According to the Kaiser Family Foundation, 160 million Americans receive their health insurance from their employers. That figure represents three out of five non-elderly individuals. Many experts argue that using employer provided health insurance eliminates the problem of adverse selection by forming an insurance pool around a non-medical issue (employment). Jayanta Bahattacharya and William Vogt are not sure this is the case. In their 2006 NBER working paper, the authors aim to test whether or not healthy individuals are more likely to receive employer provided insurance at their jobs.
Model
The model Bhattacharya and Vogt set up is a two period model. In the first period i) employer set wage and benefit levels, ii) workers are informed if they are health or sick, iii) workers choose their employer and then iv) a health shock occurs and the employee consumes the needed medical care. In the second period: i) the workers see a new health state, ii) there is an involuntary turnover rate ‘T‘ where these workers must seek new employment, iii) workers in the ‘(1-T)‘ group can elect to switch employers, and iv) a second health shock occurs and the employee consumes the needed medical care.
The individual’s utility function is:
‘Y‘ is one’s income and ‘m‘ is out of pocket medical expenses. The separable v function is utility gained from health capital. ‘H‘ is the initial health capital level and f is an increased or decreased level of health depending on health spending ‘m‘ and a random shock ‘e‘. The shock variable is distributed according to the cdf ‘F‘ which depends on whether the individual is sick or healthy. F_sick first order stochastically dominates F_well. Because of this assumption, we can prove (mathematically) the following three statements:
- The cost of care for the sick is higher than the cost of care for the well
- The Utility of the insured who are sick is lower than the utility of the insured who are well.
- The Utility of the uninsured who are sick is lower than the utility of the uninsured who are well.
There are four possible indirect utility functions in the first period.
- U_SU: E_{F_s} [U(Y-m,H+f(m,e))]
- U_SI: E_{F_s} [U(Y,H+f(m,e))]
- U_WU: E_{F_w} [U(Y-m,H+f(m,e))]
- U_WI: E_{F_w} [U(Y,H+f(m,e))]
In the second period, however, the author assumes that there is a switching cost of ‘c‘ utils if the employee decides to change jobs. This can be justified as a psychic cost to the worker or the loss of job-specific human capital. The author assumes U_WU(W-p)-c<U_WU(W) meaning that one will reduce overall utility if the person leaves a job and purchases private health insurance rather than staying at a job and using the employer provided insurance.
Equilibrium
The authors seek a symmetric subgame perfect Nash equilibrium where 1) both sick and well workers choose an employer offerreing insurance in period one and 2) neither sick nor well workers voluntarily turn over to change insurance status in period two. Working out the mathematics we find the following conclusions:
- Among people who do not receive employer provided insurance, the sick benefit the most from purchasing individual insurance: U_SI(W-p)-U_SU(W) > U_WI(W-p)-U_WU(W). Thus the only people left uninsured will be the healthy ones.
- A pooling equilibrium is more likely if there are high switching costs ‘c‘.
- A pooling equilibrium is more likely with a low exogenous turnover rate ‘T‘.
- A pooling equilibrium is more likely if the probability that a well person will become sick is high and the probability a sick person will become well is also high. Algebraically, the author say this means that ‘P_ww‘ is low.
Empirical Tests
The authors use data from the 1995-2005 March CPS as well as the Occupational Information Network and the Census Public Use Microdata Sample. They aim to test conclusions 2-4 above as well as seeing if there is any evidence of adverse selection. The data also show that industries with a high level of job specific human capital–which in this case the authors use as a proxy for ‘c‘ –have workers which are more likely to be covered by their employers. The coefficient on ‘P_ww‘ is not significantly different from zero, but the ‘P_ww‘ variable is likely not measure with accuracy. A high turnover rate ‘T‘ actually has the opposite effect than the one predicted by the authors. The measure ‘T‘ however is problematic to measure in the data since turnover empirically is a mix of exogenous and self selected turnover (quits) and thus the authors disregard any results for the variable.
Analysis
The model of this paper is elegant and produces sensible conclusions. While the homogeneity of employers greatly simplifies the model, it makes the mathematical conclusions less robust. The empirical work gives some suggestive evidence that adverse selection in employer choice of workers may be a problem. Using such a large data set adds precision to the author’s estimates, but it may be difficult to decompose the countervailing forces within each industry without having more detailed knowledge of each sector. Still, the combination of theory and empirical work is good; using other data sets to substantiate the authors claims would make their conclusions more robust.
Bhattacharya, Vogt (2006); “Employment and adverse selection in health insurance,” NBER Working Paper No. 12430.
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