Covid-19 and the Golden Rule of Social Distancing

Abstract

We derive a golden rule of social distancing when new variants of a pandemic continuously enter a country requiring permanent social distancing. The results show that high-income countries should have more social distancing while service economies should have less social distancing. Moreover, vaccination lowers the socially optimal level of social distancing. Using data on infections and death from COVID-19 we find that countries do not follow the golden rule when it comes to income and the importance of the service sector. However, consistent with our model, island economies, which have a lower rate of inflow of new infections, suffer fewer local infections and deaths.

Appendices

Appendices

1.1 Appendix 1

The derivatives are (from (1), (3) and (4)):

$$ \frac{\partial {\overset{\sim }{N}}_o}{\partial s}=-\left[A-R\right]\frac{\left[1-h(m)\right]q}{{\left[1-r\left(s,v\right)\right]}^2}\frac{\partial r}{\partial s}>0\vspace*{-1.6pc} $$

$$ \frac{\partial {\overset{\sim }{N}}_o}{\partial v}=-\left[A-R\right]\frac{\left[1-h(m)\right]q}{{\left[1-r\left(s,v\right)\right]}^2}\frac{\partial r}{\partial v}>0\vspace*{-1.3pc} $$

$$ \frac{\partial {\overset{\sim }{N}}_o}{\partial m}=\left[A-R\right]\frac{q}{1-r\left(s,v\right)}\frac{\partial h}{\partial m}>0\vspace*{-1.3pc} $$

$$ \frac{\partial {\overset{\sim }{N}}_o}{\partial q}=-\left[A-R\right]\frac{1-h(m)}{1-r\left(s,v\right)}<0\vspace*{-1.3pc} $$

$$ \frac{\partial^2{\overset{\sim }{N}}_o}{\partial {s}^2}=-\left[A-R\right]\frac{\left[1-h(m)\right]q}{{\left[1-r\left(s,v\right)\right]}^2}\frac{\partial^2r}{\partial {s}^2}-2\left[A-R\right]\frac{\left[1-h(m)\right]q}{{\left[1-r\left(s,v\right)\right]}^3}{\left[\frac{\partial r}{\partial s}\right]}^2<0\vspace*{-1.3pc} $$

$$ \frac{\partial^2{\overset{\sim }{N}}_o}{\partial s\partial v}=-\left[A-R\right]\frac{\left[1-h(m)\right]q}{{\left[1-r\left(s,v\right)\right]}^2}\frac{\partial^2r}{\partial s\partial v}-2\left[A-R\right]\frac{\left[1-h(m)\right]q}{{\left[1-r\left(s,v\right)\right]}^3}\frac{\partial r}{\partial s}\frac{\partial r}{\partial v}<0 \vspace*{-1.3pc}$$

$$ \frac{\partial^2{\overset{\sim }{N}}_o}{\partial s\partial m}=\left[A-R\right]\frac{q}{{\left[1-r\left(s,v\right)\right]}^2}\frac{\partial r}{\partial s}\frac{\partial h}{\partial m}<0\vspace*{-1.3pc} $$

$$ \frac{\partial^2{\overset{\sim }{N}}_o}{\partial s\partial q}=-\left[A-R\right]\frac{1-h(m)}{{\left[1-r\left(s,v\right)\right]}^2}\frac{\partial r}{\partial s}>0 $$

1.2 Appendix 2

The first-order derivatives of the Lagrangian are (after using (4) and that N _w = R):

$$ \frac{\mathrm{\partial \Gamma }}{\partial \lambda }={N}_w\left[y-\gamma G(s)-{c}_w\right]-{\overset{\sim }{N}}_o{c}_o\vspace*{-1.3pc} $$

$$ \frac{\mathrm{\partial \Gamma }}{\partial {c}_w}={N}_w\left[{u}^{\prime}\left({c}_w\right)-\lambda \right]\vspace*{-1.3pc} $$

$$ \frac{\mathrm{\partial \Gamma }}{\partial {c}_o}={\overset{\sim }{N}}_o\left[{u}^{\prime}\left({c}_o\right)-\lambda \right]\vspace*{-1.3pc} $$

$$ \frac{\mathrm{\partial \Gamma }}{\partial s}=u\left({c}_o\right)\frac{\partial {\overset{\sim }{N}}_o}{\partial s}-\lambda \left[{N}_w\gamma {G}^{\prime }(s)+{c}_o\frac{\partial {\overset{\sim }{N}}_o}{\partial s}\right] $$

and the second-order derivatives are (after evaluating those at maximum and rewriting):

$$\begin{array}{l} \frac{\partial^2\Gamma}{\partial {\lambda}^2}=0,\frac{\partial^2\Gamma}{\partial \lambda \partial {c}_w}=-{N}_w,\frac{\partial^2\Gamma}{\partial \lambda \partial {c}_o}=-{\overset{\sim }{N}}_o,\frac{\partial^2\Gamma}{\partial \lambda \partial s} =-\left[{N}_w\gamma {G}^{\prime }(s)+c\frac{\partial {\overset{\sim }{N}}_o}{\partial s}\right] \\[6pt] \qquad\ \frac{\partial^2\Gamma}{\partial {c}_w^2}={N}_w{u}^{\prime \prime }(c),\frac{\partial^2\Gamma}{\partial {c}_w\partial {c}_o}=\frac{\partial^2\Gamma}{\partial {c}_w\partial s}=0\end{array}\vspace*{-1pc} $$

$$ \frac{\partial^2\Gamma}{\partial {c}_0^2}={\overset{\sim }{N}}_o{u}^{\prime \prime }(c),\frac{\partial^2\Gamma}{\partial {c}_o\partial s}=0\vspace*{-1pc} $$

$$ \frac{\partial^2\Gamma}{\partial {s}^2}=\gamma {N}_w{u}^{\prime }(c){G}^{\prime }(s)\left[\frac{\frac{\partial^2{\overset{\sim }{N}}_o}{\partial {s}^2}}{\frac{\partial {\overset{\sim }{N}}_o}{\partial s}}-\frac{G^{\prime \prime }(s)}{G^{\prime }(s)}\right] $$

Hence, for the first-order conditions being necessary and sufficient for a maximum, the following must hold (second order conditions using the bordered Hessian matrix):

$$ -{N}_w{\overset{\sim }{N}}_o\left[{N}_w+{\overset{\sim }{N}}_o\right]{u}^{\prime \prime }(c)>0 $$

which always holds, and:

$$ -{N}_w{\overset{\sim }{N}}_o\left[\begin{array}{c}{\left[{N}_w\gamma {G}^{\prime }(s)+{c}_o\frac{\partial {\overset{\sim }{N}}_o}{\partial s}\right]}^2{\left[{u}^{\prime \prime }(c)\right]}^2\\ {}+\Phi \left[{N}_w+{\overset{\sim }{N}}_o\right]{u}^{\prime \prime }(c)\end{array}\right]<0 $$

where:

$$ \Phi \equiv \gamma {N}_w{u}^{\prime }(c){G}^{\prime }(s)\left[\frac{\frac{\partial^2{\overset{\sim }{N}}_o}{\partial {s}^2}}{\frac{\partial {\overset{\sim }{N}}_o}{\partial s}}-\frac{G^{\prime \prime }(s)}{G^{\prime }(s)}\right]\gtreqless 0 $$

For the second condition to hold it is necessary that:

$$ {\left[{N}_w\gamma {G}^{\prime }(s)+{c}_o\frac{\partial {\overset{\sim }{N}}_o}{\partial s}\right]}^2{\left[{u}^{\prime \prime }(c)\right]}^2+\Phi \left[{N}_w+{\overset{\sim }{N}}_o\right]{u}^{\prime \prime }(c)>0 $$

Since the first term is positive and $\left[{N}_w+{\overset{\sim }{N}}_o\right]{u}^{\prime \prime }(c)<0$, this holds for all negative Φ and positive Φ such that

$$ \Phi <-\frac{{\left[{N}_w\gamma {G}^{\prime }(s)+{c}_o\frac{\partial {\overset{\sim }{N}}_o}{\partial s}\right]}^2{u}^{\prime \prime }(c)}{N_w+{\overset{\sim }{N}}_o} $$

or iff:

$$ \gamma {N}_w{u}^{\prime }(c){G}^{\prime }(s)\left[\frac{\frac{\partial^2{\overset{\sim }{N}}_o}{\partial {s}^2}}{\frac{\partial {\overset{\sim }{N}}_o}{\partial s}}-\frac{G^{\prime \prime }(s)}{G^{\prime }(s)}\right]<-\frac{{\left[{N}_w\gamma {G}^{\prime }(s)+{c}_o\frac{\partial {\overset{\sim }{N}}_o}{\partial s}\right]}^2{u}^{\prime \prime }(c)}{N_w+{\overset{\sim }{N}}_{o.}} $$

where the right hand side of the inequality is strictly positive. Since $\frac{\partial {\overset{\sim }{N}}_o}{\partial s}>0$, $\frac{\partial^2{\overset{\sim }{N}}_o}{\partial {s}^2}<0$, γN _w u ^′(c)G ^′(s) > 0 and G ^′(s) > 0 there clearly exist G ^′′(s) ⋛ 0 ensuring that the second order conditions are fulfilled. We therefore conclude that the second condition holds, which is assumed in what follows.

1.3 Appendix 3

Taking the total difference of the first order conditions in (7)–(10) (after using (11) and (4)) with respect to the endogenous variables λ, c _w, c _o and s and the exogenous variables/parameters y, γ, m, q and v gives the following after collecting terms and rearranging:

$$ {u}^{\prime \prime }(c)d{c}_w- d\lambda =0\vspace*{-1.6pc} $$

$$ {u}^{\prime \prime }(c)d{c}_o- d\lambda =0\vspace*{-1.6pc} $$

$$\begin{array}{l} -\left[{N}_w\gamma {G}^{\prime }(s)+c\frac{\partial {\overset{\sim }{N}}_o}{\partial s}\right] d\lambda +\Phi ds={N}_w{u}^{\prime }(c){G}^{\prime }(s) d\gamma \\[8pt] -{u}^{\prime }(c){N}_w{\gamma G}^{\prime }(s)\frac{\frac{\partial^2{\overset{\sim }{N}}_o}{\partial s\partial m}}{\frac{\partial {\overset{\sim }{N}}_o}{\partial s}} dm-{u}^{\prime }(c){N}_w{\gamma G}^{\prime }(s)\frac{\frac{\partial^2{\overset{\sim }{N}}_o}{\partial s\partial q}}{\frac{\partial {\overset{\sim }{N}}_o}{\partial s}} dq-{u}^{\prime }(c){N}_w{\gamma G}^{\prime }(s)\frac{\frac{\partial^2{\overset{\sim }{N}}_o}{\partial s\partial v}}{\frac{\partial {\overset{\sim }{N}}_o}{\partial s}} dv\end{array} $$

$$\begin{array}{l} -{N}_wd{c}_w-{\overset{\sim }{N}}_od{c}_o-\left[{N}_w\gamma {G}^{\prime }(s)+c\frac{\partial {\overset{\sim }{N}}_o}{\partial s}\right] ds \\[10pt] =-{N}_w dy+{N}_wG(s) d\gamma +c\frac{\partial {\overset{\sim }{N}}_o}{\partial m} dm+c\frac{\partial {\overset{\sim }{N}}_o}{\partial q} dq+c\frac{\partial {\overset{\sim }{N}}_o}{\partial v} dv\end{array} $$

Setting dγ = dm = dq = dv = 0 and dividing through each equation with dy gives four equations in four unknown variables, i. e. $\frac{\partial \lambda }{\partial y}$, $\frac{\partial {c}_w}{\partial y}$, $\frac{\partial {c}_o}{\partial y}$ and $\frac{\partial s}{\partial y}$:

$$ {u}^{\prime \prime }(c)\frac{\partial {c}_w}{\partial y}-\frac{\partial \lambda }{\partial y}=0\vspace*{-1pc} $$

$$ {u}^{\prime \prime }(c)\frac{\partial {c}_o}{\partial y}-\frac{\partial \lambda }{\partial y}=0\vspace*{-1pc} $$

$$ -\left[{N}_w\gamma {G}^{\prime }(s)+c\frac{\partial {\overset{\sim }{N}}_o}{\partial s}\right]\frac{\partial \lambda }{\partial y}+\Phi \frac{\partial s}{\partial y}=0\vspace*{-1pc} $$

$$ -{N}_w\frac{\partial {c}_w}{\partial y}-{\overset{\sim }{N}}_o\frac{\partial {c}_o}{\partial y}-\left[{N}_w\gamma {G}^{\prime }(s)+c\frac{\partial {\overset{\sim }{N}}_o}{\partial s}\right]\frac{\partial s}{\partial y}=-{N}_w $$

Solving for $\frac{\partial s}{\partial y}$ gives:

$$ \frac{\partial s}{\partial y}=\frac{N_w\left[{N}_w\gamma {G}^{\prime }(s)+{c}_o\frac{\partial {\overset{\sim }{N}}_o}{\partial s}\right]{\left[{u}^{\prime \prime }(c)\right]}^2}{{\left[{N}_w\gamma {G}^{\prime }(s)+{c}_o\frac{\partial {\overset{\sim }{N}}_o}{\partial s}\right]}^2{\left[{u}^{\prime \prime }(c)\right]}^2+\Phi \left[{N}_w+{\overset{\sim }{N}}_o\right]{u}^{\prime \prime }(c)} $$

which gives (13). Similarly calculating $\frac{\partial s}{\partial \gamma }$, $\frac{\partial s}{\partial m}$, $\frac{\partial s}{\partial q}$ and $\frac{\partial s}{\partial v}$ gives (14)–(17). Note that the denominator in the derivatives:

$$ \Psi \equiv {\left[{N}_w\gamma {G}^{\prime }(s)+{c}_o\frac{\partial {\overset{\sim }{N}}_o}{\partial s}\right]}^2{\left[{u}^{\prime \prime }(c)\right]}^2+\Phi \left[{N}_w+{\overset{\sim }{N}}_o\right]{u}^{\prime \prime }(c) $$

is positive (see Appendix 2).

1.4 Appendix 4

Table 1 The data and their sources