Empirical Models of Demand for InsuranceRelate demand models to –rm™s problem of pricing and...

Empirical Models of Demand for Insurance

Liran Einav (Stanford and NBER)

Cowles Lunch Talk, Yale UniversitySeptember 18, 2013

Liran Einav (Stanford and NBER) ()Empirical Models of Demand for Insurance Cowles, Sept. 2013 1 / 33

Why insurance?

Insurance markets make a good target for economists

Large and importantPotential market failuresCommon government interventions via regulation or provisionLarge and detailed data sets are increasingly available

Extremely in�uential theory work on the potential for market failures(e.g., Arrow 1963; Akerlof 1970; Rothschild and Stiglitz 1976)

Early in�uential empirical work on insurance in the 1990s (e.g.,Chiappori and Salanie 2000) mostly focused on testing. Underlyingissues more subtle:

On the demand side, increasing evidence of important and relevantindividual heterogeneityOn the supply side, increased concerns about �lemon dropping�and�cherry picking�and the more general need to better understandcompetition, market power, and supply chains in these�contract/selection markets�


Plan for today�s talk

As the title suggest, I will mostly focus on demand and say very littleabout competition or market equilibrium:

To �x ideas, start with a general (fairly standard) framework ofdemand in insurance markets

Emphasize the key aspect that distinguishes demand in insurance:insurance providers directly care about who bought the product andhow he/she behaves after the purchase

Continue by describing three speci�c examples of demand models indi¤erent insurance contexts, all starting with a model of �realizedutility�

Connect these models to models that are based on indirect utility,which are more common in �traditional� IO

Relate demand models to �rm�s problem of pricing and plan design

(see also our review at Einav, Finkelstein, and Levin, 2010)



To �x ideas, start with a general (fairly standard) framework ofdemand in insurance marketsContinue by describing three speci�c examples of demand models indi¤erent insurance contexts, all starting with a model of �realizedutility�




General framework: notation

ζ: Vector of consumer characteristics (risk, preferences, income,information, etc.)

ζ = (x , ν): Useful to decompose into observables x and unobs. ν

(φ, p): Contract, i.e. coverage characteristics φ and a price(premium) p

S : Set of possible outcomes (e.g., accident)

A: Set of actions available to the consumer during coverage period(e.g., driving care)

Actions could be contingent, thus covering "ex-post moral hazard"


General framework: expected utility

Adopt a standard expected utility framework

Consumer�s valuation of a contract (φ, p) is

v (φ, p, ζ) = maxa2A ∑

s2Sπ (s ja, ζ) u (s, a, ζ, φ, p)

Let also a� (ζ, φ, p) denote the consumer�s optimal behavior, andπ� (�jφ, p, ζ) = π (�ja� (ζ, φ, p) , ζ)Convenient assumption: v (φ, p, ζ) is additively separable in p

Equivalent to CARA in the standard framework

1 Changes in the premium do not a¤ect consumer behavior a� or theoutcome probabilities π�; helps with identi�cation

2 Also leads to a natural choice of social welfare function


General framework: demand and cost

Insurer (expected) costs:

c (φ, ζ) = ∑s2S

π� (s jφ, ζ) τ(s, φ)

where insurance payments τ(s, φ) depend on outcome and coverage

Considering a set of insurance contracts J, consumer withcharacteristics ζ �nds contract j 2 J optimal i¤

v(φj , pj , ζ) � v(φk , pk , ζ) for all k 2 J

An important point I will get back to: while the primitives includeπ(�), u(�), S , and A, for many questions of interest knowledge ofv(�) and c(�) su¢ ces


The distinguishing feature of insurance markets

The key (I think) aspect which makes insurance (or credit) marketsdi¤erent from more traditional markets is that they are �selectionmarkets�

That is, unit costs depend on the composition of consumers (i.e.enrollee characteristics ζ) rather than just the quantity of consumers

Purchase and consumption/utilization are separated in timeConsumption/utilization enter the insurance company�s pro�ts

Notes:

The �rst feature is not uncommon in many �traditional�markets (e.g.,Dubin and McFadden, 1984)Even the second is sometimes important, but perhaps not as �rst orderas in insurance




Continue by describing three speci�c examples of demandmodels in di¤erent insurance contexts, all starting with a modelof �realized utility�Connect these models to models that are based on indirect utility,which are more common in �traditional� IO



Example 1: Deductible choice in auto/home insurance

Based on Cohen and Einav (2007)

Main objective: estimates of risk aversion

Auto insurance (in Israel) contracts characterized by a premium and a(per claim) deductible, (p, d)

Consider a binary choice between (pl , dl ) and (ph , dh) with dl < dhand pl > ph

Key assumptions:

No behavioral e¤ect of the contract (that is, risk invariant to coverage)Claims are a realization of a Poisson process (given the contracts, canfocus on claim counts and abstract from distribution of claim amount)Individuals know their (objective) individual-speci�c claim rate λ


Deductible choice (cont.)

Expected utility from contract φ = (p, d) over a short time period t:

v(p, d ,w ,λ,ψ) = (1� λt)u(w � pt) + (λt)u(w � pt � d)

Assume (heterogeneous) CARA utility u(x) = � exp(�ψx)

Model is convenient/transparent as one can obtain a closed-formexpression for the deductible choice:

Low Deductible , λ >ψ (pl � ph)

exp(ψdh)� exp(ψdl )

Low deductible more appealing if higher risk or higher risk aversion


Deductible choice (cont.)


Example 2: Guarantee choice in the context of annuities

Based on Einav, Finkelstein, and Schrimpf (2010)Main objective: estimate welfare consequences of suboptimal pricingdue to private information

Annuity contracts (in UK) chosen by retireesFocus on a setting with mandatory annuitization, so primary choice isthe guarantee periodContracts characterized by pairs of guarantee lengths (always 0, 5, or10 years in this market) and corresponding annuity rates z10 < z5 < z0Longer guarantees mean getting more if die quickly, but less if not

Standard annuity framework:Fully rational, forward looking, risk averse retireesRetirees with stock of wealth face stochastic mortality with mortalityhazard κt (α)Flow utility given by either utility from consumption (if alive) orbequest (if dead):

v(wt , ct ) = (1� κt ) u(ct ) + κtb(wt )


Annuity guarantee choice (cont.)

Without annuity, optimal consumption is the solution to

V NAt (wt ) = maxct�0

h(1� κt )(u(ct ) + δV NAt+1(wt+1)) + κtb(wt )

is.t. wt+1 = (1+ r)(wt � ct ) � 0

With annuity and guarantee g , modi�ed problem is

V A(g )t (w t ) = maxct�0

"(1� κt )(u(ct ) + δV A(g )t+1 (wt+1))

+κtb(wt + Zt (g))

#s.t. wt+1 = (1+ r)(wt + zt (g)� ct ) � 0

where Zt (g) =t0+g∑

τ=t

�� 11+r

�τ�tzτ(g)

�Optimal choice compares V A(0)0 (w0), V

A(5)0 (w0), and V

A(10)0 (w0)


Annuity guarantee choice (cont.)

Focus on heterogeneity in (mortality) risk and preferences (forbequest). Longer guarantee is more appealing if higher risk or greaterbequest preferences

Choose noguarantee period

Choose 10 yearguarantee period

Choose 5yearguaranteeperiod


Example 3: Health insurance coverage choice

Based on Einav, Finkelstein, Ryan, Schrimpf, and Cullen (2013)

Main objective: assess the importance of �selection on moral hazard�

Choice among �ve employer-provided PPO health insurance plans

PPO means that tradeo¤ is solely �nancial (as in two previousexamples); all choices associated with the same doctors, networks, andother restrictionsContracts characterized by the generosity of the out-of-pocket functioncj (m) and corresponding premium pjKey aspects of cj (m) are the annual (cumulative!) deductible level andthe oop maximum (although rarely binding)

Model designed to isolate three distinct determinants of anindividual�s coverage choice: health risk and risk aversion, which aresimilar to the two previous examples, as well as �moral hazard type,�which is likely more important in the context of health insurance


Health insurance choice (cont.)

An employee in a given year is characterized by

λ (monetized) health realizationFλ(�) that govern health riskψ coe¢ cient of absolute risk aversionω moral hazard type (price sensitivity)

Two period model:

Period 1: given (Fλ(�),ω,ψ), make optimal plan choice j� from a planmenu JPeriod 2: given plan j , health realization λ, and ω, make optimalutilization (spending) choice m� � 0

Note: totally abstract from the dynamics within the coverage year(which is the focus of two other papers of ours: one of which intomorrow�s IO seminar talk)


Health insurance choice: period 2

Utility in period 2 given by:

u(m;λ,ω, j) =�(m� λ)� 1

2ω(m� λ)2

�| {z }+ [y � cj (m)� pj ]| {z }

h(m� λ;ω) y(m)

Optimal spending given by

m�(λ,ω, j) = argmaxm�0

u(m;λ,ω, j)

With linear contracts (cj (m) = cjm):

m�(λ,ω, j) = λ+ω(1� cj )

so ω (�moral hazard type�) is the utilization di¤erence between fulland no insurance


Health insurance choice: period 1

An individual valuation of plans has a CARA form over period 2�srealized utility:

vj (Fλ(�),ω,ψ) =Z� exp(�ψu�(λ,ω, j))dFλ(λ)

so optimal plan choice given by

j�(Fλ(�),ω,ψ) = argmaxj2J

vj (Fλ(�),ω,ψ).

Optimal choice trades o¤ higher up-front payment for moresubsequent coverage

More coverage means both higher expected reimbursement and shedso¤ more risk

Higher coverage more attractive for �higher�Fλ(�) (risk), higher ψ(risk aversion), and higher ω (moral hazard)





Connect these models to models that are based on indirectutility, which are more common in �traditional� IORelate demand models to �rm�s problem of pricing and plan design


Connecting examples to models of indirect utility

In all three examples, the model primitive was the realized utility,while typically in IO we model valuation (or indirect utility) directly

Consider the auto insurance example. Choice given by

Low Deductible , λ >ψ (pl � ph)

exp(ψdh)� exp(ψdl )or, equivalently, the incremental �indirect utility� from upgrading to alow deductible coverage is

vi =λiψi(exp(ψidh)� exp(ψidl ))� (pl � ph)

Instead, one could directly (and perhaps more �exibly) specify v as afunction of price and deductible

vi = f (di ,LD , di ,HD ; θi )� (pl � ph)where θi is a vector of random coe¢ cients, allowed to be correlatedwith subsequent outcomes to account for possible selection.


Possible bene�ts from modeling realized utility

What is the bene�t, then, from modeling these �deeper�primitives?

1 Analyzing decision-making under uncertainty is one of the mostfundamental topics in microeconomic theory

Economic theory provides structure, guidance, and sometimesrestrictions on contract valuationsFor the same reason, the utility parameters of the �deeper�primitives(e.g., risk aversion) are often of independent interest, as they mayapply more broadlyIn contrast, economists are less likely to care about the deeper(psychological?) primitives that explain why, say, individuals valueconvertible cars





2 A model of deeper primitives allows for more guidance regarding outof sample extrapolation and for richer counterfactual exercises

Consider choices among health insurance plans with deductibles thatrange from zero to $1,000. A model of realized utility can provideguidance regarding extrapolating to, say, $2,000, and only a model ofrealized utility can tell us something about w.t.p. for, say, a plan with a�donut hole�





2 A model of deeper primitives allows for more guidance regarding outof sample extrapolation and for richer counterfactual exercises

3 Perhaps also more elegant: connection between ex-ante risk andex-post cost is more clearly and transparently speci�ed


But even in insurance, this is not always necessary

For many questions of interest (pricing or welfare from similarcontracts), however, a model of contract valuation v may su¢ ce

Indeed, Einav, Finkelstein, and Cullen (2010) take such approach

In other insurance settings, products also vary in non-�nancialaspects. Then, the appeal of a realized utility model diminishes

Consider, for example, the setting of Bundorf, Levin, and Mahoney(2012) who model health insurance choice between HMO and PPOIndeed, they revert to a model that looks much more like the�standard� indirect utility model:

vij = φj + z0i βj + f (ri + εi ;γj )� αpj + εij

with a key distinction being the importance of the correlation betweensome of the random coe¢ cients (in this case εi , as well as ri which isobserved) and insurer�s cost


Informal notes on estimation and identi�cation

Both approaches lead to estimates of demand and claims rates andhow they covary with consumer preferences. Moreover, both recoverthe essential information needed to analyze consumer surplus orexplore the implications of many policy interventionsFor either approach, source of identifying variation is key, and itscredibility should be evaluated in a given context

Much work on general identi�cation results for v , but not much workon formal identi�cation results for general cases of uInitial results in two recent papers by Perrigne and Vuong

Two aspects that are important to pay attention to in insurance:

Because o¤ers and choice sets are highly customized to individualconsumers, need disaggregated data at the choice-set level (at least),otherwise may confuse demand and supplyNot impossible, but di¢ cult to make empirical progress w/o some dataon subsequent outcomes. Simple (but loose) intuition for identi�cationis of a 2-to-2 mapping (or 3-to-3 in the context of third example)






Relate demand models to �rm�s problem of pricing and plandesign


Pricing and plan design

Easiest to illustrate issues with a monopoly provider of a singleinsurance contract, (φ, p)Normalizing consumer value from no coverage to zero, andmaintaining the quasi-linearity assumption, a consumer withcharacteristics ζ will purchase the contract if v (φ, ζ) � pShare of consumers who purchase will be:

Q(φ, p) =Z1fv (φ, ζ) � pgdF (ζ)

and the insurer�s expected costs are:

C (φ, p) =Z1fv (φ, ζ) � pgc (φ, ζ) dF (ζ)

The �rm�s problem:

maxφ,p

Π(φ, p) = p �Q(φ, p)� C (φ, p)


Pricing and plan design (cont.)

Fixing the coverage φ, the e¤ect of a small increase in price is:

dΠ(p)dp

= Q (p) +dQ(p)dp

��p �Eζ [c (φ, ζ) jv(φ, ζ) = p]

�The attractiveness of the marginal consumer relative to the averageconsumer plays a key role

Choice of φ is similar, with the added subtlety that changes incoverage may a¤ect utilization as well as selection:

dΠ(φ)dφ

=dQ(φ)dφ

��p �Eζ [c (φ, ζ) jv(φ, ζ) = p]

��Eζ

�∂c (φ, ζ)

∂φjv(φ, ζ) � p

�So an additional e¤ect of coverage features is its e¤ect on costs,potentially by inducing behavioral changes as well as mechanicallyEasy to come up with examples where the marginal consumer w.r.t. pis di¤erent from the marginal consumer w.r.t. φ


Pricing and plan design (cont.)

The types of demand models described earlier provide exactly theprimitives needed to ��ll in� these �rst order conditions

One can extend a similar analysis to an oligopolistic setting, althoughthis would raise both conceptual and computational challenges

Conceptually, even a game in which �rms compete in prices alone maynot have the convexity properties typically invoked to assure existenceor to justify an analysis based on �rst order conditions for optimalpricingComputationally, even if an equilibrium does exist and even if it can becharacterized in terms of �rst order conditions, solving numerically forthe equilibrium may be challenging


Illustrate using the deductible choice example

Recall the �rst (auto insurance) example, where individuals arecharacterized by (λi ,ψi )

λi is the individual-speci�c Poisson claim rateψi is the coe¢ cient of (absolute) risk aversion

Only λi a¤ects insurer�s pro�ts. In particular, the incremental cost(to the insurer) associated with selling a low deductible contract toindividual i is

c (dl ,λi )� c (dh,λi ) = λi (dl � dh)

thus linking demand to cost

Consider now how this a¤ects pricing decisions (next slide)


Di¤erential incentives to raise prices due to selection


Final remarks

Described recent empirical models of insurance and their applications

Did not cover results, but some initial results are somewhat surprising:general �nding that certain kinds of welfare losses from asymmetricinformation, at least in some insurance markets, may be modest

This is clearly a very early stage of the literature, so lots of room and(I think) interest for more work

Many many open topics, as well as many other insurance marketsthat haven�t been intensively explored

Underwriting and risk selection (Hendren, f�coming)Dynamic insurance provision (Hendel and Lizzeri, 2003; Handel,Hendel, and Whinston, 2013)Consumer search and switching costs (Handel, f�coming)Alternative models of consumer behavior (Barseghyan et al., f�coming)


Empirical Models of Demand for InsuranceRelate demand models to –rm™s problem of pricing and...

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