Measuring Ex-Ante Welfare in Insurance Markets

Transcription

1 Measuring Ex-Ante Welfare in Insurance Markets Nathaniel Hendren October, 207 Abstract Revealed-preference measures of willingness to pay generally provide a gold standard input into welfare analysis. But, applied to insurance markets they can be misleading. This is because they only capture the value of insurance against the portion of risk that remains at the time of making an insurance choice. This paper develops a sufficient statistics method to recover ex-ante measures of welfare that are consistent with ex-ante expected utility but can be applied to markets in which choices are made after some information has been revealed. I provide a benchmark method of implementation of this approach that requires only market willingness to pay and cost curves, as in Einav, Finkelstein, and Cullen (200), combined with a measure of risk aversion. I apply the approach to estimates from existing literature in health insurance. Relative to a criteria focused on market surplus or deadweight loss, an ex-ante expected utility criteria can generate different conclusions about the optimal size of insurance markets, the welfare cost of adverse selection, and the desirability of mandates. Introduction There is a large and growing literature using reduced-form methods to estimate the willingness to pay for insurance along with the costs these individuals impose on the insurer. These revealed preferences are then used to to normatively assess optimal insurance subsidies and mandates by constructing market surplus as the difference between individuals willingness to pay relative to their costs (Einav et al. (200); Hackmann et al. (205)). Although revealed preferences generally are a gold standard input into welfare analysis, they can be misleading when applied to insurance markets. Insurance has value from insuring against the realization of risk. But, revealed preferences for insurance only measure the value of insurance against any risk that remains at the time of choosing the insurance contract. Therefore, in adversely selected markets where individuals have some knowledge about their risk when choices are make, the average willingness to pay for insurance will generally be less than the ex-ante willingness to pay for insurance (Hirshleifer (97)). Policies Harvard University, nhendren@fas.harvard.edu. I am very grateful to Raj Chetty, David Cutler, Liran Einav, Amy Finkelstein, Mark Shepard, and Mike Whinston for helpful comments and discussions.

2 that maximize ex-ante expected utility may differ from those that maximize market surplus that is measured from revealed preference willingness to pay and costs. This goal of this paper is to provide an empirical method to recover a measure of welfare that is consistent with expected utility but that retains the empirical credibility of the Einav et al. (200) framework that uses revealed preference estimates from variation in prices. To do so, I augment the Einav et al. (200) framework with sufficient statistics that enable the researcher to move from willingness to pay that is revealed at the time of insurance market choices to an ex-ante welfare perspective. I first characterize these sufficient statistics, and then I provide assumptions under which one can estimate these statistics using the market willingness to pay and cost curves combined with a measure of risk aversion. I apply the approach to the study of low-income health insurance subsidies in Finkelstein et al. (207) and employer-provided top-up insurance in Einav et al. (200). Ishowthattheex-antewelfareperspectivecanleadtodifferentconclusionsabouttheoptimality of government subsidies and mandates. While the applications studied here involve cases where the researcher directly observes prices and willingness to pay, the methods can be applied more broadly to settings where researchers implicitly infer the value of insurance from other revealed preference choices, such as distortions in labor supply (Gallen (204)). To illustrate the distinction between ex-ante welfare and observed willingness to pay, it is useful to start with an example. This example will help to both define the problem and illustrate the proposed solution. Related literature will be discussed along the way, and I return to a discussion of the key lessons in the conclusion. 2 Stylized Example Suppose individuals have $30 dollars but face a risk of losing $m dollars, where m is uniformly distributed between 0 and 0. First, suppose willingness to pay for insurance against the realization of m is measured prior to a time when individuals have any knowledge about their risk of loss. Let D Ex ante denote this willingness to pay (or demand ), which solves u 30 D Ex ante = E [u (30 m)] () where E [u (30 m)] = R u (30 m) dm is the expected utility if uninsured. Suppose individuals have a utility function with a constant coefficient of relative risk aversion of 3 (i.e. u (c) = c and =3). In this case, it is straightforward to compute that they are willing to pay D Ex ante =5.50 for insurance against m. This insurance policy would cost the insurer E [m] =5, so that the individuals are willing to pay a markup of 0.50 over actuarially fair insurance. Full insurance generates a market surplus of $

3 Figure : Example Willingness to Pay and Cost Curves A. Before Information Revealed B. After Information Revealed W Ex Ante = $0.50 s CE = Fraction Insured (s) s CE = Fraction Insured (s) Observed Demand Average Cost No lost surplus from foregone trades Marginal Cost Demand Cost Figure, Panel A, translates this scenario using the demand and cost curve framework formalized in Einav et al. (200). The horizontal axis enumerates the population in descending order of their willingness to pay for insurance (using an index s 2 [0, ]), and the vertical axis reflects prices, costs, and willingness to pay in the market. Each individual is willing to pay $5.50 for insurance, generating a flat willingness to pay, or demand, curve of D (s) = $5.50. Becausenoone knows anything about their particular cost, each individual imposes a cost of $5 on the insurance company, generating a flat cost curve of C (s) = $5. If a competitive market were to open up in this setting, one would expect everyone (s CE = 00%) to purchase insurance at a price of $5. This allocation would generate W Ex Ante = $0.50 of welfare, as reflected by the market surplus defined as the integral between demand and cost curve. Now, what happens if some information is revealed at the time individuals decide whether to purchase insurance? For simplicity, consider the extreme case that individuals have fully learned their cost, m. Willingness to pay will equal individuals known costs, D (s) =m (s). Those who learn they will lose $0 will be willing to pay $0 for insurance against their loss; individuals who learn they will lose $0 will be willing to pay nothing. The uniform distribution of risks generates alineardemandcurvefallingfrom$0ats =0to $0 at s =. The cost imposed on the insurer by the marginal type s, C (s), will equal their willingness to pay of D (s). Therefore, the demand curve lies everywhere on top of the cost curve of the marginal types, as illustrated in Panel B. If an insurer were to try to sell insurance, they would need to set prices to cover the average cost of those who purchase insurance. Let AC ( s) =E [C (s) s apple s] denote the average cost of those with willingness to pay above D ( s). This average cost lies everywhere above the demand curve. Since no one is willing to pay the pooled cost of those with higher willingness to pay, the market would fully unravel. The unique competitive equilibrium would involve no one obtaining any insurance, s CE = 0%. 3

4 What is the welfare cost of this market unraveling? From a market surplus perspective, there is no welfare loss. Because the demand curve equals the marginal cost curve, there are no valuable foregone trades that can take place at the tie insurance choices are made. This reflects an extreme case of a more general phenomenon identified in Hirshleifer (97). The market demand curve does not capture the value of insurance against the portion of risk that has already been realized at the time insurance choices are made. As will be illustrated throughout the paper, this means that policies that maximize market surplus may not maximize ex-ante expected utility. How can one recover the ex-ante expected utility measure of welfare in equation ()? The traditional approach would require the econometrician to specify economic primitives including a utility function and information set. It would then also require measuring outcomes such as consumption (or assume proxies for consumption) to infer the ex-ante value of insurance from the model. Intuitively, if one knows the utility function, u, andthecross-sectionaldistribution of consumption (30 m in the example above), then one can use this information to compute D Ex Ante in equation (). For recent implementations of this approach, see Handel et al. (205), Section IV of Einav et al. (206), or Finkelstein et al. (206). Such an approach abandons the transparency of the Einav et al. (200) that relies on revealed preference for direct measures of willingness to pay. In contrast, the goal of this paper is to estimate D Ex Ante without knowledge of the full distribution of primitives (e.g. u and m). Moreover, I want to evaluate the welfare impact of subsidies that lead to partial insurance take-up in markets with adverse selection, but evaluate the welfare properties of these subsidies from an ex-ante welfare perspective. To accomplish these goals, I develop a sufficient-statistics approach that continues to utilize revealed preference as in Einav et al. (200) to measure the willingness to pay of the risk that remains at the time of choosing insurance. But, characterize sufficient statistics that allow one to add the additional component of the value of insurance against the portion of the risk that has been already been revealed at the time of choosing insurance. For example, I construct a new ex-ante demand curve, D Ex Ante (s), that measures the ex-ante willingness to pay for having a fraction s of the market insured. Comparing this ex-ante demand curve to the cost curves in Figure will facilitate an analysis of the impact of policies on ex-ante expected utility. To illustrate how to construct an ex-ante value of insurance, let p I denote the price of insurance and p U denote the price of being uninsured (so that p I p U is the marginal price of obtaining insurance). To begin, consider the willingness to pay for a larger insurance market using a budgetneutral shift in insurance prices that requires the total amount of money collected to equal the total cost of the insured, sp I +( s) p U = sac (s). This assumption is not essential. In the next Section I show how one can analyze non-budget neutral policies by constructing the marginal value of public funds (MVPF) as in Hendren (206) applied to a policy of using government funds to lower insurance prices. Suppose that prices are set such that a fraction s =0.5 of the population chooses to purchase 4

5 insurance, as illustrated in Figure 2, Panel A. It is straightforward to show that this corresponds to p I =6.25 and p U =.25, sothatthemarginalpriceofinsuranceis$5. Now,considerexpanding the size of the insurance market from s to s + ds by decreasing p I financed by an increase in p U. This lowers the marginal price of insurance, p I p U,byD 0 (s) ds. The resource constraint implies that the price faced by the uninsured increases by dp U = sd 0 (s) ds, andthepriceofinsurance must decrease by dp I =( s) D 0 (s) ds. This change in insurance prices will generate a transfer from the uninsured to the insured, as indicated by the blue arrow in Figure 2, Panel B. From a market surplus perspective, this transfer has no welfare impact. But, from an ex-ante expected utility perspective, these transfers have value to the extent to which the marginal utilities of income differ for the insured and uninsured. If the marginal utility of income is higher (lower) for the insured than uninsured, then lowering (raising) the price of insurance increases welfare. Accounting for these difference in marginal utilities of income between the insured and uninsured is the key to constructing ex-ante measures of welfare. Prior to learning one s willingness to pay, there is a chance s of being insured. The impact of lower insurance prices on ex-ante expected utility is given by s dp I ds E [ Insured] ds = s ( s) D 0 (s) E [ Insured] ds where E [ Insured] is the average marginal utility of income for the fraction s of the market that is insured. Conversely, the cost of having a higher price on ex-ante expected utility is given by ( s) dp U ds E [ Uninsured] ds = s ( s) D 0 (s) E [ Insured] ds where E [ Uninsured] is the average marginal utility of income for the fraction s of the market that is uninsured (for notational simplicity, I suppress the dependence of these marginal utilities on s, p I,andp U ). Summing these two effects yields the ex-ante value of expanding the size of the insurance market by ds: EA(s) =s ( s) D 0 E [ Insured] E [ Uninsured] (s) {z } E [ ] Transfer {z } Difference in Marginal Utilities (2) where I normalize by the average marginal utility of income, E [ ],togenerateamoney-metric utility measure. The first term, s ( s)( D 0 (s)), canlooselybeinterpretedasthesizeofthe blue arrow in Figure 2, Panel B. Steeper slopes of demand imply greater price changes (and thus larger transfers) one moves from s to s + ds of the market being insured. E[ Insured] E[ Uninsured] E[] The second term,,isthepercentagedifferenceinmarginalutilitiesbetweentheinsured and uninsured population. Weighting by the difference in marginal utilities recovers the ex-ante Obtaining s =0.5 would require individuals who purchase insurance to pay p I =$6.25 and those who do not purchase insurance to pay p U =$.25, sothatthemarginalpriceofinsuranceisp I p U =$5and the aggregate resource constraint holds. 5

6 value of insurance. 2 To the extent to which those choosing to buy insurance have a higher marginal utility of income, the transfer from the uninsured to the insured increases ex-ante expected utility. 3 Given EA(s) in equation (2), I define the ex-ante demand curve, D Ex Ante (s), asthesumof the willingness to pay revealed by choices, D (s), and the additional ex-ante value of insurance, EA(s): D Ex Ante (s) =D (s)+ea(s) (3) Prior to learning their willingness to pay for insurance, individuals are willing to pay D Ex Ante (s) to have prices set such that a fraction s of the market is insured. In particular, the value of having everyone insured, D Ex Ante (), isequaltod Ex ante in equation (). 4 This representation of exante willingness to pay using marginal utilities of income and the market demand and cost curves as in equation (2) will be the first main result of the paper, which will be derived in a more general setting in Section This result is akin to the Baily-Chetty condition in optimal unemployment insurance that measures the value of more generous social insurance using the marginal utility of the beneficiaries (e.g. unemployed) relative to nonbeneficiaries (e.g. employed). (Baily (978); Chetty (2006a)). Here, the beneficiaries of lower insurance prices are those who choose to purchase insurance. 3 It is also possible that those who are uninsured have a higher marginal utility of income than the insured. This could be the case if the reason for not obtaining coverage is liquidity constraints, so that those choosing to forego insurance have a higher return to other forms of spending. This is ruled out in the simple example presented in this introduction, but will be considered in the more general model in the next Section. 4 More precisely, this is true up to an approximation error resulting from the fact that the average marginal utility, E [], varieswithmarketsizes, 5 In principle, one could derive an ex-ante demand curve in equation (2) for any type of good, not just insurance. The key question is whether the decision to purchase the good reveals something about the marginal utility of income. In the context of insurance markets, one naturally expects information to be revealed over time. This makes it particularly reasonable to expect that the marginal utility of income is different for those who choose to purchase versus those who forego insurance. In this sense, the distinction between market demand and ex-ante demand is likely to e particularly important in insurance contexts with adverse selection. 6

7 Figure 2: Recovering Ex-Ante Willingness to Pay A. Marginal Increase in Fraction Insured B. Transfer from Uninsured to Insured Marginal Price (p I p U ) Lowers p I -p U by D (s)ds ds Marginal Price (p I p U ) Fraction s pay lower prices Fraction -s pay higher prices dp I = ( s)d' ( s)ds dp U = sd' ( s)ds ds Fraction Insured (s) Fraction Insured (s) Demand Marginal Cost Demand Marginal Cost C. Valuation of Transfer using Marginal Utilities D. Generating Ex-Ante Demand Marginal Price (p I p U ) & ( (( s)s( D' ( s) )) E [ u Insured] E [ u Uninsured] c c ( E [ u EA( s) = c Insured] ' ( (( s)s( D' ( s) )) c D( s) E"# D( s' ) s' > s$ ( % ) ( s) = y p I Insured Uninsured () ( ) (2) ( ) ( s) ( y p I ) + c ( y p I ) D( s) D( s' ) #$ % & Marginal Price (p I p U ) W Ex-Ante = $0.50 EA(0.5) =.5*.5*(-0)*(3/25)*(-2.5) = Fraction Insured (s) Fraction Insured (s) Demand Marginal Cost Demand 'Ex-ante' Demand, D(s)+EA(s) Marginal Cost AkeybarriertoestimatingD Ex Ante (s) is that one does not readily observe the differences in marginal utilities between the insured and uninsured. The second main result of the paper builds on the literature on optimal unemployment insurance (e.g. Baily (978); Chetty (2006a)) by providing conditions under which one can approximate the difference in marginal utilities for the insured versus uninsured using Taylor expansions of the marginal utility function. Under some additional assumptions, I write this difference in marginal utilities as a function of (i) the willingness to pay curve, D (s), and(ii)anestimateofriskaversion. To illustrate this, return to the example above. The insured have consumption of 30 p I. So, their marginal utility is given by (30 p I ), where is the marginal utility function (e.g. (c) =c if u (c) is constant relative risk aversion). The consumption of the uninsured facing known loss m (s) is given by 30 p U m(s), sothattheirmarginalutilityis (30 p U m (s)). Averaging across the uninsured with different loss sizes and using the identity D (s) =m (s), the 7

8 average marginal utility of the uninsured is given by E [ (30 p U D (s 0 )) s 0 s]. Now, consider a first order Taylor expansion to the marginal utility function of the insured around a consumption level c. This yields 30 p U D s 0 (c )+c (c ) 30 p U D s 0 c (c )+c (c ) p I p U D s 0 Similarly, the marginal utility of the insured is given by (30 p I ) (c )+c (c ) [30 p I c ] So, the difference between insured and uninsured is given by E [ Insured] E [ Uninsured] c (c ) (30 p I c ) 30 p U D s 0 c c (c ) D s 0 D (s) where p I p U = D (s) is the equilibrium price of insurance when a fraction s purchases insurance. Now, take expectations over the uninsured types, s 0,andnormalizebyE[ ] (c ), where c is the average consumption in the population. This yields an expression for the percentage difference between the marginal utility of insured and uninsured: E [ Insured] E [ Uninsured] E [ ] c D (s) E D s 0 s<s 0 (4) u where cc is the coefficient of absolute risk aversion (evaluated at c )andd(s) E [D (s 0 ) s <s 0 ] is the difference between the willingness to pay of the average uninsured person and the price, D (s) =p I (s) p U (s), when a fraction s of the market is insured. Equation (4) provides a method to estimate the ex-ante measures of welfare using the market demand curve and a measure of risk aversion. Risk aversion can either be imported from another setting, or one can infer it by comparing the markup individuals are willing to pay for insurance to the variance reduction offered by the insurance product, as will be shown in Section A. 6 In the stylized example, the coefficient of relative risk aversion is 3 and the average consumption in the population is 25. So, the coefficient of absolute risk aversion is approximately 3/25. Using equation (2), the ex-ante value of insurance from expanding the market when exactly 50% have insurance is EA(0.5) = ( 0) (3/25) (5 2.5) = From behind the veil of ignorance, individuals are willing to pay $0.75 to expand the size of the insurance market from 50% to 5% insured relative to what would be indicated by their demand curve (which equals D (0.5) = 5). 6 For example, in a CARA-Normal model the coefficient of absolute risk aversion is equal to twice the ratio of the markup individuals are willing to pay for insurance relative to the variance reduction in out of pocket expenses it provides. Appendix A provides a more general characterization for more general utility functions and risk distributions. In this simple example, there is no remaining risk that drives insurance demand. As a result, willingness to pay does not reveal anything about risk aversion; but in more realistic empirical applications one in principle can estimate this risk aversion coefficient internally. 8

9 This is illustrated in Figure 2, Panel D. Panel D of Figure 2 uses equations (2) and (4) to calculate EA(s) for all values of s 2 [0, ]. Adding this ex-ante value to the market demand curve yields the ex-ante demand curve, D Ex Ante (s) =D (s) +EA(s), depictedbythesolidredline. Ateachvalueofs, D Ex Ante (s) measures the impact on ex-ante expected utility of expanding the size of the insurance market. Integrating from s =0to s =yields the value of insuring everyone, Z 0 D Ex Ante Ex Ante (s) =5.50 = D Integrating under the ex-ante demand curve in Figure 2, Panel D, yields $5.50. Not coincidentally, this equals the integral under the demand curve in Figure, Panel A. 7 The ex-ante utility impact of having everyone insured, s =,doesnotdependonhowmuchinformationhasbeenrevealedat the time the decision to purchase insurance is made. The ex-ante demand curve provides precisely this stable normative guidance on the welfare impact of a mandate regardless of the amount of information is revealed at the time willingness to pay is measured. In contrast, market surplus does not.moreover, an ex-ante welfare perspective can lead to different welfare conclusions. From an ex-ante utility perspective, a mandate generates $5.50-$5 = $0.50 of welfare. In contrast, it generates no market surplus. The model in this section is highly stylized. There is no moral hazard, no preference heterogeneity, and the model assumed all information about costs, m, was revealed at the time of making the insurance decision. The next section extends these derivations to capture more realistic features of insurance markets encountered in common empirical applications, such as the one considered in Einav et al. (200) or Finkelstein et al. (207). 3 General Model As noted above, revealed preferences for insurance only reveal the willingness to pay for insurance against the portion of risk that remains at the time of choosing insurance. I use the demand and cost curves from the Einav et al. (200) framework to capture this value of insurance. But, I augment these with a set of sufficient statistics that capture the value of insurance against the portion of risk that has already been revealed at the time of purchasing insurance. Combining, these allow one to evaluate the impact of policies, such as subsidies and mandates, on ex-ante expected utility. 7 The approximation error from the Taylor expansion yields a difference that is noticeable only in the third decimal place 9

10 Figure 3: Timeline of Information Revelation and Insurance Purchase Ex-Ante Signal (s) Realized Event (θ) Realized New Sufficient Statistics Insurance Choice Revealed Preference (Einav, Finkelstein, and Cullen (200) While the language of this section will refer to a health insurance context, it is straightforward to amend the model to capture other insurance settings, such as unemployment insurance. Proposition will provide a generalization of equation (2). As in the stylized example, the key additional piece of information that is required to measure impacts on ex-ante expected utility is the difference in marginal utilities of income between the insured and uninsured. Section 4 will then discuss the general conditions under which one can approximate this difference using the demand and cost curves combined with a measure of risk aversion, as in equation Setup Figure 3 presents a timeline of information revelation and outlines the empirical approach. Loosely, I will use the demand and cost curves as in the Einav et al. (200) framework to measure the value of insurance against the portion of risk that has not yet been realized at the time of choosing to purchase insurance. But, I incorporate additional sufficient statistics to incorporate the value of insurance against the portion of the risk that has already been realized at the time of choosing to purchase insurance. Individuals face uncertainty over a future event, captured by a random variable. Forexample, can represent health or sickness. After learning, individualschoosetheirnon-medicalconsumption, c, andmedicalspending, m. Forexample, ifonebecomessick, s/hemaychoosetogotothe doctor for treatment. Individuals have a utility function over these choices, u (c, m; ). The event,, canaffectbothpreferencesandthebudgetconstraintthrougheffectsonincome,y ( ). There exists an insurance contract at price p I that allows individuals to obtain medical services at cost x (m; ) yielding the budget constraint c I ( )+x m I ( ); + p I apple y ( ) 0

11 Conversely, uninsured individuals must pay the full price 8 of m, yieldingabudgetconstraint c U ( )+m U ( )+p U apple y ( ) where p U is a penalty or tax paid by individuals that are uninsured. Let c I ( ),m I ( ) denote the choice of consumption and medical spending of an insured type, andlet denote the choices of an uninsured type. 9 c U ( ),m U ( ) At the time individuals make the decision to be insured or uninsured, individuals may know something about their particular type, which I denote by a signal s 2 [0, ]. For simplicity, I follow Einav et al. (200) and assume that only the relative price of insurance, p I p U affects demand. Appendix B provides a general treatment of Proposition below when demand is affected differentially by increases of p U as opposed to decreases in p I. Given s, letd ( s) denote the marginal price that a type s is willing to pay for insurance. This solves h E u y ( ) x m I ( ); D ( s) i h i p U,m I ( ); s = E u y ( ) m U ( ) p U,m U ( ); s All s such that p I p U apple D ( s) will choose to purchase insurance, whereas types s for which D ( s) >p I p U will choose to remain uninsured and pay p U. Without loss of generality, assume that s is ordered so that demand, D ( s), isdecreasingin s. Following Einav et al. (200), define the average cost imposed on the insurer when a fraction s of the market owns insurance by (5) AC (s) =E m I ( ) x m I ( ); s apple s (6) so that sac (s) is the total cost of insuring a fraction s of the market. Define C (s) to characterize how the total cost to the insurer changes as the size of the market expands, C (s) = d ds [sac (s)]. Given the assumption that individuals choices are not affected by prices p U and p 0 I,thiscostis the net difference between expenditures and out-of-pocket spending for those with signal s: C (s) =E m I ( ) x m I ( ); s = s (7) Finally, let p I (s) and p U (s) denote the prices of insurance and remaining uninsured when a 8 It is straightforward to generalize the model to allow for partial insurance, or multiple insurance contract choices. 9 The notation m I ( ) implies that m I ( ) is not a function of the price, p I.Inprinciple,thechoiceofm I ( ) could depend on p I; for example, if insurance is cheaper, individuals may make riskier choices that increase health costs later on. For now, I adopt the common assumption that m I ( ) does not depend on p I, but it is straightforward to relax this assumption: if m I depends on p I,thenthecosttoaninsurerofraising/loweringtheirpriceswouldalso include a component from the impact of these price changes on the costs of their insured pool. Similarly, I make the simplifying assumption that m U ( ) does not depend on p U. However, in contrast to the assumption that m I ( ) does not depend on p I,thisassumptioniswithoutlossofgeneralitybecauseoftheenvelope theorem: m U ( ) is fully paid by the individual so that behavioral responses of m U do not affect welfare measures of either the individual or other parties. 0 More generally, if prices do affect the cost to the insurer, this marginal cost function contains an additional term reflecting the net cost of those behavioral responses on the insurance company.

12 fraction s of the market owns insurance. By definition, these prices must be consistent with the definition of willingness to pay, D (s) =p I (s) p U (s) (8) Lastly, let G (s) denote the total cost (net of premiums collected) to the insurer of insuring a fraction s of the market by setting prices p I (s) and p U (s): G (s) = sac (s) {z } Cost of Insured [sp I (s)+( s) p U (s)] {z } Premiums Collected (9) In the case in which insurers earn zero profits, or in which the government breaks even, one can set G (s) =0so that prices p I (s) and p U (s) are then defined implicitly as solutions to equations (9) and (8). Below, I illustrate how to conduct welfare analysis for both budget neutral (G (s) =0) and non-budget neutral settings (G (s) 6= 0). 3.2 Ex-Ante Welfare The goal of this paper is to evaluate insurance market policies using an ex-ante expected utility criteria. Let W (s) denote the ex-ante expected utility when prices, p I (s) and p U (s), aresuchthat afractions of the market owns insurance. This is given by W (s) = Z s 0 E u y ( ) p I (s),m I ( ) s d s + Z s E u y ( ) m U ( ) p U (s),m U ( ); s d s (0) The first term integrates over those who choose to be insured, s apple s. The second term integrates over those who choose to be uninsured, s >s. If one observed or estimated the utility function and its arguments, one could directly measure W (s). This would be analogous to the approach to measuring welfare taken by Handel et al. (205) and Finkelstein et al. (206). Here, I instead follow the sufficient statistics approach of Einav et al. (200) and build a measure of W (s) from the willingness to pay and cost curves. To estimate the ex-ante welfare value of changing prices that change the size of the insurance market, I focus on W 0 (s) E[]. This is the ratio of the marginal utility impact of expanding the insurance market, W 0 (s), tothemarginalutilityimpactofpaying$inallstatesoftheworld,e [ ]. This measures individuals marginal willingness to pay for a larger insurance market out of their own income. To characterize W 0 (s) E[], I proceed as follows. To begin, I follow Einav et al. (200) and use the willingness to pay function, D (s), tocapturetheimpactontheutilityoftheuninsured, E u y ( ) m U ( ) p U,m U ( ); s = E u y ( ) D ( s) p U,m I ( ); s. This yields an expression for W (s) that does not require keeping track of the uninsured utility: W (s) = Z s 0 E u y ( ) p I (s),m I ( ) s d s + Z s E u y ( ) D ( s) p U,m I ( ); s d s Now, the marginal welfare impact of expanding the size of the insurance market is given by 2

13 taking the derivative with respect to s, W 0 (s) = sp 0 I (s) E y ( ) p I (s),m I ( ) s apple s () ( s) p 0 U (s) E y ( ) D ( s) p U (s),m I ( ); s s (2) The first term captures the welfare increase from lower prices for the insured (p 0 I < 0). From behind the veil of ignorance, this price reduction of p 0 I occurs with chance s and is valued using the marginal utility of income of the insured, E y ( ) p I (s),m I ( ) s apple s. The second term captures the welfare cost of having higher prices faced by the uninsured (p 0 U > 0). This price increase occurs with a chance s and is valued using the average marginal utility of income, E y ( ) D ( s) p U (s),m I ( ); s s. To characterize W 0 (s), one needs to know how prices are affected by the policy change. This in turn depends on whether the policy is budget neutral (G (s) =0) or not. I begin with budget neutral policies Budget Neutral Policies I first consider the case when the premiums collected cover the cost of the insured, as in Einav et al. (200). Combining equation () with the resource constraint in equation (9) when G (s) =0 yields the following result. Proposition. Suppose prices, p I (s) and p U (s), are set to cover the total cost of the insured so that G (s) =0, as in Einav et al. (200). Then the marginal welfare impact of expanding the size of the insurance market from s to s + ds is given by W 0 (s ) E [ ] D (s )+EA(s ) {z } D Ex Ante (s) C (s ) (3) where EA(s ) is the additional ex-ante value of expanding the size of the insurance market, EA(s )=( s ) C (s ) D (s ) s D 0 (s ) {z } Transfer from Uninsured to Insured (s ) (4) and (s) is the percentage difference in marginal utilities of income for the insured relative to the uninsured, (s) = E y ( ) p I (s),m I ( ); s apple s E E [ ] y ( ) D ( s) p U,m I ( ); s s (5) Proof. See Appendix C. Equation shows that the marginal ex-ante willingness to pay for a larger insurance market is given by the sum of D (s) +EA(s) C(s). The term D (s) C (s) is the familiar market surplus term: expanding the size of the insurance market increases ex-ante welfare to the extent 3

14 to which individuals are willing to pay more than their costs for insurance. But, in addition to this, EA(s) captures the ex-ante value of expanding the size of the market through its impact on insurance prices. Expanding the insurance market induces a transfer from uninsured to insured of size ( s )(C (s ) D (s ) s D 0 (s )). Note that this term reduces to the transfer in equation (2) when demand equals marginal cost, D (s) =C (s), asinthestylizedexample. Movingfinancial resources from the uninsured to the insured increases ex-ante welfare to the extent to which the marginal utility of income is higher for the insured than the uninsured. This difference is captured by the term (s ). The sign of (s ) In canonical models of insurance, one would expect (s ) > 0. Forexample, in the stylized example in Section 2, those who choose to purchase insurance expect to face a higher financial loss than those who remain uninsured. This means that the consumption levels of the insured are lower than those of the uninsured. Concavity of the utility function then implies that the marginal utilities of the insured are higher than the uninsured, so that (s ) > 0. But, it is also possible to have (s ) < 0. For example, could reflect a liquidity or income shock to y ( ) so that the primary driver of the decision to purchase insurance is not a higher expected cost, but rather an income shock that makes the value of medical care less than the value of additional other consumption. If the uninsured are foregoing insurance purchase because of this liquidity shock, then it is feasible that those who forego insurance have a higher marginal utility of income than those who purchased, (s ) < 0. In this case, expanding the size of the insurance market will transfer resources from the liquidity constrained to those who are less constrained, which would suggest that EA(s ) < 0. Going forward, most of the discussion will consider the benchmark case where (s ) > 0. But,thishighlightsthevalueoffutureworkonthedeterminants of the insurance purchase and the implications for (s ) The MVPF for Non-Budget Neutral Policies Proposition applied to budget-neutral policy changes where taxes and penalties from the uninsured cover the cost of subsidies to the insured. What about the case of a non-budget neutral policy that subsidizes insurance premiums using government funds that are paid by other members of the economy? To assess such policies, I follow Hendren (206) and consider the question: How much are individuals willing to pay for the policy change per dollar of total cost to the government? This magnitude is defined in Hendren (206) to be the marginal value of public funds (MVPF), and is given by MVPF (s) = W 0 (s) E[] Marginal WTP G 0 = (s) Marginal Cost The numerator is the individual s willingness to pay out of their own income for a larger insurance market. The denominator is the marginal cost to the government of expanding the size of the insurance market. Hendren (206, 204) shows how one can compare this MVPF to the MVPF of other policies affecting similar populations to study the efficiency of government policies: if the 4

15 MVPF of lowering health insurance prices is higher than the MPVF of a tax cut to a similar population (e.g. EITC in the case of low-income health insurance subsidies), then welfare can be increased by lowering health insurance prices financed by a reduction in EITC benefits. Proposition 2 provides a characterization of the MVPF for the case when uninsured individuals pay no penalty, p U (s) =0. Proposition 2. Suppose p U (s) =0. The MVPF of expanding the size of the insurance market is given by where +( s) (s) MVPF (s) = + C(s) D(s) s( D 0 (s)) (s) is the percentage difference in marginal utilities of income for the insured relative to the uninsured given by equation (5). Proof. Every dollar the government spends on additional subsidies leads to C(s) D(s) sd 0 (s) (6) dollars accruing to the insured. Hence, from behind the veil of ignorance, this generates a welfare impact of E [ Insured] From behind the veil of ignorance, $ of additional resources leads to an increase in utility of E [ ].Hence, the MVPF is given by Now, note that MVPF (s) = E [ Insured] E [ ] C(s) D(s) sd 0 (s) E [ ] = se [ Insured]+( s) E [ Uninsured] = E [ Insured]+( s)(e [ Uninsured] E [ Insured]) C(s) D(s) sd 0 (s). so that Hence, the MVPF is given by E [ Insured] E [ ] = ( s) (s) +( s) (s) MVPF (s) = + C(s) D(s) s( D 0 (s)) The denominator in equation (6), + C(s) D(s) s( D 0 (s)) reflects the marginal cost of lowering insurance prices. This cost has two components. First, lowering premiums by $ increases the cost by $ for each of the s enrollees. Second, there is an additional cost from those induced to purchase insurance by the lower prices. These enrollees pay D (s) =p I (s) but cost the insurer C (s). So,theyimpose anetcostofc (s) D (s). A$pricereductionincreasesthesizeofthemarketby D 0 (s).hence, the total cost normalized by the size of the market s, ofloweringpremiumsby$is+ C(s) D(s) s( D 0 (s)). The numerator in equation (6) reflects the willingness to pay for lower insurance premiums. An individual who has already learned s and decided to purchase insurance is willing to pay $ More generally, if there are additional behavioral responses that affect the government budget (e.g. if insurance improves health and increases taxable income, or if the subsidies distort labor supply, etc.), these would also need to be incorporated into the marginal cost of lowering premiums. 5

16 to have premiums that are $ lower. So, if welfare were not being calculated from behind the veil of ignorance, the numerator would simply by and the welfare impact would be. C(s) D(s) + s( D 0 (s)) This corresponds to the MPVF reported in Finkelstein et al. (207). But, from behind the veil of ignorance, individuals are willing to pay an additional ( s) (s) to have premiums that are $ lower. In sum, equations (3) and (6) in Propositions and 2 characterize how to measure the exante expected utility impact of insurance market policies. Conducting such an analysis requires an estimate of the difference in marginal utilities between insured and uninsured, (s). The next section discusses a benchmark path to providing such an estimate. 4 Implementation Using Market Demand and Cost Curves This section provides conditions under which one can write (s) as a function of market level demand curves combined with a measure of risk aversion analogous to equation 4 in the stylized example of Section 2. This estimate of risk aversion can either be imported from external settings (e.g. a coefficient of relative risk aversion of 3 or absolute risk aversion of 5x0 4 ), or it can be estimated internally using the relationship between the markup individuals are willing to pay and the reduction in consumption variance provided by the insurance. The assumptions required are not without loss of generality, but are common in the literature on optimal unemployment insurance (Baily (978); Chetty (2006a)). Moreover, many of these assumptions are satisfied in the structural models used to estimate the WTP for insurance (Handel et al. (205)). Thus, it provides a benchmark method to infer whether an ex-ante welfare perspective can lead to welfare conclusions that differ from a focus on market surplus. Section 4.2 shows how relaxing these assumptions is possible if one observes additional data elements. 4. Implementation Assumptions To estimate (s), consider a Taylor expansion of the utility. This will illustrate the potential sources of differences in marginal utilities between insured and uninsured. Let ȳ = E [y ( )] denote the average income of the population. And, let c =ȳ p I denote average consumption. To help illustrate the role of preference heterogeneity, assume is a uni-dimensional index, 2 R, and assume that the utility function, u (c, m; ), is continuously differentiable with respect to. Let = E [ ] denote the average in the population. To a first order Taylor approximation, the average marginal utility in the population, E y ( ) p I (s),m I ( );,isgivenbythemarginalutility of the average type, c, m,. 2 2 To see this, note that one can write y ( ) p I (s),m I ( ); as y ( ) p I (s),m I ( ); c (y ( ) p I c)+m m I ( ) m + 6

17 Using a Taylor expansion, the marginal utility of another type is given by y ( ) D ( s) p U,m I ( ); c, m, = c (y ( ) D ( s) p U (ȳ p I)) + m m I ( ) m + {z } {z } {z } Consumption Medical Spending Preferences (7) The difference in marginal utilities between the insured and uninsured depends on the average differences in consumption, medical spending, and preferences for the insured versus uninsured. E [ s apple s ] E [ s >s ] c (E [(y ( ) D ( s) p U (ȳ p I)) s apple s ] E [(y ( ) D ( s) p U (ȳ p I)) s >s ]) h +m E m I ( ) s apple s i E hm I ( ) s >s i + (E [ s apple s ] E [ s >s ]) where c, m,and are all evaluated at the average type, c, m,. To provide an estimate of (s) without significant additional data requirements, I impose additional assumptions that shut down the effects of these potential complementarities. To begin, it is certainly conceivable that spending more on medical spending could cause individuals to have a higher or lower marginal utility of income. 3 But, absent clear evidence on this, a natural benchmark is to assume that the marginal utility of consumption does not depend on the level of medical spending. Assumption. (No Complementarities/Substitutabilities between c and m)) The marginal utility function, (c, m; ) does not depend on m. This assumption implies that m m I ( ) m =0. It would be satisfied if, for example, preferences over c and m are additively separable (e.g. u (c, m; ) =a (c; )+b (m; ) for some functions a and b). It would also be satisfied in the broad class of models that assume a single consumption argument in the utility function, such as in the example in Section 2 (and the more general model of Handel et al. (205)) where c = y m and utility is only an argument of consumption, c. Next, consider the last term in equation (7). It again is certainly possible that differences in preference realizations or health shocks,, leadtodifferencesininsurancedemand. Forexample, those who choose to purchase insurance may have higher risk aversion (and thus higher values of c ). But, it is not immediately clear whether there is systematic differences between the insured and uninsured in their preferences,, thatwouldgeneratedifferencesinmarginalutilities,. 4 Thus, a benchmark assumption is to rule out a role of preference heterogeneity in generating differences in marginal utilities of consumption. where subscripts denote derivatives and c, m, and are evaluated at c, m,.hence, i i E h y ( ) p I (s),m I ( ); E hc (y ( ) p I c)+m m I ( ) m + h i = c (E [y ( )] p I c)+m E m I ( ) m + E [ ] which equals zero by the definition of c, m, and. 3 Note that the term m holds fixed the level of. Thismeansthatm is not about whether sicker or healthier have higher marginal utilities of consumption but rather conditional on a health state, does higher medical spending increase or decrease the marginal utility of consumption. 4 As discussed in Section 4.2, one potential source of a relationship between and would be if health status affects the marginal utility of income as in Finkelstein et al. (203). 7

18 Assumption 2. (Common Preferences) The marginal utility function, (c, m; ), does not depend on zero. This assumption implies that =0so that the last term in equation (7) is equal to Under Assumptions -2, the difference in marginal utilities depends only on the difference in consumption, y ( ) D ( s) p U (ȳ p I ),andthecurvatureoftheutilityfunction,c. 5 Moreover, the difference in consumption results from two components. First, there is a difference between the willingness to pay of the uninsured type, D ( s), andthepriceofinsurance,d (s) =p I p U. 6 Second, there is a potential difference between income, y ( ), andtheincomeoftheaverageinsured, ȳ. The third assumption rules out differences in marginal utilities driven by income differences. Assumption 3. (No Liquidity / Income Differences) Income does not systematically vary between insured and uninsured, ȳ = E [y ( ) s apple s] =E [y ( ) s >s]. Assumption 3 rules out liquidity effects as a primary source of variation in demand for insurance. As discussed in Section 4.2, one can incorporate liquidity effects if one is able to observe the average income levels of the insured and uninsured. Assumption 3 implies that the difference in demand between the marginal insured type, D (s )= p I p U,andtheaverageuninsuredtype,E [D (s) s s ] drives differences in consumption between the insured and uninsured. Combined with Assumptions and 2, this allows one to infer (s ) solely from the curvature of the utility function and the shape of the demand curve. Proposition 3. Suppose Assumptions -3 hold. Then, (s ) (D (s ) E [D (s) s s ]) (8) where the denotes a first-order Taylor approximation and = ucc( c, m; ) ( c, m; ) is the coefficient of absolute risk aversion evaluated at the average level of consumption, c, medical spending, m, and health status,. The ex-ante component of willingness to pay is given by EA(s ) ( s ) C (s ) D (s ) s D 0 (s ) (D (s ) E [D (s) s s ]) (9) so that it is identified from the demand and cost curves, combined with a coefficient of absolute risk aversion,. Proof. Imposing Assumptions -3 to equation (7) and dividing by the average marginal utility of income, c, m; yields the result. 5 This representation is analogous to the Baily-Chetty condition that characterizes the difference in the marginal utility of unemployed and employed using their difference in consumption multiplied by a coefficient of curvature of the utility function (Baily (978); Chetty (2006b)). 6 For example, in the example in Section 2, the willingness to pay of the average uninsured person is less than the price of insurance, p I p U, capturing the fact that the uninsured have a lower expected losses. 8

19 Equation (9) is identical to the expression in equation (4) of the stylized example. In the more general setup, it provides a benchmark method estimate important distinction between ex-ante and observed demand. 4.2 Violations of Assumptions -3 (s) and ascertain whether there is an While the general derivation of ex-ante measures of willingness to pay in Section 3 used a fairly general setup, the implementation assumptions above for measuring (s) rely on fairly strong assumptions. Here, I discuss these limitations in detail and also illustrate how one can often relax these assumptions with suitable additional empirical estimates. Assumption To begin, consider Assumption. Although this assumption is satisfied in many models of insurance that do not allow for m to be a separate argument of the utility function (e.g. Handel et al. (205)), Assumption is violated if consumption of medical spending is a substitute (or complement) to consumption. In this case, m 6=0and ucm( c, m; ) (s ) can be written as: (s ) (D (s ) E [D (s) s s ]) + m (E [m s s] m) where ucm = measures how the marginal utility of consumption varies with the level ( c, m; ) of medical spending (holding c and constant). This complementarity/substitutability of the utility function could be estimated with exogenous variation in both income and prices of medical spending, m. In particular, ucm would govern how individuals budget allocation between c and m varies if one faces higher prices for m but is compensated with an equivalent increase in income. Assumption 2 Assumption 2 would be violated if those who purchase insurance have a different marginal utility of consumption even if they have the same level of consumption. 6=0so that (s )= (D (s ) E [D (s) s s ]) + E [ s s ] In this case, One potential reason for 6=0would be if the marginal utility of consumption depended on health status. If sicker people have lower marginal utilities of income (as in Finkelstein et al. (203)), and the sick are more likely to purchase insurance, then those who purchase insurance may have lower marginal utilities of income than those who choose not to purchase insurance. Given an estimate of E [ s s ],onecouldrecover (s) in this more general setting. Heterogeneous Risk Aversion. One might have also thought that heterogeneous risk aversion could generate a violation of Assumption 2. For example, one would expect that the insured might have higher risk aversion than the uninsured. But, this does not necessarily lead to a violation of Assumption 2: it is not necessary that those with greater curvature in the utility function (second derivatives of u) also have greater(or lower) marginal utilities(first derivatives of u). For example, 9