2.2.1 Macroeconomic dynamics

In the face of the COVID-19 pandemic, China prioritized the health and well-being of its citizens over short-term economic gains. This commitment was reflected in the stringent nationwide lockdown that lasted two and a half months and ultimately succeeded in containing the virus. With the epidemic under control, economic activities gradually resumed.

Figure 1 shows the year-on-year growth rates of GDP, total consumption, online consumption, and investment from January 20, 2020 to September 28, 2020.Footnote ⁵ An advantage of using year-on-year growth rates is that they largely eliminate the effects of seasonality. The data are from the National Bureau of Statistics of China. We transform quarterly or monthly data into weekly data using the quadratic-match sum method. Appendix B provides details of this transformation. The weekly GDP growth rate, depicted by the solid line, reached its lowest point in week 7 at $-15.94\%$ . This negative growth rate persisted for 13 weeks before turning slightly positive in week 14 (April 20–27), and GDP growth remained positive thereafter. Similarly, the weekly growth rate of aggregate consumption, shown by the dashed line, bottomed out in week 5 at −27.43%. It stayed negative for 19 weeks and turned positive in week 20 (June 1–8), then continued on an upward path.

Figure 1.

The year-on-year growth rates of GDP, total consumption, online consumption, and investment from January 20, 2020 to September 28, 2020.

The solid line with circles shows the year-on-year growth rate of investment at the weekly frequency. Its trough occurred in week 7, at −37.79%. After 17 weeks of negative growth, investment became slightly positive starting week 18 (May 18–25), followed by a moderate upward trend.

The dashed line with circles represents the weekly year-on-year growth rate of online consumption. The lowest point was in week 7, at −14.79%. Growth turned positive in week 10 (March 23–30) and remained consistently positive for the rest of the period. These patterns yield key insights. First, the decline in online consumption was about half that of total consumption, implying a sharper contraction in offline spending. Second, online consumption rebounded much faster than total consumption, underscoring its stabilizing role during the crisis. The next subsections examine this role more systematically, first by analyzing the foundations of China’s online economy and then presenting empirical evidence of its buffering effects.

2.2.2 Development of the online economy before the pandemic

Prior to the COVID-19 pandemic, China’s online economy had already undergone rapid and large scale – albeit uneven – development, supported by three main factors. First, China made substantial investments in digital infrastructure. Between 2015 and 2020, these investments totaled over 1.5 trillion yuan.Footnote ⁶ According to the Digital China Development Report (2020), released by the Cyberspace Administration of China, China had become the global leader in digital infrastructure, building the world’s largest fiber-optic and 4G/5G networks.Footnote ⁷ Fixed broadband household penetration rose from 52.6% in 2015 to 96% in 2020, while mobile broadband user penetration increased from 57.4% to 108%.Footnote ⁸ In terms of 5G network construction, China ranked first globally in both speed and scale of deployment, establishing 718,000 base stations and over 200 million terminal connections by 2020. The number of internet users increased from 688 million in 2015 to 989 million in 2020, while the internet penetration rate rose from 50.3% to 70.4% over the same period.Footnote ⁹ This rapid expansion in user access, combined with extensive infrastructure investment, enabled the online economy (e.g. WFH) to operate with minimal disruption during the pandemic.

Second, China developed a sophisticated e-commerce ecosystem, characterized by vertically integrated platforms, seamless digital payments, and efficient logistics. Major firms like Alibaba and JD.com manage end-to-end supply chains – from AI-based inventory to last-mile delivery. Ubiquitous mobile payments (e.g., Alipay and WeChat Pay) facilitate over 98% of all transactions. During the pandemic, these platforms sustained over 90% operational capacity, supporting “contactless consumption” and maintaining household access to essential goods.

Third, consumer behavior reinforced this digital shift. Unlike many Western consumers who prioritize privacy, Chinese consumers prioritize convenience over absolute privacy protection. They are willing to make a controlled trade-off of their privacy rights in exchange for digital benefits. This accelerated the adoption of livestreamed shopping, telemedicine, and online education during lockdowns.

Together, these factors – digital infrastructure, platform capacities, and consumer adaptability – positioned China as the global leader in online retail. According to the United Nations Conference on Trade and Development (UNCTAD), seven countries – Australia, Canada, China, Korea, Singapore, the United Kingdom, and the United States – accounted for 65% of global Business-to-Consumer (B2C) e-commerce in 2019. China led the group in 2018–2020, with online retail sales of $\$$ 1,060.4 billion, $\$$ 1,233.6 billion and $\$$ 1,414.3 billion, respectively, amounting to over half of the total online retail sales among the seven countries.

However, the development of online retail has not been uniform across China. A joint report by the World Bank and Alibaba Group shows significant regional disparities. In 2017, online purchases accounted for 49% of retail sales in Shanghai, 44% in Beijing, 36% in Guangzhou, and 35% in Zhejiang – all coastal and economically advanced regions. In contrast, seven inland provinces reported less than 2% online retail share.Footnote ¹⁰ China’s ranking in the 2019 UNCTAD B2C E-commerce Index further highlights this discrepancy.Footnote ¹¹ While Switzerland ranked first, China placed $55^{th}.$ This relatively low ranking reflects the index’s emphasis on readiness factors such as internet access, secure servers, financial inclusion, and postal reliability – rather than the scale of e-commerce itself. For example, in 2019, China ranked only $87^{th}$ in internet penetration and $33^{rd}$ in online shopping penetration.

3.1.1 The household’s problem

Prior to the epidemic, the economy is populated by a continuum of ex-ante identical households with measure one. In period $t$ , the representative household consumes offline consumption goods $\left ( c_{t}^{out}\right )$ and online consumption goods $\left ( c_{t}^{in}\right )$ , saves by purchasing firms’ shares $\left ( s_{t}\right )$ at price $q_{t}$ , and supplies labor $\left ( n_{t}\right )$ to firms. The representative household’s lifetime utility is given by (3)

(3) \begin{equation} \sum _{t=0}^{\infty }\beta ^{t}\left \{ \phi \ln c_{t}^{out}+\left ( 1-\phi \right ) \ln c_{t}^{in}-\frac {\theta }{2}n_{t}^{2}\right \} \end{equation}

where $\beta \in \left ( 0,1\right )$ denotes the discount factor, and $\phi$ and $\theta$ capture the utility weight of offline consumption and the disutility of labor, respectively.

The household’s period budget constraint is given by

(4) \begin{equation} c_{t}^{out}+p_{h}c_{t}^{in}+q_{t}s_{t}=w_{t}n_{t}+d_{t}s_{t-1}+q_{t}s_{t-1} \end{equation}

where $p_{h}$ denotes the price of e-commerce goods, $w_{t}$ denotes the real wage and $d_{t}$ denotes the dividends on the shares owned by the household that are distributed by firms. The household spends its labor income ( $w_{t}n_{t}$ ) and the revenues from holding period $t-1$ shares ( $d_{t}s_{t-1}+q_{t}s_{t-1}$ ) on offline and online consumption and for their purchase of period $t$ shares ( $q_{t}s_{t}$ ).

For simplicity, we normalize to one the price of offline consumption goods and treat the price of online consumption goods, $p_{h}$ , as a parameter. We assume that the price of online consumption goods is a fraction of that of offline consumption goods, that is, $p_{h}\lt 1$ . We calibrate the value of $p_{h}$ in Section 4 and provide arguments why $p_{h}\lt 1$ .

Let $\lambda _{bt}$ denote the Lagrangian multiplier associated with the budget constraint (4). The household’s problem is to choose offline and online consumption, savings, and labor supply to maximize its lifetime utility (3) subject to the budget constraint (4). The first-order conditions for the representative household’s problem are:

(5) \begin{align} \left [ c_{t}^{out}\right ] :& \quad \phi \frac {1}{c_{t}^{out}} =\lambda _{bt} \end{align}

(6) \begin{align} \left [ c_{t}^{in}\right ] :& \quad \left ( 1-\phi \right ) \frac {1}{c_{t}^{in}}=p_{h}\lambda _{bt} \end{align}

(7) \begin{align} \big[ n_{t}\big] :& \quad \theta n_{t}=\lambda _{bt}w_{t} \end{align}

(8) \begin{align} \big[ s_{t}\big] :& \quad q_{t}\lambda _{bt}=\beta \lambda _{bt+1}\big( d_{t+1}+q_{t+1}\big) \end{align}

3.1.2 The firm’s problem

There is a continuum of competitive representative firms of unit measure. The representative firm produces goods using capital ( $k_{t}$ ), labor ( $n_{t}$ ), and technology ( $A_{t}$ ) according to the Cobb-Douglas production function:

(9) \begin{equation} Y_{t}=A_{t}k_{t}^{\alpha }n_{t}^{1-\alpha } \end{equation}

For simplicity, we normalize $A_{t}$ to oneFootnote ²⁰

We assume that the firm accumulates capital and that capital depreciates at the rate $\delta$ . The firm’s investment is given by

(10) \begin{equation} I_{t}=k_{t+1}-\big( 1-\delta \big) k_{t} \end{equation}

Define $d_{t}\equiv d\big( k_{t}\big) =k_{t}^{\alpha }n_{t}^{1-\alpha }-w_{t}n_{t}-\big( k_{t+1}-\big( 1-\delta \big) k_{t}\big)$ as the dividend paid by the firm to the household in period $t$ and $D\big( k_{t}\big) =\max \sum _{j=0}^{\infty }\beta ^{j}\frac {\lambda _{bt+j}}{\lambda _{bt}}d\big( k_{t+j}\big) =\max \sum _{j=0}^{\infty }\beta ^{j}\frac{u^{\prime }\big( c_{t+j}^{out}\big) }{u^{\prime }\big( c_{t}^{out}\big) }d\big( k_{t+j}\big)$ as the maximum value of the firm from period $t$ onwards.Footnote ²¹ The firm chooses capital ( $k_{t+1}$ ) and labor ( $n_{t}$ ) to maximize its value. We can write the firm’s problem in recursive form as

(11) \begin{equation} D\big( k_{t}\big) =\max _{\left \{ k_{t+1},n_{_{t}}\right \} }\left \{ k_{t}^{\alpha }n_{t}^{1-\alpha }-w_{t}n_{t}-\left ( k_{t+1}-\left ( 1-\delta \right ) k_{t}\right ) +\beta \frac {u^{\prime }\left ( c_{t+1}^{out}\right ) }{u^{\prime }\left ( c_{t}^{out}\right ) }D\left ( k_{t+1}\right ) \right \} \end{equation}

The first-order conditions for the firm’s problem read:

(12) \begin{align} \big[ n_{t}\big] :& \quad \left ( 1-\alpha \right ) \frac {Y_{t}}{n_{t}}=w_{t} \end{align}

(13) \begin{align} \big[k_{t+1}\big] :& \quad \beta \frac {u^{\prime }\big( c_{t+1}^{out}\big) }{u^{\prime }\big( c_{t}^{out}\big)}\big( \alpha \frac {Y_{t+1}}{k_{t+1}}+1-\delta \big) =1 \end{align}

In equilibrium, the representative household and the representative firm solve their optimization problems and the labor, goods, and asset markets clear.Footnote ²² Appendix F presents the solution for the steady state of the pre-infection economy, which serves as the initial point for the post-infection economy.

3.2.1 Testing and households’ beliefs about their health status

Modeling the macroeconomic effect of testing is challenging because the timing of testing and its accuracy can generate huge heterogeneity in people’s beliefs about their health status, thus inducing great heterogeneity in people’s economic decisions. As in Eichenbaum et al. (Reference Eichenbaum, Rebelo and Trabandt2022), to mitigate these sources of heterogeneity, we make some assumptions. First, we divide the population into a testing pool and a non-testing pool. In each period, the government tests $\mu$ percent of the population that has not yet been tested, resulting in a growing number of individuals in the testing pool over time depending on the value of $\mu$ . Second, we posit that those in the testing pool are tested every period until they recover or die. This assumption ensures that a person’s current economic decisions are not affected by the timing of their entry into the testing pool, but rather by his/her current health status. Third, we assume that tests are accurate, allowing for the revelation of a person’s health status. This assumption reduces the degree of heterogeneity without fundamentally changing the basic results.

We assume that each group (people outside the testing pool or people inside the testing pool) consists of four sub-groups: susceptible, infected, recovered, and deceased. The fraction of the susceptible, infected, recovered, and deceased people who are outside the testing pool are denoted by $S_{t}^{u},I_{t}^{u},R_{t}^{u}$ , and $D_{t}^{u}$ , respectively. The fraction of the susceptible, infected, recovered, and deceased people who are inside the testing pool are denoted by $S_{t}^{k},I_{t}^{k},R_{t}^{k}$ , and $D_{t}^{k}$ , respectively. At time zero, a fraction $\varepsilon$ of the population becomes infected. We assume that this information is public and that testing starts in period zero, so the initial conditions for the different groups of people are $I_{0}^{u}=\varepsilon,S_{0}^{u}=1-\varepsilon,R_{0}^{u}=0,I_{0}^{k}=S_{0}^{k}=R_{0}^{k}=0.$

Households’ beliefs about their health status. After period zero, individuals’ health status may change due to recovery or death for those infected and the possibility of infection for those susceptible. Individuals outside of the testing pool lack knowledge of their own health status and so form subjective probabilities regarding their health.

We denote individuals’ subjective probabilities regarding their health status at time $t$ , given they are alive at time $t$ ( $a_{t}$ ), as $p(s_{t}|a_{t})$ , $p(i_{t}|a_{t})$ , and $p(r_{t}|a_{t})$ for susceptibility $\left ( s_{t}\right )$ , infection $\left ( i_{t}\right )$ , and recovery $\left ( r_{t}\right )$ , respectively. It is assumed that in every period, a fraction $\pi _{r}$ of infected people recover and a fraction $\pi _{d}$ of infected people die. The subjective probability for susceptible individuals outside of the testing pool of becoming infected by the virus is denoted as $\tau _{t}^{u}$ . Appendix D details that the subjective probabilities of individuals becoming susceptible, infected, or recovered at time $t+1$ , conditional on being alive at time $t$ , are given by equations (14), (15), and (16), respectively.

(14) \begin{equation} p(s_{t+1}\mid a_{t+1}) [ 1-\pi _{d}p ( i_{t}\mid a_{t})] =( 1-\tau _{t}^{u}) p( s_{t}\mid a_{t}) \end{equation}

(15) \begin{equation} p\left ( i_{t+1}\mid a_{t+1}\right ) \left [ 1-\pi _{d}p\left ( i_{t}\mid a_{t}\right ) \right ] =\tau _{t}^{u}\,\,p\left ( s_{t}\mid a_{t}\right ) +\left ( 1-\pi _{r}-\pi _{d}\right ) p\left ( i_{t}\mid a_{t}\right ) \end{equation}

(16) \begin{equation} p\left ( r_{t+1}\mid a_{t+1}\right ) \left [ 1-\pi _{d}p\left ( i_{t}\mid a_{t}\right ) \right ] =p\left ( r_{t}\mid a_{t}\right ) +\pi _{r}p\left ( i_{t}\mid a_{t}\right ) \end{equation}

where $[ 1-\pi _{d}p( i_{t}\mid a_{t})]$ $(\!=p( a_{t+1}\mid a_{t}))$ is the probability that the individual is still alive at time $t+1$ .

We assume that at time zero the probabilities that an individual outside the testing pool is susceptible, infected, or recovered are given by $p\left ( s_{0}\mid a_{0}\right ) =1-\varepsilon$ , $p\left ( i_{0}\mid a_{0}\right ) =\varepsilon$ , $p\left ( r_{0}\mid a_{0}\right ) =0$ , respectively. In the above equations, a key variable is $\tau _{t}^{u}$ , which represents the subjective probability of susceptible persons outside the testing pool that he or she is infected by the virus.Footnote ²³ This variable plays a dual role as one of the determinants of both the subjective probabilities regarding individuals’ health states outside of the testing pool and the population dynamics across various groups of individuals. We will specify how $\tau _{t}^{u}$ is determined in the next subsection.

3.2.2 People outside the testing pool (or uncertain about health status)

The lifetime utility of an individual who is outside the testing pool at time $t$ , $U_{t}^{u}$ , is given by

(17) \begin{align} U_{t}^{u}=& u\big( c_{t}^{out,u},c_{t}^{in,u},n_{t}^{u}\big) +\left ( 1-\mu \right ) \beta \left [ 1-{\pi _{d}}p\left ( i_{t}\mid a_{t}\right ) \right ] U_{t+1}^{u} \notag \\ & +\mu \beta \left [ 1-\pi _{d}\left ( i_{t}\mid a_{t}\right ) \right ] \left [ p\left ( s_{t+1}\mid a_{t}\right ) U_{t+1}^{s}+p\left ( i_{t+1}\mid a_{t}\right ) U_{t+1}^{i}+p\left ( r_{t+1}\mid a_{t}\right ) U_{t+1}^{r}\right ] \end{align}

Here, the variables $U_{t+1}^{s},U_{t+1}^{i}$ , and $U_{t+1}^{r}$ denote the lifetime utility of a tested person who is susceptible, infected, and recovered at time $t+1$ , respectively. The parameter $\mu$ represents the fraction of people that enter the test pool (or the fraction of people who are certain about their health status). Equation (17) states that the lifetime utility of the person at time $t$ is equal to the current period utility $u( c_{t}^{out,u},c_{t}^{in,u},n_{t}^{u})$ plus the discounted expected lifetime utility from time $t+1$ . In period $t+1$ , the probability of an individual being alive and not in the testing pool is $\left ( 1-\mu \right ) \left [ 1-\pi _{d}\left ( i_{t}\mid a_{t}\right ) \right ]$ . The probabilities of an individual being alive and tested negative, positive, or recovered are given by $\mu \left [ 1-\pi _{d}\left ( i_{t}\mid a_{t}\right ) \right ] p\left ( s_{t+1}\mid a_{t}\right )$ , $\mu \left [ 1-\pi _{d}\left ( i_{t}\mid a_{t}\right ) \right ] p\left ( i_{t+1}\mid a_{t}\right )$ , and $\mu \left [ 1-\pi _{d}\left ( i_{t}\mid a_{t}\right ) \right ] p\left ( r_{t+1}\mid a_{t}\right )$ , respectively. The discounted expected lifetime utility from time $t+1$ is given by the sum of the second and the third terms in equation (17).

We assume that the period utility of the person outside the testing pool is given by

(18) \begin{equation} u\left ( c_{t}^{out,u},c_{t}^{in,u},n_{t}^{u}\right ) =\left ( 1-L_{t}\right ) \phi \ln c_{t}^{out,u}+\left ( 1-\chi L_{t}\right ) \left ( 1-\phi \right ) \ln c_{t}^{in,u}-\frac {\theta }{2}\left ( n_{t}^{u}\right ) ^{2} \end{equation}

where $c_{t}^{out,u}$ and $c_{t}^{in,u}$ denote the individual’s offline consumption and online consumption, respectively. The variable $0\lt L_{t}\lt 1$ represents lockdown shocks which reduce the utilities and, hence, demands for offline and online consumption. The parameter $0\lt \chi \lt 1$ captures the fact that lockdown shocks have relatively smaller effects on online consumption. In addition, we assume $n_{t}^{u}=\left ( 1-L_{t}\right ) n_{t}^{u}+L_{t}n_{t}^{u}$ . That is, $\left ( 1-L_{t}\right )$ fraction of labor is performed on-site and the remaining $L_{t}$ fraction of labor is performed at home.

We assume that lockdown shocks also directly affect online consumption through the presence in the utility function, for the following reason: although online consumption was more resilient than offline channels, it was not entirely insulated from lockdown-related disruptions. While certain segments of the online economy – particularly direct-distribution platforms such as JD.com, SF Express, and Hema Fresh – expanded rapidly by adopting models like “online ordering + contactless pickup lockers”, other parts of the online retail infrastructure experienced significant constraints. In particular, franchise-based delivery services (e.g., YTO, STO, and Yunda) saw substantial declines in volume – by as much as 35% or more – due to delayed resumptions of operations, cross-regional transportation bottlenecks caused by highway controls, and reduced efficiency at sorting hubs following tightened public health protocols. These operational frictions constrained consumers’ ability to access online goods and services during lockdowns.Footnote ²⁴

The budget constraint of an individual outside the testing pool is given by

(19) \begin{equation} c_{t}^{out,u}+p_{h}c_{t}^{in,u}+q_{t}s_{t}^{u}=w_{t}n_{t}^{u}+d_{t}s_{t-1}^{u}+q_{t}s_{t-1}^{u}+\Gamma _{t}^{u} \end{equation}

where $s_{t}^{u}$ is the equity share held by an individual outside the testing pool, and $\Gamma _{t}^{u}$ is the lump-sum transfers. In steady state $\Gamma _{t}^{u}=0$ . For simplicity, following Zhao (Reference Zhao2020), we assume that the same wage rate $w_{t}$ is received by households, irrespective of their place of work.

In our model, the probability of being susceptible and infected at time $t+1$ for an individual outside the testing pool depends on $\tau _{t}^{u}$ , the probability of the susceptible person becoming infected at time $t$ , which in turn, depends on the social interactions between the susceptible and infected people outside the testing pool.Footnote ²⁵ Following Eichenbaum et al. (Reference Eichenbaum, Rebelo, Trabandt and Koijen2021), we assume that social interactions occur in three ways: purchasing consumption goods, working, and engaging in non-economic activities. Thus, the probability that the virus is transmitted to a susceptible person outside the testing pool due to social interactions is given by

(20) \begin{eqnarray} \tau _{t}^{u}&=&\frac {1}{\sigma _{u}}\left [ \pi _{1}\left ( 1-L_{t}\right ) c_{t}^{out,u}\left ( 1-L_{t}\right ) I_{t}^{u}C_{t}^{out,u}\right.\nonumber\\&&\left. +\pi _{2}n_{t}^{u}\left ( 1-L_{t}\right ) I_{t}^{u}N_{t}\left ( 1-L_{t}\right ) +\pi _{3}\left ( 1-L_{t}\right ) I_{t}^{u}\left ( 1-L_{t}\right ) \right ] \end{eqnarray}

Here, $\left ( 1-L_{t}\right ) I_{t}^{u}C_{t}^{out,u}$ and $\left ( 1-L_{t}\right ) I_{t}^{u}N_{t}^{u}$ are the average consumption and hours worked of infected people, whose health status is unknown and who are subject to the lockdown restriction. In (20), the first two terms in the square brackets, $\pi _{1}\left ( 1-L_{t}\right ) c_{t}^{out,u}\left ( 1-L_{t}\right ) I_{t}^{u}C_{t}^{out,u}$ and $\pi _{2}n_{t}^{u}\left ( 1-L_{t}\right ) I_{t}^{u}N_{t}\left ( 1-L_{t}\right )$ , reflect virus transmissions that result from consumption- and work-related interactions, respectively. The parameter $\pi _{1}$ reflects the probability that the virus is transmitted as a result of shopping interactions. The parameter $\pi _{2}$ reflects the probability that the virus is transmitted as a result of work interactions. Note that in the first term, only people who do offline shopping interact with each other. Similarly, in the second term, only people who work on-site interact with each other. The term $\pi _{3}\left ( 1-L_{t}\right ) I_{t}^{u}\left ( 1-L_{t}\right )$ reflects transmissions that result from non-economic interactions, where $\pi _{3}$ reflects the probability that the virus is transmitted as a result of non-economic social interactions. Finally, $\sigma _{u}$ is a scaling parameter that is used to better match the health outcomes (e.g. the death rate and the infection rate).Footnote ²⁶

The problem of an individual outside the testing pool is to maximize his or her lifetime utility (17) subject to the budget constraint (19), the transmission function (20), and the laws of motion for the probability of being susceptible (14) and infected (15).Footnote ²⁷

The first-order conditions areFootnote ²⁸

(21) \begin{align} \left [ c_{t}^{out,u}\right ] :& \quad \frac {\left ( 1-L_{t}\right ) \phi }{c_{t}^{out,u}}-\lambda _{bt}^{u}+\lambda _{\tau t}^{u}\frac {1}{\sigma _{u}}\pi _{1}\left ( 1-L_{t}\right ) ^{2}I_{t}^{u}C_{t}^{out,u}=0 \end{align}

(22) \begin{align} \left [ c_{t}^{in,u}\right ]:&\quad \frac {\left ( 1-\chi L_{t}\right ) \left ( 1-\phi \right ) }{c_{t}^{in,u}}-p_{h}\lambda _{bt}^{u}=0 \end{align}

(23) \begin{align} \left [ n_{t}^{u}\right ] :& \quad {-\theta n_{t}^{u}}+\lambda _{bt}^{u}w_{t}+\lambda _{\tau t}^{u}\frac {1}{\sigma _{u}}\pi _{2}\left ( 1-L_{t}\right ) I_{t}^{u}N_{t}^{u}\left ( 1-L_{t}\right ) =0\end{align}

(24) \begin{align} \left [ s_{t}^{u}\right ] :& \quad q_{t}\lambda _{bt}^{u}=\beta \lambda _{bt+1}^{u}\left [ 1-\pi _{d}p\left ( i_{t}\mid a_{t}\right ) \right ] \left ( d_{t+1}+q_{t+1}\right ) \\[10pt]\nonumber \end{align}

(25) \begin{equation} \left [ p\left ( i_{t+1}\mid a_{t+1}\right ) \right ] : \left ( \begin{array}{c} \frac {dU_{t}^{u}}{dp\left ( i_{t+1}\mid a_{t+1}\right ) }-\lambda _{it}^{u}\left [ 1-\pi _{d}p\left ( i_{t}\mid a_{t}\right ) \right ] +\beta \lambda _{it+1}^{u}p\left ( a_{t+1}\mid a_{t}\right ) \left ( 1-\pi _{r}-\pi _{d}\right ) \\[8pt] +\beta \lambda _{it+1}^{u}p\left ( a_{t+1}\mid a_{t}\right ) \pi _{d}p\left ( i_{t+2}\mid a_{t+2}\right ) +\beta \lambda _{st+1}^{u}p\left ( a_{t+1}\mid a_{t}\right ) \pi _{d}p\left ( s_{t+2}\mid a_{t+2}\right ) \\[8pt] +\beta \lambda _{rt+1}^{u}p\left ( a_{t+1}\mid a_{t}\right ) \pi _{r}+\beta \lambda _{rt+1}^{u}p\left ( a_{t+1}\mid a_{t}\right ) \pi _{d}p\left ( r_{t+2}\mid a_{t+2}\right ) =0\end{array}\right ) \end{equation}

(26) \begin{equation} \left [ p\left ( s_{t+1}\mid a_{t+1}\right ) \right ] :\quad \left ( \begin{array}{c} \frac {dU_{t}^{u}}{dp\left ( s_{t+1}\mid a_{t+1}\right ) }-\lambda _{st}^{u}\left [ 1-\pi _{d}p\left ( i_{t}\mid a_{t}\right ) \right ] +\beta \lambda _{it+1}^{u}p\left ( a_{t+1}\mid a_{t}\right ) \tau _{t+1}^{u} \\[8pt] +\beta \lambda _{st+1}^{u}p\left ( a_{t+1}\mid a_{t}\right ) \left ( 1-\tau _{t+1}^{u}\right ) =0\end{array}\right ) \end{equation}

where $\lambda _{bt+j}^{u}\beta ^{j}p\left ( a_{t+j}\mid a_{t}\right )$ , $\lambda _{\tau t+j}^{u}\beta ^{j}p\left ( a_{t+j}\mid a_{t}\right )$ , $\lambda _{st+j}^{u}\beta ^{j}p\left ( a_{t+j}\mid a_{t}\right )$ , and $\lambda _{it+j}^{u}\beta ^{j}p\left ( a_{t+j}\mid a_{t}\right )$ denote the Lagrange multipliers associated with the budget constraint (19), the transmission function (20), and the laws of motion for the probability of being susceptible (14) and infected (15), respectively.

3.2.3 People inside the testing pool

Under our assumptions about testing (Section 3.2.1), people inside the testing pool know whether they are susceptible, infected or recovered in the current period, but susceptible and infected people face uncertainty about their future health status.

Susceptible people Let $\tau _{t}^{s}$ be the probability that a susceptible person becomes infected and $1-\tau _{t}^{s}$ be the probability that a susceptible person remains susceptible. The lifetime utility of a susceptible person, $U_{t}^{s}$ , is

(27) \begin{equation} U_{t}^{s}=\left ( 1-L_{t}\right ) \phi \ln c_{t}^{out,s}+{\left ( 1-\chi L_{t}\right )} \left ( 1-\phi \right ) \ln c_{t}^{in,s}-\frac {\theta }{2}\left ( n_{t}^{s}\right ) ^{2}+\beta \left [ \left ( 1-\tau _{t}^{s}\right ) U_{t+1}^{s}+\tau _{t}^{s}U_{t+1}^{i}\right ] \end{equation}

where $c_{t}^{out,s}$ and $c_{t}^{in,s}$ denote the person’s offline consumption and online consumption, respectively. The person’s labor $n_{t}^{s}$ consists of work on-site, $\left ( 1-L_{t}\right ) n_{t}^{s}$ , and work at home, $L_{t}n_{t}^{s}$ .

The budget constraint is

(28) \begin{equation} c_{t}^{out,s}+p_{h}c_{t}^{in,s}+q_{t}s_{t}^{s}=w_{t}n_{t}^{s}+d_{t}s_{t-1}^{u}+q_{t}s_{t-1}^{u}+\Gamma _{t}^{s} \end{equation}

where $s_{t}^{s}$ is the equity share held by the susceptible person, and $\Gamma _{t}^{s}$ is the lump-sum transfers. In steady state $\Gamma _{t}^{s}=0$ .

The probability that the virus is transmitted to a susceptible person due to social interactions is given by

(29) \begin{eqnarray} \tau _{t}^{s}&=& \frac {1}{\sigma _{s}}\left[ \pi _{1}\left ( 1-L_{t}\right ) c_{t}^{out,s}\left( 1-L_{t}\right) I_{t}^{u}C_{t}^{out,u}\right.\nonumber\\&&\left. +\pi _{2}n_{t}^{s}\left ( 1-L_{t}\right ) I_{t}^{u}N_{t}^{u}\left ( 1-L_{t}\right ) +\pi _{3}\left ( 1-L_{t}\right ) I_{t}^{u}\left ( 1-L_{t}\right ) \right ] \end{eqnarray}

Similar to equation (20), the three terms on the RHS of equation (29) represent the three channels through which the susceptible individual is infected by the virus: the consumption channel, the working channel, and the non-economic channel. Again, $\sigma _{s}$ is the scaling parameter which is used to better match the health outcomes.

The problem of a susceptible individual is to maximize his or her lifetime utility (27) subject to the budget constraint (28) and the transmission function (29). The first-order conditions are

(30) \begin{align} \left [ c_{t}^{out,s}\right ] :& \quad \frac {\left ( 1-L_{t}\right ) \phi }{c_{t}^{out,s}}-\lambda _{bt}^{s}+\lambda _{\tau t}^{s}\frac {1}{\sigma _{s}}\pi _{1}\left ( 1-L_{t}\right ) ^{2}I_{t}^{u}C_{t}^{out,u}=0 \end{align}

(31) \begin{align} \left [ c_{t}^{in,s}\right ] :& \quad \frac {\left ( 1-\chi L_{t}\right ) \left ( 1-\phi \right ) }{{c_{t}^{in,s}}}-p_{h}\lambda _{bt}^{s}=0 \end{align}

(32) \begin{align} \left [ n_{t}^{s}\right ] :& \quad -\theta n_{t}^{s}+\lambda _{bt}^{s}w_{t}+\lambda _{\tau t}^{s}\frac {1}{\sigma _{s}}\pi _{2}\left ( 1-L_{t}\right ) I_{t}^{u}N_{t}^{u}\left ( 1-L_{t}\right ) =0 \end{align}

(33) \begin{align} \left [ s_{t}^{s}\right ] :& \quad q_{t}\lambda _{bt}^{s}=\beta \lambda _{bt+1}^{s}( d_{t+1}+q_{t+1}) \end{align}

where $\lambda _{bt}^{s}$ and $\lambda _{\tau t}^{s}$ are the Lagrange multipliers associated with the budget constraint (28) and the transmission function (29), respectively.

Infected people In our benchmark model, the identified infected people are quarantined and their consumption is provided by the government. For simplicity, we assume that they can maintain their consumption at the steady-state level of the pre-infection economy. Therefore, $c_{t}^{out,i}=c^{out}$ , $c_{t}^{in,i}=c^{in}$ . Clearly, infected people cannot (or are not allowed to) work. Therefore, $n_{t}^{i}=0.$ Footnote ²⁹

To facilitate our counterfactual analysis, below we also present the problem of the infected people in a general case. An infected person at time $t$ could become recovered with probability $\pi _{r}$ at time $t+1$ , or remain infected with probability $1-\pi _{r}-\pi _{d}$ . The lifetime utility of an infected person, $U_{t}^{i}$ , is

(34) \begin{equation} U_{t}^{i}=\left ( 1-L_{t}\right ) \phi \ln c_{t}^{out,i}+\left ( 1-\chi L_{t}\right ) \left ( 1-\phi \right ) \ln c_{t}^{in,i}-\frac {\theta }{2}\left ( n_{t}^{i}\right ) ^{2}+\beta \left [ \left ( 1-\pi _{r}-\pi _{d}\right ) U_{t+1}^{i}+\pi _{r}U_{t+1}^{r}\right ] \end{equation}

The budget constraint is

(35) \begin{equation} c_{t}^{out,i}+p_{h}c_{t}^{in,i}+{q_{t}}s_{t}^{i}=w_{t}n_{t}^{i}+d_{t}s_{t-1}^{i}+q_{t}s_{t-1}^{i}+\Gamma _{t}^{i} \end{equation}

where $s_{t}^{i}$ is the equity share held by the infected person, and $\Gamma _{t}^{i}$ is the lump-sum transfers. In steady state $\Gamma _{t}^{i}=0$ .

The problem of an infected person is to maximize his or her lifetime utility (34) subject to the budget constraint (35). The first-order conditions are

(36) \begin{equation} \left [ c_{t}^{out,i}\right ] :\quad \frac {\left ( 1-L_{t}\right ) \phi }{c_{t}^{out,i}}-\lambda _{bt}^{i}=0 \end{equation}

(37) \begin{equation} \left [ c_{t}^{in,i}\right ] :\quad \frac {\left ( 1-\chi L_{t}\right ) \left ( 1-\phi \right ) }{c_{t}^{in,i}}-p_{h}\lambda _{bt}^{i}=0 \end{equation}

(38) \begin{equation} \left [ n_{t}^{i}\right ] :\quad -\theta n_{t}^{i}+\lambda _{bt}^{i}w_{t}=0 \end{equation}

(39) \begin{equation} \left [ s_{t}^{i}\right ] :\quad q_{t}\lambda _{bt}^{i}=\beta \lambda _{bt+1}^{i}\left ( d_{t+1}+q_{t+1}\right ) \end{equation}

where $\lambda _{bt}^{i}$ is the Lagrangian multiplier associated with the budget constraint (35).

Recovered people We assume that a recovered individual will not become infected or susceptible in the future. The lifetime utility of a recovered person, $U_{t}^{r}$ , is

(40) \begin{equation} U_{t}^{r}=\left ( 1-L_{t}\right ) \phi \ln c_{t}^{out,r}+\left ( 1-\chi L_{t}\right ) \left ( 1-\phi \right ) \ln c_{t}^{in,r}-\frac {\theta }{2}\left ( n_{t}^{r}\right ) ^{2}+\beta U_{t+1}^{r} \end{equation}

The person’s budget constraint is

(41) \begin{equation} c_{t}^{out,r}+p_{h}c_{t}^{in,r}+q_{t}s_{t}^{r}=w_{t}n_{t}^{r}+d_{t}s_{t-1}^{r}+q_{t}s_{t-1}^{r}+\Gamma _{t}^{r} \end{equation}

where $s_{t}^{r}$ is the equity share, and $\Gamma _{t}^{r}$ is the lump-sum transfers. In steady state $\Gamma _{t}^{r}=0$ .

The problem of a recovered individual is to maximize his or her lifetime utility (40) subject to the budget constraint (41). The first-order conditions read:

(42) \begin{align} \left [ c_{t}^{out,r}\right ] :& \quad \frac {\left ( 1-L_{t}\right ) \phi }{c_{t}^{out,r}}-\lambda _{bt}^{r}=0 \end{align}

(43) \begin{align} \left [ c_{t}^{in,r}\right ] :& \quad \frac {\left ( 1-\chi L_{t}\right ) \left ( 1-\phi \right ) }{c_{t}^{in,r}}-p_{h}\lambda _{bt}^{r}=0 \end{align}

(44) \begin{align} \left [ n_{t}^{r}\right ] :& \quad -\theta n_{t}^{r}+\lambda _{bt}^{r}w_{t}=0 \end{align}

(45) \begin{align} \left [ s_{t}^{r}\right ] :& \quad q_{t}\lambda _{bt}^{r}=\beta \lambda _{bt+1}^{r}\left ( d_{t+1}+q_{t+1}\right ) \end{align}

where $\lambda _{bt}^{r}$ is the Lagrange multiplier associated with the budget constraint (41).

3.2.4 Firms

We follow Zhao (Reference Zhao2020) in modeling the firm’s production function. Specifically, we assume that during the pandemic a fraction $\left ( 1-L_{t}\right )$ of labor is on-site and the remaining fraction $L_{t}$ of labor is performed at home. The production function is given by

(46) \begin{equation} Y_{t} =k_{t}^{\alpha }\left [ \left ( 1-L_{t}\right ) n_{t}+{\Phi L_{t}}n_{t}\right ] ^{1-\alpha } \end{equation}

Here, the parameter $\Phi$ $\left ( 0\lt \Phi \lt 1\right )$ measures the productivity of workers who work from home relative to workers who work on-site.Footnote ³⁰ The firm’s problem is to choose labor and capital to maximize its lifetime dividends:

(47) \begin{equation} \max \left \{ k_{t}^{\alpha }\left [ \left ( 1-L_{t}\right ) n_{t}+\Phi \left ( L_{t}\right ) n_{t}\right ] ^{1-\alpha }-w_{t}n_{t}-\left ( k_{t+1}-\left ( 1-\delta \right ) k_{t}\right ) +\beta \frac{u^{\prime }( c_{t+1}^{out,r}) }{u^{\prime }( c_{t}^{out,r}) }D( k_{t+1}) \right \} \end{equation}

where $D\left ( k_{t}\right ) =\max \sum _{j=0}^{\infty }\beta ^{j}\frac {\lambda _{bt+j}^{r}}{\lambda _{bt}^{r}}d\left ( k_{t+j}\right ) =\max \sum _{j=0}^{\infty }\beta ^{j}\frac {u^{\prime }\left ( c_{t+j}^{out,r}\right ) }{u^{\prime }\left ( c_{t}^{out,r}\right ) }d\left ( k_{t+j}\right )$ Footnote ³¹ is the maximum value of the firm and $d( k_{t}) =k_{t}^{\alpha }\left [ \left ( 1-L_{t}\right ) n_{t}+\Phi L_{t}n_{t}\right ] ^{1-\alpha }-w_{t}n_{t}-( k_{t+1}-( 1-\delta) k_{t})$ is the dividend of the firm.

The first-order conditions are given by

(48) \begin{align} \left [ n_{t}\right ] :& \quad \left ( 1-\alpha \right ) \frac {Y_{t}}{n_{t}}=w_{t} \end{align}

(49) \begin{align} \left [ k_{t+1}\right ] :& \quad \beta \frac {u^{\prime }\left ( c_{t+1}^{out,r}\right ) }{u^{\prime }\left ( c_{t}^{out,r}\right ) }\left ( \alpha \frac {y_{t+1}}{k_{t+1}}+1-\delta \right ) =1 \end{align}

3.2.5 Modeling the lockdown shocks

We construct the exogenous lockdown variable, $L_{t}$ , based on the Baidu daily within-city travel intensity index (Baidu index for short). We follow three steps to construct $L_{t}$ . In the first step, we normalize the Baidu index on January 20, 2020, to be one and calculate the corresponding index from January 21, 2020 to April 8, 2020. We then calculate the difference between one and the above calculated index from January 20 to April 8. We call the result the daily preliminary lockdown index. In the second step, we calculate the weekly average of the daily preliminary lockdown index and denote it by $l_{t}^{w}$ . In the third step, we construct the lockdown variable $\left ( L_{t}\right )$ by calculating the cumulative average of the weekly lockdown index from week 1 to week 9 as follows:Footnote ³²

(50) \begin{equation} L_{t} =\frac {1}{1+j}\sum _{t=1}^{j+1}l_{t}^{w} \qquad \textrm {j}=0,1,2,\ldots 8 \end{equation}

We use the cumulative average for two reasons. First, it captures the persistence or the memory feature of the lockdown shocks. Indeed, although the adoption and lifting of the lockdown policy may be sudden, the real and psychological impacts of this policy may be relatively long-lasting. Second, if we directly used the weekly preliminary lockdown index $\left ( l_{t}^{w}\right )$ as the lockdown variable, the model economy would recover fairly quickly in response to the epidemic, which is inconsistent with the data.

Figure 2.

The preliminary lockdown index and lockdown shocks.

In Figure 2, the dashed curve shows the weekly preliminary lockdown index $\left ( l_{t}^{w}\right )$ , while the solid curve shows the lockdown variable $\left ( L_{t}\right )$ we use for our model. Compared to the weekly index, the peak of the lockdown variable occurred about one week later and it did not die out rapidly. Specifically, the lockdown variable increased sharply after the announcement of lockdown (January 23, 2020) and peaked between week 4 and week 5 (between February 17 and February 24, 2020). As the infection rate declined, the lockdown variable declined gradually. It turns out that this modeling of the lockdown variable can better capture the behavior of the key macroeconomic variables in response to the epidemic.