The Covid-19 Recession in Germany: A Macro-Epidemiological Analysis

3.1 Epidemiological Dynamics

The SIR framework is a dynamic, epidemiological model that analyzes the propagation of an infectious disease within a given population (Kermack and McKendrick 1927). It begins by assuming that at time t = 0, an initial share of a population is infected with a contagious disease. The population is thus divided into four sub-groups: individuals that are either susceptible to the disease, currently infected with it, or that have recovered or died from it. The relative size of these sub-groups is assumed to evolve dynamically over time, as susceptible individuals eventually become infected, and infected ones, after remaining as such for some time, either recover or die. The dynamic evolution of the sub-groups sizes is described through a set of interdependent difference equations that reflect the disease’s fundamental properties regarding its contagiousness and lethality:

where S _t , I _t , R _t and D _t denote the shares of susceptible, infected, recovered, and deceased individuals in the total population, respectively; π _r and π _d are parameters describing the probabilities of individuals either recovering or dying from the disease in each period and T _t denotes the share of newly infected population members.

Following the exogenous infection of an initial share (I₀) of the population at time t = 0, equations (1)–(4) denote the laws of motion which govern the development of aggregate health conditions in the population across periods. Equation (1) simply states that next period’s share of susceptible population members is composed of the current share of susceptibles minus the proportion of newly infected in the current period. According to equation (2), the share of infected individuals in the following period is composed of the shares of already and newly infected ones in the current period minus the proportion of infected individuals that either recover or die during this period.

Equations (3) and (4) imply that the cumulative shares of recovered/deceased population members next period simply consist of the shares of recovered/deceased individuals in the current period plus the shares of newly recovered/deceased. Equations (1)–(4) are related to the development of the pandemic. But how is this related to (macro)economic decisions?

According to equation (5), new infections in the economy are generated by three additively separable terms, with each of these terms carrying a different assumption about how the pandemic is able to spread among the economy’s population:

The first term states that susceptible members of the population can be infected through consumption activities. The magnitude by which this occurs is given by the product of the aggregate consumption of susceptible individuals, S t C t S , by the total consumption of infected individuals, I t C t I , and the parameter π₁. Following Eichenbaum et al. (2022a), π₁ is related to both the time spent on consumption activities, as well as the probability of becoming infected through such.

Similarly, the second term in equation (5) relates workplace interactions to the creation of new infections. Susceptible individuals are assumed to meet infected ones at work, leading to new infections proportional to the product of the aggregate labor supply of susceptibles, S t N t S , by the aggregate labor supply of infected, I t N t I , and the parameter π₂. Here, π₂ denotes the “pure” probability of becoming infected in the workplace, as time spent on work is already captured by the fact that labor supply is expressed in hours worked.

Finally, the third term of equation (5) rationalizes that, apart from working and consuming, susceptible population members can also be infected through social interactions with infected individuals that are unrelated to economic activity. Thereby, the term S _t I _t describes the magnitude of these interactions, whereas π₃ governs the probability of infection through such meetings. The final expression of the term, 1 − μ t l , is related to potential government interventions that aim at reducing these types of interactions between individuals.

In summary, equation (5) integrates epidemiological and economic dynamics by specifying the share of newly infected individuals as a non-linear function of the aggregate consumption and labor supply of susceptible and infected individuals. The fundamental assumption that underlies this specification is that both consumption and work activities involve a degree of physical proximity between population members that facilitates the propagation of an infectious disease within the economy.

3.2 The Household’s Problem

Following Eichenbaum et al. (2022a), the model features an infinitely-lived, representative household which is normalized to measure one in size. Its members are divided into different groups based on their respective health status, with s _t , i _t , and r _t denoting the share of the household’s members that are either susceptible, infected, or recovered at time t. We assume that the household consumes and supplies labor contingent on the respective health conditions of its members. Therefore, its consumption c t s , c t i , c t r and labor supply n t s , n t i , n t r are differentiated by the respective superscripts r, i, s. Deceased individuals do not enter the household structure anymore, as they can neither supply labor nor consume.

The household as a whole is further assumed to invest in capital k _t and bonds B _t , which yield a nominal interest rate of R t k and R t − 1 b , as well as paying lump-sum taxes ψ _t and receiving the profits (ϕ _t ) of monopolistically competitive firms. Additionally, it is subjected to distortionary taxes on labor μ t n and consumption μ t c , as well as restrictions on random interactions 1 − μ t l , which altogether serve as reduced-form representations of containment policies. They will be discussed in detail in Section 3.4.

The household’s maximization problem can be written as follows:

subject to:

where β denotes the psychological discount factor of the household, θ a labor-scaling parameter, and σ the weekly depreciation rate of capital; P _t and W _t refer to the aggregate price level and the nominal wage rate, respectively, whereas x _t denotes investment and τ _t is the share of newly infected individuals in the population at each period.

The household’s budget constraint is defined by equation (6) while equation (7) defines a standard law of motion for capital. Equations (8)–(11) denote the epidemiological dynamics described in the previous section applied to the household level.

A central assumption of the model is that its representative household is not passively subjected to the pandemic, but can actively choose its development by optimizing its behavior with regard to the variables s_t+1, i_t+1, r_t+1 and τ _t . As these four variables refer to the share of infected individuals on a household level, this implies that the household is aware of the health statuses of its own members and is able to influence them through its respective behavior. This can arguably be regarded as an admissible assumption since individuals within a given household should be able to observe and coordinate their respective health conditions. Of course this is a simplifying assumption used in Eichenbaum et al. (2022a) as a “reduced-form” implicit decisions on consumption and labor of household’s members that are conscious of the increased risks associated with them.^[3]

However, in contrast to its ability to choose actions that influence epidemiological variables within itself, the individual household takes the aggregate variables S_t+1, I_t+1, and R_t+1, T_t+1 as given. Nevertheless, due to the assumption of a representative household of measure one, it follows after aggregation that s_t+1 = S_t+1, i_t+1 = I_t+1, r_t+1 = R_t+1, and τ_t+1 = T_t+1. Hence, even though epidemiological variables on the household and the aggregate level are conceptually different, they are quantitatively equivalent in the macroeconomic equilibrium.

Related to the conceptual distinction between the household and the aggregate, it should also be noted that in equation (8), I _t C ^I and I _t N ^I refer to the aggregate consumption and labor-supply of infected individuals in the economy. This implies that new infections are generated not only by the interaction of susceptible individuals with infected ones within their own household but by the interaction of susceptibles with infected ones in the entire economy.

3.3 Government

The government is assumed to finance a constant stream of public consumption (G) through its revenues from lump-sum (ψ _t ) and distortionary μ t c , μ t n taxes. Its budget constraint in nominal terms is defined as

From the assumption of a constant government consumption stream in equation (15), it follows that when revenues through distortionary taxes increase, lump-sum taxes must decrease by an equal amount. This implies that all revenues from distortionary taxes are rebated as a lump-sum to the household. Consequently, while μ t c and μ t n affect the household’s behavior by changing the relative costs of its consumption and labor-supply decisions, they leave government expenditures unaffected. This specification is similar to the one used by Eichenbaum et al. (2021).

In addition to a governmental authority directing fiscal policy measures, the model also features a monetary authority that conducts interest rate policies according to the following Taylor rule:

where R t b denotes the nominal interest rate on bonds, π _t refers to the gross inflation rate, y _t denotes real output, and y t f represents hypothetical real output in a flexible-price economy. Furthermore, R ^b and π refer to the pre-pandemic steady-state values for the interest rate on bonds and inflation, respectively, and parameters r _π and r _x govern the magnitude by which monetary policy responds to deviations in output and inflation from their corresponding (flexible-prices) steady-state values.

Following equation (16), the monetary authority adjusts the interest rate on bonds R t b relative to its target value (R ^b ) proportional to the percentage change of gross inflation (π _t ) relative to its desired value (π) and the percentage difference between actual output y _t and its hypothetical value in a flexible-price economy y t f .

We are aware of the fact that Germany has no autonomous monetary policy. However, given its size within the euro area, we consider this assumption a better approximation than for a smaller euro-area country (i.e. we assume that the ECB reacts more to German output gaps and inflation gaps than, for instance, to the Portuguese ones).

3.4 Containment Policies

We incorporate containment policies into the model through a combination of the distortionary taxes μ t c and μ t n , as well as the “tax” on social interactions μ t l . The idea of containment policies manifesting themselves in economic models as taxes follows Hebert and Curry (2022). The authors highlight that these sorts of policies impose transaction costs on agents: not only costs to fulfill the transaction itself, in the spirit of Allen (1991), but also psychological costs related to isolation – see the references therein.

As can be seen in 15, these taxes enter the government budget constraint. However, as Government spending is assumed to be constant, all revenue from these taxes is rebated to the households via a decrease on the lump-sum tax ψ _t , which is similar to the formulation of containment policies in Eichenbaum et al. (2021). This captures other fiscal and income policies, such as subsidies to households, that the government used during the pandemic. By modelling containment policies specifically as distortionary taxes on consumption and labor we assume that similar to the endogenous containment behavior of individuals, those policies affect aggregate output primarily through their consumption and labor supply effects. This property is clearly illustrated by the first-order conditions with respect to consumption and labor supply of infected and recovered individuals:

where the superscript j ∈ {i, r} refers to infected/recovered individuals. It is easy to see that an increase in μ t c , ceteris paribus, leads to a decline in the consumption of all individuals. Analogously, increases in μ t n negatively affect labor supply. Finally, these consumption and labor-supply effects are complemented by the term μ t l , which directly reduces the number of social interactions s _t I _t in equation (8).

The representation of containment policies by the variables μ t c , μ t n , and μ t l is supported by mobility data, which illustrates how containment policies affected economic and social activity in Germany. Firstly, by closing bars and restaurants or confining people to their homes (Bundesregierung 2022), individuals were physically restrained from consuming certain types of goods and services, which is reflected in the top panel of Figure 1. It shows that the implementation of general lockdowns in Germany (as marked by the orange bars) coincided with a steep decrease in consumption traffic in the retail and recreation sectors, relative to a pre-Covid baseline. As not all goods and services in the retail and recreation sector have an equivalent, non-physical substitute, these forced reductions in physical consumption traffic imply that containment policies imposed temporary quantitative restrictions on the consumption possibilities of individuals.

Figure 1:

Consumption and labor mobility in Germany: January to December 2020. Source: Google Trends.

Secondly, the bottom panel of Figure 1 shows that strict containment policies in Germany were further associated with a strong decline in workplace traffic. Since, according to Dingel and Neiman (2020), only a subset of jobs in Germany can effectively be performed from home. These forced reductions in physical workplace presence also imply a quantitative restriction on the labor hours that individuals were able to supply.^[7] Nevertheless, as containment policies in Germany also specifically restricted the number of people allowed within a 2-m radius of each other, independent of their respective location (Bundesregierung 2022), Figure 2 also implies a relationship between containment policies and restricting social interactions that were unrelated to economic activity.

Figure 2:

Social contacts in Germany: January to December 2020. Source: Google Trends.

In summary, based on their impact on economic and social mobility, containment policies in Germany can be understood as temporary quantitative restrictions which simultaneously reduce consumption, labor hours, and social activity.

Given the increase in the price of consumption goods through μ t c , households can afford less of these goods for any fixed budget which, following equations (12) and (17), reduces consumption. This mechanism is approximately equivalent to a direct quantitative restriction, as regardless of whether the household is physically restricted from buying the goods or simply cannot afford them, it is constrained in its consumption possibilities. A similar mechanism applies for μ t n : with an increase in μ t n , households have to give up relatively more leisure to achieve a certain amount of labor income. Since giving up leisure implies disutility, a rise in μ t n increases the costs of supplying labor, which ultimately reduces its supply via equations (13) and (18). Hence, the household is again constrained in its choices, as the amount of labor that was optimal prior to the tax is now “unaffordable” due to the increase in its associated utility costs. Finally, the term μ t l has a more straightforward interpretation as, since opposed to rendering some consumption-leisure choices unaffordable, it directly restricts social interactions by diminishing the term s _t I _t .

3.5 Containment Policies and the Labor Wedge

Apart from matching empirical features of containment policies, their theoretical representation as simultaneous consumption and labor-supply restrictions also fits the results of the BCA exercise. This follows from combining the first-order conditions with respect to consumption and labor supply of infected and recovered individuals into the respective single-equilibrium condition expressed by equation (19):

where U n t j , U c t j with j ∈ {i, r} denote the marginal utilities of consumption and labor supply of susceptible or infected individuals.

Equation (19) is derived analogously to equation (14). It illustrates that by introducing a distortion into the relationship between the marginal rate of substitution of consumption for labor, and the marginal product of labor, containment policies resemble a labor wedge for infected and recovered individuals.

A similar observation follows from equation (14) for susceptible individuals. However, for susceptible individuals, the frictions induced by containment policies mix with those induced by voluntary containment to form one composite labor wedge. Furthermore, given the negative effects of μ t c , μ t n , and μ t l on consumption, labor supply, and social interactions, and the direct link of infections with these variables, it follows that, within this composite wedge, exogenous and endogenous containment effects interact with each other.

Therefore, by reducing consumption, labor supply, and social interactions of susceptible individuals, containment policies directly suppress new infections through equation (8). This, in turn, stimulates the willingness of susceptibles to consume and supply labor again, since with lower infections the respective health risks associated with those activities diminish as well. Hence, once containment policies are being eased, the economy experiences a boost in consumption demand and labor supply. However, as with higher economic and social activity infections start to rise again, this boost is only short-lived and the initial stimulus it provides is eventually reversed.

In summary, the aforementioned properties of equations (14) and (19) have two important implications: (i) they demonstrate that the exogenous containment policies in the form of μ t c and μ t n introduce frictions into the model that are consistent with those highlighted by the BCA analysis; (ii) they illustrate that the consumption and labor-supply effects of endogenous containment behavior interact in complex ways with those associated with exogenous containment policies through the development of new infections in the economy.

4.1 Containment Dynamics

As containment policies in Germany closely followed the development of the pandemic, we assume that consumption, labor, and social restrictions associated with these policies evolved according to a set of AR(1) processes:

where a t μ , j , for j = c, n, l, denote exogenous innovations and ρ _t is a common time-dependent persistence parameter, assumed to follow a random walk:

where a t ρ is an innovation as well.

We set the starting date of the model simulation (t = 0) to be the first week of 2020.^[8] We introduce three shifts on μ t c , μ t n , and μ t l : on the 12th, the 45th, and the 51st week of 2020. The benchmark value for the persistence parameter for the autoregressive processes for the consumption, labor, and social restrictions is obtained by estimating an AR(1) model for the labor wedge from our BCA exercise.^[9] This implies ρ _t equal to 0.9123.^[10] The timing of all shifts was chosen to correspond to the weeks in which the German government announced and adopted severe containment measures (Bundesregierung 2022).

The model with the containment shifts is able to replicate the macroeconomic variables. However, to improve the performance related to the epidemiological variables, we need to introduce shifts on the persistence of the containment policies.^[11] We do an exercise by setting a positive shift or around half standard error on the 12th week of the simulation (reverted four weeks later), a positive shift – that corresponds to a (positive) 2 standard deviations shift, but truncated at ρ = 1 – on the 45th week, partially reverted on the 51st week of the simulation. These shifts do not affect significantly the behavior of macroeconomic variables, but they are crucial to provide a more realistic picture of the number of infections. When setting the magnitudes and the timing of the shifts, we were inspired by Eichenbaum et al. (2024), which finds that the beliefs of agents about the risks and their persistence were evolving along the pandemic period.

We then verify the adherence of these choices by examining the dynamics of both the behavior of the weekly infections as well as the Oxford Covid-19 Stringency Index (OCSI henceforth) of Hale et al. (2021) during the pandemic, which provides a composite cumulative measure for the severity of government-imposed restrictions in response to Covid-19.^[12] This aggregate interpretation of the OCSI makes it a good reference for the behavior of μ t c , μ t n , and μ t l , since, given the aggregate nature of the model, these variables also reflect the cumulative restrictions that containment policies pose to economic and social behavior, rather than referring to any specific measure in isolation.^[13]

Figure 3 presents the OCSI’s weekly average for Germany from 2020:Q1 to 2021:Q2, along with a weekly average of nationwide infections (in thousands). The two bars on the 12th and 51st weeks in Figure 3 mark the implementation of full, nationwide lockdowns in Germany following the escalation of country-wide Covid-19 infections during the first and second wave of the pandemic. Such lockdowns encompassed the nationwide closure of most of the recreational and retail sector, educational institutions, and public spaces, as well as serious restrictions on the mobility and social behavior of individuals (Bundesregierung 2022). This sharp increase in economic and social restrictions is readily reflected within the behavior of the OCSI, as the index displays a pronounced increase at these two particular dates.^[14]

Figure 3:

Lockdown stringency and daily infections. Source: OCSI and Destatis.

The model easily accommodates containment dynamics around the 12th and 51st weeks of 2020. However, dealing with the OCSI’s subtle behavior during the 45th week of 2020 requires additional analysis, a topic that can be explored in future works.

The patterns of the containment variables μ t c , μ t n , and μ t l that result from the entire shift calibration outlined above are summarized in Figure 4. As we can see, consumption, labor, and social restrictions approximate the major empirical trends of the OCSI. Hence, they are consistent with the central empirical features of containment policies in Germany between 2020:Q1 and 2021:Q2. It is important to note that the OSCI stringency index (solid black line) is the result of both exogenous shifts (in this case, actual shocks) and endogenous responses of agents. Therefore, one should not expect that consumption tax (dashed blue line), the labor tax (dashed red line), and the social restrictions (dashed magenta line) add up to something that can be mapped into the OCSI index, even though they contribute to it. Each of these three model containment policy components produces, by construction, the spikes and the sort of autoregressive dynamics after intensifying the containment as observed in the data.

Figure 4:

Model containment dynamics. Source: OSCI, Destatis, and authors’ own calculations.

4.2 SIR Parameters

In order to calibrate the epidemiological parameter that governs the weekly probability of dying from a Covid-19 infection (π _d ), we rely on the weekly reported data on Covid-19 cases and deaths in Germany provided by RKI (2022c) and RKI (2022e).

Based on these data, we compute an initial, age-weighted average of Covid-19 case fatality rates in the German population aged 10 to 69 between 2020:Q1 and 2021:Q2, which results in a value of about 0.0055. Then, the average case fatality rate is adjusted by a factor of 1 1.8 in order to transform it into an infection fatality rate. The adjustment factor of 1 1.8 is taken from the RKI’s publication on the share of under-reported Covid-19 cases in Germany (RKI 2022d) and represents an estimate of the ratio of detected to undetected Covid-19 cases based on serological samples. It is important to adjust the crude case fatality rate by this factor, as deaths are usually observed more accurately than infections, leading crude fatality rates to significantly overstate the actual probability of dying from a Covid-19 infection.

The final infection fatality rate for Germany that results from adjusting the initial case fatality rate by the number of undetected infections is approximately equal to 0.0055/1.8 ≈ 0.003. With this numerical value for the infection fatality rate at hand, we follow Eichenbaum et al. (2022a) by assuming that the average time to either recover or die from Covid-19 is about two weeks. From this assumption, it follows that π _d equals 7 14 0.003 = 0.0015 and that π _d + π _r  = 0.5, which in turn determines the weekly probability of recovering from Covid-19 as π _r  = 0.5 − 0.0015 = 0.4985.^[15]

Following Eichenbaum et al. (2022a), we calibrate the remaining epidemiological parameters π₁, π₂, and π₃ so that the relative shares of infections that result from consumption, labor, and social activities in the model population are consistent with the respective estimates of these shares obtained from the epidemiological data. However, as there is no direct source that reports Covid-19 infections in Germany based on economic activity, we obtain such benchmark estimates by matching and aggregating a broad range of micro-level data. We use the dataset of (RKI 2022b) which reports the weekly amount of Covid-19 infections in Germany that can be attributed to outbreaks in 19 distinct settings. These settings refer to a broad range of common locations where transmission of Covid-19 was likely to occur, such as educational institutions, private homes, public transport, etc.

Based on this data, we compute the total amount of infections that have occurred in each of these locations over the course of the pandemic. Then we match such time-aggregated data for every location with a variety of sources that contain information regarding the number of people that visit each location for either consumption, work, or unrelated purposes. As an example, we match the total amount of infections that resulted from outbreaks in hospitals with data on the average German patient-to-staff ratio (Destatis 2022b, 2022c). Such a ratio carries information regarding the number of people that visit hospitals for either consumption- or work-related purposes, as we assume that by going to the hospital, patients are consuming a service, whereas staff members are there to supply labor. According to this logic, we further match the total number of infections in overnight accommodations with customer-to-staff ratios (Destatis 2022a, 2022f), the number of infections in retirement homes with resident-to-staff ratios (Destatis 2022d, 2022e), etc. We follow this approach for all locations except public transport and leisure, for which there are no such ratios available. Hence, for these particular locations, we resort to ratios related to time-use instead, for example on the time that individuals spend in public transport to travel to work, relative to the time they spend commuting to consumption-related activities (Eurostat 2021).

Subsequently, after having established such matchings, we disaggregate the total amount of infections in a particular location based on the respective ratios. Since according to the patient-to-staff ratio in German hospitals approximately 95 % of all individuals in these locations follow a consumption purpose, with the remaining 5 % being there for work, we assign 95 % of all infections in hospitals as related to consumption and 5 % as related to labor. A visual summary of the results of this disaggregation procedure for every location can be found in Figure 5.

Figure 5:

Calibration of infection risk by activity. Source: RKI (2022b).

Finally, having disaggregated the number of infections in every individual location into consumption, labor, and social interactions, we simply sum the number of infections for each of these categories across all locations. Following this, we divide the total number of infections related to consumption, labor, or other interactions that result from this procedure by the total number of infections across all locations and activities to obtain the final estimates for aggregate infection shares that can be attributed to either of the three activities. The final estimates that result from this approach are that consumption and labor activities each account for about 18 % of total infections in Germany, whereas random social interactions account for the remaining 64 %. This result is very close to the findings of Eichenbaum et al. (2022a) for the US economy, which establish infection shares to be approximately 17 %, 17 %, and 66 %, respectively.

In line with Eichenbaum et al. (2022a), we map those final estimation results for the conditional transmission probabilities for Covid-19 in Germany into the model parameters by imposing that π₁, π₂, and π₃ should be such that at the beginning of the pandemic 18 % of Covid-19 transmissions are due to consumption, 18 % due to labor and 64 % due to random interactions. Hence, we set:

where C* and N* denote aggregate consumption and working hours in the pre-infection steady state.

In addition, we further impose that the values of π₁, π₂, and π₃ have to be such that, without containment policies, about 15 % of the model’s population becomes infected within the first 9 months of the pandemic. This additional assumption is necessary, since π₁, π₂, and π₃ not only govern the relative contribution of consumption, labor, and social interactions to the development of the pandemic, but also the overall speed by which the pandemic spreads among the model’s population. Since there are multiple solutions to equations (22)–(24), which all reflect a different degree of the virus’s overall contagiousness, the calibration of π₁, π₂, and π₃ requires an additional benchmark related to the transmissibility of Covid-19 to pin down a unique set of parameter values. To this end, Eichenbaum et al. (2022a) rely on a hypothetical scenario which imposes that until a postulated end of the pandemic, eventually 60 % of the total population should become infected. In contrast to this approach, we rely on a slightly different benchmark by observing that Sweden, the only industrialized country in the world that did not impose strict lockdowns during the first wave of the pandemic, reported an estimated Covid-19 seroprevalence of 15 % after the first 9 months of the pandemic (Rostami et al. 2021). Assuming that the transmission of Covid-19 in Sweden is governed by conditions that are comparable to those in Germany, this finding provides a good benchmark for the baseline calibration of the virus’s infectiousness in Germany. The parameter values of π₁, π₂, and π₃ that result from their calibration with regard to both the conditional and absolute transmission of Covid-19 in Germany are π₁ = 2,428 × 10⁻⁷, π₂ = 23,706 × 10⁻⁴, and π₃ = 0.4252.

4.3 Economic Parameters

Based on Hinterlang et al. (2023), we calibrate the labor share in the aggregate production function α to be equal to a standard value of 2 3 and the weekly depreciation rate of capital δ as 0.025 13 . The Calvo parameter ξ is equal to 0.9808, implying that the average price duration of intermediate-good firms is approximately one year, which is consistent with Alvarez et al.’s (2006) observations regarding the average price duration of European firms. Similarly, for the calibration of the markup parameter γ, we follow the estimates of Christopoulou and Vermeulen (2012)in which the average markup in Germany is equal to 1.33.

Moreover, noting that in the model’s pre-infection steady-state it holds that 1 β = 1 + R b * π * , we calibrate the discount factor β such that it reflects the average annualized real rate of return of the leading German stock index, the DAX, between 1993 and 2019. This yields an annual beta, β_annual, and, since the model is set at a weekly frequency, we compute the final value of beta as β a n n u a l 1 / 52 . This leads to a weekly discount factor β of approximately 0.9994.

Additionally, due to the low-inflation environment that existed before the onset of Covid-19, we set the weekly gross-inflation rate in the pre-epidemic steady-state π* equal to one. Furthermore, following Eichenbaum et al. (2022a), we calibrate aggregate income and working hours in the pre-epidemic steady-state, Y* and N*, according to their average, weekly values just prior to Covid-19. To this end, we compute the real values of GDP and total hours worked per working-age person of the last quarter of 2019 based on data from the OECD Economic Outlook (OECD 2021) and break them down into their weekly counterparts by dividing them by 13. This procedure results in weekly real per working-age population values for output and hours worked of Y* = 1,162 and N* = 22.46.^[16] The steady-state values of income and hours worked implicitly define the values of the aggregate productivity parameter (A) and the labor scaling parameter (θ) to be equal to 2.99793 and 0.00165, respectively. This result follows from the pre-Covid steady-state equations for Y* and N*.

Finally, we calibrate the pre-infection government consumption to output ratio (η*) according to the fraction between government final consumption expenditures and GDP in the last quarter of 2019 (OECD 2021) and set the values of the Taylor-rule parameters equal to those of Eichenbaum et al. (2022a), with r _π  = 1.5 and r _x  = 0.5. A comprehensive summary of all the aforementioned economic, as well as epidemiological parameter calibrations, can be found in table 1.

5.1 Weekly Dynamics

To evaluate the model’s ability in replicating high-frequency output dynamics, we compare the model-implied weekly growth rates of GDP, expressed as percentage deviations from the pre-pandemic steady-state, to the OECD’s tracker of weekly GDP in Germany (Woloszko 2020). Essentially, the OECD tracker estimates weekly output growth relative to its pre-Covid trend based on Google Trends search data. Therefore, it also describes the development of GDP relative to its pre-pandemic state, which facilitates clear comparability between the OECD tracker and the model-generated data.

According to Figure 6, the model-generated weekly output growth provides a very good approximation to the corresponding estimates reported by the OECD tracker. In almost all instances between the first quarter of 2020 and the second quarter of 2021, the model produces GDP growth values that are within the 95 % confidence range of the OECD tracker’s estimates.

Figure 6:

Model implied weekly GDP growth versus OECD weekly tracker. Notes: The solid blue line represents the output from the model, whereas the solid red line is the OECD weekly tracker data. The shaded area is for the 95 % confidence interval of the GDP weekly tracker. Source: OECD and authors’ own calculation.

Additionally, Figure 7 presents the comparison between the model’s output and observed data regarding the epidemiological dynamics of Covid in Germany. To this end, it presents the percentages of the model’s pre-Covid working-age population that cumulatively dies from the disease or is newly infected within a given week to those percentages reported in the epidemiological data in RKI (2022c, 2022e).

Figure 7:

Deaths and infections. Notes: The solid blue line represents the output from the model, whereas the solid black line stands for the epidemiological data from RKI (2022c, 2022e). The red line is the number of infections multiplied by 1.8 in order to account for under-reported cases.

Similarly, Figure 7 also reports a good performance of the model along its epidemiological dimension. Taking into account that actual infections were under-reported by at least a factor of 1.8 (RKI 2022d), the model produces plausible values of weekly new infections. Furthermore, it generates multiple waves which, in their respective timing, closely match those observed in the data. Finally, it also replicates the share of the German working-age population that has cumulatively died from Covid-19 until the end of 2021:Q2. The model’s only discernible shortcoming is that it misses the renewed peak of infections occurring towards the end of the second wave.

One plausible explanation for this particular limitation is that, according to RKI (2022a), the Covid mutation AlphaB.1.1 (commonly known as the British variant) became the dominant strain in Germany by the 9th week of 2021. Since this mutation was significantly more infectious than the original virus, this circumstance is likely to have caused the renewed increase in cases shortly after the first peak of the second wave. Such increase in cases, however, was met with the beginning of the Covid-19 vaccination campaign in Germany, which ultimately reduced the virus’s infectiousness again by providing at least partial immunization and breaking infection chains. Hence, the second peak of the second wave was likely the result of a back-and-forth between positive and negative shifts to the virus’s transmissibility, which is difficult to reconcile within the model’s simple SIR framework.

However, despite this shortcoming in replicating infection dynamics towards the end of the observational period, the model is still capable of explaining fluctuations in weekly GDP and epidemiological variables with substantial accuracy. Consequently, it rationalizes that output fluctuations in Germany during Covid-19 were driven by a combination of the mechanisms described below.

As the model’s response to the general lockdown in the 12th week of 2020 closely replicates the initial, sharp decline of weekly OECD tracker data in Figure 6, one might wonder: is the sharp drop in aggregate output at the onset of the pandemic primarily driven by containment policies that generated consumption constraints, labor-supply constraints, or both mechanism share more or less equal importance?^[17]

In order to answer that question, we have simulated the model considering three additional scenarios, presented in Figure 8, besides the baseline case in Figure 6: one only the consumption restrictions (the dashed blue line), another one with only labor restrictions (the red dash-dotted line), and a third additional case with only social restrictions (dotted magenta line).

Figure 8:

Model simulation with only one type of restriction. Notes: The black solid line is the output from the baseline scenario; the dashed blue line stands for the simulations with only the consumption restrictions, whereas the red dash-dotted line represents the simulation with only labor restrictions. The shaded area is for the 95 % confidence interval of the GDP weekly tracker.

The labor supply response to containment policies is the main mechanism in our model that generates not only the initial, drastic decline in economic activity, but is also related to the subsequent, V-shaped recovery in output as follows: first, economic and social restrictions associated with containment policies are lifted. In the model, this corresponds to the geometric decay in variables μ t c , μ t n , and μ t l after the initial shock. Second, as Figure 7 illustrates, containment policies lead to a significant decline in infections. This, in turn, increases the desire of susceptibles to consume and supply labor, as health risks associated with these activities are reduced. Hence, the relaxation of consumption and labor constraints is complemented by an actual willingness of individuals to engage in these activities. Working in conjunction, these two mechanisms appear to drive the fast recovery of output until around week 30 of 2020. Then, given the upturn in economic and social activity, infections start to rapidly increase as well. This, according to the model, prevents a full recovery of output towards its pre-pandemic levels, as with rising infections, susceptible individuals start to cut back on consumption and labor supply again. Moreover, in response to accelerating Covid infections and deaths, a second set of lockdown measures is implemented in weeks 45 and 51, which explains the renewed declines in output in these instances.

Following the sharp drop in GDP in week 51, the model implies that its subsequent recovery is again the result of the phasing-out process of containment policies. However, as after the second lockdown containment policies were more persistent, such recovery process is slower compared to after the first lockdown. Moreover, Figure 6 also shows that, while model-generated GDP growth still falls within the 95 % confidence range of the OECD tracker, it persistently lies above its mean estimates and close to the upper confidence bound. This tendency is the result of the model’s shortcoming in replicating the renewed spike of infections towards the end of the second wave, as by understating the number of infections, it misses the negative contribution of voluntary containment to weekly output dynamics.

5.2 Quarterly Dynamics

In addition to evaluating the model’s ability to replicate weekly output fluctuations, Figure 9 presents the output dynamics on a quarterly frequency. Moreover, it also assesses the model’s ability to rationalize quarterly consumption, labor, and investment patterns. To this end, it compares the model implied quarterly growth rates for German GDP, investment, consumption, and hours worked relative to 2019:Q4, to those based on data from the OECD’s Economic Outlook (OECD 2021).

Figure 9:

Model implied versus data-implied quarterly growth rates. Notes: The solid blue line represents the output from the model, whereas the solid red line is the data from (OECD 2021).

As previously illustrated, the weekly OECD tracker data is based on estimates which are surrounded by a potentially high degree of uncertainty. Hence, comparing the model-generated data also with quarterly GDP values based on National Accounts provides a robustness check of the model’s explanatory abilities. Furthermore, since self-mitigation and containment policies are assumed to translate themselves into output fluctuations primarily through their consumption and labor-supply effects, it is crucial that the model provides an appropriate description of these variables as well. Ideally, we would also like to analyze the model’s capabilities of explaining consumption and working hours fluctuations on a high-frequency level. However, to the best of our knowledge, there are no available indicators of weekly consumption and hours worked in Germany.

According to the upper-left panel of Figure 9, the model also accounts for the quarterly output dynamics, besides slightly understating GDP growth in 2020:Q2 and overstating it in 2020:Q4. Hence, comparing the model-generated data to empirical observations on quarterly GDP corroborates the conclusion that output fluctuations in Germany were primarily driven by the self-mitigation of households and containment policies.

This interpretation of the German Covid-19 recession is further supported by the upper-right and lower-right panels of Figure 9. The upper-right panel demonstrates that model-generated consumption patterns follow very closely those observed in the data. A similar observation can be made for hours worked in the lower-right panel, albeit a slight tendency of the model to overstate declines in working hours. These results support the model’s central assumption that both the endogenous responses of households to the pandemic and containment policies translate themselves into output fluctuations primarily via their consumption and labor-supply effects.

Finally, the lower-left panel of Figure 9 demonstrates that, while overstating the recovery of investment in 2020:Q3 to some degree and understating it in 2021:Q2, the model also provides a reasonable description of quarterly investment dynamics. As within the model’s theoretical framework, investment dynamics are primarily driven by the combination of negative supply and demand shifts resulting from the consumption and labor-supply effects of self-mitigation and containment policies, this observation further highlights the importance of these channels for explaining macroeconomic fluctuations during the German Covid-19 crisis. Moreover, since investment constitutes an integral part of aggregate output, it further supports their role in explaining output dynamics as well.

5.3 Counterfactual Analysis: Evaluating the Lockdowns

Having concluded that containment policies are responsible for large output losses in Germany during Covid-19, we further conduct a counterfactual analysis to quantify those losses relative to a no-lockdown scenario. The results of such analysis are displayed in Figures 10 and 11.

Figure 10:

Quarterly model implied growth rates without lockdowns versus actual growth rates. Notes: The solid blue line represents the output from the model without any containment policies implemented, whereas the solid red line is the data from (OECD 2021).

Figure 11:

Model-implied deaths and infections: Lockdown versus No-Lockdown. Notes: The solid blue line represents the output from the model without any containment policies implemented, whereas the solid black line stands for the epidemiological data from RKI (2022c, 2022e). The red line is the number of infections multiplied by 1.8 in order to account for under-reported cases.

Figure 10 shows that even in the absence of containment policies, the model implies a sizeable decline in consumption, hours worked, investment, and GDP. The underlying reason for this result is displayed in Figure 11. It illustrates that without binding containment policies, infections and deaths rise to a significantly higher level than with containment policies. In response to this higher level of infections, susceptibles reduce their consumption and labor supply more drastically, as the health risks associated with those activities are far more pronounced. Thus, even though individuals are not forced to drastically cut back on consumption and labor supply through business closures and stay-at-home orders, they eventually would do so voluntarily in response to escalating epidemiological dynamics, triggering a significant recession nonetheless.

This recession, however, lags behind the one that resulted from actual containment policies (see Figure 10) since the peak declines of GDP and other macroeconomic aggregates occur in 2020:Q4 instead of 2020:Q2, implying that individuals are somewhat slower to react to epidemiological developments than policymakers. Moreover, whereas Figure 10 shows that a no-lockdown solution eventually leads to a pronounced recession as well, it also shows that the respective declines in aggregate output, consumption, and hours worked still remain significantly below those that have resulted under the actual containment policies.

Therefore, the counterfactual analysis implies that containment policies in Germany substantially aggravated economic losses during the pandemic, relative to a no-intervention scenario. Nevertheless, such policies also helped to substantially reduce the toll of infections and deaths, as shown in Figure 11. A natural question that arises from these opposing developments of GDP and deaths between the simulated laissez-faire and the actual containment approach taken by the German government is which policy would have been preferable ex-post. To answer this question, we employ a simple accounting framework based on the Value-of-Life (VoL henceforth) estimates for Germany reported in Sprengler (2004).

According to Sprengler (2004) the nominal VoL in Germany in 2004 was about 1.65 million euros, which translates into a value of 1.9 million euros in 2015 prices,^[18] using the GDP deflator for adjustment. Based on this VoL estimate, we compute the monetary benefit of the reduced death toll under containment policies, compared to the laissez-faire approach, by multiplying it with the difference in total deaths between the laissez-faire approach and actual deaths reported in the data from 2020:Q1 to 2021:Q2. According to the model, the number of deaths in the working-age population in the laissez-faire case would have exceeded those in reality by about 60,253. Hence, the monetary benefit of containment policies in terms of preventing potential deaths amounts to 114.48 billion euros (60,253 × 1.9 M euros). This is around 95 % of the cumulative difference between the model-implied levels of GDP under the laissez-faire scenario and actual GDP between 2020:Q1 and 2021:Q2 – total output losses under a no-lockdown rule would have potentially been 121 billion euros. If we consider an updated VoL of 2.36 million euros, from Sweis (2022), the monetary benefit of saving lives under containment policies would be 142.45 billion euros.^[19]

Therefore, even though containment policies were indeed subject to a strong trade-off between saving lives and saving livelihoods (Kaplan et al. 2020), they appear to have been the more favourable policy option compared to a no-lockdown solution.

Of course, this result strongly hinges on the model’s assumption that, in the absence of containment policies, individuals have a strong incentive to engage in voluntary containment, as this creates a lower bound to the economic fallout that could have potentially been avoided by adopting a laissez-faire policy, which drives the positive net-effect when health benefits and GDP losses are being compared. However, the experience of Sweden, which even in the absence of strict containment policies in 2020 experienced a peak decline in quarterly GDP of around 8.8 % (relative to 2019:Q4), supports this assumption.^[20]

5.4 Limitations and Additional Specifications

Although providing a theoretical framework that accounts for crucial features of the Covid-19 crisis, such as the explicit connection between epidemiological and macroeconomic outcomes and the effects of lockdowns, our model still features a variety of abstractions that demand some further discussion. First, it abstracts entirely from the fiscal policy measures adopted by the German government in order to cushion the effects of the pandemic. Due to their unprecedented size, those measures might also have shaped German output dynamics in a non-trivial manner, rendering their omission a potentially weakening simplification.

Nevertheless, Hinterlang et al. (2023) show that the differences between the trajectory of German output with and without fiscal policy interventions is estimated to be relatively small. The authors provide a comprehensive analysis of Germany’s fiscal package employing a sectoral, large-scale DSGE model. This result provides some comfort that the potential inclusion of fiscal interventions into our model would not have altered its fundamental conclusions.

Second, the model assumes a closed economy framework, which stands in contrast to Germany’s traditionally export-reliant economy. Nevertheless, as the BCA analysis has shown, the government wedge, which is essentially a measure of frictions associated with government spending and trade dynamics, is of relatively low importance for explaining output variations. Furthermore, Bonadio et al. (2021) demonstrate, using a global-network model, that the significant output losses in Germany towards the beginning of the pandemic were caused predominantly by the effects of domestic lockdowns. Hence, while open-economy channels certainly contributed to the development of GDP in Germany during the pandemic, they appear to have been dominated by the effects of domestic shutdowns and mobility restrictions. Thus, for the sake of simplicity, we relied on a leaner structure based on the evidence pointing toward a more domestically-oriented episode.

Third, our analysis also abstracts from any type of financial frictions which might have amplified Covid-19 associated supply shifts or might have been a source of macroeconomic disturbances themselves. However, according to Eichenbaum et al. (2022a), while exhibiting a spike in response to the pandemic, financial stress in Germany was relatively low and short-lived compared to previous crises. Hence, we assume that their conclusion also holds for the German Covid-19 recession, abstracting also from financial frictions, which seems to be an admissible first-order simplification.

Finally, the model assumes that the key SIR probabilities of infection (π₁, π₂, and π₂), recovery (π _r ), and death (π _d ) remain unchanged over the period. This crucial assumption, ignoring the strong seasonal component observed in infection rates and the random emergence of new virus variants that can be either more or less transmissible, may explain why our model is much less successful in replicating new infections (especially between waves and drastically in the third wave) and consequently deaths than it is in replicating macroeconomic variables. However, it is not easy to honestly predict how important this alternative modelling option would be for the results obtained.