Academic Articles Health Insurance

Moral hazard, adverse selection and health expenditures: A semiparametric analysis

Moral hazard and adverse selection are ever-present problems in the health insurance market.  Identifying their existence and their magnitudes is difficult.  Most of the papers presented in this blog have used a reduced form approach [see Finkelstein, McGarry (2003) and Bhattacharya, Vogt (2006)].  Today we will look at a paper by Bajari, Hong and Khwaja where the authors use a structural approach.


The authors set up a two period model.  In the first time period the individuals choose the level of insurance; in the second period they choose a level of medical expenditures ‘m‘.  The individuals are assumed to have a separable utility function as follows:

  • U(c,m-q;g)=(1-g1)^{-1}*c^(1-g1) + g2(1-g3)^{-1}*(m-q)^{1-g3}
  • c=y-p-z(m)

The variables g1, g2, and g3 are parameters characterizing the consumer’s utility function; c is a consumption of non-medical goods; q is the consumers latent health status; y is an exogenous income measure, p is the insurance premium, z(m) is the copayment schedule.  The first order condition is:

  • g2(m-q)^{-g3}=c^{-g1} * [z'(m)]

This first order condition will be the cornerstone of all subsequent empirical analysis.


The authors use data from the 1996 (wave 3) Health and Retirement Study (HRS).  The HRS is a panel study which tracks individuals over age 50 for a period of 2 years.  The first step in the analysis is estimate the health insurance co-payment schedule (z(m))using data on out of pocket medical expenditures and insurance choices.  The reimbursement schedule is unique for each insurance group (eg: employer, Medicare, etc.), but not for each individual.  Since c, y, p, m are known, and z(m) is estimated, we can now try to recover the g‘s and the q variable.  The authors assume ‘y’ is exogenous.  This is unlikely since those with a poor health (high q) are unlikely to earn much money if they are continually sick.  Nevertheless, if y were to be exogenous, we can solve the FOC for q so that q_i=r(m_i,p_i,y_i,z(m_i),g). 

The authors then use three instrumental variables (x_i) to formulate a method of moments estimator where the median latent health status u_q, as well as the parameters g are estimated.  The median moment condition below is wisely employed since it is more robust to censoring at the upper and lower tails of the conditional distribution of the observables. 

  • (u_q,g)=argmin || n^{-1} SUM_i [f(x_i)*{1{r()}-0.5}] ||

The brackets “|| ||” represent the quadratic norm, ie: ||x||=x’Wx.   If the instruments are truly valid, then f(x_i) should be orthogonal to r() when r is evaluated at the true vetor g.  The instruments used are as follows:

  1. State level housing price index
  2. County level malpractice insurance component of the Geographic Practice Cost Index (GPCI)
  3. Number of establishments in a county

In order for these to be good instruments, the variables chosen must provide variation in the price of providing medical goods but be uncorrelated with the latent health distribution.  Each of these three seems to fulfill this requirement, but there could be some problems.  While higher rental prices should increase the cost of living and thus the cost of medical services, it is also possible that only healthier people are able to make sufficient income to be able to afford houses in high rent areas.  Malpractice insurance certainly effects the price of medical services and likely does not significantly effect health status.  The author mentions that any defensive medicine caused by higher malpractice insurance rates should be reflected in the price consumers pay for health insurance.  The number of establishments in a county may effect prices, but the direction is unclear.  This is likely a weak instrument. 


The authors find that the coefficient of relative risk aversion for aggregate consumption (g1) is 0.85, while the same parameter for consumption of health services (g3) is 1.52.  This shows that individuals are more risk averse with respect to health status than the aggregate consumption commodity.  The authors find that the utility weight on the consumption of health services relative to the composite good (g2) is 1.37.  Thus the medical services good is more highly valued than the composite good. The authors’ estimate of the median value of latent health status (u_q) is $4063.  For the 25th percentile, u_q has a value of $708 and for the 75th percentile the value is $11,653.  We see how the latent health variable has a significant effect on healthcare costs.

The authors can also estimate the medical expenditure elasticity at different expenditure percentiles.  The median elasticity is -0.21, but is -0.47 at the 25th percentile and -0.01 at the 75th percentile.  Bajari, et al. also subdivide the elasticities by insurance type.  They find no significant difference in elasticities between insurance groups, except for the self employed who have a higher elasticity, likely because as a group they are healthier than the other categories.  The variation in the elasticity of demand for health care occurs within plans, not between them.  

To estimate the moral hazard phenomenon, the authors examine the correlation between estimated elasticities and various individual characteristics.  The authors find that elasticity monotonically increases as self reported health status improves.  This implies that younger and healthier individuals have a more elastic response to price incentive. 

At first glance, it seems that adverse selection is a problem since we find that those who are uninsured have the best latent health status at the median.  The authors use a Kolmogorov-Smirnov test statistic to see if the uninsured are truly different than the insured.  The K-S test cannot reject the equality of the distributions across the insurance categories.  So there is no adverse selection?  The authors wisely note that this could simply be due to the fact that the categories chosen (ie: Medicare, VA/CHAMPUS, employer-provided, etc.) are too broad and that there is adverse selection present, but it is within each insurance category.  Further, since the HRS deals with individuals above age 50, it is likely that adverse selection is less of a problem since many of these individuals are eligible for public insurance (Medicare for those over 65).