Health insurance and entrepreneurship

Health insurance and entrepreneurship Raquel Fonseca Université du Québec à Montréal, CIRANO and RAND Vincenzo Quadrini University of Southern California February 11, 2015 VERY PRELIMINARY AND INCOMPLETE. PLEASE DO NOT QUOTE Abstract We will build a general equilibrium model with occupational choice, health insurance choice and size decisions. Some agents have the ability to run a business and become employers while others work only for a wage. Agents also make decisions about health insurance. In particular, entrepreneurs have to choose whether to offer health insurance to the employees, and wage workers have to choose whether to work for employers offering health insurance or for employers that do not offer health insurance. The model is calibrated to replicate some key empirical moments of the data. Workers without coverage have a better health status, lower assets and lower income. The model predicts that uninsured workers are predominantly poorer and employed by smaller firms. Furthermore, since young agents are on average healthier and they have lower incomes, the model also predicts that young workers have less coverage than older workers. The pool of insured workers contains workers with lower than average health status. This composition bias follows from the fact that individual health status are not observable and premiums cannot be dependent on individual characteristics. 1

1 Introduction The major concern at the center of the political debate over the reform of the health care system is that a large percentage of workers do not have health insurance coverage. According to some recent estimates, about 18 percent of nonelderly Americans are uninsured. See, for example, Gruber (2008). Most of the covered workers get insurance through their employers. In general, workers employed by companies that sponsor insurance tend to subscribe to the program and get coverage. Instead, workers without insurance tend to be employed by employers that do not sponsor insurance. Therefore, in order to understand why a large fraction of workers are without health insurance we have to understand why certain employers do not provide insurance and why some workers accept or choose to work for these employers. Understanding the insurance choices made by employers and employees is central to understanding the economic implications of health care reform, in particular a reform that will extend coverage to a larger percentage of workers. Thus far, the main political debate has focused on the trade-off between the financial cost of a particular reform versus the moral imperative or societal benefit of having most workers shielded from unexpected health needs. Less consideration has been given to the macroeconomic consequences of health care reform, especially for small and medium size businesses. The goal of this paper is to investigate how reforms extending health insurance coverage affect the employment choice of businesses and, by aggregation, the whole economy. We consider a general equilibrium model where agents face medical expense shocks and they differ in the likelihood of incurring medical expenses (health status). Some agents have the ability to run a business and become employers while others work only for a wage. In addition to the saving and investment decisions made by agents in typical entrepreneurial models with financial constraints, agents also make decisions about health insurance. In particular, entrepreneurs have to choose whether to offer health insurance to the employees, and wage workers have to choose whether to work for employers offering health insurance (in which case they will be covered) or for employers that do not offer health insurance (in which case they will not be covered). An insurance program sponsored by an employer allows for higher coverage for the same premium or lower per-worker premium for the 2

same coverage, but it also involves a fixed cost. Therefore, there are economies of scale in providing insurance. As a result, only employers (firms) that are relatively large find it optimal to provide insurance. Thus, the equilibrium is characterized by a fraction of workers employed by firms that do not provide insurance while the remaining fraction is employed by firms that provide insurance. Workers without coverage have a better health status, lower assets and lower income. Therefore, as in the data, the model predicts that uninsured workers are predominantly poorer and employed by smaller firms. Furthermore, since young agents are on average healthier and they have lower incomes, the model also predicts that young workers have less coverage than older workers. Another prediction of the model is that the pool of insured workers contains workers with lower than average health status. This composition bias follows from the fact that individual health status are not observable and premiums cannot be dependent on individual characteristics. The model is calibrated to replicate some key empirical moments of the data. The goal is to use the calibrated model to study several types of health care reforms such as a mandatory requirement for employers to provide health insurance or the subsidization of health insurance funded by alternative types of taxation schemes. For the moment, however, the analysis is limited to constructing the model. Before describing the model, however, we provide some empirical observations that motivate the theoretical analysis. 2 Literature review There are several studies that explore health coverage and various aspects of employment using structural models. In particular, Jeske and Kitao (2009) find that the removal of the health insurance subsidy by employer-provided groups would result in a deterioration of the health quality of such groups and a rise in premiums, which would trigger a decline in insurance coverage and, consequently, a decline in social welfare. Other papers studying the employment-based health insurance system at the macroeconomic level are Dey and Flinn (2005), DeLoach and Platania (2013), Fang and Gavazza (2011), Hsu (2013), Hansen et al. (2014) and Pashchenko and Porapakkarm (2013). Entrepreneurship has been extensively studied in different fields such as finance, corporate business, and labor economics. A few 3

studies also attempt to address the macroeconomic aspects of entrepreneurship at both the empirical and theoretical levels (see Quadrini (2009) for a survey). Some of these studies try to understand the determinants of individuals occupational choice and analyze macroeconomic issues. All the studies reviewed above which are based on structural models do not consider explicitly the role played by entrepreneurs in shaping the health insurance coverage we observe in equilibrium as well as the impact of health reforms on entrepreneurship. However, a recent literature focus on how the health insurance coverage affect firm size as Brugemann and Manovskii (2010) and Aizawa and Fang (2013). All the studies reviewed above which are based on structural models do not consider explicitly the role played by entrepreneurs (smaller firms) in shaping the health insurance coverage we observe in equilibrium as well as the impact of health reforms on entrepreneurship. Given the strong association between the characteristics of firms and health insurance coverage shown by the data, it becomes essential to link the decisions of entrepreneurs or firms to the decisions of workers. The goal of this paper is to link two existing literatures. The first literature studies the role of entrepreneurship for the industry dynamics and macroeconomic performance but ignores the importance of health insurance policies. The second literature does focus on the issue of health insurance but ignores entrepreneurship. By linking these two literatures, this project will provide new and more complete insights about the micro and macroeconomic effects of health policy reforms. 3 Some Evidence We first present some cross-sectional facts about health insurance and entrepreneurship. 3.1 Cross-sectional facts Figure 1 shows that 17 percent of the nonelderly population in the US has not insurance coverage. The data is for the year 2006 but this number is quite stable over different years. Of those insured, the predominant source of coverage, almost three-fourth, comes from employerssponsored programs. Public plans, often government-sponsored programs such as Medicaid, provides insurance for 17 percent of nonelderly 4

population while only seven percent comes from individual purchased plans. Of course, if employers-sponsored plans are the most popular form of coverage, the provision of insurance through employers must has some advantages Distribution over individual of Health Insurance purchases. CoverageSomething to which we will comment again Nonelderly below. Population, 2006 (EBRI Issue Brief No. 310) 0.7 0.6 0.60 0.5 0.4 0.3 0.2 0.17 0.17 0.1 0.07 0 Employment-based coverage Individually purchased Public coverage No health insurance Figure 1: Distribution of health insurance coverage for nonelderly population, 2006. Source: EBRI Issue Brief No. 310. Next we focus on different types of workers, specifically self-employed versus wage and salary workers. Figure 2 shows that the percentage of self-employed workers without coverage is significantly higher than wage workers. In fact, in the age group 18-64, 26 percent of self-employed workers are without insurance, compared to 17 percent for wage workers. Unsurprisingly, among the self-employed who are insured, a larger percentage of coverage comes from individually purchased plans as opposed to employment-based coverage. Figure 2 also shows that the percentage of uninsured workers is much larger for those working in the private sector (20 percent) compared to those working in the public sector (6 percent). So far we have seen that lack of coverage is especially important for individuals working in the private sector and especially those who work for themselves as opposed to wage workers. Next we concentrate on the private sector and show how the rate of coverage changes with the employer size. Figure 3 reports the type of coverage for different size classes of employers. In firms with less than 10 employees, 35 percent 5

Health insurance coverage of workers by type of coverage, Ages 18-64 100% 6% 90% 26% 17% 7% 3% 20% 80% 70% 6% 7% 4% 7% 5% 60% 50% 40% 30% 20% 19% 49% 71% 85% 69% Uninsured Public Individually purchased Employment-based coverage 10% 0% Self-employed Type of workers Wage and salary workers Public sector Private sector Employment sector Figure 2: Health insurance coverage of workers by type of coverage, Ages 18-64, 2006. Source: EBRI Issue Brief No. 310. of workers are uninsured. On the opposite side, workers employed in firms with at least 1,000 employees, the percentage of uninsured is only 13 percent. Therefore, there is a strong relation between the size of the employers and the level of coverage. This fact is somewhat related to the observation that self-employed workers have a lower rate of coverage than wage workers since self-employed are mostly owners of relatively small firms. We can now come back to the early observation that the primary source of coverage comes from employer-sponsored programs. As noted above, because employer-sponsored plans are more prevalent than individually purchased plans, the former must have some advantage over the latter plans. Tax benefits deriving from the deductability of insurance premiums is obviously an important element. On top of this, however, there must also be some other factors. One common explanation is that firms pool large number of workers and this has two effects: it reduces the per-insured administrative cost and it mitigates the adverse selection problem for health insurers. This suggests the presence of economies of scale which makes the provision of health insurance especially attractive for large employers. As we will see, we will take this approach in the construction of the theoretical model. 6

Health insurance coverage of workers in private sector Ages 18-64 100% 90% 80% 35% 29% 21% 7% 16% 6% 3% 12% 13% 7% 7% 3% 3% 70% 8% 5% 60% 8% 6% 50% 40% 30% 20% 10% 48% 57% 68% 74% 78% 77% Uninsured Public Individually purchased Employment-based coverage 10% 0% <10 10-24 25-99 100-499 500-999 >999 Firm size (number of employees) Figure 3: Health insurance coverage of workers in private sector Ages 18-64, 2006. Source: EBRI Issue Brief No. 310. Another interesting observation is that the percentage of uninsured individuals declines with family income. As shown in Figure 4, among individuals with family incomes below 20,000 dollars, more than 30 percent are without insurance. However, if we consider individuals with family income of at least 75,000 dollars, the percentage of uninsured is only 7 percent. The percentage of uninsured individuals in low income families would probably be even higher in absence of public plans often sponsored by the government. The first four pictures suggest that in order to understand why so many individuals do not have coverage requires us to understand why small firms and their owners (self-employed) choose not to provide insurance. At the same time we also need to understand why individual working for small firms are willing to work for them even if they do not provide insurance. Finally, why workers without insurance are poorer? As we will see, one approach we take to capture the lower coverage of self-employed workers and of workers employed in smaller firms is that health insurance tends to be more expensive for small employers. To corroborate this hypothesis, we now provide some evidence in favor of this hypothesis. 7

Health Insurance Covergage of Nonelderly population by Family Income, 2006 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% 35% 33% 45% 10% 10% 38% 9% 20% <$10,000 $10,000- $19,999 29% 26% 8% 37% $20,000- $29,999 23% 20% 7% 50% $30,000- $39,999 Family income 18% 15% 6% 61% $40,000- $49,999 13% 10% 6% 71% $50,000- $74,999 7% 6% 5% 82% >$74,999 Uninsured Public Individually purchased Employment-based coverage Figure 4: Health Insurance Covergage of Nonelderly population by Family Income, 2006. Source: EBRI Issue Brief No. 310. 4 The model 4.1 Overview 4.1.1 Households The economy is populated by a continuum of agents who lives for a total of I periods. They are born and become active at age 1, work for N < I periods and retire at age N + 1. They maximize the lifetime utility I E t β i 1 (c t+i 1q t+i 1 ) 1 σ, 1 σ i=1 where c is consumption and q is a variable that captures the quality of health services as described below. This specification of the utility function implies wealth effects in the quality of health services. The agent starts the period with health status h t. During the period he/she incurs medical expenses m t. Medical expenses are stochastic with probability distribution g(h t ; m t ). The probability distribution of m t for agents with higher health status stochastically dominates the distribution for agents with lower h t. There are health insurance companies that provide insurance plans 8

competitively. They have access to better quality medical services than individuals. Alternatively we could assume that they have access to lower cost for medical services than individuals for the same quality. Therefore, purchasing health insurance has two benefits: it eliminates uncertainty about medical expenses and provides higher quality health services. Given the health shock m t, the quality of health services is q t = e γ(1 λt)mt, where λ t is a dummy variable that takes the value of 1 if the agent is insured and 0 otherwise. The parameter γ captures the importance of health quality. The idea is that insured agents receive better health services and the utility impact of a medical expense shock is smaller. Since the choices of consumption and health insurance are made at the beginning of the period before knowing the medical expense m t, it will be convenient to define the expected value of current utility as U(c t, λ t, h t ) = c1 σ t 1 σ E te γ(σ 1)(1 λt)mt, where the expectation is over medical expenses. The expected utility is also a function of the health status because the probability distribution of medical expenses m t depends on h t. This specification makes clear that the value of health insurance increases as the health status deteriorates and the level of consumption increases. At each point in time, individuals are characterized by the following exogenous states: Age: Indexed by i it evolves deterministically from 1 to I. Earning ability: This is the efficiency units of labor supplied to the market for a wage. Denoted by ε i,t = µ i ξ t, is the product of two components. The first component, µ i, evolves deterministically with age and captures the age profile of earnings. The second component, ξ t, is stochastic and follows a first order Markov process which is independent of age. If the agent chooses to become an entrepreneur, the efficiency units of labor are first used in the owned business. Entrepreneurial ability: The output produced by undertaking an entrepreneurial activity is y t = z t f(k t, l t ), where z t is the entrepreneurial ability, k t is the input of capital and l t is the input of labor. The entrepreneurial ability z t evolves stochastically 9

according to a first order Markov process which is independent of the process for the earning ability. Cost of sponsoring health insurance: As described below, agents who become entrepreneurs choose whether to sponsor health insurance to their employees. This involves a fixed cost κ t. We allow this cost to differ across firms so that also some small firms will sponsor insurance. One explanation why this cost is smaller for certain firms could be the fact that the firms have access to some form of cooperative agreement among a large groups of employers. This is captured by assuming that κ t is stochastic and follows a Markov process. Health status: Denoted by h t, it evolves stochastically according to a Markov process. The health status determines the probability distribution of incurring medical expenses. The agents who have an investment opportunity (z t > 0) may choose to become entrepreneurs. If they choose to do so, they hire labor (including their own) and decide whether to sponsor health insurance for their employees. The insurance cost has two components: a fixed cost κ t and a (per-worker) premium p. When the employer sponsors health insurance, workers have access to cheaper health plans since they effectively share the fixed cost. To simplify the presentation we assume that the premium p is paid by workers. This is without loss of generality if we assume that the firm is unable to select workers with different skills. In fact, if the firm cannot control the skill composition of the hired workers, it is irrelevant whether the premium is paid by the worker or the firm. The firm, however, does pay the fixed cost κ t, which is necessary to make the health insurance accessible at the premium p. 1 Wage rates are determined in equilibrium and they depend on whether the firm sponsors insurance. The wage rate paid by firms who do not sponsor health insurance is w t (0) while the wage rate offered by firms sponsoring insurance is w t (1). In choosing whether to sponsor or not insurance, employers compare the benefit from making high 1 If the premium is paid by the firm and the firm could select workers with different skills, then it would prefer to hire high skilled workers. In fact, by choosing workers with higher skills, the firm has to hire fewer workers and pay a smaller number of premiums. What this implies is that in equilibrium firms will offer different wage rates (per unit of labor efficiency) to workers with different skills. To avoid this complication we assume that the firm cannot discriminate on the rate paid on each efficiency unit of hired labor. 10

quality insurance more affordable and paying a lower wage rate, with the payment of the fixed cost κ t. Therefore, the provision of health insurance becomes attractive only if the firm operates above a certain scale. It is important to point out that, even if firms that sponsor insurance pay a lower wage rate, the per-worker wage is not necessarily lower. As we will see, in equilibrium there will be a selection of high skilled workers employed in firms that sponsor insurance. Thus, the total wages received by these workers, w t (1)ε, are higher than the wages earned by workers without health insurance. There is borrowing and lending but the debt must be collateralized by capital. Denoting by R t the gross interest rate, the cost at time t of a claim to b t units of consumption goods in period t + 1 is b t /R t. A negative b t denotes borrowing. The collateral constraint takes the form b t /R t αk t. This constraint implies that agents can borrow only against a fraction of their investment k t. The parameter α captures the financial markets tightness. 4.2 Optimization problem Denoting by a t the beginning-of-period net worth, the agent chooses consumption, capital and bonds, facing the budget constraint a t = c t + k t + b t R t. (1) In addition to the portfolio choice, the agent makes several decisions. First, he/she chooses whether to work for an employer who sponsors health insurance or for an employer who does not provide the insurance option. The employment choice is captured by the dummy variable λ t {0, 1}. Second, an agent that chooses entrepreneurship, that is, k t > 0, has to decide whether to offer health insurance. This choice will be denoted by χ t {0, 1}. All these choices are made before knowing the medical expenses m t. Given these choices and the ex-post medical expenses, the net worth at the end of the period is x t = b t +F (z t, k t, l t ) l t w(χ t ) χ t κ t +ε i,t w t (λ t )(1 τ) pλ t (1 λ t )m t, (2) where F (z t, k t, l t ) = (1 δ)k t + z t f(k t, l t ) is the gross return from production and τ is the tax rate paid by workers. The tax revenues are used to finance social security and medical expenses incurred by senior residents covered by Medicare as specified below. 11

Notice that the only uncertainty in affecting the next period net worth are the medical expenses m t but only if the agent is not insured, that is, if he/she chooses λ t = 0. The uncertainty in medical expenses implies that the agent may end up with low net worth and defaults on the medical bill. This happens if the next period net worth falls below a minimum value a min. Therefore, the next period net worth after the default decision is equal to a t+1 = max{a min, x t }. The lower bound a min is the retained net worth in case of default. As we will see, a min determines how costly is to default: lower is the value of a min and lower is the incentive to default. Denote by s = (ε, z, κ, h) the exogenous states. The states are the earning skills, ε, the entrepreneurial skills, z, the cost of sponsoring insurance, κ, and the health status, h. We will use the variable e to denote the occupational decision: worker (e = 0) and entrepreneur (e = 1). An agent that chooses to become an entrepreneur will be insured only if he/she chooses to sponsor insurance for all employees. Therefore, λ = χ. The optimization problem faced by an agent with states s and a in the pre-retirement age i = 1,.., N, can be written as V i (s, a) = max e,χ,λ,b,k,l U(c, λ, h) + βe m,s V i+1 ( s, max{x, a min } ) (3) b R αk subject to c = a k b R [ ] x = b + e F (z, k, l) lw(χ) χκ + εw(λ)(1 τ) pλ (1 λ)m λ = χ. The problem is subject to the borrowing constraint, the budget constraint, and the law of motion for the next period net worth before the decision to default. If the end of period net worth x is smaller than the lower bound on assets, the agent defaults and gets a discount on his liabilities. 2 The problem is also subject to the constraint λ = χ 2 We could also assume that default implies a utility cost which, however, is not essential 12

which requires an entrepreneur to be covered if and only if he/she provides insurance to all employees. Notice that the expectation in the objective function is over (current) medical expenses and next period exogenous states. For an agent in the retirement age i = N +1,.., I, the optimization problem simplifies to V i (s, a) = max b 0 U(c, 1, h) + βe s V i+1(s, a ) (4) subject to c = a b R a = b + y, where y is social security income paid by the government. Since Medicare covers the medical expenses of all retirees, seniors do not face any uncertainty about m. Social security payments and health expenses for seniors are financed with taxes paid by workers as we have seen above. 4.3 Simplification of the optimization problem Agents in working ages choose 3 continuous variables: b, k, and l. However, because the productivity z is known when the agent chooses l, the input of labor is simply a function of k. In particular, given k, the input of labor solves the static first order condition F l (z, k, l) = w(χ) where the marginal productivity of labor is equalized to the wage rate. Therefore, we can write the input of labor as a deterministic and static function of capital and the insurance dummy, i.e. l(k, χ). The dummy variable χ is important since the firm has to pay different wage rates depending on the sponsoring of insurance. We can further simplify by taking advantage of the fact that z is known when the agent chooses k. Define the agent s gross savings as n = k + b/r. The budget constraint can then be written as c = a n. Once we know n, we can use the first order condition for k, together with the borrowing limit, to decompose n in k and b. for the key properties of the model. 13

Define k(χ) the optimal input of capital in absence of the borrowing limit. This solves the first order condition F k (z, k(χ), ) l( k(χ), χ) = R. Once we know n, the input of capital is determined as k(χ) if n (1 α) k(χ) k(n, χ) =, if n < (1 α) k(χ) n 1 α and the bond is equal to (n k(χ))r if n (1 α) k(χ) b(n, χ) = if n < (1 α) k(χ) αnr 1 α Because the input of labor is a function of k, which in turn is a function of n and χ, we can also write it as l(n, χ). By further taking into account that λ = χ, the optimization problem solve by an agent in the working age i = 1,.., N can be reformulated as ( ) V i (s, a) = max U(a n, χ, h) + βe m,s V i+1 s, max{x, a min } (5) e,χ,n subject to [ ] x = b(n, χ) + e F (z, k(n, χ), l(n, χ)) l(n, χ)w(χ) χκ. + εw(χ)(1 τ) χp (1 χ)m. The optimization problem is now reduced to the choice of three variables: whether to be a worker or an entrepreneur, e; whether to sponsor insurance as an employer, χ (or to work for an employer that provides insurance if the agent chooses to become a worker); and the gross savings, n. Therefore, we have only one continuous decision variable and two discrete decisions. For a retired agent the optimization problems remains as described earlier. Notice that, even if the agent does not forgone the labor income when choosing to become an entrepreneur, this does not imply that all agents with z > 0 choose to become entrepreneurs. The availability of health insurance plays an important role. In particular, if the agent is unable to operate a business at a sufficiently large scale, the firm 14

may be unable to offer health insurance and the entrepreneur will not be covered. This may discourage the creation of the business. This is one of the mechanism through which health insurance policies may affect entrepreneurship. 4.4 Steady state equilibrium A steady state equilibrium will be characterized by a distribution of agents M i (s, a) with i = 1,.., I, by a tax rate τ and by three prices: the gross interest rate, R, the wage rate paid by employers who do not sponsor insurance, w(0), and the wage rate paid by employers who sponsor insurance, w(1). The tax rate balances the government budget and the prices clear the markets for bonds and labor. 5 Quantitative analysis Agents live for a total of 13 periods where a period lasts five years. They become active at age 21-25, retire at age 66-70 and die at age 86. Consistent with the five-year period we set the discount factor to β = 0.8. The relative risk aversion parameter is set to σ = 1.5. For the moment we ignore the importance of health care quality and set γ = 1. The earning process ε i,t = µ i ξ t results form the product of two component. The first component, µ i, depends deterministically only on age while the second component, ξ t is stochastic and age independent. The first component is calibration to replicate the average earning profile of different age cohorts as reported in Diaz-Gimenez, Quadrini, Ros-Rull, and Rodrguez (2002). The second component follows a zero-mean first order autoregressive process with autoregressive parameter of 0.9 and standard deviation 0.218. The process is then approximated with a 9 states Markov chain using Tauchen s method. The production function for entrepreneurs is specified as y t = z t (kt θ lt+1 1 θ )ν. The parameters θ and ν are set to 0.35 and 0.975 respectively. Therefore, the production function is close to constant returns. The depreciation rate is δ = 0.27. The productivity variable can take two values, z t {0, 0.1}. The first realization, being zero, characterizes agents who do not have entrepreneurial skills. Agents with entrepreneurial skills, z t > 0, may or may not choose to become entrepreneurs depending on their particular states. The specific value 15

of z t is not important and different values will simply rescale of the whole economy. The entrepreneurial skills evolve according to the transition matrix [ ] 0.97 0.03 P z =. 0.05 0.95 The borrowing limit is specified as b t αk t with α = 0.8. Therefore, agents can borrow up to 80 percent of the invested capital. We now turn to the description of the health status and shocks. First we assume that there are three health status h. Higher is the health status h and lower is the probability of incurring medical expenses m. We restrict the medical expenses to take the single value m = 0.33. Therefore, all agents incur the same medical expenses but with different probabilities. This is similar to having different values of m with proper rescaling of probabilities. The health status evolves according to the transition probability matrix P h = 1.000 0.000 0.000 0.212 0.788 0.000 0.000 0.212 0.788 and the probabilities of incurring the medical expenses, for each health status, are (0.0528, 0.0305, 0.0250). In deriving these numbers we have impose several restrictions. First, the lowest health status is permanent. Second, agents with the highest (best) health status cannot transit directly to the worst status. Third, once the agent looses the highest health status, he or she will never return to it. With these assumptions we are able to formalize the idea that the health status deteriorates on average over the life cycle. Given the restrictions, we have 5 parameters to pin down. Two parameters in the transition matrix and three probabilities of incurring medical expenses conditional on each health status. The five parameters are chosen to minimize the sum of squared differences between the life cycle medical expenses observed in the data and those generated by the model. The expenses in the data and those generated by the model are plotted in Figure 5. In the steady state equilibrium the medical expenses are about 11 percent the total value of wages. The cost of sponsoring health insurance κ is assumed to take only one value which we set to 0.05. This is slightly less than half the, 16

Figure 5: Medical expenses for different age groups: PSID data and model. Data normalized by the average per-capital expenses generated by the model. average wages. Together with the net worth exemption a min in case of default (minimum consumption), this parameter is important in affecting the proportion of agents without insurance. We set a min = 0.00025 which is a very small percentage of per-capita income. With this number the fraction of uninsured workers is close to 17 percent. We now describe the government policy. Government pays for pensions and medical expenses of retirees (social security and medicare). The medical expenses are exogenous and therefore they are not under the control of the government. Pensions are the same for all retirees and they are set to 75 percent the average (steady state) earnings of agents in the last period of working life (61-65 cohort). Given the whole government expenses for pensions and medicare, the tax rate on wage incomes is determined to balance the budget and it is equal to 38.75 percent. Finally we have to fix the initial distribution over the various exogenous states for newborn agents. Newborns start with zero assets and with the highest health status. Furthermore, 10 percent of them have entrepreneurial skills. Of course this does not mean that they become entrepreneurs immediately since they do not have assets. Finally, the initial distribution of the age-independent component of 17

earnings ξ is assumed to be uniform. Starting in the next period it evolves according to the Markov process described above. 5.1 Equilibrium properties Table 1 reports some statistics for the steady state equilibrium. About 11.4 percent of agents in the pre-retirement age become entrepreneurs and 88.6 percent are wage workers. Among entrepreneurs 24.1 percent are uninsured while the percentage of uninsured workers is only 13 percent. The wage rate (per-efficiency unit of labor) is about 10 percent higher for uninsured workers. However, this does not mean that uninsured workers earn higher labor income. In fact, as we will see, uninsured workers are younger (healthier) and therefore, they earn lower wages. Since some of the agents default on their medical expenses, the health insurance premium is increased to cover the unpaid medical costs. In the model a significant fraction of agents do default on their medical expenses. As a result of this the insurance premium includes a mark-up of about 10 percent. Table 1: Steady state statistics. Fraction of wage workers 0.8861 Fraction of entrepreneurs 0.1139 Uninsured wage workers 0.1304 Uninsured entrepreneurs 0.2411 Health insurance premium 0.0123 Insurance mark-up 0.0012 Wage rate insured 0.1067 Wage rate uninsured 0.1132 Interest rate 0.1692 Tax rate on wages 0.3875 The first panel of Figure 6 plots the occupational distribution over the life-cycle. All newborn agents choose to be workers. This is a consequence of the borrowing limit. Even though some of the newborn agents have the entrepreneurial skills, they are unable to borrow to finance the input of capital which effectively prevents them from becoming entrepreneurs. As they advance in age, some of them choose 18

to become entrepreneurs later in life thanks to their savings. result, the fraction of entrepreneurs increases with age. As a Figure 6: Occupation, health insurance and incomes over the life-cycle. The second panel of Figure 6 plots the percentage of workers and entrepreneurs that are uninsured. As can be seen, this percentage is very high for young agents but then it declines later in life. As will be shown shortly, the main mechanism that generates this life-cycle pattern is related to the health status composition of the population. As agents become older, the health status tends to deteriorate on average. Consequently, health insurance becomes more valuable for older cohorts (since the insurance premium does not depend on age). The last two panels show that the wage income increases on average with age. This follows directly from the calibration of the skill process where in addition to the stochastic component, ξ t, there is a deterministic component that increases with age, µ i. This feature is important for the model to generate another empirical regularity, that is, the fact that the percentage of workers with health insurance coverage increases with income and wage earnings. In fact, since young workers have lower incomes and fewer have health insurance, we have 19

that health coverage increases with income. We now move to Figure 7 which plots the health status distribution of workers and entrepreneurs over the life-cycle, separately for those with and without health insurance. As we conjectured above, agents without health insurance have higher health status, and therefore, they face lower probability of incurring medical expenses. Since the insurance premium is not differentiated by health status, agents who expect high medical expenses (low health status) prefer jobs with sponsored insurance. Figure 7: Health status distribution over occupation and insurance coverage over the life-cycle. We conclude this section by observing that the model also replicates the firm-size dependence of health insurance coverage. Although in a very stylized way, small firms do not provide insurance because of the fix cost κ. However, once the production scale reaches a certain level, it becomes optimal to pay the fix cost and hire at the lower wage rate. Still, the earnings of workers employed in large firms are higher than the earnings of those employed by small firms. This is because the average skills of workers employed by larger firms is higher. 20

6 Conclusion We have developed an occupational choice model where employers choose whether to sponsor insurance and workers choose whether to work for employers who sponsor insurance. The model is consistent with several features of the data. In particular, workers have higher health coverage that self-employed; health coverage increases with age and income; larger firms are more likely to sponsor insurance. At this stage we have only constructed the model. The next step is to use the model to study the impact of different policy reforms that directly or indirectly affect the equilibrium number of workers with insurance coverage. This will be the objective of future research. References Aizawa, N. and Fang, H. (2013). Equilibrium labor market search and health insurance reform. NBER Working Papers 18698, National Bureau of Economic Research, Inc. Brugemann, B. and Manovskii, I. (2010). Fragility: A quantitative analysis of the us health insurance system. Mimeo, University of Pennsilvania. DeLoach, S. and Platania, J. (2013). The Macroeconomic Consequences of Financing Health Insurance. International Advances in Economic Research, 19(2):107 129. Dey, M. S. and Flinn, C. J. (2005). An equilibrium model of health insurance provision and wage determination. Econometrica, 73(2):571 627. Fang, H. and Gavazza, A. (2011). Dynamic inefficiencies in an employment-based health insurance system: Theory and evidence. American Economic Review, 101(7):3047 77. Hansen, G. D., Hsu, M., and Lee, J. (2014). Health insurance reform: The impact of a medicare buy-in. Journal of Economic Dynamics and Control, 45(C):315 329. Hsu, M. (2013). Health insurance and precautionary saving: A structural analysis. Review of Economic Dynamics, 16(3):511 526. 21

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