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Human Development at Risk: Economic Growth with Pollution-Induced Health Shocks

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Abstract

Risks to human health stemming from polluted air, water, and soil are substantial, especially in the rapidly growing economies. The present paper develops a theoretical framework to study an endogenously growing economy which is subject to pollution-induced health shocks with the health status being an argument of the welfare function. Pollution, arising as a negative externality from production, adversely and randomly affects the regeneration ability of a human body leading to a decline in the overall health status of the population. We include two types of uncertainty surrounding the health status: continuous small-scale fluctuations, driven by the Wiener process, and large-scale shocks or epidemics, driven by the Poisson process. We derive closed-form analytical solutions for the optimal abatement policy and the growth rate of consumption. Devoting a constant fraction of output to emissions abatement delivers the first-best allocation. This fraction is an increasing function of total factor productivity, polluting intensity of production, and damage intensity of both continuous and jump-type shocks. A higher frequency of jumps also calls for more vigorous abatement policies. By contrast, the optimal growth rate of the economy is decreasing in the frequency and intensity of shocks and in the polluting intensity of output. The efficiency of abatement technology has, in general, an ambiguous bearing on both the growth rate and on the abatement share due to the opposing forces of the direct and indirect effects.

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Notes

  1. 88% of those premature deaths occurred in low- and middle-income countries.

  2. A recent study found that air pollution in Europe costs no less than USD 1.6 trillion a year in diseases and deaths, causing approximate 600,000 premature deaths (WHO and OECD 2015).

  3. These authors use an overlapping generations model with individuals living for two periods. The young generation is never sick while the old faces the probability of being in an unhealthy state, reducing its consumption; health status does not enter the utility function. The paper focuses on savings as a substitute for lacking health insurance and on steady-state analysis with a constant capital stock.

  4. Empirical evidence is strongly supportive of this argument. Mullahy and Portney (1990) find a significant negative impact of air pollution on health. Bloom et al. (2004) estimate a production function using work experience and health, showing that ceteris paribus good health has a positive, sizable, and statistically significant effect on aggregate output. Using a range of micro- and macroeconometric setups, Bleakley (2010) empirically assesses how diseases depress development of human capital and global income, confirming high benefits of improving health capital. Using pooled data for 21 OECD countries over the past two centuries, Madsen (2012) shows that health has been highly influential for the quantity and quality of schooling, innovations and growth. The effects of major health shocks on economic development has also been debated in the empirical literature. Various studies find that the broad diffusion of diseases is detrimental to development, see e.g. Percoco (2015) for the case of Spanish flu. Conversely, Acemoglu and Johnson (2007) suggest that mortal diseases and pandemics have a positive impact on the economy, because lower labor supply after the shocks increases real wages improving living conditions. Acemoglu and Robinson (2012) argue that an extreme health shock like the Black Death may be considered as a “critical juncture in history,” disrupting the existing economic balance in society and possibly create room for institutional improvements.

  5. Explicit treatment of endogenous mortality and/or fertility would require a different preference structure which would complicate the analysis considerably, without adding to the main topic of pollution-induced health shocks. It is intuitive that if exposure to pollution also reduces productive life span of an individual, there is an even stronger incentive to abate than in our baseline model.

  6. Pautrel (2012) extends the analysis of pollution effects on health by showing that the relationship between environmental taxation and economic growth as well as between environmental tax and lifetime welfare are both inverted U-shaped.

  7. Jouvet et al. (2010) analyze optimal policies when longevity is negatively influenced by pollution but positively affected by health expenditures.

  8. Including risk and pollution but not human health, Soretz (2007) analyzes efficient emission taxation within a stochastic model of endogenous growth. The author derives that a linear capital income tax joint with a linear abatement subsidy build an efficient public policy scheme.

  9. Uncertainty and health is studied by several authors. Cropper (1977) adds uncertainty to the Grossman model, assuming that illness occurs if the stock of health falls below a critical sickness level, which is random. Selden (1993) and Chang (1996) find that, given uncertainty, risk-averse individuals make larger investments in health than they would in its absence; compared to a model with perfect certainty, the expected value of the stock of health is larger and the optimal quantities of gross investment and health inputs are also larger in a model with uncertainty. Hugonnier et al. (2013) study different types of stochastic processes in health development but they consider neither economic growth nor the environment which is the focus of our contribution.

  10. We do not include natural or technology shocks which are unrelated to environmental flow pollution.

  11. See Hammitt (2002) and Bleichrodt and Quiggin (1999) for the foundations of utility functions including health. See Finkelstein et al. (2013) for empirical evidence.

  12. Including human capital as a separate input would require adding another stock variable which would complicate the model considerably. We can show in our simple framework that output growth is negatively affected by shocks to human health even when the health status is not an argument in the production function.

  13. We are thankful to an anonymous referee for pointing out that the assumption \(\varepsilon <1\) can be dropped if a slightly different utility function is adopted, namely \(U(c,H)=\frac{(cH^{\beta })^{1-\varepsilon }}{1-\varepsilon }\). The solution involves a term which determines the sign of the second derivative of utility with respect to health. With our initial utility function this term is given by \((\beta -1)/(1-\varepsilon )\), and with the alternative utility function it becomes \(\beta (1-\varepsilon )-1\). The difference in assumptions is consequential for restriction on the sign of \(U_{hh}\): The latter expression requires a joint restriction on \(\beta \) and \(\varepsilon \), making one parameter a function of the other, while the former expression requires separate, independent restrictions on \(\beta \) and \(\varepsilon \).

  14. The formulation is analoguous to the modeling of spillovers from average human capital in the seminal model of Lucas (1988).

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Appendix

Appendix

1.1 Derivations of Section 2

1.1.1 Model Setup

The Bellman equation reads

$$\begin{aligned} \rho V(K_{t},H_{t})=\max _{c_{t}}\left\{ U(c_{t},H_{t})+\frac{1}{dt}{\mathbb {E}}_{t}dV(K_{t},H_{t})\right\} . \end{aligned}$$

Given the process (3) and applying the Ito’s Lemma, the Bellman equation for our specific problem becomes

$$\begin{aligned} \rho V(K,H)=\max \left\{ U(c,H)+V_{k}[(1-m_{t})Y_{t}-c_{t}]+\frac{1}{2} V_{hh}R^{2}(q)H^{2}\right\} , \end{aligned}$$

The first-order conditions with respect to the control and the state variables are

$$\begin{aligned}&c :\;U_{c}-V_{k}=0, \end{aligned}$$
(25)
$$\begin{aligned}&m:\;-V_{k}Y+\frac{1}{2}V_{hh}2R\left( q\right) R^{\prime }\left( q\right) \left( -\mu Y^{\prime }\left( K\right) \right) H^{2}=0, \end{aligned}$$
(26)
$$\begin{aligned}&K :\;\rho V_{k}=V_{kk}\left[ \left( 1-m\right) Y-c\right] +V_{k}\left( 1-m\right) Y^{\prime }\left( K\right) \nonumber \\&\qquad \qquad +\frac{1}{2}\left[ V_{hhk}R^{2}\left( q\right) +V_{hh}2R\left( q\right) R^{\prime }\left( q\right) q_{k}\right] H^{2}, \end{aligned}$$
(27)
$$\begin{aligned}&H \;:\;\rho V_{h}=U_{h}+V_{kh}\left[ \left( 1-m\right) Y-c\right] +\frac{1}{2} \left[ V_{hhh}H^{2}+V_{hh}2H\right] R^{2}\left( q\right) . \end{aligned}$$
(28)

We guess and subsequently verify that the value function is of the form

$$\begin{aligned} V(K,H)=\frac{xK^{1-\varepsilon }}{1-\varepsilon }H^{\beta }, \end{aligned}$$

where x is a constant to be determined. Then Eq. (26) implies that the equilibrium emissions concentration is as stated in Eq. (9):

$$\begin{aligned} q^{*}=\frac{1-\varepsilon }{\beta (1-\beta )\mu \delta ^{2}} \end{aligned}$$

and hence the optimal fraction of output to be spent on abatement is given by Eq. (10):

$$\begin{aligned} m^{*}=\frac{\phi }{\mu }-\frac{1-\varepsilon }{A\beta (1-\beta )(\mu \delta )^{2}}. \end{aligned}$$

Equations (25) and (27) allow us to obtain the law of motion for the optimal consumption rate. First, we compute the differential of \( V_{k}\) using Ito’s Lemma:

$$\begin{aligned} dV_{k}= & {} \left\{ V_{kk}[(1-m)Y-c]+\frac{1}{2}V_{khh}(R(q)H)^{2}\right\} dt+V_{kh}R(q)Hdz \nonumber \\= & {} V_{k}\left\{ \rho -(1-m)Y^{\prime }(K)\right\} dt+V_{kh}R(q)Hdz, \end{aligned}$$
(29)

where the second line follows after substituting Eq. (27). Inverting the marginal utility function to solve for c, we obtain \(c=U_{c}^{-\frac{1}{\varepsilon }}H^{\frac{\beta }{\varepsilon }}=f(U_{c},H)\). Both arguments of the f(., .) function are stochastic processes with \(dU_{c}\) given by (29) and dH by (3). Hence, to find dc we again need to apply Ito’s Lemma:

$$\begin{aligned} dc= & {} \left\{ \frac{\partial f}{\partial U_{c}}U_{c}\left[ \rho -(1-m)Y^{\prime }(K)\right] +\frac{1}{2}\left[ \frac{\partial ^{2}f}{\partial U_{c}^{2}}V_{kh}^{2}+ \frac{\partial ^{2}f}{\partial H^{2}}+2\frac{\partial ^{2}f}{\partial U_{c}\partial H}V_{kh}\right] R^{2}(q)H^{2}\right\} dt \\&+\,\left[ \frac{\partial f}{ \partial U_{c}}V_{kh}+\frac{\partial f}{\partial H}\right] R(q)Hdz \end{aligned}$$

Substituting the expressions for the partial derivatives of U, f and Y, we obtain:

$$\begin{aligned} \frac{dc}{c} =\frac{1}{\varepsilon }\left\{ (1-m)A-\rho +\frac{1}{2}\beta (\beta -1)(\delta q)^{2} \right\} dt. \end{aligned}$$
(30)

Note that \(\frac{\partial f}{ \partial U_{c}}V_{kh}+\frac{\partial f}{\partial H}=0\) and thus the stochastic element disappears from the optimal growth rate. Finally, we substitute the solutions for m and q into (30) to obtain the expression in Eq. (11) in the text.

The value function and Eq. (25) imply that c is proportional to K. Substituting the optimal controls into the HJB equation, we verify that our guess of the value function is indeed correct and x is a constant which depends on the parameters of the model

$$\begin{aligned} x=\left\{ \frac{1}{\varepsilon }\left[ \rho -(1-\varepsilon )A\left( 1-\frac{ \phi }{\mu }\right) -\frac{(1-\varepsilon )^{2}}{2(\mu \delta )^{2}\beta (\beta -1)}\right] \right\} ^{-\varepsilon }. \end{aligned}$$

1.1.2 Comparative Statics

Follow from taking the partial derivatives in (10):

$$\begin{aligned} \frac{\partial m^{*}}{\partial A}= & {} \frac{1-\varepsilon }{A^{2}\beta \left( 1-\beta \right) \left( \mu \delta \right) ^{2}}>0, \\ \frac{\partial m^{*}}{\partial \phi }= & {} \frac{1}{\mu }>0, \\ \frac{\partial m^{*}}{\partial \delta }= & {} \frac{2\left( 1-\varepsilon \right) }{A\beta \left( 1-\beta \right) \mu ^{2}\delta ^{3}}>0; \\ \frac{\partial m^{*}}{\partial \mu }= & {} \frac{1}{\mu }\left[ \frac{ 2\left( 1-\varepsilon \right) }{A\beta \left( 1-\beta \right) \left( \mu \delta \right) ^{2}}-\frac{\phi }{\mu }\right] \gtrless 0, \\ \frac{\partial m^{*}}{\partial \beta }= & {} \frac{\left( 1-\varepsilon \right) \left( 1-2\beta \right) }{ A\beta ^{2}\left( 1-\beta \right) ^{2}\left( \mu \delta \right) ^{2}}\gtrless 0\Leftrightarrow \beta \lessgtr \frac{1}{2}. \end{aligned}$$

Follow from taking the partial derivatives in (11):

$$\begin{aligned} \frac{\partial g^{*}}{\partial A}= & {} \frac{1}{\varepsilon }\left( 1-\frac{ \phi }{\mu }\right)>0, \\ \frac{\partial g^{*}}{\partial \phi }= & {} -\frac{A}{\varepsilon \mu }<0, \\ \frac{\partial g^{*}}{\partial \delta }= & {} -\frac{\left( 1-\varepsilon ^{2}\right) }{\beta \left( 1-\beta \right) \mu ^{2}\delta ^{3}} <0; \\ \frac{\partial g^{*}}{\partial \beta }= & {} \frac{ \left( 1-\varepsilon ^{2}\right) \left( 2\beta -1\right) }{2[\beta \left( 1-\beta \right) \mu \delta ]^{2}}\gtrless 0 \Leftrightarrow \beta \gtrless \frac{1}{2}, \\ \frac{\partial g^{*}}{\partial \mu }= & {} \frac{A}{\varepsilon \mu }\left[ \frac{\phi }{\mu }-\frac{\left( 1-\varepsilon ^{2}\right) }{\beta \left( 1-\beta \right) \left( \mu \delta \right) ^{2}}\right] > 0. \end{aligned}$$

1.2 Derivations of Section 3

Proof of Proposition 2

Rewriting Eq. (16) as

$$\begin{aligned} \lambda \beta ' \sigma \mu (1-\sigma q_{e}^{*})^{\beta ' -1}=1-\varepsilon -\beta ' (1-\beta ')\delta ^{2}\mu q_{e}^{*}. \end{aligned}$$
(31)

one can immediately see, recalling that \(\beta '\in (0,1)\), that the left-hand side is an increasing and convex function of \(q_{e}^{*}\) with the origin at \(\lambda \beta ' \sigma \mu \), while the right-hand side is a decreasing linear function of \(q_{e}^{*}\) with the origin at \(1-\varepsilon \). Thus a solution exists if and only if \(1-\varepsilon \geqslant \lambda \beta ' \sigma \mu \). Also not that \(LHS=0\) at \(q=q^{*}\), hence \(q_{e}^{*}<q^{*}\).

The optimal trend consumption growth rate is obtained in a similar way as in Sect. 2, by applying the Ito’s Lemma on the differential of \(V_{k}\) and substituting for terms from Eq. (14). To see that \(g^{*}_{e}<g^{*}\), recall that

$$\begin{aligned} g^{*}=\frac{1}{\varepsilon }\left\{ A\left( 1-m^{*}\right) -\rho +\frac{1}{2} \beta ' (1-\beta ' )(\delta q^{*})^{2}\right\} \end{aligned}$$
(32)

and

$$\begin{aligned} g_{e}=\frac{1}{\varepsilon }\left\{ A\left( 1-m^{*}_{e}\right) -\rho +\frac{1}{2} \beta ' (1-\beta ' )(\delta q^{*}_{e})^{2}+\lambda \left[ (1-\sigma q^{*}_{e})^{\beta '}-1\right] \right\} . \end{aligned}$$
(33)

We have established that the optimal mitigation expenditure is larger with two sources of uncertainty, hence the first term inside the curly braces of (32) is larger than the respective term in (33). Emissions concentration is larger in the first scenario, hence the second term  (32) is larger than the corresponding term in (33). Finally \((1-\sigma q^{*}_{e})\) is smaller than unity and hence the last term in  (33) is negative. Therefore, a term by term comparison proves the claim. \(\square \)

One can verify that our initial guess of the value function is correct by substituting the optimal controls into HJB. The constant factor is given by

$$\begin{aligned} {\tilde{x}}=\left( \frac{1}{\varepsilon }\right) ^{-\varepsilon }\chi \left\{ \rho -(1-\varepsilon )(1-m^{*}_{e})A-\frac{1}{2}\beta ' (1-\beta ')(\delta q^{*}_{e})^{2}-\lambda \left[ (1-\sigma q^{*}_{e})^{\beta '}-1\right] \right\} ^{-\varepsilon }. \end{aligned}$$

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Bretschger, L., Vinogradova, A. Human Development at Risk: Economic Growth with Pollution-Induced Health Shocks. Environ Resource Econ 66, 481–495 (2017). https://doi.org/10.1007/s10640-016-0089-0

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