downloaded data on consumption, investment and price indeces from the following three links: https://ec.europa.eu/eurostat/databrowser/view/namq_10_gdp/default/     The variables displayed are real GDP, real consumption of non-durables and services, real …xed investment, purchases of durables goods, the consumer price index, and the core consumer price index. In all cases, we graph percent deviations of the variables from their values at the beginning of the recession. Figure   Note: Shaded areas indicate NBER recessions (peak to trough). See text for data sources.
3 Table 1 reports the percentage decline from peak to trough for real GDP, real consumption of non-durables and services, and real …xed investment plus purchases of durables goods for the Covid recession, the Great Recession, and the average of the other post-war recessions. For the real variables, troughs are speci…c to the variables in question. For the nominal variables (the headline and core CPI) we compute the percentage change from peak to trough in real GDP.             Table 7 reports the average peak-to-trough change in real GDP, investment, consumption, and the consumer price index across our six economies in the Covid recession. We see that the average decline in all real variables is about the same. This fact contrasts sharply with the relative behavior of these variables in other recessions. Finally, Figure 4 depicts …nancial stress indices for France, Germany, Italy, and the UK. 10 As in the U.S., there was a modest rise in the level of …nancial stress that quickly subsided probably as a result of the response of governments and central banks to the crisis. 10 We could not …nd a comparable index for Canada. We also abstract from containment measures introduced in various countries. We do so for two reasons. First, during the epidemic, there was positive comovement between consumption, investment, and output both in periods with and without containment. Second, modeling those measures would considerably complicate the analysis without changing our basic message. The reason is that containment acts as an additional negative shock to the demand for consumption and supply of labor (see e.g. Eichenbaum, Rebelo, and Trabandt (2021)).

Tables
The economy is initially in a steady state where all people are identical. The population is then divided into four groups: susceptible (people who have not yet been exposed to the virus), infected (people who have been infected by the virus), recovered (people who survived the infection and acquired immunity), and deceased (people who died from the infection). We denote the fraction of the initial population in each group by S t , I t , R t , and D t , respectively. The variable T t denotes the number of newly infected people.
At time zero, a fraction " of the population is infected by a virus: The rest of the population is susceptible to the virus: Social interactions occur at the beginning of the period (infected and susceptible people meet). Then, changes in health status unrelated to social interactions (recovery or death) occur. At the end of the period, the consequences of social interactions materialize and T t susceptible people become infected.
As in Eichenbaum, Rebelo and Trabandt (2021), we assume that susceptible people can become infected in three ways: purchasing consumer goods, working, and through random interactions unrelated to economic activity. The number of newly infected people is given by the transmission function: 11 The variables C s t and C i t represent the consumption of a susceptible and infected person, respectively. The variables N s t and N i t represent hours worked of a susceptible and infected person, respectively. The number of newly infected people that results from consumptionrelated interactions is given by 1 (S t C s t ) (I t C i t ). The terms S t C s t and I t C i t represent total consumption of susceptible and infected people, respectively. The parameter 1 re ‡ects both the amount of time spent in consumption activities and the probability of becoming infected as a result of those activities.
The number of newly infected people that results from interactions at work is given by The terms S t N s t and I t N i t represent total hours worked by susceptible and infected people, respectively. The parameter 2 re ‡ects the probability of becoming infected as a result of work interactions.
Susceptible and infected people can meet in ways unrelated to consuming or working.
The number of random meetings between infected and susceptible people is S t I t . These meetings result in 3 S t I t newly infected people. The number of susceptible people at time t + 1 is given by: The number of infected people at time t + 1 is equal to the number of infected people at time t plus the number of newly infected people (T t ) minus the number of infected people who recovered ( r I t ) and the number of infected people who died ( d I t ): Here, r is the rate at which infected people recover from the infection and d is the probability that an infected person dies.
The number of recovered people at time t + 1 is the number of recovered people at time t plus the number of infected people who just recovered ( r I t ): Finally, the number of deceased people at time t + 1 is the number of deceased people at time t plus the number of new deaths ( d I t ): People have rational expectations, so that they are aware of the initial infection and understand the laws of motion governing population health dynamics.
Final good producers Final output, Y t , is produced by a representative, competitive …rm using the technology: The variable Y i;t denotes the quantity of intermediate input i used by the …rm.
Pro…t maximization implies the following demand schedule for intermediate products: Here, P i;t denotes the price of intermediate input i in units of the …nal good.
The price of output is given by: .

Intermediate goods producers
Intermediate good i is produced by a monopolist using labor, N i;t , and capital, K i;t , according to the technology: The pro…ts of intermediate-good producer i at time t are: .
Monopolist i chooses its price subject to Calvo (1983) style price-setting frictions. With probability 1 the …rm reoptimizes P i;t . With probability , P i;t = P i;t 1 . The …rm chooses its optimal time-t price,P t , to maximize: subject to the demand curve (7). Here, b t+j is the Lagrange multiplier of the household problem associated with the nominal budget constraint and mc t denotes the real marginal cost at time t: where W t and R k t are the nominal wage rate and rental rate of capital, respectively.
Households For tractability, we assume that people are organized into households, each of which has a continuum of identical members. This household structure introduces limited sharing of health risks. Without the household structure, the asset holdings of a person would depend on how long they had a particular health status. So, as time goes by, we would have to keep track of an increasing number of types of people.
At time zero, a household has a continuum of measure one of family members. The household maximizes its lifetime utility: subject to the budget constraint: nominal bond holdings, and R b t the interest rate on nominal bonds. The law of motion for the stock of capital is: The number of newly infected people is given by: The …rst-order conditions for c s t , c i t and c r t are: Here, b t and t are the Lagrange multiplier on the household budget constraint and the transmission function (11), respectively. The …rst-order conditions for n s t , n i t and n r t are: The …rst-order condition for k t+1 is: The …rst-order conditions for s t+1 , i t+1 , r t+1 , and t are: Monetary policy The monetary authority controls the nominal interest rate, R b t . It chooses this rate according to the following Taylor-type rule: where Y f t , is output in a ‡exible-price version of the economy. The variables , and R b t are the steady-state values of the rate of in ‡ation and the nominal interest rate, respectively.
Fiscal policy Real government spending, G, is constant over time and is …nanced with lump-sum taxes, t .
Equilibrium In equilibrium, the market for goods and hours worked clear, households and …rms solve their maximization problems.
The fraction of people in the family who are susceptible, infected and recovered is the same as the corresponding fraction in the population: Labor demand is equal to labor supply: The demand for …nal goods equals the …nal goods supply: where K t is the aggregate supply of capital and C t , X t and G are aggregate consumption, investment, and government expenditures, respectively. Consumption and investment are given by: The law of motion for the aggregate capital stock is: In equilibrium, the market for physical capital clears Since nominal bonds are in zero net supply, in equilibrium: The appendix contains the list of model equilibrium conditions.

Parameter values
Each time period corresponds to a week. We assume that it takes on average 14 days to either recover or die from the infection. Since our model is weekly, we set r + d = 7=14.
Based on data for South Korea for people younger than 65 years, we choose the mortality rate to be 0:2 percent which implies d = 7 0:002=14. With this value for the mortality rate, the model can account for the peak-to-trough decline in GDP during the Covid recession in the U.S.
We set 1 , 2 , and 3 to 2:5684 10 7 , 1:5936 10 4 , and 0:4997, respectively. These values imply that in the beginning of the epidemic 1=6 of the virus transmissions come from consumption, 1=6 come from work and 2=3 come from non-economic activities: Here, C and N denote consumption and hours worked in the pre-epidemic steady state, respectively. We choose the level of 1 so that the Kermack  This share corresponds roughly to the average share of investment in GDP in the U.S.

economy.
We assume that = 0:98 so that prices change on average once a year. The coe¢ cients in the Taylor rule are = 1:5 and x = 0:5=52.

The impact of an epidemic
In this section, we discuss the impact of an epidemic in the model. Our parameterization of the transmission function (1) implies that an epidemic can be thought of as giving rise to negative aggregate demand and aggregate supply shocks. The aggregate demand shock arises because susceptible people reduce their consumption to lower their probability of being infected. A simple way to see this e¤ect is to consider the …rst-order condition for c s t : Recall that b t > 0 is the Lagrange multiplier on the household budget constraint and t < 0 is the Lagrange multiplier on t . Other things equal, the larger is 1 I t C I t the lower is c s t . The negative aggregate supply shock arises because susceptible people reduce their hours worked to lower their probability of becoming infected. To see this e¤ect, recall the …rst-order condition for n s t : Other things equal, the larger is 2 I t N I t the smaller is n s t . Working in tandem, aggregate demand and supply shocks generate a prolonged recession. However, the qualitative and quantitative responses of consumption, hours worked and investment depend very much on which shock dominates.
The previous intuition about demand and supply shocks is suggestive about the …rst-order e¤ects of the epidemic. There are other general equilibrium e¤ects that must be considered.
As it turns out, those e¤ects do not overturn the intuition based on demand and supply shocks.

Epidemics as a shock to the demand for consumption
To isolate the e¤ect of the epidemic on consumption demand, we set 2 to zero so that hours worked do not a¤ect the probability of a susceptible person becoming infected. We calibrate 1 to 6:3897 10 7 so that 1=3 of the infections at the beginning of the epidemic are driven by consumption (see equation (16)).

Susceptibles Infected Recovered
The intuition for the results in Figures 5 and 6 is that the infection acts as a negative shock to susceptible people's demand for consumption. The household reduces c s t to lower the probability of susceptible people becoming infected. Consistent with this intuition, the path for c s t is the mirror image of the path for I t . The health status of infected and recovered people is not a¤ected by exposure to the virus. So, their consumption demand does not shift down in response to movements in I t .
As a result, the household does not reduce c i t and c r t . In fact, they rise by a modest amount. To understand this response, note that the income of the household does not fall by very much. But c s t falls by a very large amount. The household uses a small part of the savings from the earnings of susceptible people to fund a small rise in c i t and c r t . Figure 5 shows that the household uses most of those savings to …nance a massive increase in investment. By building up the capital stock, the household makes it possible for c s t to rise once infections start to decline without large increases in n s to work declines. So, there is a small initial fall in hours worked. After a delay, hours worked rise, re ‡ecting the increase in the marginal product of labor associated with the build up of capital.
In sum, when 2 = 0, the epidemic generates a mild recession. But, with this parameterization, the model cannot rationalize two key features of the COVID-19 recession: the large drop in output and the positive comovement between investment and consumption. 12 It is possible that in a model with sticky wages in which employment is demand determined one could rationalize the features of the covid recession with 2 = 0. However, there is strong evidence that infections happen through interactions in the workplace, i.e 2 is positive (see e.g. Barbieri et al. (2020) and OSHA). 13

Epidemics as a shock to the supply of labor
To isolate the e¤ect of the epidemic on the supply of labor, we set 1 to zero. With this parameterization, consumption does not a¤ect the probability of a susceptible person becoming infected. We calibrate 2 to 3:1871 10 4 so that 1=3 of the infections at the beginning of the epidemic (equation (17)) are driven by hours worked. and there is a large fall in investment (79 percent from trough to peak). 12 These declines in measures of economic activity occurred before lockdowns were imposed, as well as in countries like Sweden and South Korea, and U.S. states that did not impose lockdowns (see Andersen Figure 8 shows that c s t , c i t , and c r t all decline by the same small amount. In contrast, hours worked by di¤erent types of people respond very di¤erently: n s t falls by 34 percent from peak to trough, while both n i t and n r t rise by 6 percent from trough to peak.

Susceptibles Infected Recovered
As discussed above, when 1 = 0, the infection acts as a negative shock to susceptible people's supply of labor. The household cuts back on n s t to reduce the probability of susceptible people becoming infected. Consistent with this logic, the reduction in n s t mirrors the path for I t .
The household has an incentive to smooth consumption over time because consuming does not increase anyone's probability of becoming infected. Infected and recovered people are not a¤ected by exposure to the virus. So, to smooth consumption over time and across people, the household increases n i t and n r t . The income of susceptible people falls dramatically. But their consumption does not, so their savings turn sharply negative. The household …nances that dissaving by a massive decline in investment. In e¤ect, investment allows the household to smooth consumption and hours worked in response to a transitory fall in n s t . In sum, when 1 = 0, the epidemic causes a large recession. But, with this parameterization, the model cannot rationalize a key feature of the COVID-19 recession: the large observed decline in consumption.

Epidemics as a shock to the demand for consumption and the supply of labor
In our benchmark calibration, both 1 and 2 are positive. So an epidemic acts like a negative shock to both consumption demand and labor supply. 14 Figure 9 displays the total impact of the epidemic on key macro variables. This …gure shows that the model captures the salient features of the epidemic recession. There is a large drop in output, consumption, investment, and hours worked with peak to trough declines of 8, 9, 12, and 12 percent, respectively. The drop in consumption re ‡ects the fall in consumption demand by susceptible people. The large fall in investment re ‡ects the importance of the labor supply shock. As in the data, the epidemic recession is associated with mild de ‡ation.

Real Interest Rate
Notes: x-axis in weeks. GDP, consumption, hours and investment in percent deviations from initial steady state. Inflation, nominal and real interest rates in percent. Infected and deaths in percent of initial population.  Figure 10 shows the individual responses of consumption and hours worked of people with di¤erent health statuses. We see that susceptible people dramatically reduce their consumption and hours worked to reduce the probability of becoming infected. The labor income of susceptible people drops by more than their consumption. The resulting negative savings are feasible because infected and recovered people work more and consume roughly the same amount as they did before the epidemic.

Susceptibles Infected Recovered
To understand the last result, recall that the health status of infected and recovered people is not a¤ected by exposure to the virus. So, the household wants to keep their consumption relatively smooth while increasing their labor supply as part of the risk-sharing arrangement within the family.
Finally, Figure 9 also displays the impact of an epidemic in a version of the model in which prices are ‡exible. We see that the recession is slightly larger when prices are sticky.
The key di¤erence between the two models concerns in ‡ation. The ‡exible price model predicts a larger fall in prices than the sticky price model.

Conclusion
We analyze the e¤ects of an epidemic in an NK model. We show that this model can rationalize the positive comovement of consumption, investment, and output and the moderate de ‡ation observed in the recessions associated with the Covid epidemic in six developed countries. A natural next step is to embed an epidemiological model into a DSGE model to evaluate the various policy interventions implemented during the Covid recession. .
The …rst-order conditions for optimal price setting are: Finally, the Taylor rule is given by: Here, y f t is ‡exible price output which can be computed using equations 1) 31) setting = 0.
In equations 1) 31)~ b t is the scaled Lagrange multiplier, i.e.~ b t = b t P t : We solve the nonlinear equilibrium equations 1) 31) as well as their ‡exible price version using a gradient-based two-point boundary-value algorithm.