Intervention uncertainty, household health, and pandemic

Abstract

This study builds a policy choice model wherein household health status responds to the lockdown during the COVID-19 pandemic. Considering an exogenous policy-decision date, the model implies that the government should maintain the current policy if the perceived effects on infection are below a certain threshold. Specifically, the threshold is determined by policy uncertainty and household concerns regarding health service provision, which further controls the announcement effects of the lockdown. Higher policy uncertainty and concerns regarding health services will diminish the positive impact of the lockdown on household health status.

1. Introduction

After the outbreak of COVID-19, countries worldwide have proposed various measures to combat the virus, such as reallocated medical resources, lockdowns, and other social distancing measures (Alfano and Ercolano, 2020). Existing research on the pandemic has already thoroughly examined its public health implications and negative economic outcomes, including the trade-offs between mortality reduction and GDP and consumption (Hall et al., 2020, Thunström et al., 2020, Baek et al., 2020).

Deviating from existing literature, this study presents a novel framework to examine the government’s optimal choice to maximize households’ health status during the pandemic, considering that the government only follows its own perceived belief. This has not been studied in previous research.1 We accommodate this setting by assuming that both the government and households update their beliefs regarding the pandemic through Bayesian learning processes and that the government makes decisions based on their beliefs. Finally, we use a simple game-theory framework to express the model’s primary intuition.

Moreover, we incorporate the health costs incurred by the significant reallocation of medical resources; herein, we assume that households will be more concerned regarding their health status in the future owing to health risks related to the reallocation of medical resources, even though the reallocation is required for the lockdown to deal with the pandemic.2 Our objective is reflecting the government’s leading role when attempting to make a trade-off and decide whether to maintain the status quo or take action. Thus, we assume that the health losses of society can be fully recognized by policymakers but not by households—with notably significantly higher healthcare expenditure and issues arising from resource reallocation.

Thus, the main trade-off in our setting is lowering the infection rate versus resolving households’ health service concerns. Lowering the infection rate means that implementing a lockdown for the pandemic significantly reduces the infection rate, thereby increasing households’ health status when the pandemic ends. However, the health costs of policy change will generate health risks, undermining the health status of households. In particular, health costs reflect the reallocation of medical resources toward fighting the pandemic, precipitating various health risks to households’ health status owing to the negative effect on the healthcare system.3 We define this phenomenon as households’ health service concern to characterize the potential negative effects of health risks on households’ health status. Moreover, the government’s policy choice must resolve health service concerns.

Our results demonstrate that the government should replace the existing policy with a lockdown when the perceived effects of the current policy on infection are sufficiently unfavorable. This suggests that the government’s posterior mean concerning the effects of the current policy is lower than the threshold based on Bayesian learning, which decreases health costs and policy uncertainty. Moreover, the announcement effects of the lockdown on a household’s health status are often positive, which lowers households’ health service concerns. In an extreme scenario wherein households are infinitely concerned regarding health services, the announcement effects will be zero. Meanwhile, announcement effects also result in a decrease in policy uncertainty, which further controls the upper bound of the effects. This result is intuitive because a household with a high health service is extremely averse to the negative effects of implementing a lockdown to combat the pandemic. This is because a lockdown might reduce the provision of other health services that individuals can receive and, consequently, increase the potential for psychological problems. These health risks diminish the announcement effects of the lockdown on households’ health status to zero. Similarly, higher policy uncertainty also causes an aggregate negative effect on households’ health status through, for example, delayed diagnosis, a decline in research on other diseases, or latent health issues caused by the lockdown.

Our main contribution is enriching the literature on COVID-19-related policies by providing a model to address the trade-off between the immediate impact and health risk of lockdown; our study adds to the work of Caulkins et al. (2021), Bandyopadhyay et al. (2021) and Miclo et al. (2022). Furthermore, regarding its theoretical contribution, this study contributes to the literature on political economy learning models developed by Callander, 2011, Ales et al., 2014 and Strulovici (2010). Specifically, Callander (2011)’s channel of voters is through repeated elections. Strulovici (2010) conducted an experiment wherein voters updated their beliefs regarding their utilities, and Ales et al. (2014) examined the relationship between the political and business cycles wherein policymakers fail to commit. By contrast, our study aims to analyze the optimal policy choice of the government during the pandemic through learning and its dynamic effects on households’ health status. Existing studies on policy uncertainty have demonstrated that the effects of such uncertainty are often irreversible (Bloom, 2009, Bloom et al., 2018). Moreover, prior research has demonstrated that policy uncertainty significantly impacts social welfare, various key macroeconomic indices, and financial markets (Gomes et al., 2008, Drazen and Helpman, 1990). The reactions of stock markets to government policy choices have been studied theoretically and empirically (Pastor and Veronesi, 2012, Julio and Yook, 2012, Lensman and Troshkin, 2021). However, our study differs from prior research in that we focus on the policy choice of a government during a pandemic and its effects on households’ health status instead of analyzing fiscal policy.

The remainder of this paper is organized as follows: Section 2 outlines the benchmark model without an exogenous policy change date. Section 3 introduces a simple game theory framework to express our main intuition. Finally, Section 4 concludes the study.

2. Model

2.1. Setting

The environment that we consider is an economy with a continuum of households indexed by $j\in [0,1]$. Let the pandemic period be a finite interval, denoted as $[0,t_0]$. To simplify our analysis, a household $j$ only values its health status in each period, given by $H_t^j$. At the pandemic’s beginning, we assume that all the households’ health statuses are the same, normalized as $H_0^j=1$ for all $j\in [0,1]$. During the pandemic, the health status of households is affected by the virus’ spread. Let $d{i}_t^j$ be the infection rate of household $j$. We focus on the virus’ impact; therefore, it is the only factor in our model that affects households’ health status. Moreover, we further assume that the virus’ spread linearly influences households’ health status. Thus, the evolution law of the household $j$’s health status is expressed as follows:

$$d{H}_t^j=H_t^{j}d{i}_t^j$$ (1)

Additionally, the increment in household $j$’s health status during the pandemic interval $[0,t_0]$ evolves based on the following process:

$$d{i}_t^j=(a_t-\mu )dt+\delta d{W}_t+{\delta }_{1}d{W}_t^j$$ (2)

where $\mu$ is the average infection rate of the virus for all households, $\delta$ measures the aggregate volatility of the infection, ${\delta }_1$ measures the idiosyncratic volatility of the infection, and $a_t$ is the government policy variable that measures the effects of government policy on the average infection rate of the virus.4 In particular, $a_t=0$ indicates that the government policy is ineffective in stopping the virus’ spread.

As households only value their health status, the utility function of an arbitrary household $j$ in period $t\in [0,t_0]$ is defined as:

$$U\left(H_t^j\right)=\frac{{\left(H_t^j\right)}^{1-\theta }}{1-\theta }$$ (3)

where $\theta >1$ measures the households’ health service concern, which is common for all households. In our framework, the effects of the government policy at $t$ are denoted as $a_t$. At the beginning of the pandemic, it is set to $a_0$. Moreover, we assume that the government decides whether to implement lockdown at an exogenous time $t^{\ast }\in [0,t_0]$.5 The government has only two policy choices: maintaining the current policy or implementing a lockdown. Hence, a policy change indicates that a lockdown is implemented at $t^{\ast }$. We model $a_t$ as a jump process.

$$a_t=\left\{\begin{array}{c}{a}_0,t\le t^{\ast }\hfill \\ a_0\left(\text{Old}\right)\text{or }{a}_1\left(\text{New}\right),t>t^{\ast }\hfill \end{array}\right)$$

where $a_0$ corresponds to the effects of government policy on the pandemic at the beginning and $a_1$ denotes the impact of lockdown after the policy change date $t^{\ast }$. As previously discussed, this change only affects households’ average infection rate after the policy change date.

Our framework’s information friction arises because of the unobservability of $a_0$ and $a_1$. This information friction is realistic because the policy’s impact is often not completely consistent with the thoughts of policymakers and households before its implementation. In the following presentation, we use $i=g$ to index variables corresponding to the government’s belief and $i=h$ for the households. Moreover, the government usually provides more precise information than households. In most countries, before the scale and virulence of the pandemic became clear, people tended to believe that the virus was not serious. Therefore, we assume that households and the government exhibit their corresponding beliefs about $a_0$ before the policy change, as follows:

$$a_0^g\sim N(0,{\omega }_g^2),a_0^h\sim N(0,{\omega }_h^2),{\omega }_g<{\omega }_h$$ (4)

where $a_0^g$ and $a_0^h$ denote the prior beliefs of the government and households about $a_0$, which are independent of each other. Both prior distributions follow normal distributions centered at 0. Simultaneously, the government’s beliefs are more precise than the households’ beliefs, as indicated by ${\omega }_g<{\omega }_h$. After the policy change date $t^{\ast }$, the beliefs of the government and households regarding the $a_1$ are given by two independent normal distributions:

$$a_1^g\sim N(\alpha \mu ,{\omega }_{1,g}^2),a_1^h\sim N(\alpha \mu ,{\omega }_{1,h}^2),{\omega }_{1,g}<{\omega }_{1,h}$$ (5)

Here, $\alpha \in [0,1]$. To simplify our analysis, we assume that both priors of $a_1$ are normally distributed. The government and households share the same prior mean denoted by $\alpha \mu$. However, the government’s prior information is more precise than the households’ prior information, characterized by ${\omega }_{1,g}<{\omega }_{1,h}$. Additionally, $\alpha$ measures how effective the households and government believe the lockdown will be. Finally, all prior beliefs are common knowledge to both the government and households.

We assume that the government only uses its own prior beliefs as the basis for its decision-making. This assumption is natural because the government’s behavior typically has its own logic and is not directly affected by households. Regarding the households’ problems, their perceived health status was only directly affected by their beliefs regarding the policy impact. Meanwhile, the common knowledge assumption indicates that households understand the learning process and prior government beliefs, which are used to form their beliefs regarding the probability of policy change.

Regarding the government’s decision problem, its primary target is maximizing the health status of households at the end of the pandemic, denoted as $t_0$. This specification is realistic because, when a pandemic occurs in the real world, the government’s ultimate target is ensuring that the number of households that can survive the pandemic is maximized by its policy. Following this logic, the government’s decision problem at the policy change date $t^{\ast }$ is given by:

$$max\left[E_{t^{\ast },g}[\frac{H_{t_0}^{1-\theta }}{1-\theta }\left|Old\right],E_{t^{\ast },g}\left[\frac{{\left(\tau H_{t_0}\right)}^{1-\theta }}{1-\theta }\right|New]\right]$$ (6)

where $E_{t^{\ast },g}\left[\right]$ is the expectation operator conditional on the government’s information set up to time $t^{\ast }$, and $H_T={\int }_0^1{H}_T^{j}dj$ denotes the aggregate health status of households at the end of the pandemic period. $\tau$ is the health cost due to the policy change, expressed in terms of health status.6 We assume that $\tau$ follows a lognormal distribution with the mean $-\frac{{\omega }_{\tau }^2}{2}$ and variance ${\omega }_{\tau }^2$:

$$e=log\left(\tau \right)\sim N(-\frac{{\omega }_e^2}{2},{\omega }_e^2)$$ (7)

This implies that the expected value of $\tau$ is 1. Moreover, $e$ follows the normal distribution specified in Eq. (7). If lockdown is implemented at the policy change date $t^{\ast }$, the government can observe $\tau$, but the households cannot. Therefore, ${\omega }_e$ can be interpreted as policy uncertainty from the perspective of households, which can generate a jump in households’ health status at $t^{\ast }$.

In this framework, the government is quasi-benevolent in maximizing households’ health status; instead, it might seek quick success and benefits to stop the spread of the virus. Although these two goals do not conflict under most circumstances, the objective of maximizing health can be affected by stochastic factors. The government may not be entirely benevolent for numerous reasons, one of which is the delayed diagnosis of cancer (Maringe et al., 2020). Within our framework, everything related to policy change is compressed into the health costs, $\tau$. Thus, a stochastic $\tau$ reflects that the policy-making process is not exhaustive and is difficult to predict accurately, suggesting that the health benefits of some groups cannot be guaranteed. Nevertheless, this simple setting allows us to examine the effects of government policy on household health status during the pandemic

2.2. The government’s learning process

As discussed earlier, the exact level of $a_t$ is neither observed by the household nor the government. At the beginning of the pandemic, they had corresponding prior beliefs regarding $a_0$. The learning in our framework follows a Bayesian approach: the government and households update their beliefs regarding $a_t$ based on their observations of the economy’s aggregate infection rate, which are the same for both. In summary, learning in this economy can be interpreted using the following lemma:

Lemma 1

The economy’s aggregate infection rate is an aggregation of the individual infection rate given Eq. (2) over households $j\in [0,1]$

$$d{i}_t=(a_t-\mu )dt+\delta d{W}_t$$ (8)

The government and households’ posterior distributions of $a_t$ based on the prior distributions of $a_0^i$ and $a_1^i$ , where $i=g,h$ , is given by:

$$a_t^i\sim \left\{\begin{array}{c}N({\hat{a}}_t^i,{\hat{\omega }}_{t,i}^2),t\le t^{\ast }\hfill \\ N({\hat{a}}_{1t}^i,{\hat{\omega }}_{1t,i}^2),t>t^{\ast }\hfill \end{array}\right)$$

For the interval before the policy change $[0,t^{\ast }]$ , the posterior mean and variance: ${\hat{a}}_t^i$ and ${\hat{\omega }}_{t,i}^2$ evolve as follows:

$$d{\hat{a}}_t^i=\frac{{\hat{\omega }}_{t,i}^2}{\delta }d{\tilde{W}}_t$$$${\hat{\omega }}_{t,i}^2=\frac{1}{\frac{1}{{\omega }_i^2}+\frac{t}{{\delta }^2}}$$ (9)

where $d{\tilde{W}}_t$ is an expectation error defined as:

$$d{\tilde{W}}_t=\frac{d{i}_t-E\left[d{i}_t\right]}{\delta },t\in [0,t_0]$$

For the interval after $[t^{\ast },t^0]$ , if there is no policy change, meaning $a_t=a_0,t\ge t^{\ast }$ , ${\hat{a}}_{1t}^i$ and ${\hat{\omega }}_{1t,i}^2$ still evolve in the same way as in Eq. (9) . The government and households’ priors regarding $a_t$ will be Eq. (5) when policy change occurs at $t^{\ast }$ . Thereafter, the posterior mean ${\hat{a}}_{1t}^i$ and variance ${\hat{\omega }}_{1t,i}^2$ , where $i=g,h$ , is as follows:

$$d{\hat{a}}_{1t}^i=\alpha \mu +\frac{{\hat{\omega }}_{1t,i}^2}{\delta }d{\tilde{W}}_t$$$${\hat{\omega }}_{1t,i}^2=\frac{1}{\frac{1}{{\omega }_{1,i}^2}+\frac{t-t^{\ast }}{{\delta }^2}}$$ (10)

The idiosyncratic shocks $d{W}_t^j$s are canceled out in the evolution of the aggregate infection rate. Regarding updating beliefs during decision-making, the government updates its beliefs about the policy effect based on the learning process according to $i_t$. In particular, if $i_t$ is higher than the expected value, the posterior belief about $a_t$ also increases. Without a policy change, the policy volatility on average infection will continue decreasing as more information is disclosed. However, if lockdown $a_1$ is implemented at $t^{\ast }$, the volatility will experience a jump at $t^{\ast }$ because the policy change resets the government and households’ prior beliefs from $N({\hat{a}}_{t^{\ast }}^i,{\hat{\omega }}_{t^{\ast },i})$ to $N(\alpha \mu ,{\omega }_{1,i}^2)$, where $i=g,h$.

Compared with the existing research on pandemic policy within a static framework without information frictions, Lemma 1 characterizes households’ belief-updating process in a dynamic setting, which enables us to analyze the optimal policy choice of the government during the pandemic.

2.3. Lockdown

The policy change date $t^{\ast }$ is exogenous to both government and households. The policy’s true impact on the average infection rate changes from $a_0$ to $a_1$ when the government decides to implement a lockdown at $t^{\ast }$. The corresponding evolution of the posterior beliefs of $a_t$ for the government and households also changes based on Lemma 1. According to Eq. (6), the government’s decision to change policy depends on the value of the two conditional expectations based on its information set. A lockdown would change the average infection rate in the economy. The two conditional expectations depend on two different processes of households’ aggregate health status as perceived by the government. Thus, the aggregate health status of households perceived by the government and households at the end of the pandemic can be expressed in a general analytic form based on the following lemma:

Lemma 2

Households’ aggregate health status perceived by the government and households with and without a policy change at the end of the pandemic $t_0$ can be expressed as

$$H_{t_0}=\left\{\begin{array}{c}{H}_{t^{\ast }}exp[({\tilde{a}}_1-\mu -\frac{{\delta }^2}{2})(t_0-t^{\ast })+\delta (W_{t_0}-W_{t^{\ast }})],New\hfill \\ H_{t^{\ast }}exp[({\tilde{a}}_0-\mu -\frac{{\delta }^2}{2})(t_0-t^{\ast })+\delta (W_{t_0}-W_{t^{\ast }})],Old\hfill \end{array}\right)$$

where ${\tilde{a}}_0=a_0^i$ , ${\tilde{a}}_1=a_1^i$ with $i=g,h$ reflect the different beliefs of the government and households about $a_0$ , and $a_1$ , $H_{t_0}={\int }_{j=0}^1{H}_{t_0}^{j}dj$ and $H_{t^{\ast }}={\int }_{j=0}^1{H}_{t^{\ast }}^{j}dj$ denote the perceived households’ aggregate health status at $t_0$ and $t^{\ast }$ based on the corresponding beliefs of $a_0$ and $a_1$ being considered.

Lemma 2 is crucial because it provides the evolution of the households’ actual aggregate health status. Therefore, we can solve the optimal decision problem of policy change by comparing the two expectations conditional on the government’s information set presented by Eq. (6). The results are summarized in the following proposition:

Proposition 1

During the pandemic interval $[0,t_0]$ , the government prefers the lockdown, the true impact of which is $a_1$ at the exogenous policy change date $t^{\ast }$ if 7

$${\hat{a}}_{t^{\ast },g}\le \overline{a}\left(e\right)$$ (11)

where $\overline{a}\left(e\right)$ is given by

$$\overline{a}\left(e\right)=e+\alpha \mu +\frac{(1-\theta )(t_0-t^{\ast })({\omega }_{1,g}^2-{\hat{\omega }}_{t^{\ast },g}^2)}{2}$$ (12)

Proposition 1 is among our main results because it provides an analytical threshold for the posterior mean of the government policy variable $a_t$ at the policy change date $t^{\ast }$. Specifically, the government changes its policy when Inequality (11) holds. This means that conditional on the government’s information set at $t^{\ast }$, the lockdown is considered more attractive for raising the household aggregate health status at the end of the pandemic.

Regarding Eq. (12), we can interpret its economic intuition by decomposing it into three parts. The first term corresponds to the costs associated with the policy. As previously mentioned, $e>0$ indicates that the policy delivers the benefits. The higher the $e$ is, the more likely it is that Inequality (11) will be true, and the possibility of policy change will increase. The second term represents the prior belief regarding the effectiveness of the lockdown in combating the pandemic, and a higher $\alpha$ indicates that the government believes that the lockdown will be more effective in lowering the infection rate, thus making it more likely that the lockdown will be implemented. The third term represents the change in volatility when a lockdown is implemented. $\theta >1$ indicates that higher volatility in beliefs about the lockdown will decrease the probability of policy change.

The government only observes the health costs associated with policy change $e$. Thus, the households cannot be certain whether a lockdown will be implemented after $t^{\ast }$. However, households can observe the evolution of the government’s beliefs because of our common knowledge assumption. Household belief regarding $e$ follows Eq. (7), implying that they expect $\overline{a}\left(e\right)$ to be similar to $\overline{a}\left(0\right)$ if $\tau$ is expected to be close to 1. The learning process implies that the volatility of government policy, as perceived by all agents, will increase when policy change occurs (${\omega }_1>{\hat{\omega }}_{t^{\ast },i}$, $i=g,h$), thereby leading to higher health risks. Hence, $\overline{a}\left(0\right)$ is negative. Intuitively speaking, the old policy will be replaced by lockdown if its impact on households’ health status, as perceived by the government, is too negative during the pandemic. In this case, the positive effect of the lockdown on households’ health status, as perceived by the government, is higher than the negative effects arising from higher volatility, policy change cost, and people’s pessimism about the lockdown. Regarding the posterior mean of $a_0$ as perceived by the government, it can be negative at $t^{\ast }$ while the prior is invariably 0, suggesting that the observation of the average infection rate is sufficiently low.

Proposition 1 has important policy implications, suggesting that the government is more likely to implement a lockdown if the existing policy is ineffective in keeping households’ health status at an acceptable level, conditional on its information set. This result is consistent with measures adopted by several countries during the pandemic.

2.4. Households

The previous section described the framework built to explain how the government implemented the lockdown during the pandemic and to examine its implications for infection rates from the government’s perspective. We now consider households’ perceived health status during the pandemic with government intervention. First, we suppose that households can choose to quit the labor market and stay at home during the entire pandemic to guarantee their health status before the pandemic ends without being affected by infection, as defined by Eq. (4).8 However, we assume that none of the economies chooses to do so. Therefore, an arbitrary household $j$’s stochastic discount factor on its health at time $t<t_0$ is given by

$${\gamma }_t=E_{t,h}\left[\frac{H_{t_0}^{-\theta }}{\iota }\right]$$ (13)

where $E_{t,h}\left[\right]$ represents the expectation conditional on the households’ belief at time $t$, $\iota$ is the multiplier associated with the choice problem between completely quitting the labor market and choosing to be subject to the infection rate given by Eq. (4). Thus, we define the discounted health status of household $j$ at time $t\in [0,t_0]$ as:

$$P_t^j=E_{t,h}\left[\frac{{\gamma }_{t_0}{H}_{t_0}^j}{{\gamma }_t}\right]$$ (14)

2.5. Households’ perceived health status

Households’ perceived aggregate health status jumps at $t^{\ast }$ when the lockdown is implemented. The properties of this jump are determined by the uncertainty in the government’s decisions and policy choices. Combined with Eq. (14), the households’ perceived discounted health status at $t^{\ast }$ can be characterized by the following lemma:

Lemma 3

Let $t^{\ast }+$ be the instant immediately after the announcement of the government’s decision regarding policy changes. An arbitrary household $j$ ’s expectation of the perceived discounted health status at $t^{\ast }$ is given by:

$$P_{t^{\ast }}^j=\pi P_{t^{\ast }+}^{j,New}+(1-\pi )P_{t^{\ast }+}^{j,Old}$$ (15)

where $\pi$ is equal to:

$$\pi =\frac{q\left({\hat{a}}_{t^{\ast },g}\right){\gamma }_{t^{\ast }+}^{New}}{q\left({\hat{a}}_{t^{\ast },g}\right){\gamma }_{t^{\ast }+}^{New}+(1-q\left({\hat{a}}_{t^{\ast },g}\right)){\gamma }_{t^{\ast }+}^{Old}}$$ (16)

And $P_{t^{\ast }+}^{j,New}$ , $P_{t^{\ast }+}^{j,Old}$ and $q\left({\hat{a}}_{t^{\ast },g}\right)$ are given by:

$$P_{t^{\ast }+}^j=\left\{\begin{array}{c}\frac{exp[(-\theta (\alpha -1)\mu +\frac{\theta (\theta +1){\delta }^2}{2})(t_0-t^{\ast })+\frac{{\theta }^2{(t_0-t^{\ast })}^2{\omega }_{1,h}^2}{2}]}{\iota H_{t^{\ast }+}^{\theta }},New\hfill \\ \frac{exp[(-\theta ({\hat{a}}_{t^{\ast },h}-\mu )+\frac{\theta (1+\theta ){\delta }^2}{2})(t_0-t^{\ast })+\frac{{\theta }^2{(t_0-t^{\ast })}^2{\hat{\omega }}_{t^{\ast },h}^2}{2}]}{\iota H_{t^{\ast }+}^{\theta }},Old\hfill \end{array}\right)$$

$$q\left({\hat{a}}_{t^{\ast },g}\right)=1-N({\hat{a}}_{t^{\ast },g}-\alpha \mu -\frac{(1-\theta )}{2}(t_0-t^{\ast })({\omega }_{1,g}^2-{\hat{\omega }}_{t^{\ast },g}^2);$$

$$-\frac{{\omega }_e^2}{2},{\omega }_e^2)$$

$q\left({\hat{a}}_{t^{\ast },g}\right)$ denotes the probability that the government will implement lockdown to handle the pandemic at $t^{\ast }$ conditional on its policy information set when the posterior mean of the original policy’s effects on infection is ${\hat{a}}_{t^{\ast },g}$ at $t^{\ast }$. Furthermore, expectation weight $\pi$ defined in Lemma 3 and $q\left({\hat{a}}_{t^{\ast },g}\right)$ are positively correlated. This implies that as people’s belief that policy change will occur increases, the more weight they assign to $P_{t^{\ast }+}$ in the calculation of their perceived discounted health status at $t^{\ast }$. Lemma 3 also presents the expression for households’ perceived discounted health status at $t^{\ast }$ and $t^{\ast }+$. Thus, we can feature the jump in households’ perceived health status due to the government’s policy announcement at $t^{\ast }$ in the following proposition:

Proposition 2

Let $F{\left({\hat{a}}_{t^{\ast }}\right)}^j=\frac{P_{t^{\ast }+}^{j,New}-P_{t^{\ast }}^j}{P_{t^{\ast }}^j}$ be the expected jump ratio of household $j$ ’s perceived health status after the government changes the policy at $t^{\ast }$ . This jump ratio is the same across all the households and can be expressed as:

$$F({\hat{a}}_{t^{\ast },g},{\hat{a}}_{t^{\ast },h})=F{({\hat{a}}_{t^{\ast },g},{\hat{a}}_{t^{\ast },h})}^j=\frac{(1-q\left({\hat{a}}_{t^{\ast },g}\right))R\left({\hat{a}}_{t^{\ast },h}\right)(1-M\left({\hat{a}}_{t^{\ast },h}\right))}{q\left({\hat{a}}_{t^{\ast },g}\right)+(1-q\left({\hat{a}}_{t^{\ast },g}\right))R\left({\hat{a}}_{t^{\ast },h}\right)M\left({\hat{a}}_{t^{\ast },h}\right)}$$ (17)

where $q\left({\hat{a}}_{t^{\ast },g}\right)$ is the same as defined in Lemma 3 , $R\left({\hat{a}}_{t^{\ast },h}\right)$ and $M\left({\hat{a}}_{t^{\ast },h}\right)$ are denoted as:

$$M\left({\hat{a}}_{t^{\ast },h}\right)=exp[-(\alpha \mu -{\hat{a}}_{t^{\ast },h})(t_0-t^{\ast })-\frac{(1-2\theta ){(t_0-t^{\ast })}^2({\omega }_{1,h}^2-{\hat{\omega }}_{t^{\ast },h}^2)}{2}]$$$$R\left({\hat{a}}_{t^{\ast },h}\right)=exp[-\theta ({\hat{a}}_{t^{\ast },h}-\alpha \mu )(t_0-t^{\ast })+\frac{{\theta }^2{(t_0-t^{\ast })}^2({\hat{\omega }}_{t^{\ast },h}^2-{\omega }_{1,h}^2)}{2}]$$ (18)

where $N(a;b,c)$ denotes the value of the distribution function of a normal distribution with mean $b$ and variance $c$ at point $a$ .

Proposition 2 characterizes the two forces working on $F({\hat{a}}_{t^{\ast },g},{\hat{a}}_{t^{\ast },h})$ when the government implements the lockdown at $t^{\ast }$. The implementation of lockdown implies that the prior beliefs of the government and households regarding policy effects are now $N(\alpha \mu ,{\omega }_{1,i}^2)$ instead of $N({\hat{a}}_{t^{\ast }},{\hat{\omega }}_{t^{\ast },i}^2)$ with $i=g,h$. Compared with the original policy, all agents tend to believe that the lockdown will exhibit a stronger effect in lowering the virus infection rate.9

Therefore, the lockdown exerts an upward pressure on $F({\hat{a}}_{t^{\ast },g},{\hat{a}}_{t^{\ast },h})$. Meanwhile, the implementation of the lockdown can also drive households’ stochastic discount factor because the lockdown is more uncertain than the current policy.10 These two forces exhibit opposite effects on the jump ratio of households’ perceived aggregate health status at $t^{\ast }$. The total effect depends on the calibration and learning processes of the government and households. We present the relationship between the jump ratio of households’ perceived aggregate health status $F({\hat{a}}_{t^{\ast },g},{\hat{a}}_{t^{\ast },h})$ and households’ relative health concern $\theta$ in the following proposition:

Proposition 3

Let households’ relative health concern $\theta$ be in $(1,\infty )$ and $2{(t_0-t^{\ast })}^2-(t_0-t^{\ast })>0$ . When $\theta \to \infty$ ,

$$F({\hat{a}}_{t^{\ast },g},{\hat{a}}_{t^{\ast },h})\to -1,\forall {\hat{a}}_{t^{\ast },g},{\hat{a}}_{t^{\ast },h}$$ (19)

When $\theta \to 1$ , the expected jump ratio of households’ perceived aggregate health status, conditional on the households’ belief about $a_t$ at $t^{\ast }$ , will converge as follows:

$$E_{t^{\ast },h}\left[F({\hat{a}}_{t^{\ast },g},{\hat{a}}_{t^{\ast },h})\right]\to (1-q\left({\hat{a}}_{t^{\ast },g}\right))(exp[-\alpha \mu (t_0-t^{\ast })]-1)>0$$ (20)

Proposition 3 is a natural application of Proposition 2 that examines how policy effects vary with household health service concerns. When the government implements a lockdown to tackle the pandemic at $t^{\ast }$, the downward pressure from the lockdown dominates when households are extremely concerned regarding health services ($\theta \to \infty$), implying the jump ratio of households’ perceived aggregate health status will be negative. However, the jump ratio will be positive when the households’ are relatively unconcerned regarding the health risks ($\theta \to 1$), suggesting that—owing to a lower infection rate caused by the lockdown—the positive effects on households’ perceived health status will dominate.

2.6. A harmful lockdown on pandemic

Proposition 3 implies that the implementation of lockdown at $t^{\ast }$ could negatively impact on households’ perceived health status when they are extremely concerned regarding health services ($\theta \to \infty$). This corresponds to the situation wherein households are highly concerned regarding health risks, such that the volatility generated by the lockdown could worsen their health status. This is because these households are likely to believe that after the lockdown’s implementation (e.g., a complete lockdown), the government may not have sufficient financial resources to support the healthcare system at the same level as before the pandemic. Moreover, a degraded health system may harm their health status, even though a lockdown helps stop the virus’ spread. To further understand the situation wherein the effects of lockdown are negative, we propose the following proposition:

Proposition 4

Based on our common knowledge assumption, we can find a threshold of the government’s posterior mean of the original policy effects, conditional on the households’ belief, $\overline{a}\left({\hat{a}}_{t^{\ast },h}\right)$ , such that:

$$F({\hat{a}}_{t^{\ast },g},{\hat{a}}_{t^{\ast },h})\le 0,\forall {\hat{a}}_{t^{\ast },g}\ge \overline{a}\left({\hat{a}}_{t^{\ast },h}\right)$$ (21)

where $\overline{a}\left({\hat{a}}_{t^{\ast },h}\right)$ is equal to

$$\overline{a}\left({\hat{a}}_{t^{\ast },h}\right)=\alpha \mu +\frac{(1-2\theta )(t_0-t)({\omega }_{1,h}^2-{\hat{\omega }}_{t^{\ast },h}^2)}{2}=\overline{a}$$ (22)

Proposition 4 demonstrates that households’ perceived aggregate health status will decline owing to the lockdown implemented by the government if the original policy does not perform too badly in preventing the spread of the virus. Another interesting finding is that the threshold is independent of households’ belief ${\hat{a}}_{t^{\ast },h}$, which is consistent with the real-world fact that the government makes policy choices based on its own belief. Consequently, the positive effects of the lockdown in lowering the infection rate are not sufficiently significant to compensate for the health risks to future medical services.

Combining the results of Proposition 1, Proposition 4, the lockdown implemented by the government may negatively affect households’ perceived aggregate health status if the government’s posterior mean of the original policy’s impact on the infection rate satisfies the following inequality:

$$\overline{a}\le {\hat{a}}_{t^{\ast },g}\le \overline{a}\left(e\right)$$ (23)

Inequality (23) further confirms that only in a situation where ${\hat{a}}_{t^{\ast },g}<\overline{a}$ can the government’s implementation of lockdown positively affect households’ perceived aggregate health status. However, a possibility exists that the set of ${\hat{a}}_{t^{\ast },g}$ satisfying Inequality (23) is empty. We describe the relationship between $\overline{a}$ and $\overline{a}\left(e\right)$ in the following proposition:

Proposition 5

(a) The difference between $\overline{a}\left(e\right)$ and $\overline{a}$ is the increase in households’ prior variance of the lockdown ${\omega }_{1,h}$ .

(b) The set of ${\hat{a}}_{t^{\ast },g}$ satisfying Inequality (23) is not empty if it is expensive for the government to implement lockdown $\tau >1$ .

(c) If $\tau <1$ , government can benefit from implementing lockdown. The set of ${\hat{a}}_{t^{\ast },g}$ satisfying Inequality (23) is empty when $e<\overline{e}$ where

$$\overline{e}=\frac{\theta (t_0-t^{\ast })({\omega }_{1,h}^2-{\hat{\omega }}_{t^{\ast },h}^2)}{2}$$ (24)

Proposition 5 presents a high ${\omega }_{1,h}$, indicating that households believe the lockdown makes their future health status more volatile. An intuitive explanation is that households in this economy think that the lockdown can lower the aggregate infection rate but might not be effective in maintaining their health due to medical resources being reallocated to deal with the pandemic.11 The health risk channel described above will diminish the positive effects of the lockdown on perceived health status through the infection rate channel. Thus, a government that changes its policy under a higher ${\omega }_{1,h}$ is more likely to induce a negative $F({\hat{a}}_{t^{\ast },g},{\hat{a}}_{t^{\ast },h})$, leading to a set of ${\hat{a}}_{t^{\ast },g}$ satisfying Inequality (23) becoming larger with ${\hat{a}}_{t^{\ast },h}$ being given.

As implementing a lockdown usually means that the government has to cut its expenditure on numerous public services—including the healthcare system, pensions, and several other services that are essential for household health; $e$ in the expression $\overline{a}\left(e\right)$ usually tends to be positive in real life. A positive $e$ indicates health costs associated with policy changes in terms of household health status, which implies a trade-off arising from policy changes. Therefore, a negative $F({\hat{a}}_{t^{\ast },g},{\hat{a}}_{t^{\ast },h})$ will be realized if the government changes policy when the current policy does not perform too badly in preventing the virus’ spread.

Alternatively, conditional on ${\hat{a}}_{t^{\ast },h}$, a health benefit ($e<0$) in terms of households’ health associated with policy change could make the set of ${\hat{a}}_{t^{\ast },g}$ satisfying Inequality (23) empty.12 However, this is not consistent with most countries’ policy practice. Consequently, we focus on the case in which Proposition $\left(b\right)$ of Proposition 5 is true.

2.7. Announcement effects of the lockdown

We used the jump ratio of households’ perceived aggregate health status to measure the effects of the lockdown’s announcement, which are defined as the expected value $F\left({\hat{a}}_{t^{\ast }}\right)$ of the information set of households when a policy change occurs.

$$An=E_{t^{\ast },h}\left[F({\hat{a}}_{t^{\ast },g},{\hat{a}}_{t^{\ast },h})\right|{\hat{a}}_{t^{\ast },g}\le \overline{a}\left(e\right)]$$ (25)

This announcement effect is calculated using the households’ information set before the realization of $e$.13 Prior beliefs and learning processes are common knowledge; hence, Eq. (25) only includes integration over $e$ using the distribution defined in Eq. (7). The behavior of Eq. (25) is summarized in the following lemma:

Lemma 4

The announcement effect is upper bounded by the following expression:

$$E_{t^{\ast },h}\left[F({\hat{a}}_{t^{\ast },g},{\hat{a}}_{t^{\ast },h})\right|{\hat{a}}_{t^{\ast },g}\le \overline{a}\left(e\right)]\le \tilde{\Psi }({\mu }_z-v_1)-\tilde{\Psi }\left({\mu }_z\right)$$ (26)

where function $\tilde{\Psi }\left(z\right)$ , and variables $z$ ${\mu }_z$ , $v_1$ are defined as:

$$z=(1-\theta )({\hat{a}}_{t^{\ast },g}-\alpha \mu )(t_0-t^{\ast })-\frac{{(\theta -1)}^2{(t_0-t^{\ast })}^2({\omega }_{1,h}^2-{\hat{\omega }}_{t^{\ast },h}^2)}{2}$$$$\tilde{\Psi }\left(z\right)=\frac{N\left(z\right)(1-N\left(\overline{z}\right))}{1-N\left(z\right)+\left(N\left(z\right)exp\left(z\right)\right)}$$$${\mu }_z=-\alpha (1-\theta )(t_0-t^{\ast })\mu -\frac{{(\theta -1)}^2{(t_0-t^{\ast })}^2({\omega }_{1,h}^2-{\hat{\omega }}_{t^{\ast },h}^2)}{2}$$$$v_1=(t_0-t^{\ast }){({\omega }_h^2-{\hat{\omega }}_{t^{\ast },h}^2)}^{\frac{1}{2}}$$ (27)

The intuition behind Lemma 4 is straightforward. Specifically, as households become increasingly concerned regarding health services, the upper bound of the announcement effect of lockdown is lower, which is consistent with our argument about households’ concerns regarding risks to their future health status.

2.8. Dynamics lockdown effects

This section uses households’ stochastic discount factor and perceived discounted health status, as defined in Section 2.4, to examine the dynamic effects of the lockdown. Although the government in this economy still needs to learn to update its beliefs regarding the policy effect on the rate of infection, no policy change occurs after $t^{\ast }$. Thus, we focus primarily on the government’s behavior before $t^{\ast }$. First, we characterize the properties of households’ stochastic discount factors in the following lemma:

Lemma 5

During the period $[0,t^{\ast }]$ , households’ stochastic discount factor is equal to:

$${\gamma }_t=H_t^{-\theta }\Sigma ({\hat{a}}_{t^{\ast },g},{\hat{a}}_{t^{\ast },h}),t\in [0,t^{\ast }]$$ (28)

where $\Sigma ({\hat{a}}_{t^{\ast },g},{\hat{a}}_{t^{\ast },h})$ is

$$\Sigma ({\hat{a}}_{t^{\ast },g},{\hat{a}}_{t^{\ast },h})=q_t^{New}{\sigma }_t^{New}+(1-q_t^{Old}){\sigma }_t^{Old}$$ (29)

where ${\sigma }_t^{New}$ , ${\sigma }_t^{Old}$ , $q_t^{New}$ and $q_t^{Old}$ are defined as

$${\sigma }_t^{New}=exp[\theta \mu (t_0-t)-\theta \alpha \mu (T-t^{\ast })-\theta \left({\hat{a}}_{t,h}\right)(t^{\ast }-t)+\frac{{\theta }^2((t_0-t^{\ast }){\omega }_{1,h}^2+(t^{\ast }-t){\hat{\omega }}_{t,h}^2)}{2}+\frac{\theta (1+\theta )(t_0-t){\delta }^2/2}{2}]$$$${\sigma }_t^{Old}=exp[-\theta ({\hat{a}}_{t,h}-\mu )(t_0-t)+\frac{{\theta }^2{(t_0-t)}^2{\hat{\omega }}_{t,h}^2}{2}+\frac{\theta (1+\theta )(t_0-t){\delta }^2}{2}]$$$$q_t^{New}=N(\overline{a}\left(0\right);{\hat{a}}_{t,g}-\theta {\hat{\omega }}_{t,g}^2(t^{\ast }-t)+\frac{{\omega }_c^2}{2},{\hat{\omega }}_{t,g}^2-{\hat{\omega }}_{t^{\ast },g}^2+{\omega }_c^2)$$$$q_t^{Old}=1-N(\overline{a}\left(0\right);{\hat{a}}_{t^{\ast },g}-\theta [{\omega }_{1,g}^2(t_0-t)-{\hat{\omega }}_{t^{\ast },g}^2(t_0-t^{\ast })],{\hat{\omega }}_{t,g}^2-{\hat{\omega }}_{t^{\ast },g}^2+{\omega }_c^2)$$ (30)

Lemma 5 characterizes households’ stochastic discount factor before $t^{\ast }$ by ${\gamma }_t$ and presents the dynamics of ${\gamma }_t$ during the same period. To draw implications according to the evolution of ${\gamma }_t$, we summarize its properties for $0<t<t^{\ast }$ as follows:

Lemma 6

Define the evolution of households’ stochastic discount factor before $t^{\ast }$ as:

$$d{\gamma }_t={\gamma }_t(-{\omega }_{\gamma ,t}d{\hat{W}}_t+{\Xi }_{\gamma }{1}_{t=t^{\ast }})$$ (31)

where $1_{t=t^{\ast }}$ is an indicator function equaling 1 if $t=t^{\ast }$ and 0; otherwise, ${\hat{W}}_t$ follows the definition provided in Lemma 1 and ${\omega }_{\gamma ,t}$ is equal to:

$${\omega }_{\gamma ,t}=\theta \delta -\frac{{\hat{\omega }}_{t,h}^2{\Sigma }_{{\hat{a}}_{t^{\ast },h}}({\hat{a}}_{t^{\ast },g},{\hat{a}}_{t^{\ast },h})}{\delta \Sigma ({\hat{a}}_{t^{\ast },g},{\hat{a}}_{t^{\ast },h})}$$

where ${\Sigma }_{{\hat{a}}_{t^{\ast },h}}({\hat{a}}_{t^{\ast },g},{\hat{a}}_{t^{\ast },h})$ is the derivative of $\Sigma ({\hat{a}}_{t^{\ast },g},{\hat{a}}_{t^{\ast },h})$ with respect to ${\hat{a}}_{t^{\ast },h}$ . Finally, ${\Xi }_{\gamma }$ is given by

$${\Xi }_{\gamma }=\left\{\begin{array}{c}{\Xi }_{\gamma }^{New}=\frac{(1-q\left({\hat{a}}_{t^{\ast },g}\right))(1-R\left({\hat{a}}_{t^{\ast },h}\right))}{q\left({\hat{a}}_{t^{\ast },g}\right)+(1-q\left({\hat{a}}_{t^{\ast },g}\right))R\left({\hat{a}}_{t^{\ast },h}\right)},New\hfill \\ {\Xi }_{\gamma }^{New}=\frac{q\left({\hat{a}}_{t^{\ast },g}\right)(R\left({\hat{a}}_{t^{\ast },h}\right)-1)}{q\left({\hat{a}}_{t^{\ast },g}\right)+(1-q\left({\hat{a}}_{t^{\ast },g}\right))R\left({\hat{a}}_{t^{\ast },h}\right)},Old\hfill \end{array}\right)$$

where $q\left({\hat{a}}_{t^{\ast },g}\right)$ and $R\left({\hat{a}}_{t^{\ast },h}\right)$ are given by Proposition 2 . ${\Xi }_{\gamma }^{New}>0$ and $\Xi {\gamma }^{Old}<0$ iff

$${\hat{a}}_{t^{\ast },g}>\tilde{a}=\frac{\theta (t_0-t^{\ast })({\hat{\omega }}_{t^{\ast },h}^2-{\omega }_{1,h}^2)}{2}+\theta \alpha \mu$$

Lemma 6 indicates a jump in households’ stochastic discount factor at policy change date $t^{\ast }$. This is because ${\gamma }_t$ is a martingale before $t^{\ast }$. The expected value of the jump is invariably zero, implying that jumps with and without a policy change have different signs. Moreover, there is a threshold $\tilde{a}$ for ${\hat{a}}_{t^{\ast },g}$ such that when ${\hat{a}}_{t^{\ast },g}\le \tilde{a}$, ${\Xi }_{\gamma }^{New}$ is positive, ${\Xi }_{\gamma }^{Old}$ is negative. Using the results of the households’ stochastic discount factor, the households’ perceived discounted health status before $t^{\ast }$ can be characterized by the following lemma:

Lemma 7

Before the policy change date $t^{\ast }$ , an arbitrary household $j$ ’s perceived discounted health status is equal to

$$P_t^j=\frac{H_t^j\Theta ({\hat{a}}_{t^{\ast },g},{\hat{a}}_{t^{\ast },h})}{\Sigma ({\hat{a}}_{t^{\ast },g},{\hat{a}}_{t^{\ast },h})}$$ (32)

where $\Theta ({\hat{a}}_{t^{\ast },g},{\hat{a}}_{t^{\ast },h})$ is defined as

$$\Theta ({\hat{a}}_{t^{\ast },g},{\hat{a}}_{t^{\ast },h})={\tilde{q}}_t^{New}{\tilde{\sigma }}_t^{New}+(1-{\tilde{q}}_t^{Old}){\tilde{\sigma }}_t^{Old}$$

where ${\tilde{q}}_t^{New}$ and ${\tilde{q}}_t^{Old}$ are

$${\tilde{\sigma }}_t^{New}=exp[-(1-\theta )\mu (t_0-t)+(1-\theta )\alpha \mu (t_0-t^{\ast })+(1-\theta ){\hat{a}}_{t,h}(t^{\ast }-t)-\frac{\theta (1-\theta ){\delta }^2(t_0-t)}{2}+\frac{{(1-\theta )}^2((t^{\ast }-t){\hat{\omega }}_{t,h}^2+(t^{\ast }-t){\omega }_{1,h}^2)}{2}]$$$${\tilde{\sigma }}_t^{Old}=exp[-(1-\theta )\mu (t_0-t)-\frac{\theta (1-\theta ){\delta }^2(t_0-t)}{2}+(1-\theta ){\hat{a}}_{t,h}(t^{\ast }-t)+\frac{{(1-\theta )}^2{\hat{\omega }}_{t,h}^2{(t_0-t)}^2}{2}]$$$${\tilde{q}}_t^{New}=N(\overline{a}\left(0\right);{\hat{a}}_{t,g}+(1-\theta ){\hat{\omega }}_{t,g}^2(t^{\ast }-t)+\frac{{\omega }_e^2}{2},{\hat{\omega }}_{t,g}^2-{\hat{\omega }}_{t^{\ast },g}^2+{\omega }_e^2)$$$${\tilde{q}}^{Old}=1-N(\overline{a}\left(0\right);{\hat{a}}_{t,g}+(1-\theta )[{\hat{\omega }}_{t,g}^2{(t_0-t)}^2-(t_0-t^{\ast }){\hat{\omega }}_{t^{\ast },g}^2]+\frac{{\omega }_e^2}{2};{\hat{\omega }}_{t,g}^2-{\hat{\omega }}_{t^{\ast },g}^2+{\omega }_e^2)$$ (33)

Lemma 7 presents an expression of an arbitrary household $j$’s perceived health status before the policy change date $t^{\ast }$, which can be considered an expectation of the households’ perceived health status over two different policy decision situations. However, we are more interested in the dynamics of households’ perceived health status around policy change date $t^{\ast }$, which is summarized in the following lemma:

Lemma 8

Before $t^{\ast }$ , the dynamics of an arbitrary household $j$ ’s perceived health status is equal to

$$d{P}_t^j=P_t^j({\mu }_{h,t}dt+{\omega }_{h,t}d{\hat{W}}_t+{\delta }_{1}d{W}_t^j+{\Xi }_P{1}_{t=t^{\ast }})$$ (34)

where ${\omega }_{H,t}$ and ${\mu }_H$ are equal to

$${\omega }_{P,t}=\delta +{\hat{\omega }}_{t,h}^2(\frac{{\Theta }_{{\hat{a}}_{t,h}}({\hat{a}}_{t^{\ast },g},{\hat{a}}_{t^{\ast },h})}{\delta \Theta ({\hat{a}}_{t^{\ast },g},{\hat{a}}_{t^{\ast },h})}-\frac{R_{{\hat{a}}_{t,h}}\left({\hat{a}}_{t,h}\right)}{\delta R\left({\hat{a}}_{t,h}\right)})$$$${\mu }_P={\omega }_{\gamma ,t}{\omega }_{P,t}$$ (35)

where ${\omega }_{\gamma ,t}$ is defined in Proposition 5 , and the jump factor in the evolution ${\Xi }_H$ is equal to

$${\Xi }_P=\left\{\begin{array}{c}F({\hat{a}}_{t^{\ast },g},{\hat{a}}_{t^{\ast },h}),New\hfill \\ F({\hat{a}}_{t^{\ast },g},{\hat{a}}_{t^{\ast },h})M\left({\hat{a}}_{t^{\ast },h}\right)+M\left({\hat{a}}_{t^{\ast },h}\right)-1,Old\hfill \end{array}\right)$$

Lemma 8 indicates that the jump in households’ discounted health status at $t^{\ast }$ can differ depending on the government’s decision regarding whether to implement a lockdown to stop the infection. The features of the jump when policy changes, ${\Xi }_P^{New}$, have already been discussed in Proposition 2, and we characterize the properties of the expected jump at $t^{\ast }$ in the following proposition:

Proposition 6

The expected jump in households’ perceived discounted health status at $t^{\ast }$ is given by

$$E_{t^{\ast }}\left[{\Xi }_P\right]=q\left({\hat{a}}_{t^{\ast },g}\right){\Xi }_P^{New}+(1-q\left({\hat{a}}_{t^{\ast },g}\right)){\Xi }_P^{Old}$$ (36)

(1) $E_{t^{\ast }}\left[{\Xi }_P\right]$ is positive only when

$${\hat{a}}_{t^{\ast },g}>\tilde{a}\text{ or }{\hat{a}}_{t^{\ast },g}<\overline{a}$$ (37)

where $\tilde{a}$ is defined in Lemma 6 and $\overline{a}$ is defined in Proposition 4

(2) ${\Xi }_P^{Old}>0$ if and only if ${\hat{a}}_{t^{\ast },g}>\overline{a}$

Proposition 6 implies that the government implemented the lockdown because it believes that this policy will exhibit a transitory positive effect on households’ health status around the announcement date. Households’ health service concerns can also affect the magnitude of jumps in their perceived discounted health status when the government’s decision is revealed. In particular, the probability of implementing lockdown converges to zero as $\theta$ approaches $\infty$. This means $E_{t^{\ast }}\left[{\Xi }_P\right]$ approaches zero. On the contrary, lockdown will almost indubitably be introduced when $\theta \to 1$, leading $E_{t^{\ast }}\left[{\Xi }_P\right]$ to converge to some positive value.

Corollary 1

(1) When households’ health service concern goes to $\infty$ , $E_{t^{\ast },h}\left[{\Xi }_P\right]$ goes to zero for all ${\hat{a}}_{t^{\ast },g}$ .

(2) When households’ health service concern goes to 1, $E_{t^{\ast },h}\left[{\Xi }_P\right]$ will converge to a non-negative number determined by ${\hat{a}}_{t^{\ast },g}$ . It will converge to zero when $({\hat{a}}_{t^{\ast },h}-\alpha \mu )(t_0-t^{\ast })-\frac{(1-2\theta )(t_0-t^{\ast })({\omega }_{1,h}^2-{\hat{\omega }}_{t^{\ast },h}^2)}{2}$ .

Corollary 1 indicates that if households are extremely concerned regarding future health services, the welfare loss caused by health risks to their future health status will dominate. There was nearly no jump in households’ perceived discounted health status on the policy change date. However, this jump will be significantly positive if households are not highly concerned regarding health services, in which case the positive effect associated with the sharp decline in the infection rate outweighs future uncertainty regarding their perceived health status.14

3. A simple game theory interpretation

Moreover, we can interpret our key results regarding government policy choices using a simple game-theory framework with learning. To simplify our analysis, we focus on the government in this game. Regarding the pandemic period $[0,t_0]$, we classify it into two types: a serious pandemic $\omega =h$ and a normal pandemic $\omega =l$. The exogenous policy decision date is $t^{\ast }\in [0,t]$, which is the only period during which the government can switch policies. The signal received by the government is denoted by $s_t$ in each period, which corresponds to the aggregate infection rate in the benchmark setting. The government chooses to implement lockdown in response to the pandemic, $a=h$, or to maintain the current policy, $a=l$. For tractability, we assume $h>l$. The policy’s impact is set to depend on whether $a$ equals $\omega$. Therefore, we use the distribution $p\left(\omega \right)$ to feature the government’s prior beliefs regarding the lockdown $h$’s impact. The learning process is characterized by updating of the government’s belief regarding the policy’s impact, according to signal $s_t$. The posterior belief in each period is given by $p\left(\omega \right|s_t)$. Finally, there is also a policy cost $c(h-l)$ if the government chooses to switch from the prevailing policy $a=l$ to $a=h$.

As elucidated before, the government aimed to maximize households’ health status at the end of the pandemic. Therefore, the objective function of the government at $t=0$ is given by:

$$W_{\beta }\left(0\right)={\beta }^{t_0}I(a=\omega )-{\beta }^{t^{\ast }}c(h-l)I(a_{t^{\ast }}=h)$$ (38)

where $\beta$ is the discount factor, and $I\left[\right]$ is an indicator function equal to 1 if the condition in the bracket is satisfied and 0 otherwise. The idea behind this simple setting is that the households’ health status will be maximized if a correct policy is implemented. The lockdown will result in a cost for households’ health status because this switch might cause all medical resources to be concentrated on dealing with the pandemic, resulting in undertreating other diseases. Therefore, we face the same trade-off as in our benchmark setting: lowering the virus’ infection rate versus increasing households’ concerns regarding other medical services. Regarding the government’s policy choice problem, policymakers must compare the two conditional expectations to decide whether to implement a lockdown $h$.

$$max\left\{E_{t^{\ast },g}[W_{\beta }\left(t^{\ast }\right)|a_{t^{\ast }}=h],E_{t^{\ast },g}\left[W_{\beta }\left(t^{\ast }\right)\right|a_{t^{\ast }}=l]\right\}$$ (39)

where $E_{t^{\ast },g}$ is the expectation operator conditional on the government’s information set at $t^{\ast }$. With some technical assumptions concerning the government’s prior beliefs, we generate the following result regarding the government’s policy decision at time $t^{\ast }$, which is similar to the result in Proposition 1:

Proposition 7

Suppose $h>l$ , $c>0$ . Considering that the government’s learning process regarding the pandemic is characterized by $p\left(\omega \right|s_t)$ , the Bayesian Nash Equilibrium for the government is implementing lockdown at the policy change date $t^{\ast }$ , if and only if, $p\left(\omega \right|s_{t^{\ast }})$ satisfies the following condition

$$p(w=l\left|s_{t^{\ast }}\right)\le p(w=h|s_{t^{\ast }})-{\beta }^{t^{\ast }-t_0}c(h-l)$$ (40)

The intuition behind Proposition 7 is similar to Proposition 1 in our benchmark setting. In particular, it is only when the government believes that the possibility of a normal pandemic is extremely low or that the current policy $a=l$ is not sufficiently effective to implement a policy change. The second term on the right-hand side captures the trade-off that even though $a=l$ is highly effective for stopping the virus’ spread, such a policy change is usually accompanied by a reallocation of medical resources towards the pandemic, which will increase households’ concerns regarding other medical services. This channel features health risks and negatively pressures households’ health status. However, although this analysis has produced valuable insights, further analysis of the dynamics of households’ health status requires a richer framework in game theory. We leave this problem to future research.

4. Concluding remarks

This study is the first to examine the government’s optimal policy choice during the pandemic to address the trade-off between lowering the current infection rate vs. resolving households’ health service concerns. We assume that the government’s policy choice aims to maximize households’ health status at the end of the pandemic based on its perceived belief. The lockdown will generate health risks to households’ health status owing to the reallocation of medical resources, which will cause problems including delayed diagnosis and a decline in other health research owing to the crowding-out effect.

Our model predicts that the government should implement a lockdown to stop the virus’ spread when the perceived policy effect on the infection rate is lower than a certain threshold. Specifically, this threshold is determined by the net health gains from the change, policy uncertainty, and household health service concerns. Moreover, the upper bound of the jump in households’ perceived health status when the lockdown is implemented is further controlled by policy uncertainty and health service concerns: The higher the policy uncertainty, the lower the upper bound. Higher health service concerns diminished the lockdown’s jump effect. Intuitively, this model can be applied to the context of most governments facing a disaster, such as the COVID-19 pandemic, for the first time, by answering the following questions: Why did some governments refuse to adopt the lockdown (net health gains of a change)? Why did some countries control the spread of COVID-19 quickly (policy uncertainty)? Why do people in different countries disagree on similar policies (health service concerns)?

Manuscript handled by Editor Raouf Boucekkinne

Appendix A. Supplementary data

The following is the Supplementary material related to this article.

MMC S1

The Supplementary data contains all the proofs of the lemmas and propositions.

Data availability

No data was used for the research described in the article.