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The Review of Economic Studies
, Volume Advance Article – Mar 28, 2018

47 pages

/lp/ou_press/analysing-the-effects-of-insuring-health-risks-on-the-trade-off-FY6REYjwlk

- Publisher
- Oxford University Press
- Copyright
- © The Author(s) 2018. Published by Oxford University Press on behalf of The Review of Economic Studies Limited.
- ISSN
- 0034-6527
- eISSN
- 1467-937X
- D.O.I.
- 10.1093/restud/rdy017
- Publisher site
- See Article on Publisher Site

Abstract This article quantitatively evaluates the trade-off between the provision of health-related social insurance and the incentives to maintain good health through costly investments. To do so, we construct and estimate a dynamic model of health investments and health insurance in which the cross-sectional health distribution evolves endogenously and is shaped by labour market and health insurance policies. A no wage discrimination law in the labour market limits the extent to which wages can depend on the health status of a worker, and a no prior conditions law outlaws higher insurance premia for individuals with worse health status. In the model, the static gains from better insurance against poor health induced by these policies are traded off against their adverse dynamic incentive effects on household efforts to lead a healthy life. In our quantitative analysis, we find that it is optimal to insure 80% of labour market-related income risk (70% if a no prior conditions law is also present). Providing full insurance is strongly suboptimal, however, since at high levels of consumption insurance, the negative dynamic incentive effects on health effort and thus the population health distribution in the long run start to dominate the short-run consumption insurance gains. 1. Introduction In this article, we study the impact of social insurance policies aimed at reducing a household’s exposure to health-related risk in health care and labour markets. We model and quantitatively examine the trade-off between the benefits of greater insurance against health risks in these two contexts, and the resulting reduction in the incentives of households to maintain their health. In many countries, government policies or regulations restrict the extent to which health insurance companies can condition health insurance premia on pre-existing differences in the health conditions of individuals. We will call such legislation that insures individuals against health-dependent expenditure risk on health insurance premia a no prior conditions law. For example, in Switzerland health insurance is compulsory and premia charged by private insurance companies cannot depend on the health status of the insured. In the U.S., the Patient Protection and Affordable Care Act (ACA henceforth) contains a provision that requires health insurers to offer the same insurance premium to all applicants of the same age and location without regard to gender or pre-existing health conditions. Similarly, in the context of the labour market, across a wide set of countries, regulations are in place that limit the extent to which employers can condition a worker’s compensation on (the change in) her health status. Examples of such regulations, which we label wage non-discrimination laws in this article, include special employment protection, job quotas and wage subsidies for the disabled in Germany. For the U.S., the Americans with Disabilities Act Amendments Act (ADAAA) of 2008 significantly broadened the restrictions on employers imposed by the Americans with Disabilities Act (ADA) from 1990.1 The size of health-related risks and the importance of potential incentive effects from no prior conditions laws and wage non-discrimination legislation rest on three empirical observations that we document for the U.S. in Section 2. First, a better health status increases a worker’s productivity and thus her labour earnings to a sizable degree. Second, while somewhat less important than the impact on earnings, a good health status reduces the chances of getting acutely sick and thus reduces health expenditure risk to a significant degree. Third, households can affect the evolution of their health status over time by taking costly (in terms of resources or utility) actions such as exercising and abstaining from smoking, and over time, this has an appreciable impact on the health distribution of the population. Based on these observations, the main argument of the article is that it is crucial to study no prior conditions laws in the health insurance market and no wage discrimination laws in the labour market jointly, since their interaction provides social insurance against health-related risks, but negatively impacts health effort incentives. To quantitatively analyse the impact of these policies, we construct a dynamic life-cycle model with endogenous and stochastically evolving health. A person’s health status is an individual state variable that determines both a household’s productivity at work and the likelihood that it is subject to an adverse health-related shock. This shock in turn affects productivity and can be offset by medical expenditures for which individuals can purchase health insurance.2 Health status itself is persistent and changes stochastically over time, and its evolution is affected by the household’s effort choice to maintain its health. Since health-related social insurance policies reduce an individual’s economic incentives to maintain her health, a trade-off between the provision of social insurance and private incentives emerges, rendering the adoption of these policies a non-trivial policy design question. In addition, the endogenous demand for medical expenditures and thus private insurance contracts respond to the public policy regime. We use the model as a theoretical and quantitative labouratory for the study of no prior conditions and no wage discrimination social insurance policies. To isolate the social insurance benefits and incentive costs of both policies, we first study the static and the dynamic impact of idealized versions of both policies that fully insure, at no administrative costs, the health-related income and expenditure risks, respectively. The static analysis holds the population health distribution fixed and focuses on the equilibrium health insurance contract and the provision of consumption insurance against adverse health status by the policies. In contrast, the key aspect of the dynamic analysis is the impact the policies have on individuals’ incentives to maintain their health, and the interaction this creates between the health distribution of the population and the costs of health insurance and the productivity of the workforce. After evaluating the idealized versions of these policies, we also assess the insurance benefits and incentive costs of no wage discrimination laws that are only partially effective in insuring health-related income risk, and might require resource costs to be enforced. This last analysis reflects our view that, whereas strong forms of no prior conditions laws that rule out health-condition-based insurance premia are ubiquitous across countries,3 the degree of labour income insurance through no wage discrimination type legislation varies significantly across countries and over time (as in the U.S. when the ADAAA replaced the largely ineffective original ADA) and is unlikely as complete and cost-effective as our idealized version of the no wage discrimination law models it.4 To empirically implement our quantitative analysis, we first estimate and calibrate the model to U.S. PSID and MEPS data to match key statistics on labour earnings, medical expenditures and physical exercise levels. We then use the model as a quantitative labouratory to evaluate the consequences of the different policy options. Although, in the U.S. context, we think of the ACA as providing the no prior conditions legislation and the ADAAA as the broad motivation for the no wage discrimination policy, our focus is on the specific role of these classes of policies in providing social insurance against health-related income and health insurance premium risk as well as their negative incentive effects. Ours is clearly not intended to be a comprehensive study of all other aspects and provisions of the ACA or health-related labour market policies such as the ADAAA. At the same time, given the prevalence of similar policies in other countries, the relevance of our analysis is not limited to the U.S. context either. Our quantitative analysis reveals that it is optimal, from an ex-ante lifetime utility perspective, to insure 80% of labour market-related income risk (and 70%, if a no prior conditions law is also present). On the other hand, providing full insurance, by combining a fully effective no wage discrimination and a no prior conditions law, is strongly suboptimal, since at high levels of consumption insurance, the negative dynamic incentive effects on health effort and thus the population health distribution in the long run start to dominate the short-run, static consumption insurance gains. Thus, overall, health-related risks in the labour and health insurance market call for strong social insurance, and both policies studied in this article are welfare improving relative to the competitive equilibrium. The optimal extent of insurance remains large even when we consider empirically plausible resource costs associated with the implementation of the no-wage-discrimination policy. However, even without any resource costs, complete insurance is never optimal. At the broadest level, our article positively and normatively evaluates the trade-off between the provision of income insurance and the distortion of incentives induced by social insurance policies, but shifts the focus to health-related (insurance) policies. On this general level, our work therefore builds upon the large literature assessing this trade-off for other social insurance policies, such as (among others) unemployment insurance (e.g.Shavell and Weiss (1979), and Hopenhayn and Nicolini (1997)), disability insurance (e.g.Golosov and Tsyvinski (2007), and Low and Pistaferri (2015)) and progressive income taxation (e.g.Mirrlees (1971)) and the large literatures that followed these contributions. The health-related social insurance policies we study provide income and thus consumption insurance benefits in much the same way as in the general social insurance literature cited above. On the other hand, the dynamic health effort incentive effects are specific to the health policies we model. They impact the future evolution of individual health status, and through it, future earnings capabilities, and therefore the distribution of income and health in the economy as a whole. The article thus connects and contributes to three specific literatures at the intersection of quantitative macroeconomics and health economics. First, on the modelling side, our structure with endogenous effort choice and the dynamic and endogenous evolution of health status builds upon the strand of the quantitative macroeconomics literature that has endogenized the evolution of health status over the life cycle. Second, it builds upon the empirical literature investigating the impact of health or diseases on earnings, and the determinants of the dynamics of health, since our theoretical mechanism crucially relies on these links. Finally, since in the U.S., the economy to which we calibrate our model, the social insurance we model is motivated by selected provisions in the ACA and the ADAAA, we contribute to the literature evaluating these health-related policies. Starting with the seminal works of Grossman (1972) and Ehrlich and Becker (1972), the first literature starts with the insight that health is in part an investment good whose evolution can be actively impacted by investing effort or resources. Our model, in which dynamic (and stochastic) health transitions can be influenced by costly health effort, directly builds upon this tradition. A recent model-based quantitative literature has used related dynamic models with dynamic health updates to study the macroeconomic and distributional implications of health, health insurance and health care policy reforms.5 Within this first literature, the most closely related papers are Brügemann and Manovskii (2010) and Jung and Tran (2016), since they also study the effects of the ACA on health insurance coverage and macroeconomic aggregates. Aizawa and Fang (2015) extend the labour market mobility model with the employer-sponsored health insurance constructed by Dey and Flinn (2005) to study the effects of the ACA. As with Brügemann and Manovskii (2010) and Jung and Tran (2016), neither of these last papers is concerned with the incentive effects on health efforts and thus health transitions induced by regulation in both the labour and the health insurance markets (and crucially, their interaction) that we formalize in our model. The second empirical strand of the literature that this article builds upon studies the impact of general health or specific diseases on earnings, and on the determinants of the dynamics of health. The empirical literature estimating the impact of health on income (see, e.g.Bartel and Taubman (1979), Mitchell and Butler (1986), Cawley (2004) and Currie and Madrian (1999) for a summary) finds a positive impact of health on earnings. Concerning the dynamics of health transitions, Pijoan-Mas and Rios-Rull (2014) document an important impact of socioeconomic status (most importantly, an education dependence that we also permit in our model). Moreover, many studies in economics and the medical literature (e.g.Colman and Dave (2012), Booth et al. (2012)) find that individual behaviour is a significant determinant of health status changes over time, which is the key premise of our model.6 Finally, there is evidence that such health behaviour responds to economic incentives. Bhattacharya et al. (2011) employ data from a Rand health insurance experiment to show that access to health insurance leads to increases in body mass and obesity because insurance insulates people from the impact of their excess weight on their medical expenditure costs. Charness and Gneezy (2009) use experimental data to show that individuals’ gym attendance responds significantly and persistently to financial incentives.7 Finally, in addition to the papers that evaluate the impact of the ACA in the U.S. in structural dynamic models, our study complements a third strand of the literature that estimates the effect of the original ADA legislation in the U.S. from 1990 on employment, wages and labour hours of the disabled (see, e.g.Acemoglu and Angrist (2001), and Deleire (2000, 2001)). Most find that the employment rate of the disabled was lower after the ADA. Deleire (2001) documents that, compared to 1984, the earnings gap between the disabled and the non-disabled fell significantly in 1993, some of which may be attributable to the ADA. These studies focus on evaluating the impact of the ADA on the labour market performance of the disabled. On the other hand, we interpret the ensuing Amendments Act as a shift in government policies towards providing more insurance against broader health-related risks and we study its impact on the health behaviour of workers. The article is organized as follows. In Section 2, we set out the empirical facts justifying our modelling approach. We describe the model and implementation of the two policies in Section 3. The theoretical analysis of the static and dynamic version of the baseline model is contained in Section 4. In Section 5 we describe how we augment the model to map it into the data, as well as our estimation and calibration procedure. Section 6 presents the main quantitative results of the policy analysis, of both the idealized and the partial insurance versions of the policies. Robustness analyses are contained in Section 7, and Section 8 concludes. Proofs and details of the quantitative analysis are relegated to the Supplementary Appendix. 2. Motivating Empirical Facts Our theoretical model is built on three premises: first, that a good health status increases a worker’s productivity and thus labour earnings; second, that a good health status reduces the chances of getting acutely sick and thus reduces expected health expenditures; and third, that workers can affect the dynamic evolution of their health statuses by taking costly actions. The public policies under study then provide additional social insurance against labour income risk and health expenditure risk, but also have an adverse impact on the dynamic incentives to lead healthy lives. The purpose of the article is to qualitatively and quantitatively evaluate this trade-off. Before we turn to this task, we first want to document the empirical plausibility of the three basic premises on which our argument is being built. In this section we focus on raw correlations motivating our analysis; when we estimate the model in Section 5, we take into account that part of these correlations could be driven by observable and unobservable household heterogeneity. In Table 1 we use PSID data from 1999 to 2009, sort individuals into four health groups ranging from (self-reported) fair to excellent health and document that better health is associated with significantly higher labour income.8 Labour income is strongly increasing in health: for example, mean (median) labour income among those reporting excellent health is $$116\%$$$$(84\%)$$ larger than for individuals with fair health. As documented in section 5, this health-income gap shrinks to 67% when we control for other observable differences across individuals and therefore remains economically very significant.9 Table 1 Labour income by health status Health status Labour income, if positive Mean St. Dev. Median Fair 32,752 29,211 26,483 Good 45,970 46,615 36,665 Very good 55,541 79,465 41,604 Excellent 70,826 129,021 48,695 All 55,075 867,289 40,797 Health status Labour income, if positive Mean St. Dev. Median Fair 32,752 29,211 26,483 Good 45,970 46,615 36,665 Very good 55,541 79,465 41,604 Excellent 70,826 129,021 48,695 All 55,075 867,289 40,797 Table 1 Labour income by health status Health status Labour income, if positive Mean St. Dev. Median Fair 32,752 29,211 26,483 Good 45,970 46,615 36,665 Very good 55,541 79,465 41,604 Excellent 70,826 129,021 48,695 All 55,075 867,289 40,797 Health status Labour income, if positive Mean St. Dev. Median Fair 32,752 29,211 26,483 Good 45,970 46,615 36,665 Very good 55,541 79,465 41,604 Excellent 70,826 129,021 48,695 All 55,075 867,289 40,797 Second, in Table 2 we exploit data from the 1997 to 2002 waves of the Medical Expenditure Panel Survey (MEPS) to document the negative correlation between health status and health expenditures. Individuals are asked to self-report their health status on the same scale as in the PSID, and in the table, we display the mean and median health expenditures across health status groups. We use the MEPS for medical expenditures, as this data set reports individual-level medical expenditures, whereas the PSID reports total medical expenditures only at the household level. Also, expenditures in the MEPS include out-of-pocket payments and payments by private insurance, Medicaid, Medicare and all other sources. Table 2 Medical expenditure by health status Medical expenditure Mean St. Dev. Median Fair 5,821 13,043 1,977 Good 2,344 6,118 733 Very good 1,601 3,861 558 Excellent 1,227 2,872 363 All 2,157 6,172 599 Medical expenditure Mean St. Dev. Median Fair 5,821 13,043 1,977 Good 2,344 6,118 733 Very good 1,601 3,861 558 Excellent 1,227 2,872 363 All 2,157 6,172 599 Table 2 Medical expenditure by health status Medical expenditure Mean St. Dev. Median Fair 5,821 13,043 1,977 Good 2,344 6,118 733 Very good 1,601 3,861 558 Excellent 1,227 2,872 363 All 2,157 6,172 599 Medical expenditure Mean St. Dev. Median Fair 5,821 13,043 1,977 Good 2,344 6,118 733 Very good 1,601 3,861 558 Excellent 1,227 2,872 363 All 2,157 6,172 599 The table shows very significant differences in mean and median health expenditures across individuals with different health statuses: those with fair health spend on average 4.7 times as much as those in the highest health category. In section 5, we document that differences in observable characteristics across health groups are partially responsible for the expenditure gaps, but most of the gap persists after controlling for them (e.g. the mean expenditure ratio between the best and worst health groups drops, from 4.7 to 2.9). The final, and perhaps most novel, premise of our model is that health status is endogenous and its stochastic evolution can be affected by an individual’s effort to lead a healthy life. This premise receives support in the raw data, as Table 3 displays. In this table we summarize, again using data from the PSID, how the dynamics of the health status of individuals is impacted by their effort to lead healthy lives.10 Table 3 Effort and health dynamics over six years Health All Bad Initial Health Good Initial Health Eff $$<$$ Avg. Eff $$\geq$$ Avg. Eff $$<$$ Avg. Eff $$\geq$$ Avg. Eff $$<$$ Avg. Eff $$\geq$$ Avg. Worsened 0.35 0.30 0.29 0.28 0.39 0.30 Unchanged 0.50 0.52 0.54 0.50 0.48 0.53 Improved 0.15 0.18 0.17 0.22 0.13 0.17 Health All Bad Initial Health Good Initial Health Eff $$<$$ Avg. Eff $$\geq$$ Avg. Eff $$<$$ Avg. Eff $$\geq$$ Avg. Eff $$<$$ Avg. Eff $$\geq$$ Avg. Worsened 0.35 0.30 0.29 0.28 0.39 0.30 Unchanged 0.50 0.52 0.54 0.50 0.48 0.53 Improved 0.15 0.18 0.17 0.22 0.13 0.17 Table 3 Effort and health dynamics over six years Health All Bad Initial Health Good Initial Health Eff $$<$$ Avg. Eff $$\geq$$ Avg. Eff $$<$$ Avg. Eff $$\geq$$ Avg. Eff $$<$$ Avg. Eff $$\geq$$ Avg. Worsened 0.35 0.30 0.29 0.28 0.39 0.30 Unchanged 0.50 0.52 0.54 0.50 0.48 0.53 Improved 0.15 0.18 0.17 0.22 0.13 0.17 Health All Bad Initial Health Good Initial Health Eff $$<$$ Avg. Eff $$\geq$$ Avg. Eff $$<$$ Avg. Eff $$\geq$$ Avg. Eff $$<$$ Avg. Eff $$\geq$$ Avg. Worsened 0.35 0.30 0.29 0.28 0.39 0.30 Unchanged 0.50 0.52 0.54 0.50 0.48 0.53 Improved 0.15 0.18 0.17 0.22 0.13 0.17 The three rows of the table display the share of individuals whose health status, over a six-year interval, either declines, stays the same or improves, such that each row sums to 1. The first two columns document that individuals with effort above the cross-sectional average are more likely to retain or improve their health statuses. The differences in health dynamics across two effort groups are significant at a 5% confidence level (using a $$\chi^2$$ test). The remaining four columns display that the positive association of effort and health transition persists once we control for initial health status.11 The data presented here suggest a strong role for health status in the determination of income and medical expenditures, and a key role for individual effort in the dynamic updating of health status. We now build a model based on these three premises to evaluate the trade-off between incentive costs and the insurance benefits of social insurance policies in the labour and health insurance market. 3. The Model Time $$t=0,1,2,\ldots T$$ is discrete and finite and the economy is populated by a cohort of a continuum of individuals of mass 1. Since we are modelling a given cohort of individuals, we will use time and the age of households interchangeably. We think of $$T$$ as the end of the working life of the age cohort under study. 3.1. Endowments and preferences Households are endowed with one unit of time, which they supply inelastically to the market. They are also endowed with an initial level of health $$h$$ and we denote by $$H=\{h_{1},\ldots ,h_{N}\}$$ the finite set of possible health levels. Households value current consumption $$c$$ and dislike the effort $$e$$ that helps maintain their health. We will assume that their preferences are additively separable over time, and that they discount the future at time discount factor $$\beta .$$ We will also assume that preferences are separable between consumption and effort, and that households value consumption according to the common period utility function $$u(c)$$ and value effort according to the period disutility function $$q(e).$$ We will denote the probability distribution over the health status $$h$$ at the beginning of period $$t$$ by $$\Phi _{t}(h),$$ and denote by $$\Phi _{0}(h)$$ the initial distribution over this characteristic. Assumption 1. The function $$u$$ is twice differentiable, strictly increasing and strictly concave. The function $$q$$ is twice differentiable, strictly increasing, strictly convex, with $$q(0)=q^{\prime }(0)=0$$ and $$\lim_{e\rightarrow \infty }q^{\prime }(e)=\infty .$$ 3.2. Health and production technology Let $$\varepsilon$$ denote the current health shock. In every period, households with current health $$h$$ with probability $$1-g(h)$$ draw a health shock $$\varepsilon \in (0,\bar{\varepsilon}]$$ that is distributed according to the probability density function $$f(\varepsilon ).$$ With probability $$g(h)$$ the household draws no shock (i.e.$$\varepsilon =0$$). Assumption 2. $$f$$ is continuous, $$g$$ is twice differentiable, strictly increasing and strictly concave. An individual’s health status evolves stochastically over time, according to the Markov transition function $$Q(h^{\prime },h;e),$$ where $$e\geq 0$$ is the exercise level by the individual. We impose the following assumption on $$Q$$. Assumption 3. $$\frac{ \partial Q (h^{ \prime } ;h ,e)}{ \partial e}$$ is increasing in $$h^{ \prime }$$ and $$\frac{ \partial ^{2}Q (h^{ \prime } ;h ,e)}{ \partial e^{2}}$$ is decreasing in $$h^{ \prime }.$$ The first assumption implies first-order stochastic dominance. The second assumption implies that the impact of effort on the transition matrix shrinks as $$e$$ gets large. Also note that by assumption, health transitions depend on the effort $$e$$ by individuals, but not on medical expenditures.12 An individual with health status $$h$$, current health shock $$\varepsilon$$ and health expenditures $$x$$ produces $$F(h,\varepsilon -x)$$ units of output. Assumption 4. $$F$$ is continuously differentiable in $$(h,y)$$, strictly increasing in $$h,$$ satisfies $$F(h,y)=F(h,0)$$ for all $$y\leq 0,$$ and $$F_{2}(h,y)<0$$ and $$F_{2}(h,\bar{\varepsilon})<-1.$$ Moreover, $$F_{22}(h,y)<0$$ for all $$y>0$$ and $$F_{12}(h,y)\geq 0$$. Figure 1a displays the production function $$F(h,.),$$ for two different levels of the current health status. Holding health status $$h$$ constant, output is decreasing in the uncured portion of the health shock $$\varepsilon -x,$$ and the decline is more rapid for lower levels of health $$(h_1<h_2).$$Figure 1b displays the production function as a function of health expenditures $$x,$$ for a fixed level of the shock $$\varepsilon ,$$ and shows that expenditures $$x$$ exceeding the health shock $$\varepsilon$$ leave output $$F(h,\varepsilon -x)$$ unaffected (and thus are suboptimal). Furthermore, a reduction of the shock $$\varepsilon$$ to a lower level, $$\varepsilon_1$$, shifts the point at which health expenditures $$x$$ become ineffective to the left. Figure 1 View largeDownload slide Production function by health status and health shock. (a) $$F(h,\varepsilon-x)$$ by health status ($$h$$) $$h_1< h_2$$; (b) $$F(h,\varepsilon-x)$$ for fixed health shock ($$\varepsilon$$) $$h_1< h_2$$ and $$\varepsilon_1< \varepsilon_2$$. Figure 1 View largeDownload slide Production function by health status and health shock. (a) $$F(h,\varepsilon-x)$$ by health status ($$h$$) $$h_1< h_2$$; (b) $$F(h,\varepsilon-x)$$ for fixed health shock ($$\varepsilon$$) $$h_1< h_2$$ and $$\varepsilon_1< \varepsilon_2$$. The assumptions on the production function $$F$$ imply that health expenditures can offset the impact of a health shock on productivity, but not raise an individual’s productivity above what it would be if there had been no shock. In addition, the last assumption on $$F$$ that $$F_{12}\geq 0$$ implies that the negative impact of a given net health shock $$y$$ is lower the healthier a person is.13 The assumption $$F_{2}(h,\bar{\varepsilon})<-1$$ insures that if an individual is hit by the worst health shock, the cost of treating this health shock, at the margin, is smaller than the positive impact on productivity (output) this treatment has. Figure 2 summarizes the time line within period $$t$$. Households enter the period with individual health status $$h$$. In the population of age $$t$$, the cross-sectional distribution of health is given by $$\Phi _{t}(h)$$. Observing $$h$$, firms then offer wages $$w(h)$$ and health insurance contracts $$\{x(\varepsilon ,h),P(h)\}$$ to households with health status $$h$$ which these households accept. Next, the health shock $$\varepsilon$$ is drawn according to the distributions $$g,f$$, and then resources according to $$x=x(\varepsilon ,h)$$ are spent on health. Now production and consumption takes place. Finally, at the end of the period, individuals make health effort choices $$e$$, and then the new health status $$h^{\prime }$$ of a household is drawn according to the health transition function $$Q(h^{\prime }|h;e)$$, which, together with $$\Phi _{t}(h)$$, determines the new cross-sectional distribution $$\Phi _{t+1}(h^{\prime })$$ at the beginning of the next period. Figure 2 View largeDownload slide Timing of the model. Figure 2 View largeDownload slide Timing of the model. 3.3. Market structure without government A large number of production firms in each period compete for workers. Firms observe the health status of a worker $$h$$ and then, prior to the realization of the health shocks, they compete for workers of type $$h$$ by offering a wage $$w(h)$$ that pools the risk of the health shocks and bundle the wage with an associated health insurance contract (specifying health expenditures $$x(\varepsilon ,h)$$ and an insurance premium $$P(h)$$) that breaks even. Perfect competition for workers of type $$h$$ requires that the combined wage and health insurance contract maximizes the period utility of the household, subject to the firm breaking even.14 In the absence of government intervention, a firm specializing in workers of type $$h$$ then offers a wage $$w^{CE}(h)$$ (where $$CE$$ stands for competitive equilibrium) and a health insurance contract $$\{x^{CE}(\varepsilon,h),P^{CE}(h)\}$$ solving \begin{eqnarray} U^{CE}(h) &=&\max_{w(h),x(\varepsilon ,h),P(h)} u\left( w(h)-P(h)\right) \label{Static1} \\ \end{eqnarray} (1) \begin{eqnarray} s.t. && P(h) =g(h)x(0,h)+(1-g(h))\int_{0}^{\bar{\varepsilon}}f(\varepsilon )x(\varepsilon ,h)d\varepsilon \label{Static2} \\ \end{eqnarray} (2) \begin{eqnarray} && w(h) =g(h)F(h,-x(0,h))+(1-g(h))\int_{0}^{\bar{\varepsilon}}f(\varepsilon )F(h,\varepsilon -x(\varepsilon ,h))d\varepsilon. \label{Static3} \end{eqnarray} (3) Note that the source of risk in the competitive equilibrium is the health status risk associated with $$h$$. This risk stems from the dependence of both wages $$w(h)$$ and health insurance premia $$P(h)$$ on $$h,$$ and these are exactly the sources of consumption risk that government policies to prevent wage discrimination and prohibit prior health conditions from affecting insurance premia are designed to tackle. Also note that, since $$w(h),x(\varepsilon ,h),P(h)$$ are chosen to maximize $$U^{CE}(h)$$ for each health status $$h$$ separately, problem (1)–(3) is equivalent to maximizing $$\sum_{h}\Phi (h)U^{CE}(h)$$ subject to constraints (2) and (3). 3.4. Government policies We now describe how we model the no prior conditions law and the no wage discrimination legislation. 3.4.1. No prior conditions law The purpose of a no prior conditions law is to prevent insurance companies from differentially pricing insurance based upon health status.15 To be successful, regulation must lead to a pooling equilibrium in which all individuals obtain insurance, and obtain it at a price that is independent of $$h$$. The best such regulation in addition ensures that the equilibrium health insurance schedule $$x(\varepsilon ,h),$$ given the constraints, is efficient. We now describe the regulations sufficient to achieve this goal. For the no prior conditions law to be effective, the government must prevent a separating equilibrium in which insurance companies use the health expenditure schedule $$x(\varepsilon ,h)$$ to select their desired health types, given that they are barred from conditioning their premia $$P$$ on $$h$$ directly. Therefore, to achieve pooling in the health insurance market, the government must regulate the health expenditure schedule $$x(\varepsilon ,h).$$ To give the legislation the best chance of being beneficial, we assume that the government regulates the health expenditure schedule$$x(\varepsilon ,h)$$efficiently. Furthermore, since risk pooling is limited if some household types $$h$$ do not buy insurance, in the benchmark model we assume that all individuals are forced to buy insurance. Given this regulation and a cross-sectional distribution of workers by health type, $$\Phi ,$$ the health insurance premium $$P$$ charged by competitive firms, given the set of regulations spelled out above, is determined by \begin{equation} P=\sum_{h}\left[ g(h)x(0,h)+(1-g(h))\int f(\varepsilon )x(\varepsilon ,h)d\varepsilon \right] \Phi (h), \label{NoPriorP} \end{equation} (4) where $$x(\varepsilon ,h)$$ is the expenditure schedule regulated by the government. This schedule is chosen to maximize $$\sum_{h}u(w(h)-P)\Phi (h),$$ with wages $$w(h)$$ determined by (3). 3.4.2. No wage discrimination law The objective of the government is to prevent workers with a lower health status $$h$$, and hence lower productivity, being paid less, to insure them against health status risk. However, if a production firm is penalized for paying workers with low health status $$h$$ low wages, but not for preferentially hiring workers with a favourable health status $$h$$, then a firm can circumvent the no wage discrimination law. Therefore, to be effective, such a law must penalize both wage discrimination and hiring discrimination by health status. In the benchmark model, we analyse the case where the policy is fully effective (by the threat of punishment) in achieving the goal of preventing differential hiring and compensation.16 Under this legislation firms, in their hiring decisions, take as given the economy-wide wage $$w$$ for a representative worker, which, by perfect competition and zero profits, is given by \begin{equation} w=\sum_{h}\left\{ g(h)F(h,-x(0,h))+(1-g(h))\int_{0}^{\bar{\varepsilon}}f(\varepsilon )\left[ F(h,\varepsilon -x(\varepsilon ,h))\right] d\varepsilon \right\} \Phi (h). \label{no-wage w-star} \end{equation} (5) Note that this wage depends upon the health expenditure schedule $$x(.).$$ If firms can discriminate between workers with different health insurance contracts they can effectively circumvent the no wage discrimination law. Therefore, as with the no prior conditions law we need to assume that the government regulates the health insurance market to insure that the no wage discrimination law is fully effective. The regulation again determines the extent of coverage by health type, $$x(\varepsilon,h),$$ subject to the requirement that the offered health insurance contracts break even, either health type by health type (in the absence of a no prior conditions law) or in expectation across health types (in the presence of the no prior conditions law). Under the no wage discrimination law, consumption is given by $$c(h) = w - P(h)$$ in the absence of a no prior conditions law, and by $$c(h)= w - P$$ in its presence, with $$P(h)$$ and $$P$$ in turn given by equations (2) or (4), respectively.17 4. Theoretical Analysis of the Model Given the timing assumptions and the restriction to static health insurance contracts, the model analysis can be separated into a static and a dynamic part. In the static problem, wages and optimal health insurance are chosen for a given distribution $$\Phi$$. The dynamic problem determines health effort choice $$e$$, which leads, through the transition function $$Q(h^{\prime },h;e),$$ to a stochastic update of individual health status and thus a new health distribution $$\Phi ^{\prime }.$$ We first analyse the static problem before turning to the dynamics in section 4.2. 4.1. Static analysis We now characterize static competitive equilibrium allocations. We first establish the efficient benchmark by analysing the solution to the social planner problem, to highlight the source of inefficiency in the competitive equilibrium (insufficient consumption insurance). The key result is that, statically, the combination of both policies provides full consumption insurance in the regulated market equilibrium, and thus restores the full efficiency of the market outcome. 4.1.1. Social Planner’s Problem Given an initial cross-sectional distribution over health status in the population $$\Phi (h)$$, the social planner maximizes ex-ante (prior to the realization of $$h$$) household utility (or, as an alternative interpretation, utilitarian social welfare). The planner’s problem is given by: \begin{equation*} U^{SP}(\Phi )=\max_{x(\varepsilon ,h),c(\varepsilon ,h)\geq 0}\sum_{h}\left\{ g(h)u(c(0,h))+(1-g(h))\int f(\varepsilon )u(c(\varepsilon ,h))d\varepsilon \right\} \Phi (h) \end{equation*} subject to the economy-wide resource constraint: \begin{eqnarray*} &&\sum_{h}\left\{ g(h)c(0,h)+(1-g(h))\int f(\varepsilon )c(\varepsilon ,h)d\varepsilon +g(h)x(0,h)+(1-g(h))\int f(\varepsilon )x(\varepsilon ,h)d\varepsilon \right\} \Phi (h) \\ &\leq &\sum_{h}\left\{ g(h)F(h,-x(0,h))+(1-g(h))\int f(\varepsilon )F(h,\varepsilon -x(\varepsilon ,h))d\varepsilon \right\} \Phi (h). \end{eqnarray*} We summarize the solution to the static social planner’s problem in the following proposition, whose proof follows directly from the first-order conditions and Assumption 4. Proposition 5. The solution to the static social planner’s problem $$\{c^{SP}(\varepsilon ,h),x^{SP}(\varepsilon ,h)\}_{h\in H}$$ satisfies $$x^{SP}(\varepsilon ,h) =\max \left[ 0,\varepsilon -\bar{\varepsilon}^{SP}(h)\right],$$ where the cut-offs $$\{\bar{\varepsilon} ^{SP}(h)\}$$ satisfy $$-F_{2}(h,\bar{\varepsilon}^{SP}(h))=1$$, and consumption is given by \begin{equation} c^{SP}(\varepsilon,h) = c^{SP}=\sum_{h}\left[ g(h)F(h,0)+(1-g(h))\int_{0}^{\bar{\varepsilon}}f(\varepsilon )\left[ F(h,\varepsilon -x^{SP}(\varepsilon ,h))-x^{SP}(\varepsilon ,h)\right] d\varepsilon \right] \Phi (h). \label{StaticFB1} \end{equation} (6) The optimal cut-off $$\{\bar{\varepsilon}^{SP}(h)\}$$ is increasing in $$h,$$ and strictly so if $$F_{12}(h,y)>0.$$ The social planner finds it optimal to provide full consumption insurance not only against adverse health shocks $$\varepsilon$$ but also against bad health status, as consumption $$c^{SP}$$ is independent of $$h.$$ The optimal health expenditure allocation is chosen to maximize the net output contribution $$F(h,\varepsilon -x(\varepsilon ,h))-x(\varepsilon ,h)$$ of a worker with characteristics $$(\varepsilon ,h),$$ which gives rise to the cut-off rule and comparative statics in the proposition. The efficient allocation is depicted in Figure 3. As shown in the proposition, optimal medical expenditures take a cut-off rule: small health shocks $$\varepsilon <\bar{\varepsilon}^{SP}(h)$$ are not treated at all, but larger shocks are fully treated up to the threshold $$\bar{\varepsilon}^{SP}(h).$$ The medical expenditures are displayed in Figure 3b for two different initial levels of health $$h_{1}<h_{2}$$: below the $$h$$-specific threshold $$\bar{\varepsilon}^{SP}(h),$$ expenditures are zero, and then rise one for one with the health shock $$\varepsilon.$$ The determination of the threshold itself is displayed in Figure 3a. It shows that under the assumption that the impact of health shocks on productivity is less severe for healthy households,18 the equilibrium features better insurance for less healthy households, reflected in a lower threshold for $$h_{1}$$ than for $$h_{2},$$ that is, $$\bar{\varepsilon}^{SP}(h_{1})<\bar{\varepsilon}^{SP}(h_{2}).$$ The equilibrium health expenditure policy function leads to a net-of-health-treatment production function $$F(h,\varepsilon -x^{SP}(\varepsilon ,h))$$ shown in Figure 3c. Figure 3 View largeDownload slide Optimal medical expenditures and production. (a) Production function, $$h_1<h_2$$; (b) Optimal medical expenditures; (c) Net-of-health-treatment production. Figure 3 View largeDownload slide Optimal medical expenditures and production. (a) Production function, $$h_1<h_2$$; (b) Optimal medical expenditures; (c) Net-of-health-treatment production. 4.1.2. Competitive equilibrium without and with policy As described in Sections 3.3 and 3.4, the equilibrium wage and health insurance contracts solve, depending on the policy regime $$i\in \{CE,NP,NW,B\}$$ in place \begin{equation} U^{i}(\Phi )=\max_{w(h),x(\varepsilon ,h),P(h)}\sum_{h}u\left( w(h)-P(h)\right) \Phi (h) \label{CEstatobj} \end{equation} (7) subject to equations (2) or (4) for premia and equations (3) or (5) for wage. In the unregulated competitive equilibrium, policy regime $$i=CE,$$ both the health insurance premium and the wage depend on the individual health status $$h$$ of the worker, as in equations (2) and (3). The no prior conditions legislation $$(i=NP),$$ replaces constraint (2) with (4), and the no wage discrimination law $$(i=NW),$$ (3) with (5). Finally, with both laws in place $$(i=B)$$, both wages and health insurance premia (and thus individual consumption) are independent of health status $$h,$$ as constraints (4) and (5) indicate. We now turn to the theoretical characterization of the competitive equilibrium under the different policy configurations, focusing specifically on the sources of the inefficiency of the laissez-faire competitive equilibrium and how the policies correct them. Characterization of the unregulated equilibrium ($$\boldsymbol{i=CE}$$): We now characterize the competitive equilibrium in the absence of policy interventions to isolate the sources of inefficiency in the market solution. Proposition 6. The unique equilibrium health insurance contract is given by $$x^{CE}(\varepsilon ,h) =\max [ 0,\varepsilon -\bar{\varepsilon}^{CE}(h)]$$, where the cut-offs satisfy \begin{equation} -F_{2}(h,\bar{\varepsilon}^{CE}(h))=1 \label{CEfocx} \end{equation} (8)and premia are given by equation (2). Equilibrium wages are determined by equation (3) evaluated at the expenditure profile $$x^{CE}(\varepsilon,h)$$, and consumption is $$c^{CE}(\varepsilon ,h) =c^{CE}(h)=w^{CE}(h)-P^{CE}(h)$$. The intuition for the equilibrium health expenditure schedule is simple. For each health status $$h$$, the household cares only about net compensation $$w(h)-P(h).$$ By comparing equations (2) and (3), we observe that for each $$\varepsilon$$ realization, the marginal cost (in terms of the consumption good) of spending an extra unit of $$x$$ is $$1,$$ and the marginal benefit is $$-F_{2}(h,\varepsilon -x(\varepsilon ,h)).$$ The optimal health expenditure schedule equates the two as long as the resulting $$x(\varepsilon ,h)$$ is interior and features $$x(\varepsilon ,h)=0$$ if for a given $$\varepsilon$$ the benefit of spending the first unit falls short of the cost of $$1.$$ Note that the equilibrium health insurance contract has the flavour of deductibles observed in reality (but here the worker pays for $$\varepsilon < \bar{\varepsilon}^{CE}(h)$$ not with out-of-pocket expenditures but with reduced productivity). It follows directly from Propositions 5 and 6 that the equilibrium health expenditure allocation is efficient: $$x^{CE}(\varepsilon ,h)=x^{SP}(\varepsilon ,h)$$ for all $$(\varepsilon ,h)$$. This last observation shows that in the static case the only source of inefficiency of the competitive equilibrium stems from the inefficient consumption insurance against adverse prior health conditions $$h$$ (which was complete in the social planner’s solution) since aggregate production and thus consumption are identical in the equilibrium and efficient allocation. The equilibrium allocation of health expenditures is efficient because the firm bundles the determination of wages and the provision of health insurance, and thus internalizes the positive effects of health spending $$x(\varepsilon ,h)$$ on worker productivity. Thus the allocation is ex-post (treating individuals with different $$h$$ as different types) Pareto-efficient but features insufficient consumption insurance across health types from an ex-ante perspective (relative to the social planner’s solution, which implements complete consumption insurance against health status, $$c^{SP}(h)=c^{SP}$$). Note that while it follows trivially from our assumptions that the worker’s net pay, $$w^{CE}(h)-P^{CE}(h),$$ is increasing in $$h,$$ it is not necessarily true that his gross wage, $$w^{CE}(h),$$ is increasing in $$h$$ as well, since equilibrium health expenditures are decreasing in health status. In Supplementary Appendix D, we provide a sufficient condition for the gross wage schedule to be monotonically increasing in health status.19 Given the results in this subsection it is plausible to expect that, statically, policies that prevent wages $$w^{CE}(h)$$ and insurance premia $$P^{CE}(h)$$ from depending on health status will restore the full efficiency of the policy-regulated competitive equilibrium. We show next that this is indeed the case, providing a normative justification for the two policy interventions within the static version of our model. Competitive equilibrium with a no prior conditions law ($$\boldsymbol{i=NP}$$): First, consider government intervention in the health insurance market. The objective of the government is to prevent consumption risk induced by health insurance premium risk, replacing (2) with (4). The next proposition characterizes the resulting regulated equilibrium allocation. Proposition 7. The equilibrium health expenditures under a no prior conditions law satisfies, for each $$\tilde{h}\in H$$, $$x^{NP}(\varepsilon ,\tilde{h})=\max [0,\varepsilon -\bar{\varepsilon}^{NP}(\tilde{h})],$$ with cut-offs uniquely determined by \begin{equation} -F_{2}(\tilde{h},\bar{\varepsilon}^{NP}(\tilde{h}))=\frac{\sum_{h}u^{\prime }(w^{NP}(h)-P^{NP})\Phi (h)}{u^{\prime }(w(\tilde{h})-P^{NP})}. \label{NPfocx} \end{equation} (9) The equilibrium wage, for each health status $$h,$$ is given by equation (3), and the health insurance premium is determined by equation (4), evaluated at the no prior conditions expenditure schedule $$x^{NP}(\varepsilon,h)$$. The health expenditure levels are no longer efficient for each health type (as they were in the competitive equilibrium) but provide additional partial consumption insurance against initial health status across health types by adjusting the cut-off levels $$\bar{\varepsilon}^{NP}(h),$$ in the absence of direct insurance against health-induced low wages. As shown in the next proposition, it is efficient to over-insure households with bad health status and under-insure those with good health status, relative to the first-best. Proposition 8. Let $$\tilde{h}$$ be the health status whose marginal utility of consumption is equal to the population average, i.e. for $$\tilde{h}$$, $$u^{\prime }(w(\tilde{h})-P)= \sum_{h}u^{\prime }(w(h)-P)\Phi (h)$$ so that $$-F_{2}(\tilde{h},\bar{\varepsilon}^{NP}(\tilde{h}))=1.$$ Then, $$\bar{\varepsilon}^{NP}(h)<\bar{\varepsilon}^{SP}(h)$$ for $$h<\tilde{h}$$; $$\bar{\varepsilon}^{NP}(h)=\bar{\varepsilon}^{SP}(h)$$ for $$h=\tilde{h}$$; and $$\bar{\varepsilon}^{NP}(h)>\bar{\varepsilon}^{SP}(h)$$ for $$h>\tilde{h}$$. The cut-offs $$\bar{\varepsilon}^{NP}(h)$$ are strictly monotonically increasing in health status $$h$$. This feature of the optimal health expenditure with a no prior conditions law also indicates that mandatory participation in the health insurance contract is an important part of government regulation, since, in the allocation described above, healthy households cross-subsidize the unhealthy in terms of insurance premia and they are given a less generous health expenditure plan (higher thresholds) than the unhealthy. Competitive equilibrium with a no wage discrimination law ($$\boldsymbol{i=NW}$$): Now we turn to the effects of government regulation on the labour market. The allocative consequences of the law are summarized in the following proposition, whose proof follows from the first-order conditions of program (7). Proposition 9. The equilibrium health expenditures under a no wage discrimination law alone satisfies, for each $$\tilde{h}\in H$$, $$x^{NW}(\varepsilon ,\tilde{h})=\max [ 0,\varepsilon -\bar{\varepsilon}^{NW}(\tilde{h})],$$ with cut-offs determined by \begin{equation} -F_{2}(\tilde{h},\bar{\varepsilon}^{NW}(\tilde{h}))=\frac{u^{\prime }(w^{NW}-P^{NW}(\tilde{h}))}{\sum_{h}u^{\prime }(w^{NW}-P^{NW}(h))\Phi (h)}. \label{NWfocx} \end{equation} (10) The equilibrium wage is determined by equation (5) and the health insurance premium is given by, for each $$h,$$ equation (2), evaluated at the health expenditure profile $$x^{NW}(\varepsilon,h)$$. Unlike in the no prior conditions case, we cannot establish monotonicity in the cut-offs $$\bar{\varepsilon}^{NW}({h})$$. Under a no prior conditions law, the regulatory authority partially insures consumption of the unhealthy by allocating higher medical expenditure to them. Under a no wage discrimination law instead, there are two opposing forces. On one hand, a one unit increase in medical expenditure $$P(h)$$ is more costly to the unhealthy, since the marginal utility of consumption is higher for this group. On the other hand, production efficiency calls for higher medical expenditure for the unhealthy, given our assumption of $$F_{12}\geq 0$$ (as was the case for the no prior conditions law). Thus the cut-offs $$\bar{\varepsilon}^{NW}(h)$$ need not be monotone in $$h$$.20 Competitive equilibrium with both policies ($$\boldsymbol{i=B}$$): Combining both a no wage discrimination law and a no prior conditions legislation restores the efficiency of equilibrium, since both policies jointly provide full consumption insurance against bad health $$h,$$ and the assumed efficient regulation of the health insurance market ensures that the health expenditure schedule is efficient as well. This is the content of the next proposition, whose proof follows trivially from the fact that maximizing (7) subject to (4) and (5) is equivalent to the social planner’s problem analysed in Section 4.1.1. Proposition 10. The unique competitive equilibrium allocation in the presence of both a no wage discrimination and a no prior conditions law implements the socially efficient allocation in the static model. 4.1.3. Summary of the analysis of the static model The competitive equilibrium implements the efficient health expenditure allocation but does not insure households against initial health conditions. A no prior conditions law and a no wage discrimination law provide partial, but not complete, insurance against this risk. The health expenditure schedule is distorted when each policy is implemented in isolation, relative to the social optimum, as the government provides additional partial consumption insurance through health expenditures. Only both laws in conjunction implement a fully efficient health expenditure schedule and full consumption insurance against initial health conditions $$h,$$ and therefore restore the static first-best allocation. 4.2. Analysis of the dynamic model Having characterized consumption allocations within a period, we now turn to the full dynamic model. Since there is no aggregate risk, the sequence of cross-sectional health distributions $$\{\Phi _{t}\}_{t=0}^{T}$$ is deterministic. Furthermore, conditional on $$\Phi _{t}$$ today, the health distribution tomorrow is completely determined by the effort choice $$e_{t}(h)$$ of households (or the social planner), so that the cross-sectional health distribution evolves as: \begin{equation} \Phi _{t+1}(h^{\prime })=\sum_{h}Q(h^{\prime };h,e_{t}(h))\Phi _{t}(h). \label{ALOM2} \end{equation} (11) Under each policy, given a sequence of aggregate distributions $$\{\Phi _{t}\}_{t=0}^{T}$$ we can solve the dynamic maximization problem of an individual household for optimal effort decisions $$\{e_{t}(h)_{h\in H}\}_{t=0}^{T}.$$ For this, in this section we assume that the continuation utility after retirement is independent of health status (and normalized to zero): for all $$h\in H$$, $$v_{T+1}(h)=0$$. We relax this assumption in our empirical implementation. A sequence of optimal effort choices in turn implies a new sequence of aggregate distributions through (11). Solving competitive equilibria then amounts to iterating on the sequences $$\{\Phi _{t},e_{t}\}.$$ Within each period, the timing of events follows exactly that of the static problem in the previous section. 4.2.1. Constrained social planner’s problem As a point of comparison for equilibrium allocations (without and with policies), we first again study the solution of a planner choosing constrained-efficient allocations. Statically, the planner can provide full consumption insurance against initial health conditions, as could both policies. In the dynamic model with endogenous effort choice, an unconstrained planner in addition could dictate effort choices, whereas both policies under consideration can impact effort choice only indirectly, through changing the economic consequences of worse health outcomes. Thus it is more instructive for comparison to study a constrained planner’s problem in which the social planner has to respect the intertemporal optimality condition with respect to household effort choice $$\{e_{t}(h)\}$$, given the age- and health-dependent consumption allocation $$\{c_{t}(h)\}$$ chosen by the planner. We think of these constraints as emerging from the inability of the planner to observe household effort choices: if a certain effort $$e_{t}(h)$$ is desired by the planner, it has to be induced by a consumption allocation that makes providing that effort individually rational, given the health-dependent consumption allocations from tomorrow onward. For comparability with the competitive equilibrium and its static contract, we also restrict the social planner to allocations that only depend on current age and health $$(t,h).$$ Let $$V_{t}(h)$$ denote the expected lifetime utility for a household with current age $$t$$ and health status $$h$$, given recursively by \begin{equation*} V_{t}(h)=u(c_{t}(h))-q(e_{t}(h))+\beta \sum_{h^{\prime }}Q(h^{\prime };h,e_{t}(h))V_{t+1}(h^{\prime }), \end{equation*} with exogenous terminal conditions $$\{V_{T+1}(h^{\prime })\equiv 0\}.$$ The social planner solves \begin{eqnarray} \max_{\{c_{t}(h),e_{t}(h)\}} V(\Phi _{0})& = &\sum_{h}V_{0}(h)\Phi _{0}(h) \hspace{1cm} s.t. \\ \end{eqnarray} (12) \begin{eqnarray} \sum_{h}c_{t}(h)\Phi _{t}(h) & \leq & \sum_{h}\left[ g(h)F(h,0)+(1-g(h))\int f(\varepsilon )\left[ F(h,\varepsilon -x(\varepsilon ,h))-x(\varepsilon ,h)\right] d\varepsilon \right] \Phi_t (h)\qquad \label{aggrcsp} \\ \end{eqnarray} (13) \begin{eqnarray} q^{\prime }(e_{t}(h)) &= & \beta \sum_{h^{\prime }}\frac{\partial Q(h^{\prime };h,e_{t}(h))}{\partial e_{t}(h)}V_{t+1}(h^{\prime }), \label{effortSP} \end{eqnarray} (14) and the general law of motion given in (11). Equation (13) represents the aggregate resource constraint, where $$x(\varepsilon ,h)=x^{SP}(\varepsilon ,h)$$ is the efficient health expenditure schedule characterized in Section 4.1.1. The second constraint is the incentive constraint on effort. It equates the marginal utility cost of effort today, $$e_{t}(h),$$ to the marginal benefit of better health from tomorrow on, given a consumption allocation chosen by the planner and encoded in $$\{V_{t+1}(h^{\prime })\}$$. Equation (14) demonstrates the trade-off for the constrained social planner also present in the evaluation of both policies. Statically, it is optimal for the planner to provide full consumption insurance. Although she can certainly implement such an allocation, it would lead to identically zero effort in all periods; see equation (14). Since the marginal cost of providing effort at $$e_{t}(h)=0$$ is zero (Assumption 1) and the benefit for health transitions and thus net production and average consumption is positive (on account of Assumptions 3 and 4), starting from the full consumption insurance and zero effort allocation, a marginal increase in effort is welfare improving (since the consumption insurance losses are second order when starting at the full consumption insurance allocation). More formally, the constrained efficient allocation is characterized in the following proposition, proved in Supplementary Appendix B. Proposition 11. The constrained efficient allocation $$\{c_{t} (h) ,e_{t} (h)\}$$ has zero effort in the last period and full consumption insurance in the first period: $$e_{T} (h) =0$$ and $$c_{0} (h) =c_{0} , \forall h\text{.}$$ Furthermore, assume $$Q$$ is $$iid$$21 and positive: $$Q (h^{ \prime } ;h ,e) =Q (h^{ \prime } ;\tilde{h} ,e) >0 , \forall (h^{ \prime } ,h ,\tilde{h} ,e)\text{.}$$ Then effort is positive in all periods but the last: $$e_{t} (h) >0 , \forall h ,t <T\text{,}$$ and has imperfect consumption insurance with consumption $$c_{t} (h)$$ strictly increasing in $$h$$ in all but the first period. This result will be in contrast to the outcome under both policies (see Proposition 12), which features full consumption insurance and zero effort, and therefore results in an inefficient allocation. 4.2.2. Dynamic competitive equilibrium without and with policy In the presence of wage and health insurance policies, households of different health types $$h$$ interact, since the population health distribution $$\Phi _{t}$$ determines the pooled wage and health insurance premium and the health expenditure cut-off $$\bar{\varepsilon}^{i}(h;\Phi _{t}).$$ The dynamic program in policy regime $$i\in \{CE,NW,NP,B\}$$ is: \begin{equation} v_{t}^{i}(h;\Phi ) = U^{i}(h,\Phi )+\max_{e_{t}(h)}\left\{ -q(e_{t}(h))+\beta \sum_{h^{\prime }}Q(h^{\prime };h,e_{t}(h))v_{t+1}^{i}(h^{\prime },\Phi ^{\prime })\right\} \hspace{1cm} {\rm where} \label{dynce1} \end{equation} (15) \begin{eqnarray} U^{i}(h,\Phi )&=&\max_{x^{i}(\varepsilon ,h,\Phi ),w^{i}(h,\Phi ),P^{i}(h,\Phi )}u(w^{i}(h,\Phi )-P^{i}(h,\Phi )) \label{dynce2} \\ \end{eqnarray} (16) \begin{eqnarray} P^{i}(h;\Phi ) &=&(1-g(h))\int_0^{\bar{\varepsilon}} f(\varepsilon )x^{i}(\varepsilon ,h;\Phi )d\varepsilon, i\in \{CE,NW\} \label{dynce3} \\ \end{eqnarray} (17) \begin{eqnarray} P^{i}(\Phi ) &=&\sum_{h}\left[ (1-g(h))\int_{0}^{\bar{\varepsilon}}f(\varepsilon )x^{i}(\varepsilon ,h;\Phi )d\varepsilon \right] \Phi (h)\text{ if }i\in \{NP,B\} \label{dynce4} \\ \end{eqnarray} (18) \begin{eqnarray} w^{i}(h;\Phi ) &=&g(h)F(h,0)+(1-g(h))\int_{0}^{\bar{\varepsilon}}f(\varepsilon )F(h,\varepsilon -x^{i}(\varepsilon ,h;\Phi ))d\varepsilon \text{ if }\{CE,NP\} \label{dynce5} \\ \end{eqnarray} (19) \begin{eqnarray} w^{i}(\Phi ) &=&\sum_{h}\left\{ g(h)F(h,0)+(1-g(h))\int_{0}^{\bar{\varepsilon}}f(\varepsilon )F(h,\varepsilon -x^{i}(\varepsilon ,h;\Phi ))d\varepsilon \right\} \Phi (h), i\in \{NW,B\} \label{dynce6}\qquad{} \end{eqnarray} (20) and $$x^{i}(\varepsilon ,h;\Phi )$$ is the equilibrium health expenditure allocation from Section 4.1.2 under policy regime $$i.$$ Section 4.1.2 showed that the equilibrium health expenditure allocation was given by the simple cut-off rule $$x^{i}(\varepsilon ,h;\Phi )=\max \left[ 0,\varepsilon -\bar{\varepsilon}^{i}(h,\Phi )\right]$$, with policy-dependent cut-offs in equations (8), (9) and (10). These cut-offs depend on the cross-sectional health distribution $$\Phi$$ through average marginal utilities of the economy. Thus, how $$x^{i}(\varepsilon ,h;\Phi )$$ and $$w^{i}(h;\Phi ),P^{i}(h;\Phi )$$ and consumption $$c^{i}(h,\Phi )$$ depend on the cross-sectional health distribution $$\Phi$$ varies across policy regime $$i$$. Note that for the household to solve its dynamic programming problem, it only needs to know the sequence of potentially $$h$$-contingent wages and health insurance premia $$\{w_{t}^{i}(h),P_{t}^{i}(h)\},$$ but not necessarily the sequence of d