COVID-19 pandemic, health risks, and economic consequences: Evidence from China.

This paper introduces the Susceptible-Infectious-Recovered model into the Bewley-type incomplete market model and uses it to study the impact of the coronavirus disease (COVID-19) pandemic on China's macroeconomics. The calibrated model predicts that the average propensity to consume household wealth will decline, while the demand for money will increase, and these predictions are consistent with the data. Monetary policy is effective because it provides enough liquidity for households to buffer health risks. Monetary stimulus is more effective in an economy with greater health risks and consumption uncertainty. Counterfactual experiments show that abandoning the containment policy too early would avoid a sharp drop in output and employment in the short term, but it would greatly increase mortality and ultimately lead to a decline in social welfare.

Introduction

Coronavirus disease 2019 (COVID-19) is an infectious disease caused by severe acute respiratory syndrome coronavirus. It was first identified in December 2019 in Wuhan, Hubei, China, and has resulted in an ongoing pandemic. As of August 30, 2020, more than 25 million cases were reported across 188 countries and territories, resulting in more than 843,000 deaths. The World Economic Outlook, updated by the International Monetary Fund in June, predicts that the global economy will shrink sharply by 4.9% in 2020, which is far more severe than the 2008–09 financial crisis.

Due to differences in population structure, residents' health levels, medical systems, and epidemic prevention policies, the impact of the COVID-19 pandemic on each country is different. As a country with a large population and a developed economy, China has successfully resisted the COVID-19 epidemic and the economy is gradually recovering. Fig. 1 plots the number of newly confirmed cases in China as well as the share of cases in Hubei province from January 20 to July 29. It is clear that most of the newly confirmed cases were in Hubei province. The spread of COVID-19 was quickly contained within Hubei province and the number of newly confirmed cases in Hubei dropped to zero on March 19 for the first time since the outbreak of the pandemic.1

Fig. 1

Spread of COVID-19 in China. The figure plots newly confirmed COVID-19 cases in China (left y-axis) and the proportion of newly confirmed cases in Hubei province (right y-axis). The statistics include the number of cases in Hong Kong, Macau, and Taiwan. The confirmed cases in Hubei province on February 12–19 included clinically diagnosed cases. Source: CEIC database.

One of the most pressing issues at present is to predict the economic consequences of COVID-19 and design policies to reduce the loss of life and wealth caused by the pandemic. Many economic studies combine representative agent macroeconomic models and epidemiological models to provide policy recommendations to curb the spread of COVID-19 and reduce the damage to the economy and health. Atkeson (2020) uses the Susceptible-Infectious-Recovered (SIR) model developed by Kermack and McKendrick (1927) to predict the progression of COVID-19 in the United States over the next 12–18 months. Eichenbaum, Rebelo, and Trabandt (2020) extend the classic SIR model to study the equilibrium interaction between economic decisions and epidemic dynamics. One of the important features in their model is that economic activity affects the rates of infection. Therefore, the competitive equilibrium is not Pareto optimal because people who are infected with the virus do not fully internalize the effect of their consumption. This leaves room for the government to intervene. Krueger, Uhlig, and Xie (2020) distinguish goods by the degree to which they can be consumed at home rather than in a social (and thus possibly contagious) context. They argue that the severity of the economic crisis in Eichenbaum et al. (2020) is much smaller if individuals can endogenously adjust the sectors in which they consume or work.

A few quantitative studies emphasize heterogeneity across households. Glover, Heathcote, Krueger, and Rios-Rull (2020) study the heterogeneous impact of government mitigation policies on different age groups and sectors. They investigate the optimal mitigation and redistribution policy by a utilitarian equal-weights social planner. Kaplan, Moll, and Violante (2020) conduct a quantitative analysis of the trade-offs between economic and health outcomes associated with alternative policy responses to the COVID-19 pandemic, with a focus on distributional implications. Similar to Glover et al. (2020) and Kaplan et al. (2020), this paper also features heterogeneous risks faced by households. We focus on the heterogeneity of health and analyze the impact of this heterogeneity on social welfare and the effect of monetary policy.

There are huge differences in the level of health between people, and the level of health is closely related to households decision-making.2 Finkelstein, Luttmer, and Notowidigdo (2013) find that a one-standard-deviation increase in the number of chronic diseases is associated with a 10%–25% decline in the marginal utility of consumption relative to the marginal utility when the individual has no chronic diseases. The outbreak of COVID-19 has exposed the differences in health between people. According to the World Health Organization (WHO), most people (about 80%) can recover from COVID-19 without special treatment, and for most people (especially children and young people), the disease caused by COVID-19 is usually minor. However, for some people, it can cause serious illness.3 Therefore, when studying the impact of the COVID-19 pandemic on economic activities and households' behavior, it is important to take into account the heterogeneity of health status.

This paper introduces the off-the-shelf epidemiological SIR model from Kermack and McKendrick (1927) into an analytically tractable Bewley-type incomplete market model to quantify the impact of the COVID-19 pandemic on China's macroeconomics. The model is based on the framework of Wen (2009), Wen (2015), and Wen and Dong (2017), which is a stochastic general equilibrium version of the model in Bewley (1983) and Lucas (1980). The key friction in the Bewley-Lucas model is the no-short-sale constraint on nominal balances, so agents cannot completely smooth idiosyncratic shocks by engaging in mutually lending and borrowing directly among themselves.

The main mechanism of the model is as follows. The impact of COVID-19 has worsened the health of infectious households and reduced their marginal utility. This has made them choose to reduce consumption and increase their demand for money, to withstand the potential increase in liquidity demand in the future (after recovery). As the only households in the model facing mortality risk, infectious households need to obtain the highest liquidity premium. However, it is not only the infectious households that increase their demand for money in the model. For the remaining households (households that are susceptible or recovered), the expected decline in the inflation rate (regardless of whether the government maintains a constant money supply growth rate or temporarily increases it) makes them also increase their demand for money. Finally, all households increase their demand for money to buffer health risks. When the nominal interest rate falls, the money demand of infectious households grows fastest, because their liquidity premium responds most to the nominal interest rate.

To quantify the economic impact of COVID-19 and the importance of health risks, we calibrate the model to match the development of the epidemic and economic losses in Hubei province, China. The benchmark model predicts that as more people are infected and nominal interest rates fall, households' demand for money will increase, especially for infectious households, and the ratio of consumption to wealth will decrease. Similar to Eichenbaum et al. (2020) and Krueger et al. (2020), in our model, COVID-19 affects the aggregate economy by simultaneously affecting supply and demand. However, unlike their model, our model also emphasizes the impact of the COVID-19 pandemic on capital investment and money demand.

We conduct some counterfactual experiments to study the role of monetary policy and containment policies in China. The complete abolition of the government's containment policy would prevent a sharp decline in production, but it would also increase the death toll by 22 times and cause substantial losses in social welfare.4 In another counterfactual experiment, we study the impact of expansionary monetary policy on economic activity and social welfare. Robustness analysis finds that monetary policy stimulus to economic activities and social welfare is proportional to the variance of the country's household health distribution.

The main contributions of this paper are as follows:

First, the paper analyzes for the first time how COVID-19 interacts with heterogeneous health risks and affects economic activity. We find that households' health risks have a quantitatively important impact on money demand. The paper analyzes an additional transmission mechanism of monetary policy: monetary policy can help households during the COVID-19 pandemic to meet their money demand and reduce the risk of future liquidity shortages. Therefore, monetary policy will be more effective in an economy where households face greater health risks and consumption uncertainty.

Second, the structural model allows us to quantify the impacts of the COVID-19 pandemic on household consumption, investment, money demand, and labor supply. So far, as far as we know, the existing macro literature does not pay much attention to capital investment and money demand (Kaplan et al. (2020) is an exception). This paper complements the literature in this field and can be used to analyze the short-term dynamics as well as the long-term impacts of the COVID-19 pandemic.

Third, after taking heterogeneity into account, the welfare effects of policies can be evaluated more accurately. We find that the welfare gains of monetary policy are positively correlated with the health risks faced by households.

Various other empirical papers study the spread of COVID-19 and its impact on the Chinese economy. Fang, Wang, and Yang (2020) quantify the causal impact of human mobility restrictions, finding that the lockdown was very effective, and provide estimates of diffusion under different scenarios. Qiu, Chen, and Shi (2020) study the impacts of social and economic factors on the transmission of COVID-19 in China and, through counterfactual analyses, quantify the effects of different public health measures in reducing the number of infections. Chen, He, Hsieh, and Song (2020a) and Chen, He, Hsieh, and Song (2020b) document that the impacts of the lockdown in China on various economic activities were immediate and dramatic. The official statistics suggest that there has been a quick recovery in manufacturing; however, small businesses were hit much harder. (Dai et al., 2020) document the impact of coronavirus on small and medium-size enterprises (SMEs), using data from the Enterprise Survey for Innovation and Entrepreneurship in China and follow-up interviews. Zhang and Wang (2020) propose how SMEs can resume production without compromising pandemic control in China.

The rest of the paper is organized as follows: Section 2 describes the spread of the pandemic in China and the government's policy response. Section 3 builds a structural model and uses it to analyze the impact of the COVID-19 pandemic on the macro economy. This section also presents the results of counterfactual experiments and robustness tests. Section 4 concludes.

The COVID-19 pandemic in China

In this section, we first summarize several key time points in the development of the COVID-19 pandemic in China. Then we describe the government's response to COVID-19, including fiscal and monetary policies. Finally, we describe the changes in some macroeconomic variables during the epidemic, including changes in the money market and the real economy.

Early responses to the COVID-19 in China

On December 27, 2019, Hubei Provincial Hospital of Integrated Chinese and Western Medicine reported cases of pneumonia of unknown cause to the Wuhan Jianghan Center for Disease Prevention and Control. Preliminary laboratory test results showed that they were cases of a viral pneumonia. On January 3, 2020, The Wuhan City Health Commission issued an Information Circular on Viral Pneumonia of Unknown Cause, reporting a total of 44 cases. Two days later, China sent a situation update to the WHO. The WHO released its first briefing on cases of pneumonia of unknown cause in Wuhan.

On January 9, the National Health Commission's expert evaluation team made a preliminary judgment that a new coronavirus was the cause. Immediately after that, Wuhan began to test all relevant cases admitted to local hospitals to screen for the new coronavirus. Three days later, the China Center for Disease Control and Prevention submitted to the WHO the genome sequence of the novel coronavirus (2019-nCoV), which was shared globally.

On January 19, after careful examination, a high-level national team of senior medical and disease control experts determined that the new coronavirus was spreading between humans. The next day, the National Health Commission held a press conference announcing that the virus could transmit from human to human.

Containing the spread of COVID-19

The situation became most pressing with the rapid increase in newly confirmed cases in China. On January 22, the State Council Information Office held its first press conference on the novel coronavirus.

On January 23, Wuhan decided to close the city's outbound routes at its airports and railway stations at 10 a.m. The Ministry of Transport issued an emergency circular suspending passenger traffic into Wuhan from other parts of the country by road or waterway. On the same day, many other cities in Hubei province also closed outbound traffic. From January 23 to 29, all provinces and equivalent administrative units on the Chinese mainland activated a level-1 public health emergency response.

In response to the epidemic, the government mobilized additional medical personnel from all over the country to Hubei, initiated large-scale screening, extended the Spring Festival holiday, postponed the start of school, and established 16 mobile hospitals.5

On February 19, for the first time in Wuhan, newly cured and discharged cases outnumbered newly confirmed ones. On February 21, all provinces began to downgrade their public health emergency response levels and gradually lift traffic restrictions. On March 11, the daily increase in the number of domestic cases on the Chinese mainland dropped to single digits.

On March 25, Hubei lifted outbound traffic restrictions and removed all health checkpoints on highways across the province except in Wuhan. With the exception of Wuhan, work and life gradually returned to normal in the whole province, and people could now leave Hubei if they had a “green” health code to show that they were not infected. On April 8, Wuhan lifted its 76-day outbound traffic restrictions and local work and daily life began to return to normal.

Fiscal and monetary policy reaction

This subsection discusses some of the fiscal and monetary policies that the government and central bank put in place. Since the outbreak, the Chinese government has introduced many tax relief measures. These include cutting the value-added tax, consumption tax, and corporate and individual income taxes, as well as waiving employers' payments to various social insurance schemes.

The State Administration of Taxation and the Ministry of Human Resources and Social Security issued guidance allowing enterprises outside/within Hubei province to make catch-up employer social security contributions within a period of three/five months following the containment of the COVID-19 outbreak without adversely affecting employee rights to social security benefits.6

Under the pandemic, local governments faced great fiscal pressure due to the increased fiscal expenditures needed to fight against the pandemic and the decreased economic activities as a result of suspension of work and production. To support local governments, the Ministry of Finance allowed them to retain 5% more tax revenue from March to June 2020, which was estimated to increase total local revenue by RMB 110 billion. The Ministry of Finance also issued 1 trillion yuan in special treasury bonds from mid-June to the end of July. All the funds raised were transferred to the local governments and required direct access to the city and county grassroots to support the local governments in fulfilling the “six guarantees” task and fighting the epidemic.7 The Ministry of Finance disclosed that the anti-pandemic special Treasury bonds were mainly invested in infrastructure construction and anti-pandemic-related expenditures.8

To cushion the economic blow of the coronavirus outbreak, the People's Bank of China (PBoC) adopted several expansionary monetary policies, including lowering the reserve requirement rate, loan prime rate (LPR), and re-lending and rediscount rates. These policies effectively reduced the financing cost of business, especially for SMEs.

On February 3 and 4, the PBoC pumped in 1.7 trillion yuan through open market operations. On February 20, the PBoC lowered the one-year LPR by 10 basis points and the five-year LPR by 5 basis points. On March 13, the PBoC said the reserve requirement rate would be lowered by 50 to 100 basis points for all banks that meet certain criteria for lending to SMEs and private companies, under its inclusive finance program. Joint-stock banks that meet other criteria will receive an additional reduction of 100 basis points in their reserve requirement rate. On April 3, the PBoC announced that it would cut the targeted reserve requirement rate by 0.5 percentage point on April 15 and again on May 15, amounting to a full 1 percentage point cut. The PBoC also announced that it would lower the interest rate paid for excess reserves from 0.72% to 0.35%.

Fig. 2 exhibits some facts about the money market in China during the pandemic. The growth rate of the money supply, measured by M1 or M2, has grown at a much faster pace than last year. The nominal interest rate, measured by the interbank lending interest rate, decreased rapidly until April 2020 and then gradually came back. The core Consumer Price Index, which excludes food and energy prices, came down by 0.7 percentage point from January to March 2020 and then gradually increased.9

Fig. 2

The Money Market during the COVID-19 Pandemic in China. Panel (a) plots the year-on-year growth rate of the money supply (M1), including currency in circulation (M0) and bank demand deposits. Panel (b) plots the year-on-year growth rate of the money supply (M2) and aggregate financing. The scale of aggregate financing refers to the amount of funds obtained by the real economy (i.e., non-financial enterprises and individuals) from the financial system. It includes four parts: financial institution on-balance sheet business, including RMB and foreign currency loans; financial institution off-balance sheet business, including entrusted loans, trust loans, and undiscounted bank acceptance bills; direct financing, including domestic stocks and corporate bonds of non-financial companies; and other projects, including small loan company loans, industrial fund investment, and so forth. Panel (c) plots the monthly average of the one-month interbank lending interest rate. Panel (d) plots the year-on-year growth rate of the core Consumer Price Index (excluding food and energy prices). Data source: WIND.

Fig. 3 plots the impact of COVID-19 on the real economy. Panel a shows that the largest fall in the annual growth rate of the cumulative gross domestic product (GDP) was in the first quarter of 2020. In the first quarter, the total GDP of Hubei province fell by nearly 40%, far exceeding the 6.8% decline in national GDP. Panel b shows that investment in fixed assets fell sharply, across the country and in Hubei province. The decline in fixed asset investment was more than twice the decline in GDP. Panel c plots the response of household consumption. Compared with the same period last year, household nominal consumption expenditure in Hubei province fell by 15% in the first half of the year, and the growth rate of national household consumption expenditure was about zero.

Fig. 3

Characteristics of the Real Economy during the COVID-19 Pandemic in China. Panel (a) plots the year-on-year growth rate of Hubei's and China's cumulative GDP. Panel (b) plots the year-on-year growth rate of cumulative investment in fixed assets. To avoid the impact of the Spring Festival, this statistical information is not reported for January of each year. Panel (c) plots the cumulative growth rate (at current prices) of household expenditure in Hubei province and China. Panel (d) plots the annual change in the average propensity to consume household disposable income. Panel (e) plots the year-on-year growth rate of cumulative private sector demand deposits. Panel (f) plots the year-on-year growth rate of new urban jobs created and the urban unemployment rate in urban areas. Data source: WIND.

Fig. 3, panel d, plots the year-on-year change in the average propensity to consume household income. Whether in Hubei province or the whole country, the average propensity to consume dropped by about 6 percentage points in the first half of 2020.

Fig. 3, panel e, shows that the number of new urban jobs created in February fell by 40% year-on-year and then slowly recovered. But as of the end of June, the amount of new employment was far from reaching the level during the same period last year. In stark contrast to the sharp decline in new urban jobs, the urban unemployment rate rose by only 1 percentage point in February and then slowly declined. This situation is reminiscent of the characteristics of China's employment volatility, as described by Storesletten, Zhao, and Zilibotti (2019). Since the agriculture sector absorbed a large portion of the reduced urban employment, the overall unemployment rate is not high.

Fig. 3, panel f, plots the nominal growth rate of demand deposits in the private sector in 2019 and January-July 2020.10 Except for January, the growth rate of demand deposits in 2020 is significantly higher than the level during the same period in 2019.

Finally, we summarize these stylized facts in the following points:

A substantial increase in the stock of money was accompanied by a decline in nominal interest rates and inflation rates.

Investment fell the most, followed by total output and household consumption.

The average propensity to consume household income has dropped sharply.

Demand deposits in the private sector rose significantly.

The number of newly created jobs in urban areas has fallen sharply, but the urban unemployment rate has increased slightly.

Later, we will build a model to explain facts 1–5, but our model will remain silent on the unemployment rate in fact 5.

Model

In the previous section, we introduced the spread of COVID-19 in China and the response of the country's macroeconomic variables. In this section, we construct a general equilibrium structural model to quantify the impact of COVID-19 on the economy and conduct some counterfactual experiments.

SIR model in discrete time

The SIR model developed by Kermack and McKendrick (1927) is widely used to describe the spread of COVID-19 (Atkeson (2020); Eichenbaum et al. (2020); Krueger et al. (2020); Glover et al. (2020)). We assume that the birth rate in the model is zero, and the initial total population size is standardized to 1. The households in the model include four types: susceptible S , infectious I , recovered R and deceased D . Given the initial distribution of the population {S 0,  I 0,  R 0,  D 0}, the dynamic adjustment of population composition is given by the following nonlinear equations: π reflects the average number of susceptible people contacted by each infected individual during the infection process. S of these contacts is susceptible to infection. Γ ∈ [0, 1] is a proxy variable for the government's containment policy: individuals must work in a company with probability 1 − Γ, and they must work from home with probability Γ.

We assume that the infection only occurs when people work in a company, so that the spread of the epidemic has nothing to do with the number of people working from home.11 Glover et al. (2020) consider a similar government containment policy (they call it a mitigation policy). Different from their assumption, we assume that people working from home still earn the same wage rate, while in Glover et al. (2020), people who are forced to stay home do not earn wage income and need to rely on government subsidies to survive.12

Therefore, will become the newly infected population in the next period. Let π and π denote the possibility of recovery and the possibility of death, respectively. The implicit assumption of the SIR model is that recovered individuals are immune to the COVID-19 virus.13

Firm sector

The production function in the model has a standard Cobb-Douglas form:

The total labor input N consists of two parts: workers working onsite (proportion 1 − Γ) and workers working from home (proportion Γ). Parameter A measures the productivity of workers who work from home relative to workers who work on site. When α = 0, the production function becomes a special case of Krueger et al. (2020). Following Glover et al. (2020), the elasticity of labor substitution between different types of labor is assumed to be infinite.

The following first-order conditions are obtained by solving the profit maximization problem of the firm: where W represents the hourly wage rate, and r represents the capital rental rate.

Households sector

We extended the setting of Wen (2015) in modeling the household sector, so that we can easily aggregate heterogeneous households.

The surviving households in each period are divided into three different categories, denoted by j, j= S, I, R. Households in category j randomly sample their health level θ according to their health distribution (the cumulative distribution function is F (θ )) in each period. The larger is θ , the better is health. Without loss of generality, we assume that the health distributions of the infected and recovered are the same, F  = F .14

The health-level distribution function satisfies the following conditionswhere

Condition 7 indicates that the average health level of infectious households is lower than that of the other two types of households.

Households' utility is given bywhere health status θ affects the households' marginal utility. This assumption follows the study by Finkelstein et al. (2013), which finds that the marginal utility declines as health deteriorates. The disutility of working ψ is inversely correlated with health status, ψ(θ ) = ψ(θ )−, α > 0 . To derive the analytical solutions, we set α = 1:

On the one hand, good health reduces the cost of work. On the other hand, good health increases the marginal utility of consumption and leisure. When α = 1, the two effects cancel each other out and the marginal utility of leisure becomes independent of the level of health.

The timing of economic activities in the model is as follows. At the beginning of period t, households will be sampled and randomly become one of the j = S, I, R, D. Households j begins with amount of real cash holdings of and k amount of capital stocks. For simplicity, we assume that the government will collect the wealth of deceased households to finance its own consumption G , which does not generate any utility flows to households.

We assume that households decide their labor supply n and capital stock k in the next period before they know their own health level θ and whether they will be infected with COVID-19 in period t + 1. Such a hypothesis can greatly simplify the solution of the model, and it is a bit close to reality: many infectious households have no symptoms at first.

When making a decision, households regard the possibility of infection, the possibility of recovery, and the possibility of death as exogenously given. When the government implements a containment policy, households must work onsite with probability Γ and work from home with probability 1 − Γ.No matter where they work, their wage rate is W . After households draw their health levels θ , they continue to make decisions on consumption c and cash holdings m .

We can formulate the households' problem recursively. First, we define the households' real wealth X as the sum of cash holdings , labor income W n , capital income (r  + δ)k , wealth gains from the sale of capital (1 − δ)k  − k , and transfers from the government tr .15

The problem of susceptible households can be written assubject to the following constraints:where denotes the expectation without knowing the health level θ . We assume that the probability of susceptible households being infected depends not only on the total probability π (1 − Γ)I , but also on the individual's health level θ , where Eφ (θ ) = 1 and , that is, healthy households are less likely to be infected.16 1 + τ is the time-varying consumption tax, which is collected by the government to cover the medical expenses of infectious households.17

Similarly, we can write down the problem of infectious households as:subject to the budget constraintwhere χ represents the per capita medical expenses of infectious households. We treat the medical expenses of infectious households as part of government expenditures, which are completely funded by government transfers, i.e., χ  = tr .18 Another interpretation is that the government maintains a medical insurance system that collects medical insurance premiums τ from everyone and pays the infectious households. Susceptible households and recovered households have no medical expenses, so the transfer payment from the government is zero. The budget of the medical insurance fund is balanced period by period:

We assume that the probability of recovery for the infectious households depends not only on the aggregate probability π , but also on the individual health level θ , where Eφ (θ ) = 1 and , i.e., healthy households are more likely to recover.

In the end, the recovered households solve the optimization problem:subject to the constraint

Recursive competitive equilibrium

where

Given the exogenous monetary supply {M ,  μ } ∞, government containment policy {Γ} ∞, medical insurance system {τ ,  tr } ∞, initial capital stocks K 1, and initial composition of households {S 0,  I 0,  R 0,  D 0}, the recursive competitive equilibrium consists of the sequence of allocations {G ,  X ,  c ,  k ,  m ,  n } ∞, prices {P ,  r ,  W } ∞, and threshold health levels {θ ∗} ∞, j = S, I, R, such that.

Given prices {P } ∞, households decisions {c ,  X ,  m ,  k ,  n } ∞, j = S, I, R solve households' optimization problems 8, 10, and 13.

The central bank's budget constraint holds:

The government's budget is balanced each period

The medical insurance budget holds

All markets clear:

Money supply equals money demand:

The supply of capital equals the aggregate demand for capital:

The supply of labor equals the aggregate demand for labor:

The supply of goods equals the demand for goods:

In the model, because money does not enter the utility function and is not required as a medium of exchange, there exists equilibrium where money is not valued, i.e., P  =  +  ∞ . In the following analysis, we focus on the case 0 < P  <  +  ∞ . Following Wen (2015), we focus on the case when money is used as a buffer against health risks for households.

We have the following proposition characterizing the solution to the households' problem.

The decision rules for consumption, money demand, and real wealth are determined bywhere the cutoff θ ∗, j = S, I, R. is determined by the following equations: where

Proof. See Appendix 5.1.

Proposition 2 describes the solutions. Eqs. (16), (17) are Euler's equations about money. ϱ(θ ∗) Eq. (19) measures the rate of return to money, which we refer to as the liquidity premium (Wen 2010a).

Take the Eq. (16) as an example. The left side represents the opportunity cost of holding one more unit of real wealth, and the right side measures the income from holding one more unit of real wealth. It consists of two parts: . The first term represents the marginal utility of consumption when the liquidity constraint is not binding, and its probability of occurrence is F (θ  ≤ θ ∗). The second term represents the marginal utility of consumption when the liquidity constraint is binding, and the probability is F (θ  > θ ∗). Therefore, ϱ(θ ∗) can be regarded as the rate of return to money or the liquidity premium. As long as the probability of being constrained is positive, ϱ(θ ∗) is strictly greater than 1. The intuition is that the option value of a dollar is greater than 1, because it provides liquidity when there is an urgent need to increase consumption.

As in Wen (2015), optimal consumption and money demand are piecewise linear functions of wealth. The threshold level θ ∗ depends on the real interest rate, inflation rate, and policy variables. Households whose health level is below the threshold level choose to hold a positive amount of cash, because when they suffer a health shock in the future, their liquidity constraint may tighten. We further use the first-order condition to obtain the following corollary:

Given π  > 0 and the condition E (θ ) < E (θ ) = E (θ ) holds, then

Proof. See Appendix 5.2.

Corollary 3 states that the infectious households will choose zero amount of capital stock in the next period. They also choose a smaller target level of wealth and are more likely hit by the liquidity constraint than the other two groups are. To understand why the infectious households behave differently, we focus on Euler Eqs. (20), (21). For j = S, R

Therefore, we havewhich indicates that the expected return to holding money equals the expected return to investing one more unit of capital for the household. For the infectious households, this is because

Hence,

Compared with the other two types of households, money brings higher expected returns to infectious households Eq. (22). Infectious households do not want to invest in capital, but some of them want to hold money. To persuade some infectious households to hold money, the liquidity premium of money must make up for the loss due to the risk of death. However, susceptible households and recovered households do not have this risk of death, so the liquidity premium of their money holdings is low.

In equilibrium, the expected capital gains will equal the liquidity premium of money held by susceptible households or recovered households, but it will be less than the liquidity premium of infectious households. When the nominal interest rate decreases, compared with the other two types of households, the money demand of infectious households increases the fastest. Infectious households have the lowest health threshold and will be more likely to hit the liquidity constraint. Expansionary monetary policy will help ease households' liquidity constraint, especially for infectious households.

Aggregation

Now we calculate the aggregate variables in the model economy. Aggregate consumption C is given bywhere

Aggregate money demand is given bywhere

Aggregate labor supply N is given bywhere the threshold levels {θ ∗} ∞, j = S, I, R are determined by Eq. (16), (17); the degenerate wealth distribution {X } ∞, j = S, I, R is given by Eq. (15); ϱ(θ ∗), j = S, I, R is defined in Eq. (19).

Steady states

Pre-COVID Steady State Before the outbreak of COVID-19, all households are susceptible. Suppose the constant money growth rate is given by ϖ > β − 1, i.e., μ  = ϖM , then the steady-state inflation rate will be ΔP /P  = ϖ. We normalize the price level P ∗ = 1. From the Euler equation, the steady-state interest rate equals r ∗ = 1/β − 1. The containment policy is not implemented and Γ∗ = 0. Because r  + δ = α(K /N ), we can solve for the steady-state capital-labor ratio The steady-state wage rate is and the capital-output ratio is given by (K/Y)∗ = βα/(1 − β + δβ). The consumption-output ratio is (C/Y)∗ = 1 − δβα/(1 − β + δβ). The liquidity premium of money is given by ρ (θ ∗) = (1 + ϖ)/β and we can solve for θ ∗. Therefore,although the growth rate of money does not affect the key ratios, such as capital/income (K/Y) and consumption/income (C/Y), it determines the levels of consumption, output, capital stock, and labor supply.

Post-COVID Steady State In the new steady state, we assume that all time-varying coefficients in the SIR model will converge: limπ  = π , limπ  = π , limπ  = π and the containment policy has been abandoned, limΓ = 0. Because

Therefore, if limS  = S ∗ > 0, then limΔS  → 0, which implies that limI  = 0, and limR  +  lim D  = 1 − S ∗.

We assume that the steady-state growth rate of money after COVID-19 is still ϖ. Therefore, the steady-state inflation rate is also ϖ. The new steady-state interest rate will return to its pre-crisis level. The economy has the same K/Y and C/Y ratios as before COVID-19.

Due to death, the total population declines. Although per capita consumption and output return to their steady-state levels before the COVID-19 outbreak, total consumption and output are smaller.

Calibration

In this subsection, we use Chinese data to calibrate the parameters of the model. The results of the calibration are shown in Table 1 .

Table 1

Calibrated parameters.

	Parameters	Value	Target
α	Capital income share	0.53	Bai and Qian (2010)
A	Relative productivity	0.06	−39% drop in output
δ	Depreciation rate	0.05∗7365	Bai and Qian (2010)
ψ	Disutility of labor	1.0	normalization
β	Discount factor	0.87^7365	(K/Y)∗ = 2.7
D0	Initial death	1.01e−10	Hubei's CDC data
I0	Initial infection	4.03e−6	Hubei's CDC data
R0	Initial recovery	4.22e−7	Hubei's CDC data
h¯	Health cost function	530	See text
κ	Health cost function	0.28	See text
σ	CDF of S/R households	2.0	See text
η	CDF of S/R households	1	Normalization
σI	CDF of I households	2.0	See text
ηI	CDF of I households	0.68	See text
ϖ	Steady-state money growth	1.025^7365−1	See text

We calibrate the parameters of the SIR model based on the recorded numbers of infected people, those who died, and recovered people in Hubei province from January 21 to April 8, 2020. The start date of the sample corresponds to the date when Hubei started to lock down the city.

We normalize the number of households of each type j = S, I, R, D based on the total population of Hubei province (59.27 million). Using the SIR model Eqs. (1), (2), (3), (4), we directly calculate the time series {π (1 − Γ),  π ,  π } based on the data. We assume that the long-term steady-state values of these parameters equal the values on April 8, 2020.

Fig. 4 , panel a, plots the actual data. By construction, the calibrated SIR model exactly matches the time path from January 21 to April 8. Fig. 4, panel b, plots the estimated parameters, where we need more information to separate the impact of the containment policy on the labor force 1 − Γ from the parameters π . We can see that (1 − Γ)π falls dramatically from week 1. An important reason is that Hubei's early nucleic acid testing capabilities could not keep up, resulting in the actual number of infections being higher than the official statistics. Over time, the nucleic acid testing capabilities greatly improved, thereby reducing the gap between the actual number of infections and the official figures.19 We do not intend to perform detailed modeling of the screening process for COVID-19 (such as nucleic acid detection), but take this probability π as exogenously given.

Fig. 4

Calibration of the SIR Model. In panel a, we normalize the number of households of each type by the total population of Hubei province (59.27 million). In panel b, we use Eqs. (1), (2), (3), (4) to directly calculate the time series π(1 − Γ), π, π.

To separate 1 − Γ from π , we use the Baidu within-city travel intensity index in Chen et al. (2020a) as a proxy for 1 − Γ. Fig. 5 , Panels a and b, plot the Baidu weighted within-city travel index for cities in Hubei province and cities outside Hubei Province, respectively. The first week here refers to the first week of the Chinese New Year.

Fig. 5

Baidu Within-City Traval Intensity Index and Calibrated Γ.We normalize the travel index from week 0 to week 1, where the definition of week 0 is the last week before each lunar year. The date of the Chinese New Year changes each year (respectively, on January 24, 2020 and February 4, 2019, that is, there is a difference of 11 days between the two). When drawing the travel intensity graph, we adjusted the difference in dates of the Chinese New Year.

Three days before the Chinese New Year in 2020, the lockdown of Hubei province began. Starting from the first week of the Chinese New Year, we can see a significant decline in the travel intensity index. There was hardly any decline in the travel intensity index for the same period in 2019.

The travel intensity index in week 1 relative to week 0 is measured by Γ. The estimated Γ is plotted in Fig. 5, panel c. Since Hubei citizens could leave Hubei freely after April 8, 2020, we assume that Γ linearly decreases to 0 from week 9 to week 12.

The calibration of the production function Y  = K ((1 − Γ)N  + AΓ N )1− involves two steps. First, we calibrate the capital income share α to 0.53, to match the capital income share estimated Bai and Qian (2010).20 Second, we calibrate the productivity parameter A= 0.06, to match the 39% decline in production in Hubei province in the first quarter of 2020.

One period in the model is one week. We set the annual capital depreciation rate at 5%, as in Bai and Qian (2010); therefore, the weekly depreciation rate . We set the annual discount factor to 0.87, to match the fixed-price capital-output ratio, which was 2.7 in 2012 (Cheremukhin, Golosov, Guriev, and Tsyvinski (2015)). Therefore, the weekly discount factor becomes .

According a the report by the State Council Information Office in China, as of May 31, a total of 162.4 billion yuan had been allocated for epidemic prevention and control funds from all levels of finance across the country.21 Assuming that the actual utilization rate of the national epidemic prevention and control funds was 50%, then the expenditure on epidemic prevention and control funds allocated to each person infected with COVID-19 was 1.4 million RMB.22 This is equivalent to 54 times the per capita consumption expenditure of Chinese residents in 2019.

In the model, we assume that the per capita medical expenditure of infectious households isand κ < 1 indicate that medical resources are public goods, and their production and provision have economies of scale.23 We calibrated the two parameters and h at the same time, so that: 1. household consumption in the model has an average decline of −15% in the first two quarters; and 2. per capita expenditure of people infected with COVID-19 is equivalent to 54 times the national per capita consumption level. The results of the calibration are and κ = 0.28.

We assume that health risks follow the Pareto distribution. The cumulative distribution functions of the health risks of susceptible and recovered households are given by

According to the definition of ϱ(θ ∗), H (θ ∗), and D (θ ∗), j = S, I, R Eq. (19), (24), (23)), we have the following analytical solution:

To simplify the problem, we assume that the health distribution functions of recovered and susceptible households are exactly the same: σ  = σ  = σ, η  = η  = η. Then we normalize η to 1 and calibrate the rest of the parameters σ, σ , η using empirical data.

We use the panel data from a nationally representative, biannual longitudinal household-level survey, the China Households Panel Studies 2010–2018 (CFPS). We first calculate household per capita non-health consumption (household consumption after deducting health expenses) using data from the CFPS.24 The CFPS data records the self-reported health levels of the households, with values ranging from 1 to 5 (larger values indicate lower levels of health). We divide families into a good health group (health level between 1 and 4) and a poor health group (health level 5). We keep households whose heads are over age 16 years and discard observations that lack household registration information or health status. In the end, we have a sample of 11,604 observations (5372 households), of which the heads of 860 households reported poor health status.

We perform panel regression on the logarithmic non-health consumption of these two groups of households. The regression model is specified as:where c denotes per capita non-health consumption. I { denotes the indicator for poor health. X represents the households' total net worth. Z is a vector of controls. α denotes the households fixed effects.

The regression results are shown in Table 2 . It can be seen from the table that bad health shocks have caused a decline in non-health-related consumer spending. Compared with the results of simple ordinary least squares regression (column 1), after controlling for household fixed effects (column 2), the impact of health on consumption becomes smaller. In the last column, we control a large number of household characteristics (occupation, household registration status, years of education, household size, work status, and so forth) and finally obtain that the impact of poor health on non-health consumption is about −7.2%.

Table 2

Regression Results on Non-health Consumption.

Variables	(1)	(2)	(3)
	Log non-health consumption
Bad health	−0.156*** (0.0267)	−0.119*** (0.0246)	−0.0718*** (0.0226)
Log total asset	0.196*** (0.00492)	0.147*** (0.00528)	0.148*** (0.00484)
Age, age2	YES	YES	YES
Year fixed effect	YES	YES	YES
Households fixed effect	NO	YES	YES
Other controls	NO	NO	YES
Observations	11,806	11,806	11,604
R-squared	0.29	0.30	0.49
Number of households	5423	5423	5372

Standard errors are in parentheses.

***p < 0.01, **p < 0.05, *p < 0.1

Next, we obtain the error term from panel regression, and calculate the variance of the error term ε grouped by health status. The variance of the good health group is 0.23, and the variance of the poor health group is 0.20. Although the difference is not large, the assumption that the two variances are the same can be rejected at the 95% confidence level. This level is comparable to results for the U.S. consumption variance data (Blundell, Pistaferri, and Preston (2008)).

Therefore, we chose σ = 2.0 to match the variance of the logarithmic consumption of susceptible households in steady state to 0.23. We chose σ  = 2.0 so that the consumption variance of the infectious households in the model in steady state (facing 2% mortality rate) is 0.20. Finally, we chose η  = 0.68 so that the average log consumption level of infectious households in steady state is 0.072 lower than that of susceptible households. Given these calibrated parameters, which means that the health level of the infectious households is on average 68% of that of the susceptible and recovered.

We assume that the central bank set the weekly growth rate of money to be in the steady state. Hence, the steady-state inflation rate is ϖ.

After the outbreak of the COVID-19 pandemic, the country adopted a series of relief measures, as described in Section 2.3. In the benchmark model, we used the actual year-on-year growth rate of China's M2 as the growth rate of the money supply in the model.25 We assume that the currency growth rate remains unchanged after reaching 11.6%. We assume that in the first period of the pandemic, residents fully expected the government's future currency growth rate.

We use the reverse-shooting algorithm to solve the entire transitional dynamics of the model. See Section 5.3 for details.

Simulation

Fig. 6, Fig. 7 exhibit the numerical simulation results. Fig. 6, panel a, plots the model's predicted path of the composition of the population. In the limit, the deceased account for 5.44 × 10−3% of the population in Hubei. Fig. 6, panel b, plots the year-on-year growth rate of money supply in the model. As a result of monetary stimulus, the total amount of money increases by about 11.6%. By construction, the baseline model fits the COVID-19 dynamics and nominal money stocks perfectly.

Fig. 6

Simulation Results of Baseline Model (I). D = deceased; I = infectious; R = recovered; SS = steady state.

Fig. 7

Simulation Results of Baseline Model (II) In panel a, I = infectious households; R = recovered households; S = susceptible households. In panel d, C = consumption; I = investment; N = employment; Y = output. APC = average propensity to consume; SS = steady state.

Fig. 6, panel c, plots the responses of the inflation rate and nominal interest rate. The money growth rate is about 2.2% in week 2, but the inflation rate is only about 0.3%, which is far less than the money growth rate. This is mainly due to two reasons. First, because households fully anticipated the increase in the government's currency, prices had already risen before the increase in the money supply. The inflation rate in the first week was as high as 8.3% (not shown in the figure), which was still less than the increase in the total money stock. Second, although there is no price stickiness, money is not neutral in the model. After the first period, the expected inflation rate continues to fall, and the nominal interest rate also falls. This will increase household demand for money, ease liquidity constraints, and stimulate consumption. Recall stylized fact 1, summarized in Section 2.3; our model successfully predicts the declines in the inflation rate and nominal interest rate. However, the model over-predicts the decline in the inflation rate in the first quarter, which may be related to the lack of price stickiness in the model. The monthly nominal interest rate in the first quarter falls by approximately 0.20%. But in the data, the nominal interest rate dropped rapidly, by 1%, and then slowly recovered.

Fig. 6, panel d, shows that after the outbreak of the COVID-19 pandemic, real interest rates first fell and then rose. The marginal products of labor and capital have been reduced due to the government's containment policy. When the containment policy gradually faded out, the real interest rate gradually returned to the previous steady-state level.

Fig. 7, panel a, shows that the cumulative money demand per capita of the three types of households has increased, and the money demand of infectious households has increased the fastest. The reason is that the expected decline in inflation will lead to a decline in the liquidity premium, which requires more households to hold money to achieve this. The money demand of infectious households is growing faster for two reasons. First, only infectious households face the risk of death, so their money holdings have the highest liquidity premium and their target wealth level is the lowest (see Corollary 3). These factors together lead to a small amount of money demand from infectious households at the beginning. Second, when the nominal interest rate drops, the liquidity premium of infectious households drops even more, which causes the threshold value of holding money θ ∗ to rise faster. As a result, the money demand of infectious households grows faster.26

Fig. 7, panel b, plots the velocity of money, which is defined as The decline in output and the increase in money demand together cause a decline in the velocity of money circulation.

We compare the average propensity to consume in the model with the actual data (stylized fact 3, summarized in Section 2.3). In the data, the consumption-to-disposable income ratio decreases by about 6% after the COVID-19 shock. In our closed economy model, S = I, and the investment-to-output ratio falls both in the data and in the model. Since households own all the capital, the aggregate household saving rate in our model must decrease. Therefore, unless there is a clearer distinction between corporate savings and household savings, our model cannot explain the declining consumption-to-income ratio. Instead, we focus on the consumption-to-wealth ratio. The average consumption-to-wealth ratio in the steady-state model is 11%, and in the CFPS data it is 16%. The two are relatively close, which is not our direct calibration target. Fig. 7, panel c, shows that the average consumption-to-wealth ratio has dropped by 3 percentage points, which accounts for 50% of the actual decline in the data. The decline in consumption is mainly due to the distortion of consumption caused by the increase in medical expenditures. Another reason is that infectious households have increased money demand and reduced consumption. The model's prediction of the decline in average propensity to consume is only half of the actual decline. This may be due to other reasons not captured by the model. For example, in the data, households' risks also come from other aspects, such as idiosyncratic income risk. Persistent downward income risks can lead to a further decline in consumption.

Fig. 7, panel d, plots the growth rate of per capita output, consumption, investment, and employment relative to the steady state. The government's containment policies reduced the marginal output of labor and capital, resulting in a sharp decline in employment and investment in the economy. The expansionary monetary policy temporarily increased the growth rate of money, causing the expected inflation rate to fall, and to a certain extent reduced the negative impact of the containment policies on employment and investment. Our model successfully predicts the responses of output and consumption in the first quarter, which are our calibration targets. Quantitatively, the model over-predicts the decline in investment (the decline in investment in the first quarter in the model was −140%, while in the data it was −80%), which may be related to the absence of investment adjustment costs.27

Policy experiment

In this section, we investigate the results of four policy experiments and compare them with the benchmark model. In experiment A, we keep the growth rate of money at the steady-state level (μ  = ϖ), but we do not change the government's containment policy. In experiment B, we temporarily increase the growth rate of money, so that the stock of money increases by 40%. More specifically, we assume the following exogenous money growth process:where we set μ 1 = 0.088 and ρ = 0.75 such that the stock of money will increase by 40% permanently. Experiments C and D are equivalent to abolishing the government's containment policy under the conditions of experiments A and B, respectively. The results of the numerical simulations are shown in Table 3 .

Table 3

Results of the Counterfactual Experiments and Robustness Checks.

Model	Benchmark Case: σ = σI = 2.0
	Government Policy		YoY Changes Relative to Pre-COVID SS (%)					Components of ΔXt (%)				Death Rate (%)	Welfare (%)
	ΔMM^∗	Γt	Yt	Nt	Ct	Invt	Xt	Money	Labor	Capital	Transfer	Dt	Δ
Benchmark	11.6%	>0	−9.24	−7.99	−5.83	−30.20	1.57	1.70	−0.55	−0.09	0.50	0.0054	−0.74
Exp. A	ϖ	>0	−9.27	−8.05	−5.82	−30.50	0.54	0.68	−0.55	−0.08	0.50	0.0054	−0.80
Exp. B	40%	>0	−9.15	−7.83	−5.87	−29.34	3.33	3.47	−0.55	−0.09	0.50	0.0054	−0.62
Exp. C	ϖ	=0	0.57	−1.88	−9.70	63.71	4.19	3.93	0.03	−1.09	1.32	0.117	−1.27
Exp. D	40%	=0	0.70	−1.64	−9.75	64.97	7.04	6.79	0.04	−1.10	1.32	0.117	−1.10
Robustness Check I: σ = σI = 1.5
	Government Policy		YoY Changes Relative to Pre-COVID SS (%)					Components of ΔXt (%)				Death Rate (%)	Welfare (%)
Model	ΔMM^∗	Γt	Yt	Nt	Ct	Invt	Xt	money	labor	capital	transfer	Dt	Δ
Benchmark	11.6%	>0	−9.21	−7.57	−4.77	−36.48	3.17	3.23	−0.19	0.01	0.12	0.0054	−0.46
Exp. A	ϖ	>0	−9.28	−7.71	−4.76	−37.08	1.50	1.55	−0.19	0.01	0.12	0.0054	−0.74
Exp. B	40%	>0	−9.02	−7.22	−4.83	−34.81	6.45	6.51	−0.18	0.00	0.12	0.0054	0.20
Exp. C	ϖ	=0	0.49	−1.17	−6.86	45.55	4.71	4.65	0.01	−0.27	0.32	0.117	−0.95
Exp. D	40%	=0	0.78	−0.65	−6.92	48.01	9.79	9.72	0.02	−0.27	0.32	0.117	−0.02
Robustness Check II: σ = σI = 2.5
	Government Policy		YoY Changes Relative to Pre-COVID SS (%)					Components of ΔXt (%)				Death Rate (%)	Welfare (%)
Model	ΔMM^∗	Γt	Yt	Nt	Ct	Invt	Xt	money	labor	capital	transfer	Dt	Δ
Benchmark	11.1%	>0	−9.23	−8.18	−6.49	−26.04	0.80	1.00	−0.97	−0.28	1.05	0.0054	−0.71
Exp. A	ϖ	>0	−9.25	−8.22	−6.48	−26.23	0.11	0.31	−0.97	−0.28	1.05	0.0054	−0.73
Exp. B	40%	>0	−9.17	−8.09	−6.52	−25.50	1.78	1.99	−0.97	−0.29	1.50	0.0054	−0.67
Exp. C	ϖ	=0	0.65	−2.27	−11.47	75.18	4.03	3.49	0.07	−2.28	2.75	0.117	−1.64
Exp. D	40%	=0	0.73	−2.13	−11.51	75.97	5.73	5.19	0.07	−2.29	2.75	0.117	−1.58

Since the government's containment policy has not changed in both cases, the death rate in experiment A has not changed. Compared with the benchmark model, experiment A shows that the 11.6% increase in the stock of money does not have a strong stimulus effect on output, employment, and consumption. The 11.6% increase in the stock of money brought a 0.03 percentage point increase in output, 0.3 percentage point increase in fixed investment, and 1.03 percentage points increase in real household wealth X .28 Becausewe can decompose the growth of total household wealth into four parts: the growth of monetary assets , the growth of labor income W n , the growth of capital gains (1 + r )k  − k , and the growth of transfer income tr . We find that the main increase in household wealth comes from the growth of monetary assets, followed by government transfers. Household labor income and capital gains have suffered losses. This is consistent with stylized fact 4 summarized in Section 2.3.

For further analysis of the welfare effects of the policy, we follow the approach of Wen, 2015 and Glover et al., 2020, and constructed a utilitarian social welfare function, in which all households have equal weights.

Let the social welfare function under the monetary policy {μ ,  π } and the containment policy {Γ} be W({μ ,  π }, {Γ})

We calculate the level of social welfare on the transition path from the initial steady state to the new steady state. For comparison, we use consumption variation (CV) to measure the welfare loss of various models relative to the steady state before the COVID-19 outbreak.

Since all households in the steady state before the epidemic are susceptible, the social welfare function can be expressed aswhere c represents the allocation of consumption in the steady state.

We denote the consumption variation by Δ, which can be calculated as the solution to the following equation:

After some algebraic derivation, we can finally solve Δ, which is given by the following expression

We calculate social welfare according to Eq. (28), and the result is shown in the last column of Table 3. The welfare loss in the benchmark model caused by the COVID-19 pandemic is about −0.74 percentage point. Policy experiment A shows that if the monetary policy in the benchmark model were not implemented, social welfare would have been reduced by an additional 0.06 percentage point.

In policy experiment B, we temporarily increase the growth rate of money, and the stock of money increases by 40%. Not surprisingly, compared with the results of policy experiment A, a substantial increase in the stock of money is a greater stimulus for consumption, investment, output, employment, and household wealth. Although the model shows that a temporary increase in the growth rate of money helps to increase social welfare, a permanent increase in the growth rate of money will bring about a permanent increase in the inflation rate, which will reduce the level of social welfare (Wen (2015)).

In policy experiment C, the government abolishes the containment policy, but the increase in infectious households leads to huge social costs. The number of infected people increases by 35 times, and the death rate increases by 22 times. Although we assume that there are economies of scale in medical treatment, due to rising medical expenses, distortions in household consumption are further aggravated and household consumption is further reduced. Due to the slow change in capital stock, the sharp drop in consumption in the short term causes households to increase investment temporarily. When the number of infections drops, investment will drop again.

After weighing economic output and mortality, the social welfare loss of experiment C has greatly increased. These calculations do not take into account the possibility of aggravating the spread of COVID-19 due to the abolition of the containment policy (Eichenbaum et al. (2020), Krueger et al. (2020)), nor the additional value of survival (Glover et al. (2020)). Therefore, our calculations can be interpreted as a conservative estimate of welfare loss.

The effectiveness of monetary policy

A key variable in the benchmark model is σ, whose size determines the health risks that households face. Following Wen (2009) and Wen (2015), we tried different values of σ as a robustness test. The results are presented in Table 3. We choose σ = 1.5 from Wen (2009) and σ = 2.5 from Wen (2015) respectively as our robustness test. These two values also correspond to the two endpoints of the interval in the literature (Clementi and Gallegati (2005); Levy and Levy (2003); Nirei and Souma (2007)). To simplify the discussion, we keep the sigma of all households equal, σ = σ . The results are shown in Table 3.

When σ = σ  = 1.5, the social welfare of policy experiments B and D improves relative to the level pre-COVID-19, and the effectiveness of monetary policy increases. This is because when sigma is small, a small drop in the interest rate will trigger a strong increase in household money demand. If the government can increase the money supply accordingly, it will enable those households to ease their liquidity constraints.

To illustrate this point, we plot the demand for money as a function of the nominal interest rate in Fig. 8 . Lucas (2000) interprets the downward-sloping curve as the “money demand” curve, which is the inverted velocity in our model.

Fig. 8

Money Demand and the Nominal Interest Rate. We assume that the real interest rate is fixed at the steady-state level and only changes the level of inflation. For ease of comparison, the vertical axis is the logarithmic currency-output ratio after removing the mean. The first period of our model is one week, and here we add up the data and convert it into quarterly data.

Fig. 8 shows that when the σ rises, the absolute value of the slope becomes smaller, which means that the elasticity of money demand with respect to interest rates becomes smaller. This suggests that monetary policy will be more effective if households face greater health risks (given the same decline in the nominal interest rate, output increases more relative to the real money balance when σ is smaller). If we further use ordinary least squares to estimate the elasticity of money demand, we get elasticities of −9 and −12, respectively (corresponding to σ = 2.5 and σ = 1.5, respectively). If money demand is converted into annual data, the elasticities are estimated to be 2 and 3, respectively.

We could estimate the elasticity of China's money demand to interest rates and use it to recalibrate the σ. The regression model is specified as:

The dependent variable is the currency-output ratio. The money supply M is measured by the stock of M2, and nominal output P Y is quarterly GDP. Here we can overwhelmingly reject the null hypothesis of a unit root at all common significance levels. The interest rate i comes from the weighted interbank lending interest rate (three months). The sample interval is selected from 2011 to the present, to exclude the impact of the economic crisis in 2008–09. The estimation results, in Table 4 , show that the elasticity using China's quarterly data since 2011 is between −7.4 and − 5.0. If we use this moment condition alone to estimate σ, ignore its impact on the variance of household consumption, then the smallest value we can get for σ is 3.7. This value is a large outlier in the literature. The fact that monetary demand is less elastic with respect to interest rates may be related to the insufficient progress of China's interest rate marketization.

Table 4

The Elasticity of Money Demand to Interest Rates.

Variables	(1)	(2)	(3)
	logM_tP_tY_t
log(1 + it)	−5.893*** (1.818)	−7.379** (2.743)	−4.985** (2.430)
L.logM_tP_tY_t		0.123 (0.172)	−0.196 (0.170)
L.log(1 + it)		2.894 (2.891)	2.529 (2.472)
Time, time2	NO	NO	YES
Observations	38	37	37
R-squared	0.226	0.259	0.492

Standard errors are in parentheses.

***p < 0.01, **p < 0.05, *p < 0.1

Conclusion

This paper introduced the existing epidemiological SIR model into an incomplete market model and used it to study the impact of the COVID-19 pandemic on China's macroeconomics. We calibrated the model to match the decrease in actual output and the variance in household consumption. The benchmark model predicts that as more and more people are infected, the average propensity to consume household wealth will drop significantly, while the total demand for money will increase. The COVID-19 pandemic has led to a decline in the health of infectious families, and families have increased their cash reserves to alleviate these health risks. Monetary policy will be more effective in the economy with greater household consumption uncertainty. Expansionary monetary policy can mitigate the negative impact of the pandemic on the economy to a certain extent. Abandoning the containment policy too early will avoid a sharp drop in output and employment in the short term, but it will also increase the mortality rate by 22 times and ultimately lead to a decline in social welfare. Robustness analysis found that the effectiveness of monetary policy is affected by household health risks. If households generally face greater health risks, the central bank will increase the provision of liquidity, which will greatly improve social welfare.

Appendix

Prof of Proposition 2

Write down the Bellman equation as

The FOCs arewhere and the envelop conditions yield

Similarly, we haveand Envelop conditions yield

andand the envelop conditions yield

Susceptible households

Case A: v  = 0.

From the FOC w.r.t m

Note that the exact form of f (θ ) will not affect the FOC. We solve for optimal consumptionwhere we defineand X denotes the same amount of wealth level held by the all Susceptible households. The intuition is that all susceptible households make decision about X before the realization of health shock, and all of them face the same distribution of idiosyncratic shocks. They will adjust their labor supply to achieve the same optimal level of wealth.

Because m  ≥ 0 implies (1 + τ )c  ≤ X , we must have θ  ≤ θ ∗.

Case B: v  > 0. (1 + τ )c  = X , i.e.,

In this case, we have θ  > θ ∗.

Combining case A and case B, we can write down the optimal consumption as:

Therefore,

Hencewhere

Thus, for Susceptible households, the decision rules for consumption, money demand, and real wealth are given by

Becausethe cutoff θ ∗ is determined by

θ ∗ is independent of individual history.

The FOC w.r.t. n can also be simplified to

Infectious Households Similarly, we consider the problem of the Infectious households,

Case A:

ν  = 0, m  > 0, θ  ≤ θ ∗. In this case, the households will hold money as a store of value

Again, the exact form of f (θ ) do not affect the FOC. We define

and

Case B:

m  = 0, v  > 0, θ  > θ ∗. Households consumption (1 + τ )c  = X , in this case, the marginal utility of consumption

The above two cases implies that

Because

Hencewhere

The decision rules for consumption, money demand, and real wealth are given by

Becausewhere the cutoff θ ∗ is determined by

The FOC w.r.t. n can also be simplified as

Recovered Households In the end, for the recovered households.

Case A:

ν  = 0, m  ≥ 0, θ  ≤ θ ∗. In this case, the households will hold money as a store of value

Again, we define

In this case

Case B:

v  > 0, m  = 0, θ  > θ ∗. Households consumption (1 + τ )c  = X , in this case, the marginal utility of consumption

The above two cases implies that

Because

Hencewhere

Thus, for the recovered households, the decision rules for consumption, money demand, and real wealth are given,where all recovered households supply labor to achieve the same level of X before the realization of θ . Becausewhere the cutoff θ is determined by the following equation

The FOC w.r.t. n becomes

Proof of Corollary 3

First of all, using the Envelop conditions, the FOC w.r.t k becomes

Plugging in the FOC w.r.t. n into the above equation yields,

Similarly, we have

and

Because π  > 0, χ  ≥ 0, χ  ≥ 0, χ  ≥ 0, we must have χ  = χ  ≡ 0, χ  > 0, ∀ i, i.e., the infectious households will always choose k  = 0.

To see why the S households and I households choose the same level of capital, i.e., k  = k , let's assume that investment in capital incurs some adjustment cost, ϕ = ηk 2, η > 0. Then

When we can see that k only depend on the aggregate variables: the wage growth and the interest rate, not the individual history. The results holds if η → 0. Hence, we have 0 < k  = k .

From

Given that π  > 0,and the condition E (θ ) < E (θ ) = E (θ ) holds, we must have

Proof. Fromwe must have

Algorithm to solve the model

In order to solve the transition, we assume a long enough transition period T = 500, after which the economy will reach the new steady states.

We first compute the S , I , R for the transition period using the (1), (2), (3), (4).

Pluginto the Euler equations and get

We can use backward shooting to solve the whole transitional path.

We make an initial guess about k , N , which have to be close to the final steady state. Then we use the Eq. (30), we solve for {k ,  W } .

Given the exogenous money growth rate {1 + μ } , we solve for the P 1, {π }1

Guess P 1, and solve for π 2 such that using the following conditions

After solving π 2, we update P 2, and solve for π 3, π 4, . …

Check if π  → ϖ under the initial guess P 1, if not, update P 1 in step 2.a

Solve for {C ,  G } using

and

From the resource constraint K  + G  + C  = K (1 − δ) + Y ,divide both sides by N

Hence, given the guess for N , k , we can solve for N , N , ……N 1.

Check whether k 1 ∗ N 1 equals K 1 = K , if not, update initial guess in step 1

Funding

This project was supported by the National Science Fund for Young Scholars of China (Grant No. 71603011).