For instance, individuals may have convex utility for some outcomes and they may weight probability as in prospect theory. The other is that not insuring is the best option given risk attitude, which differs from the ubiquitous risk aversion that is assumed when estimating the gains from insurance. The apparent inconsistency has two possible causes. 2009 Gross and Notowidigdo 2011 Shigeioki 2014 Limwattananon et al. This low demand does not square with estimates of large welfare gains from insuring low-income populations that are obtained under assumptions that individuals (i) have accurate perception of their medical expenditure risks and (ii) are risk averse (Pauly et al. 1997 Bundorf and Pauly 2006 Levy and DeLiere 2008) and willingness to pay tends to be below the expected value of the insurance (Finkelstein et al. In the US, take-up of subsidized health insurance by low-income households has also been far from complete (Currie and Gruber 1996 Chernew et al. 2016 Pettigrew and Mathauer 2016 Wagstaff et al. Many of those not covered through mandatory insurance for salaried employees or fully subsidized insurance for the poor do not voluntarily insure, even at highly subsidized premiums (Thornton et al. WTP is reduced further by factors other than risk perception and attitude.ĭespite widespread exposure to risks of substantial medical expenses in low- and middle-income countries, demand for health insurance is often low. Convex utility in the domain of losses pushes mean WTP below the fair price and the subsidized price, and the transformation of probabilities into decision weights depresses the mean further, at least using one of two specific decompositions. This is not explained by downwardly biased beliefs: both the mean and the median subjective expectation are greater than the subsidized price. We find that the mean stated WTP of the uninsured is less than both the actuarially fair price and the subsidized price at which public insurance is offered. To apply this approach, we elicit WTP, subjective distributions of medical expenditures and risk attitude (utility curvature and probability weighting) from Filipino households in a nationwide survey. To help explain it, this paper introduces a decomposition of the stated willingness to pay (WTP) for insurance into its fair price and three behavioral deviations from that price due to risk perception and risk attitude consistent with prospect theory, plus a residual. This is puzzling from the perspective of expected utility theory. So we will invert the series for f( x) = Γ( x+2) – 1 and then adjust the result to find the inverse of Γ( x).Despite widespread exposure to substantial medical expenditure risk in low-income populations, health insurance enrollment is typically low. The equations above assume we’re working in a neighborhood of 0 and that our function is 0 at 0. Suppose we want a series for the inverse of the gamma function near 2. (More on rising and falling powers here.) Example Where the B‘s are exponential Bell polynomials, We have g 1 = 1/ f 1 (things start out easy!) and for n ≥ 2, Then you can compute the series coefficients for g from the coefficients for f as follows. (Compare with ordinary and exponential generating functions explained here.) The coefficients here are k! times the ordinary power series coefficients. Note that this isn’t the power series per se. Assume f and its inverse g have the series representations We introduce exponential Bell polynomials to encapsulate some of the complexity, analogous to introducing Bernoulli numbers above.Īssume the function we want to invert, f( x), satisfies f(0) = 0 and its derivative satisfies f‘(0) ≠ 0. starting with the power series for a function and computing the power series for its inverse, is going to be complicated. The example above suggests that inverting a power series, i.e. Reading the equation from right to left, it says a complicated sequence has a simple generating function! Computing the coefficients To a calculus student, this is bad news: a simple, familiar function has a complicated power series. It is possible to write the sum in closed form, but this requires introducing the Bernoulli numbers which are only slightly less mysterious than the power series for tangent. There’s no obvious pattern to the coefficients. For example, the coefficients in the series for arctangent have a very simple patternīut the coefficients in the series for tangent are mysterious. As a student, one of the things that seemed curious to me about power series was that a function might have a simple power series, but its inverse could be much more complicated.
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