Deductible and welfare in the U.S. health insurance market : a two-sided duopoly approach

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Deductible and welfare in the U.S. health

insurance market: a two-sided duopoly approach

M.P.A. (Milan) Karsten

Student number: 10276734 Date of final version: March 9, 2017 Master’s programme: Econometrics

Specialization: Free Track

Supervisor: dhr. dr. D. (Dávid) Kopányi Second reader: prof. dr. J. (Jan) Tuinstra

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Abstract

We construct a two-sided duopoly model that allows to analyse the welfare effects of the High-Deductible Health Plan (HDHP), a new type of insurer that is rapidly gaining market share in the U.S. health insurance market. The HDHP competes with one other health plan that does not include a deductible in its insurance contract. As the agents on either side of the market are affected by the agent on the other side, we incorporate the externalities of both sides. We find that it is best for total welfare when the HDHP is an HMO in terms of the physician diversity, although this is at the expense of the policyholders and goes along with higher health care costs. Total welfare decreases when the HDHP is a PPO in terms of the physician diversity, although the total health care costs decrease and the policyholders may benefit. We furthermore show that the found equilibria can be supported by learning methods and are stable under learning.

Statement of Originality

This document is written by Student Milan Karsten who declares to take full responsibility for the contents of this document. I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it. The Faculty of Economics and Business is responsible solely for the supervision

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1 Introduction

There seems to be a permanent debate on health care. How should countries design their health care systems? What role should the government perceive in these health care systems? Is the privatization of public health care systems a desirable trend? How can we cope with increasing health care expenditures?

One of the most commonly used policy instruments that intends to cut health care costs is the deductible excess. Policymakers hope that policyholders lower their health care consumption as they are more aware of the costs they induce and do more to prevent inducing health care costs. The High-Deductible Health Plan (HDHP) is the extreme version of this policy instrument. The minimum annual deductible excess of an HDHP is $1300 per person or $2600 per family. US taxpayers enrolled in an HDHP can open a Health Savings Account (HSA), allowing tax-free deposits to be used for future medical expenses. Therefore, since the introduction in 2006, the HDHP has become extremely popular. We can see this in Figure 11. Figure 1 shows the health plan enrollment for covered workers over the past thirty years. The rise of the HDHP has come at the expense of the Preferred Provider Organisation (PPO) and the Health Maintenance organisation (HMO). PPOs typically offer wide access to health care providers at a high cost, while the HMO offers limited access to health care, but low insurance premiums.

The question asked in this paper is how socially desirable the presence of the HDHP in the market is. We analyse the effects of a deductible excess on the welfare of the three agents in health insurance markets: the health plans, the policyholders and the health care providers. The health care providers are represented by physicians. We build on the model of Bardey and Rochet (2010). Bardey and Rochet study the competition between a PPO and an HMO. Policyholders value the quality of and access to health care, which is represented by the number of affiliated physicians each health plan offers. As HDHPs can be PPOs or HMOs in terms of access to and quality of health care, we distinguish two models. One in which the PPO competes with an HDHP, where the HDHP offers a low diversity of physicians. The other model assumes the HDHP is the firm that offers a high diversity of physicians, and that the HDHP competes with a HMO. As the equilibrium can not be calculated fully algebraically, numerical methods were used. As such, we simulate what happens to the equilibrium and welfare of the agents when a deductible is set for many different parameter

1

The table is taken from the Employer Health Benefits Survey (2016) published yearly by the Kaiser Family Foundation (KFF).

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Figure 1: Market shares of US health plans over the past 30 years

values, and compare this situation to when there was no deductible. When the deductible is set to zero both extensions of this paper are equal to the model of Bardey and Rochet (2010), allowing to effectively compare all models. As the effect can not be determined algebraically and has to be simulated, we can for most determinants only make conjectures of market and agent behaviour. The stability of the equilibria is furthermore analysed by gradient learning.

When the HDHP is the firm offering the lower diversity of physicians, we find that increasing the deductible increases the combined welfare of all agents. The market share of the HDHP increases. We expect that the PPO and HDHP both have increased profits as a result of the deductible. The physicians that represent the health care providers benefit: More physicians are hired by both health plans and the fee-for-service rates paid to the physicians increase. As a higher diversity of physicians is offered, we expect an increase in the average quality of health care. Total policyholder welfare is expected to decrease for most reasonable parameter values and the decisive parameters are identified.

If the HDHP is the health plan offering the higher diversity of physicians, we find that the HDHP does not always have an incentive to set a positive deductible. When the HDHP does set a positive deductible, total welfare is expected to decrease. The profits of the HMO are expected to always decrease. The health care providers are expected to earn lower fee-for-service rates and are

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policyholders may increase and we identified parameters that affect this. The results with learning indicate that the health plans can learn the equilibrium choices, and that the equilibria are stable under gradient learning.

The paper is organised as follows. Section 2 discusses the relevant literature. In Section 3 the model assumptions and the two-sided market effects and externalities are discussed. Section 4 derives the equilibrium in the case the HDHP is the firm offering the lower amount of physicians, while Section 5 analyses comparative statics of the model. The equilibrium of the second model is derived in Section 6, and we look at comparative statics in Section 7. Section 8 applies gradient learning on both models and discusses stability. In Section 9 the conclusions are discussed.

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2 Related Literature

There have been numerous papers on health insurance markets, and even more on two-sided market effects. The paper most relevant for this thesis is the paper of Bardey and Rochet (2010). Bardey and Rochet investigate competition between a PPO and an HMO in a two-sided framework. The two health plans compete for policyholders on one side and for physicians on the other side. PPOs typically offer a higher diversity of physicians than HMOs, but also ask higher premiums. The authors find that even though PPOs tend to attract policyholders with a higher risk, they can be more profitable due to the reduced fee-for-service rates paid to the physicians as they have a higher market share. Bardey and Rochet conclude that the outcome of the competition depends on two effects: The demand and the adverse selection effect. The demand effect is influenced by the extent to which policyholders value the diversity of physicians offered, while the adverse selection effect depends on the distribution of health risks. This result provides insights to the rise of the PPO at the cost of the HMO that was apparent from 1996 to 2005 (see Figure 1). Bardey and Rochet conclude that it could be that the demand effect dominated, which led to higher profits for the PPO, and drove the HMO partially out of the market.

HDHPs can be either PPOs or HMOs in terms of the amount of physicians they are affiliated with. Therefore we look at two different models. In the first one the PPO is the firm with the higher amount of affiliated physicians, in that case the HDHP will offer the lower diversity of physicians. Hence the HDHP will serve the demand of policyholders with a low probability of illness. The policyholders with a low probability of illness want to benefit from the lower premiums offered by the HDHP. Moreover, they obviously do not mind an insurance contract that specifies a deductible excess as much as policyholders with a high probability of illness. Therefore it is clear which policyholders want to sign a health insurance contract with each health plan. In the second model the HDHP is the firm with the higher number of affiliated physicians, and competes an HMO that offers a lower diversity of physicians. Now policyholders with a high probability of illness may prefer the HDHP, even though it comes with a deductible excess. Therefore it is somewhat less intuitive which health plan policyholders prefer.

Although there has not been extensive analytical research on copayments in health insurance markets, this paper is not the first. One paper that has done research on this topic is that of Bardey, Cremer, and Lozachmeur (2016). The authors analyse how insurance companies will design their

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copayment and coinsurance contracts in different models of imperfect competition. The coinsurance rate is defined as a percentage of total medical expenditure. The copayment rate is a fixed per-product rate, which is the similar to the deductible in this paper. By effectively setting both rates the authors show that insurers can control the health care provider’s price and extract its surplus. This ’optimal’ contract comprises the lowest possible level of the copayment rate together with the highest level of the coinsurance rate such that the constraint that the health care providers break even is binding. The insurance contact that the HDHP offers in this paper can be seen as a contract with a zero coinsurance rate and a copayment rate equal to the deductible excess. The model of Bardey, Cremer, and Lozachmeur does however not incorporate the quality aspect of health care. Policyholders choose the health plan only based on the price, coinsurance and copayment rate. Therefore the model does not take the adverse selection effect and demand effect of the model of Bardey and Rochet (2010) into account.

Boilley (2013) analyses a model that is very similar to the model of this paper. The same three agents interact: physicians, health plans and policyholders. The health plans, however, are also exogenously horizontally differentiated based on their location, which is not the case in this paper. Boiley’s goal is to distinguish market behaviour into different effects, such that the forces that drive the equilibrium can be analysed more theoretically. The author finds that the resulting Nash equilibria can be explained by two standard effects of product differentiation and two externalities of the network effects. The price competition effect drives firms to become differentiated, while the market share effect drives firms to play similar strategies. The negative effect of the network externalities corresponds to the higher costs when a higher diversity of physicians is offered. This effect is called the negative spending effect. The positive externality that results from the network effects is the effect that allows the health plans to set higher premiums as the health plans offer a higher diversity of physicians. It is called the earning effect. We will see that these effects are also present in this paper.

Baranes and Bardey (2015) analyze the equilibrium outcome of competition between multiple HMOs and conventional insurers in a two-sided model. Policyholders that choose an HMO can only receive health care from one health care provider. Conventional insurers, however, offer access to all providers that are not affiliated to an HMO. Therefore when the number of HMOs and the vertical integration increase, the diversity of physicians offered by the conventional insurers decreases. The authors conclude that the equilibrium allocation only depends on the number of HMOs when health

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care providers sign exclusivity contracts with their HMO. In addition they find that the interplay between risk segmentation and the price paid to the health care providers may induce ambiguous results: vertical integration may decrease health insurance premiums and may increase policyholder utility, depending on the marginal treatment costs.

Bardey and Bourgeon (2011) consider a two-sided model where two health plans compete for policyholders on one side and for two health care providers on the other side. The health care providers are horizontally differentiated with respect to their location and are partially altruistic. The authors show that the policyholders are better off under competition between conventional insurers compared to competition between Managed Care Organisations (MCOs). Under MCO competition, the horizontal differentiation at the provider level allows the health plans to ask higher mark-ups on premiums. On top of that the horizontal differentiation relaxes competition, resulting in a lower level of health care quality in equilibrium. The model used in this paper is very similar, although the focus is on the welfare effects of a deductible and we consider different health plans.

We will use components of the welfare analysis as done in Jofre-Bonet (2000). Jofre-Bonet analyzes the welfare effects of government intervention in different markets. Although this paper analyzes the welfare effects of a new type of insurer, the HDHP, the approach is very similar.

Brooks, Dor, and Wong (1997) analyse which factors affect the bargaining power of hospitals in an empirical study. They find that greater HMO penetration in a U.S. state tends to increase hospital bargaining power, which feels rather counter intuitive. We will analyse whether an increase in HMO market share will indeed increase fee-for-service rates, although the approach from this paper is not empirical but theoretical.

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3 Model preliminaries

The set up of the model is similar to that of Bardey and Rochet (2010) (see Section 2). It comprises three types of agents that interact.

• Two health plans, the PPO and HDHP. The health plans compete for policyholders on one side and physicians on the other side.

• Policyholders. Policyholders choose to sign a health insurance contract with one of two health plans. They have a probability of θ to become ill. This probability is heterogeneous, dis-tributed on the interval (0,1) by density f. The cumulative distribution function is denoted by F. The model will be built for an arbitrary distribution function F, which will later be specified and analysed.

• Physicians. Physicians choose to be affiliated with either of the health plans.

3.1 The health plans and policyholders’ utility

Policyholders value the diversity of physicians that the two health plans offer when they become ill. More physicians is assumed to imply better (access to) health care. We will assume a mass of physicians that we normalise to 1. Denote the physician diversity offered by a health plan by J_i where i = PPO, HDHP. The utility of the policyholder is hence increasing in the number of physicians of the chosen health plan and decreasing in the premium paid. Whereas the model of Bardey and Rochet assumed full insurance, we introduce a deductible excess that has to be paid when ill. Only the insurance contract of the HDHP includes a deductible. The resulting expected utility of a policyholder with probability of illness θ is hence denoted by

EUθ(PPPO, JPPO) = u(w − PPPO) + θλJPPO (3.1)

EUθ(PHDHP, JHDHP, DHDHP) = u(w − PHDHP− θDHDHP) + θλJHDHP (3.2)

where λ is the policyholder’s preference for diversity and w is the wealth of the policyholder before signing a health insurance contract. We assume λ is homogeneous across policyholders. When the deductible is higher than the unit cost of treatment, the policyholder would be better off by not signing a health insurance contract, but directly contacting a physician when ill. However,

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we assume that the consumers can only contact physicians through the health plans meaning that the deductible could be higher than the unit cost of treatment.2 As in Bardey and Rochet (2010) we assume the premium paid is not too large compared to the wealth of the policyholder such that u(.) can be taken as linear. As explained in the previous section, HDHPs can be PPOs or HMOs in terms of the amount of physicians that they are affiliated to. For this first model we assume that the number of physicians affiliated to the HDHP is equal to the number of physicians affiliated to an HMO. Hence the number of physicians affiliated to the HDHP is lower than the PPO: JPPO > JHDHP. To be competitive, the HDHP has a lower premium than the PPO: PPPO> PHDHP.

The demands for both health insurers are based on the distribution of the probability of illness. A policyholder with probability of illness θ will prefer the PPO over the HDHP when:

Uθ(PPPO, JPPO) > Uθ(PHDHP, JHDHP, DHDHP) (3.3)

We assume that the market for health insurance on the policyholder side is covered. For the US health insurance market this is almost the case as according to Zammitti, Cohen, and Martinez (2016) the uninsured rate is estimated at 8,6% in 2016 and was expected to decrease, although the new Republican administration led by Donald Trump might invert this trend. When the market for health insurance on the policyholders side is covered, the policyholder that is exactly indifferent be-tween the two health plans can be found. The probability of illness corresponding to this indifferent policyholder is defined by

˜

θ = PPPO− PHDHP λ(JPPO− JHDHP) + DHDHP

(3.4)

I will from now on refer to this probability of illness as the indifferent probability. Note that the indifferent probability is independent of the wealth of the policyholder. Policyholders with a high probability of illness choose the PPO, hence the PPO faces a demand equal to 1 − F(˜θ). Similarly the resulting demand for the HDHP is F(˜θ).

2_{Health plans that include a deductible excess usually also offer lower premiums. Adding a discount parameter to}

the premium would thus be a more realistic representation of the price of the HDHP. However, it turns out that this addition does not change the equilibrium prices, physician quantities and profits as the HDHP just sets a net price. This discount variable is thus left out.

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3.2 Physicians

As in Bardey and Rochet (2010) we assume that the market for the physicians is not covered such that JP P O + JHDHP < 1, and not all physicians are affiliated to one of two health plans. The

net profit level offered by health plan i we denote by Φ_i. The number of physicians that choose to be affiliated with health plan i equals Ji = Φ_δi, hence δ captures the sensitivity of supply of

the physicians to the net profit levels that the health plans offer. The net profit level equals the expected level of activity per physician times the profit margin. The profit margin is equal to the fee-for-service rate R_iminus the unit cost of treatment (marginal cost). The net profit levels of the physicians are given by

ΦPPO= RPPO− c JPPO Z 1 ˜ θ θdF(θ) (3.5) ΦHDHP= R_HDHP− c J_HDHP Z θ˜ 0 θdF(θ) (3.6)

Hence we assume that the health plans set the remuneration of their physicians in such a way that they attract exactly the preferred diversity of physicians Ji. Note that when a health plan

increases its market share and thus its expected activity level, the health plan can offer a lower fee-for-service rate to attract the same amount of physicians. This is the externality of policyholders’ side of the market. We have assumed that policyholders single-home, i.e. they can choose to be affiliated with only one health plan. According to Rochet and Tirole (2003) single-homing end users in two-sided markets are advantageous to the intermediate agent in the market, as it increases the bargaining power compared to a situation in which end-users multi-home. This would imply that the bargaining power of the health plans is rather high and that they would be able to pay low fee for service rates, and hence a low value for δ. We find that the total profit of the physicians affiliated to a health plan is equal to

ΦPPOJPPO= δJ2PPO= (RPPO− c)

Z 1 ˜ θ θdF(θ) (3.7) ΦHDHPJHDHP = δJ2HDHP= (RHDHP− c) Z θ˜ 0 θdF(θ) (3.8)

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4 Model 1: PPO-HDHP competition

The two health plans maximize the profit functions given by

ΠPPO= [1 − F(˜θ)]PPPO− δJ2PPO− c

Z 1 ˜ θ θdF(θ) (4.1) ΠHDHP= F(˜θ)PHDHP− δJ2HDHP+ (DHDHP− c) Z θ˜ 0 θdF(θ) (4.2)

Note that the HDHP has additional incomes of the deductible times the expected activity level. Taking first order derivatives with respect to P_PPO and P_HDHP we obtain3:

∂ΠPPO ∂PPPO = 1 − F(˜θ) + [c˜θ − PPPO]f(˜θ) ∂ ˜θ ∂PPPO = 0 ∂ΠHDHP ∂PHDHP = F(˜θ) + [PHDHP+ (DHDHP− c)˜θ]f(˜θ) ∂ ˜θ ∂PHDHP = 0 Using that ∂ ˜θ ∂PPPO = − ∂ ˜θ ∂PHDHP = 1 λ(JPPO− JHDHP) + DHDHP we find P_PPO= c˜θ+λ(JPPO− JHDHP) + DHDHP 1 − F(θ)˜ f(˜θ) (4.3) P_HDHP = (c − DHDHP)˜θ+λ(JPPO− JHDHP) + DHDHP F(θ)˜ f(˜θ) (4.4)

Taking first order derivatives of the profit functions with respect to J_PPO and J_HDHP we obtain

∂ΠPPO ∂JPPO = −[PPPO− c˜θ]f(˜θ) ∂ ˜θ ∂JPPO − 2δJ_PPO = 0 ∂ΠHDHP ∂JHDHP = [PHDHP+ (DHDHP− c)˜θ]f(˜θ) ∂ ˜θ ∂JHDHP − 2δJHDHP = 0

Using again that

3_{The profit functions were plotted and it was concluded that the found equilibrium prices and physician diversities}

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∂ ˜θ ∂JPPO = − ∂ ˜θ ∂JHDHP = −λθ λ(JPPO− JHDHP) + DHDHP we find J_PPO = λ 2δθ(1 − F (˜˜ θ)) (4.5) J_HDHP = λ 2δ ˜ θF (˜θ) (4.6)

Plugging the physician numbers back into equation (4.3) and (4.4) we find that the prices are defined by PPPO= c˜θ+ " λ2 2δ ˜ θ[1 − 2F(˜θ)] + DHDHP # 1 − F(˜θ) f(˜θ) (4.7) PHDHP = (c − DHDHP)˜θ+ " λ2 2δ ˜ θ[1 − 2F(˜θ)] + DHDHP # F(˜θ) f(˜θ) (4.8)

Plugging these values back in equation (3.4) we find that ˜θ solves the equation

g(˜θ, DHDHP) = ˜ θDHDHP λ2 2δθ[1 − 2F (˜˜ θ)] + DHDHP + 1 − 2F (˜θ) f (˜θ) − ˜θ = 0 (4.9)

From equation (4.9) we find that by adding a deductible to the model multiple indifferent probabilities ˜θ can co-exist on the interval (0,1). The number of different ˜θ on the interval (0,1) depends on the distribution of the probability of illness, the deductible and the ratio λ_δ2. This has substantial implications for the complexity of the model as we will see in Section 5.1. In a similar way as in Bardey and Rochet (2010) the mark-ups on net premiums including deductibles are calculated to obtain a simplified representation of the equilibrium profits. The mark-ups become:

P_PPO− ˜θc = " λ2 2δθ[1 − 2F(˜˜ θ)] + DHDHP # 1 − F(˜θ) f(˜θ) (4.10) P_HDHP− (c − D_HDHP)˜θ = " λ2 2δθ[1 − 2F(˜˜ θ)] + DHDHP # F(˜θ) f(˜θ) (4.11)

Plugging the found equilibrium values of the premiums and physician diversity in the profits of the insurers we obtain the equilibrium profit functions:

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ΠPPO = " λ2 2δθ[1 − 2F(˜˜ θ)] − λ2 4δθ˜ 2_f(˜_{θ) + D} HDHP # [1 − F(˜θ)]2 f(˜θ) − c Z 1 ˜ θ (θ − ˜θ)dFθ (4.12) ΠHDHP = " λ2 2δθ[1 − 2F(˜˜ θ)] − λ2 4δθ˜ 2_f(˜_{θ) + D} HDHP # F(˜θ)2 f(˜θ) + (DHDHP− c) Z θ˜ 0 (θ − ˜θ)dFθ (4.13)

Note that when D_HDHP = 0, the equilibrium profits simplify to the profits of the the model of Bardey and Rochet as

˜

θ = 1 − 2F(˜θ) f(˜θ)

when DHDHP= 0. The optimal value for the deductible can now be found by backward induction.

For the welfare analysis measures of policyholder welfare are required. Total policyholder welfare is defined as the integral of a simplified version of the policyholder utility given by equation (3.1) and (3.2) over all policyholders. We assume again that u(.) is linear and we do not incorporate the policyholders’ initial wealth w, as the initial wealth of the policyholders does not affect the change in utility. We find that the total policyholder welfare is given by

TU = Z 1 ˜ θ f (θ)(θλJPPO− PPPO)dθ + Z θ˜ 0 f (θ)(θ(λJHDHP− DHDHP) − PHDHP)dθ (4.14)

I define average policyholder utility as the total utility of the policyholders of a health plan divided by the total number of policyholders of the health plan. This measure can give better insight in the utility of policyholders per health plan as total utility per health plan changes as market share changes. Hence the average policyholder utility is given by

AUPPO = R1 ˜ θ f (θ)(θλJPPO− PPPO)dθ 1 − F (˜θ) (4.15) AU_HDHP= Rθ˜ 0 f (θ)(θ(λJHDHP− DHDHP) − PHDHP)dθ F (˜θ) (4.16)

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5 Equilibrium analysis model 1

As mentioned before, the addition of the deductible excess increases the complexity of the model as multiple indifferent probabilities on the interval (0,1) can exist. This implies that we can no longer find the closed form solution for ˜θ. Hence we can not find the deductible resulting in the maximum profit for the HDHP analytically. This has to be done by analysing profits for multiple values of the deductible.

5.1 Distribution of the probability of illness

To be able to numerically calculate the indifferent probability and the optimal value for the de-ductible it is necessary to make an assumption about the distribution of the probability of illness. A flexible choice is the iso-elastic probability distribution F(θ) = θ. Figure 2a shows the pdf of the probability of illness for different values of . For = 1 this distribution is the uniform distribu-tion. For = 2 the distribution is right-skewed meaning that most of the policyholders have a high probability to become ill. A right-skewed distribution for the probability of illness is hence more advantageous for the PPO, as it offers a higher number of affiliated physicians. A more left-skewed f(θ) for = 1₂ is advantageous to the HDHP, as more policyholders have a low probability of illness, meaning they do not value the physicians as much and want to benefit from the lower premium of the HDHP. The expected value of θ given f(θ) is defined by

E[θ|f(θ)] = Z 1

0

θf(θ)dθ = + 1

This expected value can be used to argue a value for that best describes the U.S. health insurance market. According to the National Center for Health Statistics (2014) 83.2% of U.S. adults had reported to have had contact with at least one health care professional in the past year. For children this rate is even higher: 92.4%. This rate is lowest at 76,8% for adults of age 18-44. When the adults get older, the rate increases. As the average age is rising, we can conclude that is increasing. Assuming ’having had contact with a health care professional in the past year’ defines being ill, would approximately be equal to 5. Hence one could argue that a very left-skewed distribution of illness (low ) is not reasonable to assume.

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(a)Distribution of probability of illness for different values of

(b) g(˜θ, DHDHP), = 2, DHDHP = 10, λ = 60, δ = 20,c =

30.

Figure 2: Illustration of f(θ) and g(˜θ, DHDHP).

5.2 Conditions on ˜θ

In equilibrium there are three conditions that constrain ˜θ. First we assume that the physicians’ market is not covered, J_PPO+ JHDHP < 1. As the functional form of the physician diversities in

equilibrium is the same as in the model of Bardey and Rochet (only the definition of ˜θ is slightly changed), the equilibrium physician diversities can be plugged in which can be rearranged to find the same condition

˜ θ < 2δ

λ (5.1)

The second condition requires that the market share of the HDHP is smaller than the PPO, as the number of physicians affiliated to the HDHP is smaller than the PPO. This results in the condition

˜ θ < 1

2 (5.2)

The third condition is that the price of the PPO is larger than the HDHP: P_PPO > PHDHP.

This condition is always satisfied in equilibrium as long as (5.2) holds. By plugging in the prices from (4.7) and (4.8) we find

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DHDHPθ > [λ(J˜ PPO− JHDHP) + D]

2˜θ_{− 1}

˜θ−1 (5.3)

The left hand side is always non-negative, and as ˜θ < 1₂ and JPPO > JHDHP we have that the

right hand side is always negative. Hence this condition does not impose more restrictions on ˜θ.

5.3 The indifferent probability of illness

Having made an assumption on the probability of illness, the indifferent probability can now be calculated for different representations of F(θ). Plugging in F(˜θ) = ˜θ_{, equation (4.9) is given by}

g(˜θ, DHDHP) = ˜ θDHDHP λ2 2δθ[1 − 2˜˜ θ] + DHDHP + 1 − 2˜θ ˜θ−1 − ˜θ = 0 (5.4)

For the graphs in this section we will use parameter values of δ = 20, λ = 60 and c = 30. To give more insight in the indifferent probability defined by (5.4), Figure 2b plots this function for = 2 and D_HDHP= 15. From equation (5.4) it is clear that ˜θ = 2−1 is always a solution. This solution is

also an upper bound for ˜θ resulting from condition (5.2), as F (2−1) = 1

2, and hence not valid as the

indifferent probability. As g(˜θ, DHDHP) is positive for very small positive θ, there is another root for

θ on the interval (0, 2−1) when the derivative is positive at θ = 2− 1

. For sufficiently small D_HDHP

this is the case. An indifferent probability on the interval could also exist when the derivative is negative. This is the case when g(˜θ, DHDHP) is not convex on the interval. However the simulations

will show that there will never exist multiple valid ˜θ on the interval (0, 2−1). For certain parameter

values there is a ˜θ on the interval ∈ (0, 2−1), but condition (5.1) is not satisfied. These conditions

lead to Lemma 1.

Lemma 1

The indifferent probability of illness ˜θ must lie in the interval

0, min 2−1,2δ λ ! (5.5)

to satisfy all conditions.

Figure 3 depicts the effects of the deductible on the indifferent probability and the physician diversities for different values of . Note that the range of the deductible-axis for the three graphs is different as the maximum indifferent probability that satisfies all conditions is different for different

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(a) = 0.5 (b) = 1 (c) = 2

Figure 3: Indifferent probability and physician diversities for different representations of the distri-bution of illness.

(a) = 0.5 (b) = 1 (c) = 2

Figure 4: Prices of the two health plans for different representations of the distribution of illness.

(see Lemma 1). In the case = 2 an increase in the deductible leads to an increase in the indifferent probability. As policyholders with the introduction of the deductible now have to pay when ill, one would expect that more policyholders would prefer to join the PPO and that the indifferent probability would go down. However, it turns out that this is not the case. Figure 4 shows the prices of the two health plans for different representations of the distribution of illness. The difference between the two prices increases as the deductible increases. For = 2 the HDHP lowers its price while the PPO increases its price. Meanwhile, the HDHP significantly increases the number of affiliated physicians4, while the number of affiliated physicians of the PPO only changes slightly.

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For all values of the difference between the physician diversity offered by the two health plans decreases as the deductible increases. The net effect is an increase in the indifferent probability. The intuition behind this is as follows: As a result of the increased deductible the PPO will ceteris paribus have an increased demand. To maximize profits it is optimal for the PPO to increase its price, which drives the indifferent probability up. The HDHP now earns a deductible each time one of its policyholders gets ill. When there is little policyholder mass nearby it is profitable to increase its price ( = 2). However when there is more policyholder mass nearby ( = 0.5), it is best to lower its price to increase demand.5 By using the implicit function theorem one can prove that the indifferent probability is always increasing in D_HDHPwhen D_HDHP= 0 as long as (2+)θ+ > 1 (see appendix A). To check whether the indifferent probability also increases when the condition is not met (for valid combinations of ˜θ and < 1), we calculate how the indifferent probability changes for many different parameter combinations. Varying λ, δ ∈ (0.1, 100), c ∈ (0, 50), ∈ (0.1, 0.9), 80000 different parameter combinations were analysed. For 83,9% of the simulations a valid ˜θ was found. For all of these simulations the derivative of ˜θ with respect to DHDHPwas positive when DHDHP = 0.

Hence we can conclude that no parameter values exist for which the condition does not hold and that a small increase in D_HDHP when D_HDHP= 0 will always increase the indifferent probability.

Proposition 1. A small increase in the deductible, compared to when the deductible is zero, will increase the indifferent probability, as both health plans will adjust prices and physician amounts. This adjustment will more than offset the direct change of the indifferent probability due to the change in the deductible.

Taking the derivative of the equilibrium physician diversities we find

∂JPPO ∂DHDHP = λ 2δ(1 − ( + 1)˜θ ₎ ∂ ˜θ ∂DHDHP (5.6) ∂JHDHP ∂DHDHP = λ 2δ( + 1)˜θ ∂ ˜θ ∂DHDHP (5.7)

For the HDHP we find that the derivative has the same sign as _D∂ ˜θ

HDHP. For the PPO this

is only the case when ˜θ < (₊₁1 )1.6 To check whether parameter combinations of ˜θ and exist 5

For some parameter values the HDHP even sets a price below zero. This would mean that policyholders earn a premium, but have to pay when they get ill, for an amount that is more than the cost of the treatment.

6

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that violate this condition we vary λ, δ ∈ (0.1, 100), c ∈ (0, 50), = 1₂, 1, 2, 4, such that 32000 different parameter combinations are analysed. For 88,1% of the simulations a valid ˜θ was found. No parameter combinations violated the condition. Therefore we expect that the average quality level of health care offered by both health plans increases when the deductible increases.

Conjecture 1. A small increase in the deductible, compared to when the deductible is zero, will at least initially increase the physician diversity offered by both health plans.

5.4 The health plans

Figure 5 depicts the profits of the two health plans for different . It can be seen that the PPO is worst off in the case the iso-elastic distribution function of the probability of illness is given by F (θ) = θ12, which is in line with the discussion from Section 5.1. The maximum profit for the

HDHP is not reached within the domain for = 1 and = 1₂. In these cases the HDHP can not increase the deductible further as this will mean that the total number of physicians affiliated exceeds 1 (condition (5.1)). Note further that the profits of the PPO are negative for = 1₂ and = 1 as the optimal prices for the PPO are much lower than the unit cost of treatment. For lower values of the unit cost of treatment the profits are positive for all . Interestingly, increasing the deductible initially increases profits for both firms. Hence the HDHP has an incentive to set a positive deductible. For some parameter values the profit of the HDHP reaches its maximum within the domain of the valid indifferent probability, and for many it does not.

We can see in Figure 5c that the HDHP makes much higher profits than the PPO when the deductible is zero. Increasing the deductible helps the competitor to make much larger profits, while the HDHP only marginally increases its own profits. We have however assumed that the HDHP is only interested in profits, and not in market share, such that the HDHP will set the deductible at maximum profits.

The profits initially seem to increase for all illustrated parameter values. We analyse whether this is always the case. Taking the derivative of the profits of the PPO and HDHP with respect to the deductible we find an expression that can be calculated numerically. These can be found in Appendix B. Again we will run the simulations for many different parameter combinations. We vary λ, δ ∈ (0.1, 100), c ∈ (0, 50), = 1₂, 1, 2, 4, such that 32000 different parameter combinations are

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(a) = 0.5 (b) = 1 (c) = 2

Figure 5: Profits of the two health plans for different representations of the distribution of illness.

both profits with respect to the deductible was positive when D_HDHP = 0. Hence we can conclude that a small increase in D_HDHP when D_HDHP= 0 will increase profits of both firms.

Conjecture 2 An increase in the deductible, compared to when there is no deductible, will at least initially increase profits of both health plans.

5.5 Policyholders and physicians

Figure 6 depicts the total and average utility of policyholders of both health plans defined by equa-tion (4.14), (4.15) and (4.16). For the parameter values used, the average utility of a policyholder from joining both health plans is negative. A negative average utility is not conflicting with model assumptions as we assume that all policyholders are obliged to take health insurance. The policy-holders of the PPO are on average better off than the policypolicy-holders of the HDHP, as they have a higher average utility. It is also seen that the average utility initially decreases as the deductible increases.

As before, the derivative of the total utility (equation (4.14)) with respect to the deductible is calculated for the same 32000 parameter combinations. Table 1 shows the average parameter values for which the total utility of the policyholders increases and decreases when the deductible increases. It is found that the total utility of the policyholders decreases for 97.6% of parameter combinations. It only increases when the ratio λ_δ is high, c is low and = 1₂. The intuition behind this is as follows. When the unit costs of treatment c are low, the change in prices given by (4.7) and (4.8)

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(a) = 0.5 (b) = 1 (c) = 2

Figure 6: Average utility of policyholders of the two health plans for different representations of the distribution of illness. λ δ λ_δ c ∂TU ∂DHDHP > 0: 62.22 42.07 1.48 2.69 0.50 ∂TU ∂DHDHP < 0: 47.63 56.28 0.85 25.53 1.85

Table 1: Parameter averages for which the total policyholder utility increases and decreases in D_HDHP.

with respect to a change in ˜θ is lower compared to when c is high. As the increase in deductible increases the indifferent probability (Proposition 1), the prices increase by a smaller amount when c is low. Hence the increase in physician amount (which is valued by the policyholders) outweighs the change in prices. The increase in physician amount only outweighs the increase in prices when is small and λ is high. When = 2 for example, Figure 3c shows that the physician diversity offered by the PPO does not increase as much when the deductible increases from zero. Simultaneously the price of the HDHP (Figure 4c) decreases, but as the HDHP has a low market share, this effect is offset by the lower increase of the physician amounts of the PPO. When the preference for diversity parameter λ is high, the increase in physician diversity is more beneficial to the policyholders. When δ is low, the physicians are not too expensive for both health plans such that the health plans will choose to increase the physician diversity offered more substantially.

A low unit cost of treatment might not be a very reasonable assumption as health care costs have been increasing rapidly. Moreover, as discussed in Section 5.1, a very left skewed probability

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(a) = 0.5 (b) = 1 (c) = 2

Figure 7: Fee-for-service rates of the two health plans for different representations of the distribution of illness.

of illness (low ) is not a reasonable assumption. As the HDHP has an incentive to set a positive deductible, we argue that the presented model predicts that the utility of the policyholders decreases.

Conjecture 3 A small increase in the deductible, compared to when the deductible is zero, will initially decrease the total utility of the policyholders for most reasonable parameter values.

The last type of agent of the model are the physicians. In Figure 7 we can see the fee-for-service rates. As c is set to 30, physicians affiliated to both health plans have positive profit margins. Note that the firm that offers a lower for-service rate still attracts physicians. We can see that the fee-for-service rates offered by both health plans increase with the deductible for all values of , and more rapidly for the HDHP. One would thus expect that the deductible is advantageous for the physicians. The total physician profits are given by equation (3.7) and (3.8). Calculating the derivative of the total physician profits with respect to D_HDHP for the same 32000 different parameter combinations we find that a small increase in the deductible compared to a model without the deductible will indeed always increase the total physician profits.

Conjecture 4 A small increase in the deductible, compared to when the deductible is zero, will initially increase the total profits of the physicians.

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6 Model 2: HDHP-HMO competition

As discussed, HDHPs can be PPOs or HMOs in terms of the amount of physicians that they are affiliated to. In the previous section we analysed the model where the HDHP is an HMO in terms of the amount of physicians affiliated. This and the next section discuss the case where the HDHP is a PPO in terms of the amount of physicians affiliated. The HDHP competes with an HMO. The assumptions of Section 3 are now slightly changed.

6.1 Changed assumptions

The number of physicians affiliated to the HDHP is higher than the HMO: J_HDHP > JHMO. The

HDHP is assumed to have a higher premium than the HMO: PHDHP > PHMO. We furthermore

assume the same utility function for the policyholders as in the first model. The utility of a policyholder with probability of illness θ is hence denoted by

EU_θ(PHDHP, JHDHP, DHDHP) = u(w − PHDHP− θDHDHP) + θλJHDHP (6.1)

EU_θ(PHMO, JHMO) = u(w − PHMO) + θλJHMO (6.2)

where DHDHP is the deductible paid when ill and affiliated with the HDHP, and w is the initial

wealth of the policyholder. The demands for both health insurers are based on the distribution of the probability of illness. Again we make the assumption that the distribution of illness is given by the iso-elastic probability distribution F(˜θ) = ˜θ. A policyholder with probability of illness θ will prefer the HDHP over the HMO when:

U_θ(PHDHP, JHDHP, DHDHP) > Uθ(PHMO, JHMO) (6.3)

We still assume that the market for health insurance on the policyholder side is covered. When the market for health insurance on the policyholders side is covered, the policyholder that is exactly indifferent between the two health plans can be found. The ’indifferent probability’ corresponding to this indifferent policyholder is now defined by

˜

θ = PHDHP− PHMO

λ(JHDHP− JHMO) − DHDHP

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We see that the indifferent probability has almost the same functional form as before, only the deductible has a sign change. If the deductible becomes too large compared to the difference in physician diversities, the denominator becomes negative. Hence we require that λ(J_HDHP−J_HMO) − DHDHP > 0.

The demand of the HDHP is now 1 − F(˜θ), and the HMO faces a demand of F(˜θ). Note that in this model the indifferent policyholder is somewhat more ambiguous. As before policyholders with a low probability of illness would prefer a health plan that offers a low diversity of physicians and a low premium, which would make these policyholders choose the HMO. However these same policyholders also mind paying the deductible less than policyholders with a high probability of illness, which attracts them to the HDHP.

6.2 Equilibrium

Now the HDHP and HMO maximize the profit functions given by

ΠHDHP = [1 − F(˜θ)]PHDHP− δJ2HDHP+ (DHDHP− c)

Z 1

˜ θ

θdF(θ) (6.5)

ΠHMO= F(˜θ)PHMO− δJ2HMO− c

Z θ˜

0

θdF(θ) (6.6)

Again the HDHP has additional incomes of the deductible times the expected activity level. Taking first order derivatives with respect to P_HDHP and P_HMO we obtain:

∂ΠHDHP ∂PHDHP = 1 − F(˜θ) + [(c − DHDHP)˜θ − PHDHP]f(˜θ) ∂ ˜θ ∂PHDHP = 0 ∂ΠHMO ∂PHMO = F(˜θ) + [PHMO− c˜θ]f(˜θ) ∂ ˜θ ∂PHMO = 0 Using that ∂ ˜θ ∂PHDHP = − ∂ ˜θ ∂PHMO = 1 λ(JHDHP− JHMO) − DHDHP we find PHDHP= (c − DHDHP)˜θ+λ(JHDHP− JHMO) − DHDHP 1 −θ˜ ˜θ−1 (6.7)

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P_HMO = c˜θ+λ(JHDHP− JHMO) − DHDHP

θ˜

(6.8)

Taking first order derivatives of the profit functions with respect to J_HDHP and J_HMOwe obtain

∂ΠHDHP ∂JHDHP = −[PHDHP− (c − DHDHP)˜θ]f(˜θ) ∂ ˜θ ∂JHDHP − 2δJ_HDHP = 0 ∂ΠHMO ∂JHMO = [PHMO− c˜θ]f(˜θ) ∂ ˜θ ∂JHMO − 2δJHMO= 0

Using again that

∂ ˜θ ∂JHDHP = − ∂ ˜θ ∂JHMO = −λθ λ(JHDHP− JHMO) − DHDHP we find J_HDHP = λ 2δ ˜ θ(1 − ˜θ) (6.9) J_HMO = λ 2δθ˜ +1 _(6.10)

Note that the physician amounts are of the same functional form as before, only the definition of ˜θ is slightly changed. Plugging the physician numbers back into equation (6.7) and (6.8) we find that the prices in equilibrium are given by

P_HDHP= (c − DHDHP)˜θ+ " λ2 2δθ[1 − 2˜˜ θ _{] − D} HDHP # 1 − ˜θ ˜θ−1 (6.11) P_HMO = c˜θ+ " λ2 2δ ˜ θ[1 − 2˜θ] − DHDHP # ˜_θ (6.12)

Plugging these values back in equation (6.4) we find that ˜θ solves the equation

g(˜θ, DHDHP) = −˜θDHDHP λ2 2δθ[1 − 2˜˜ θ] − DHDHP + 1 − 2˜θ ˜θ−1 − ˜θ = 0 (6.13)

Again we can have multiple indifferent probabilities on the interval (0,1). The mark-ups on net premiums including deductibles are calculated to specify a simplified representation of the equilibrium profits. The mark-ups become:

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P_HDHP− (c − D_HDHP)˜θ = " λ2 2δθ[1 − 2˜˜ θ _{] − D} HDHP # 1 − ˜θ ˜θ−1 (6.14) P_HMO− c˜θ = " λ2 2δθ[1 − 2˜˜ θ _{] − D} HDHP # ˜_θ (6.15)

Plugging the found equilibrium values of the prices and physician diversity in the profits of the insurers we obtain the equilibrium profits:

ΠHDHP = " λ2 2δ ˜ θ[1 − 2˜θ] −λ 2 4δ˜θ +1_{− D} HDHP # [1 − ˜θ]2 ˜θ−1 + (DHDHP− c) Z 1 ˜ θ (θ − ˜θ)dFθ (6.16) ΠHMO = " λ2 2δ ˜ θ[1 − 2˜θ] −λ 2 4δ˜θ +1_{− D} HHDP # ˜_θ − c Z θ˜ 0 (θ − ˜θ)dFθ (6.17)

Note that all the equations are very similar to the first model, although the model will turn out to behave substantially different.

7 Equilibrium analysis model 2

In a similar way as in Section 5, this section will analyse the effects of the deductible excess on all agents of the model. The HDHP offers a higher diversity of physicians than the HMO.

7.1 Conditions on ˜θ

The first two conditions of Section 5.1 still hold. As a result, Lemma 1 still holds. However, the third condition that states the price of the HDHP is higher than the price of the HMO changes. The condition is now given by

−D_HDHPθ >˜ " λ2 2δ ˜ θ[1 − 2˜θ] − DHDHP # 2˜θ− 1 ˜θ−1 (7.1)

This condition can be binding for ˜θ. As stated in the previous section, we have an additional condition constraining ˜θ, as it is required that the denominator of ˜θ is positive. In equilibrium, this condition results in

λ2

2δθ[1 − 2˜˜ θ

_{] − D}

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Figure 8: Illustration of g(˜θ, DHDHP), = 2, DHDHP = (0, 10), λ = 60, δ = 20, c = 30

Condition (7.2) is also a necessary but not sufficient condition for (7.1), as 2˜θ− 1 < 0. From the expression for g(˜θ, DHDHP), given by equation (6.13), it can be seen that the first term now

has a negative sign. Figure 8 depicts g(˜θ, DHDHP) for two different values of the deductible. When

DHDHP = 0, g(˜θ, DHDHP) is equal to the first model (Figure 2b) and the model of Bardey and

Rochet (2010). When D_HDHP > 0, we can see two vertical asymptotes where the denominator of the first term of g(˜θ, DHDHP) equals zero. These asymptotes correspond to the values of ˜θ for which

condition (7.2) is binding. Outside this interval, condition (7.2) is not satisfied and any ˜θ is not valid.

Lemma 2 Any indifferent probability ˜θ must satisfy Lemma 1 and condition (7.1) to satisfy all assumptions in equilibrium.

7.2 The new indifferent probability of illness

We can see in Figure 8 that between the two asymptotes g(˜θ, 10) lies below g(˜θ, 0). Therefore for these parameter values ˜θ decreases as the deductible increases. Looking at Appendix A, we can see that when D_HDHP = 0 the partial derivative of ˜θ with respect to DHDHP is equal to minus the partial

derivative of ˜θ with respect to DHDHP of the first model. Hence we can conclude that _∂D_HDHP∂ ˜θ < 0

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c ∈ (0, 50), ∈ (0.1, 0.9), 72000 different parameter combinations were analysed. For 83,9% of the simulations a valid ˜θ was found. Note that the percentage of simulations that resulted in a valid ˜θ is the same as in the previous model, as for D_HDHP = 0 the indifferent probability ˜θ is equal. For all of these simulations the derivative of ˜θ with respect to DHDHP was negative when DHDHP = 0.

Hence we can conclude that no parameter combinations exist for which Lemma 2 holds and the condition (2 + )θ+ > 1 is violated. Therefore a small increase in DHDHP when DHDHP = 0 will

always decrease the indifferent probability.

This again sounds counter-intuitive as we would expect that when the deductible increases, the indifferent probability increases as well. Some policyholders of the HDHP would prefer to sign a contract with the HMO as the deductible of the HDHP has increased. But, as in the previous model, the two firms choose to adjust prices and physician’s diversity offered substantially, which offsets this direct effect of the deductible.

Conjecture 5. A small increase in the deductible, compared to when the deductible is zero, will decrease the indifferent probability, as both health plans adjust prices and physician diversities. This adjustment will more than offset the direct change of the indifferent probability due to the change in the deductible.

As in the first model, we know that ∂JHMO

∂DHDHP has the same sign as

∂ ˜θ

∂DHDHP when DHDHP = 0

(see (5.6) and (5.7)). Hence the physician diversity offered by the HMO will decrease as a result of the deductible. The physician diversity of the HDHP will increase for ˜θ < (₊₁1 )1. Again the

simulations show this condition is never satisfied. Therefore we conclude that the average health care quality offered by the health plans is expected to decrease.

Conjecture 6. A small increase in the deductible, compared to when the deductible is zero, will at least initially decrease the physician diversity offered by both health plans.

7.3 The health plans

Figure 9 depicts the profits of the two health plans for different values of . For these parameter values the firm offering the higher diversity of physicians, the HDHP, is better off when is high, as more policyholders have a high probability of illness. We see that the HMO never benefits from

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(a) = 1 (b) = 2 (c) = 6

Figure 9: Profits of the two health plans for different representations of the distribution of illness. λ = 60, δ = 20, c = 30

an increase in the deductible. The HDHP can however benefit from an increase of the deductible. Again we vary λ, δ ∈ (0.1, 100), c ∈ (0, 50), e = 1₂, 1, 2, 4, such that 32000 different parameter combinations are used. For 88,1% of the simulations a valid ˜θ was found.7 The results are not as conclusive as in the first model. The HDHP increases profits by raising D_HDHP when D_HDHP = 0 in only 35.4% of total simulations. Table 2a shows the average parameter values of the simulations with increasing and decreasing profits. The table comprises only the cases in which a valid ˜θ was found. We can see that ∂πHDHP

∂DHDHP > 0 when the unit cost of treatment c is low, and the ratio

λ δ

and are high. The intuition behind this is as follows. When the deductible increases, ˜θ decreases (Lemma 2). This implies that the HDHP has a greater market share and that the HDHP has to pay the physicians more for the treatment of its policyholders (the last term of equation (6.16)). These increased payments are obviously lower when the unit costs of treatment are low.

To see how the ratio λ_δ affects the profitability of the deductible for the HDHP, we will now provide a more extensive explanation. J_HDHP and J_HMO were both found to decrease for a small positive change of the deductible compared to the situation without deductible. This can be ex-plained as follows. When the HDHP increases the deductible some policyholders will prefer to switch to the HMO, increasing the market-share of the HMO. Then it is optimal for the HMO to save slightly on the physicians affiliated, which drives the HMO’s market-share slightly down again. Hence ∂JHDHP

∂DHDHP < 0 for DHDHP = 0. When λ is high, policyholders value the diversity of physicians 7

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λ δ λ_δ c

∂πHDHP

∂DHDHP > 0: 68.84 49.25 1.40 17.07 2.01

∂πHDHP

∂DHDHP < 0: 33.93 60.46 0.56 30.34 1.69

(a) Parameter averages for which the profits of the HDHP increase and decrease in DHDHP.

λ δ λ_δ c

∂TU

∂DHDHP > 0: 42.61 57.58 0.74 28.67 1.63

∂TU

∂DHDHP < 0: 67.95 49.88 1.36 11.33 2.50

(b) Parameter averages for which the total policyholder utility increases and decreases in DHDHP.

Table 2: Summary of parameters affecting the profitability of the HDHP and total policyholder utility when the deductible increases.

largely. The HDHP will hence gain more market-share when the HMO lowers the physician diversity offered compared to a situation where λ is low. It is thus more profitable for the HDHP to lower the physician diversity offered, saving on physician costs.8 Therefore an increase in the deductible is more profitable when λ is high. When δ is high, the physicians are not very sensitive to the remuneration offered by the health plans and the change in J_HDHP as a result of the change in deductible is lower. Therefore the HDHP will be able to save less on the physician diversity offered. The distribution of the probability of illness also affects the profitability of the deductible for the HDHP. Interestingly, Table 3 shows that for a higher value of a lower ratio λ_δ is required such that a valid ˜θ is found.9 Still it is the case that on average the HDHP benefits more from increasing the deductible when is high. When is high, the expected number of treatments for which the HDHP has to pay is high, such that an increase in the deductible increases incomes very substantially.

We can thus conclude that the HDHP does not always have an incentive to set a positive deductible. Again, low unit costs of treatment might not be very reasonable to assume, such that we would not expect the HDHP to set a positive deductible in many situations.

Conjecture 7. A small increase in the deductible, compared to when the deductible is zero, will decrease the profit of the HMO, and can increase or decrease the profit of the HDHP, depending on the ratio λ_δ, and the unit costs of treatment.

8_{This is the reversed negative spending effect as defined by Boilley (2013).}

9_{This is the result of condition (5.1) that requires that the total number of affiliated physicians is smaller than 1.}

This condition can in equilibrium be rewritten as λ_δ < 2_θ˜, and ˜θ is generally higher when is high. Therefore when

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λ δ λ_δ c

= 1₂: 49.28 53.24 0.93 25 = 1: 48.44 55.16 0.88 25 = 2: 47.50 56.95 0.84 25 = 4: 46.48 58.64 0.79 25

Table 3: Parameter averages that resulted in a valid ˜θ, for different . DHDHP= 0. We see a lower

ratio λ_δ when increases.

7.4 Policyholders and physicians

By running calculations using the same 32000 parameter combinations, we find that total policy-holder utility can increase or decrease when the deductible increases. Table 2b depicts the average parameter values for both cases. For 78.8% of the parameter combinations the utility of the policy-holders increases when the deductible increases. We can see that c, the ratio λ_δ and differ when the policyholders’ utility increases or decreases when the deductible goes up. The intuition behind this is as follows. Increasing the deductible will decrease the indifferent probability (Lemma 2). Thereby prices of both firms, given by (6.11) and (6.12), will be lowered by a larger amount when the unit costs of treatment are high. Hence a high c also positively affects the utility of policyholders when the deductible increases. As discussed both health plans lower their physician diversity offered when the deductible increases. When the policyholders’ preference for diversity λ is high, the policyhold-ers’ utility suffers more when the physician diversity offered is lower. When δ is high, the decrease in physician’s diversity offered as a result of the increased deductible is lower. Hence a high ratio

λ

δ positively affects the change in utility due to the deductible. As stated before, a low implies a

relatively low λ_δ ratio in the simulations (Table 3). However when is low, most of the policyholders have a low probability of illness, such that an increase in the deductible will not result in large deductible payments to the HDHP. This effect outweighs the lower λ_δ ratio. As a result a low is beneficial to the policyholders when the HDHP increases the deductible. Looking at Table 2 we can see that the average parameter values for which the HDHP wants to set a positive deductible are almost equal to the average parameter values for which the total policyholder utility is decreasing in the deductible. However we find that for 19.1% of parameter combinations for which a valid ˜θ

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This will come at the cost of the physicians and the HMO. We conclude that the policyholders could benefit from HDHP in the market.

Conjecture 8. A small increase in the deductible, compared to when the deductible is zero, can increase or decrease total policyholder utility, depending on c, the ratio λ_δ and .

The same calculation as before shows that the total physician profits decrease as the result of the increased deductible when DHDHP = 0. Fee-for-service rates offered by both health plans

decrease, and we had already found that the physician numbers go down. The physicians are hence disadvantaged from an increase in the deductible. Note that these results are exactly opposite from the results of the first model in Section 5. 10

Conjecture 9. A small increase in the deductible, compared to when the deductible is zero, decreases the total physician profits.

7.5 Policy implications of the two models

We can conclude that it is better for the policyholders when the HDHP is a PPO in terms of the physician diversity (model 2) than when the HDHP is an HMO in terms of the physician diversity (model 1). In the first model the utility of the policyholders is almost always expected to decrease when the deductible increases, and the HDHP always has an incentive to set a positive deductible. In the second model the HDHP wants a positive deductible in only 35.4% of the parameter com-binations. In 19.1% of the parameter combinations the HDHP sets a positive deductible and the total utility of the policyholders is expected to increase. When the HDHP does not set a positive deductible the model is equal to that of Bardey and Rochet (2010), and the total policyholder utility is not affected.

As discussed, a deductible excess is intended to reduce total health care costs. When the HDHP is a PPO in terms of the physician diversity and sets a positive deductible this is the case. However

10_{As the indifferent probability goes down, the HMO has a decreased market share. Similar to Brooks, Dor, and}

Wong (1997), we hence find that a lower HMO penetration is accompanied by a decrease in fee-for-service rates. We do note that the study of Brooks, Dor, and Wong concerned the U.S. health insurance market from 1988-1992 in which no HDHP was present.

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this goes along with a lower average quality of health care. If the HDHP is an HMO in terms of the physician diversity we expect the total health care costs to increase, although the average quality of health care increases as well.

As the payments of one agent in the model are incomes of another agent, the only variable that affects the total welfare of all agents is the physician diversity. In the first model the physician diversity is expected to increase, such that the total welfare will increase. In the second model the physician diversity offered decreases, such that total welfare will decrease.

Summarising we can say that it is best for total welfare when the HDHP is an HMO in terms of the physician diversity, although this is at the expense of the policyholders and goes along with higher health care costs. Total welfare decreases when the HDHP is a PPO in terms of the physician diversity and sets a positive deductible, although the total health care costs decrease and the policyholders may benefit.

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8 Stability analysis

In the last sections we have analysed how the equilibria change when the deductible increases. In practice however, the health plans are not fully aware of the optimal choice for their price, physician diversity and deductible. As such, we analyse whether the health plans can learn the equilibrium choices over time. Figure 1 in the introduction depicts that there have been large shifts in the market shares of the firms. The HDHP has won a lot of market share partly at the expense of the PPO in recent years, and this shows that the market might not have reached its equilibrium composition yet. Therefore in this section we will analyze whether firms can learn the equilibrium choices and whether the equilibria of the model are stable under learning methods.

8.1 Gradient learning setup

To analyse this we will use gradient learning. With gradient learning the health plans change their price and physician diversity by small steps of size η in the direction of the slope of the profit function at the current price and physician diversity. Thus, they adjust their choices in small steps in such a way that is expected to give a higher profit. This does require the assumption that the health plans know the derivative of their profit function at the current price and physician diversity, which is a strong assumption in practice.11 The two health plans have naive expectations: They assume that the other firm will set the price, physician diversity and deductible in the next period equal to that of the previous period. Gradient learning is implemented in a similar way as in Anufriev, Kopányi, and Tuinstra (2013).

1. DHDHP and are set.

2. Random values are drawn for λ and δ from a uniform distribution on (0,100) and for c on (0,80). The equilibrium ˜θ* following from equation (4.9) and (6.13) is calculated. If the ˜θ* does not satisfy the conditions from Section 5.1 for the first model or Section 7.1 for the second model, repeat this step until a ˜θ* is found that is valid.

3. Two prices are drawn from a uniform distribution on (0,80). The higher of the two prices is assigned to health plan with the higher price. Hence the starting prices satisfy the condition

11_{It might be the case that the health plans know past profits for two different combinations of prices and physician}

diversity that are very similar, such that they can estimate the derivative of their profit function. The health plans could also estimate this by market research.

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(a) PPPO, PHDHP (b) JPPO, JHDHP

Figure 10: Illustration of a typical time series using gradient learning and fixed parameters, λ = 60, δ = 20, c = 30, = 2 and DHDHP= 27.7.

P_{P P O,1}> PHDHP,1 or PHDHP,1> PHM O,1.

4. Two physician diversities are drawn from range (0,1). When the sum of both is smaller than one the higher physician diversity is assigned to the health plan with the higher physician diversity. Therefore J_{P P O,1}> JHDHP,1 or JHDHP,1> JHM O,1 is satisfied.

5. The indifferent probability ˜θ1 that results from the drawn prices and physician diversities is

calculated. If ˜θ1 does not satisfy the conditions from Section 5.1 or 7.1, start over at step 3.

6. When t > 1 prices and physician diversities are updated according to    Pi,t J_i,t   =    Pi,t−1 J_i,t−1   +    η1 η2       ∂πi(Pt−1,Jt−1,Dt−1) ∂Pi,t−1 ∂πi(Pt−1,Jt−1,Dt−1) ∂Ji,t−1   

7. Repeat until |P_i,t − P_i,t−1| < T_P and |J_i,t − J_i,t−1| < T_J or until a maximum number of simulations is reached.

8.2 Optimal learning parameters

To find the optimal stopping criterion and step size the performance is demonstrated using the same parameters as in Section 5.3: λ = 60, δ = 20, c = 30 and = 2. DHDHP is set to 27.7, the value of

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(a) PPPO, PHDHP (b) JPPO, JHDHP

Figure 11: Frist 1000 periods of a time series that did not converge using gradient learning and fixed parameters, with step size 20 times as large. λ = 60, δ = 20, c = 30, = 2 and D_HDHP = 27.7.

for a typical time series. After 19951 periods the change in prices and physician diversities is smaller than the thresholds and the simulation is stopped. Note that after the first period the ˜θtcan violate

the conditions of Section 5.1 or 7.1. We will find that the equilibrium is eventually reached in most cases, even though in most simulations the conditions are not satisfied during some periods.

The thresholds that were used are T_P= 10−5 and T_J= 10−6. The optimal values for the step parameters are η1 = 10−2 and η2 = 5 · 10−3. These thresholds and step parameters perform well as

we find high convergence rates and the equilibrium is reached reasonably swiftly. Furthermore the final prices and physician diversities are very close to the equilibrium values. The step parameters and thresholds used in this illustration will therefore be used for all simulations of the first model. Higher step parameters induce diverging 2-cycles or other unstable behavior. Figure 11 shows the first 1000 simulations with the same parameter values as in Figure 10 but with the step size 20 times as large. We can see that around period 250 the prices and physician diversity change to a NaN value. With step parameters 20 times as large we often observe this, although the equilibrium is sometimes reached.

8.3 Simulations model 1: PPO-HDHP competition

In Table 4 we can see the convergence rates of the first model for different and D_HDHP. For each combination 50000 simulations were run, and the maximal number of periods for each run was 20000. The missing values in the table are combinations for D_HDHP and for which no valid

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Convergence rate = 0.5 = 2 = 6 D_HDHP = 0 93,4% 95,1% 94,9% D_HDHP = 10 - 96,8% 96,0% D_HDHP = 20 - 93,3% 88,6% DHDHP = 30 - 85,2% 81,6% DHDHP = 60 - - 61,5%

Table 4: Model 1: convergence rates for different combinations of and DHDHP.

= 0.5 = 2 = 6 λ δ Price diff λ δ Price diff λ δ Price diff DHDHP = 0 All 1,59 2,02 1,08 5,18 0,94 6,34 DHDHP = 0 Not converged 0,10 0,15 0,76 2,51 0,70 3,37 DHDHP = 10 All - - 1,49 10,41 1,13 11,50 DHDHP = 10 Not converged - - 0,91 7,03 1,13 3,74 DHDHP = 20 All - - 1,68 14,02 1,21 15,57 DHDHP = 20 Not converged - - 1,33 5,60 1,07 4,95 DHDHP = 30 All - - 1,85 17,05 1,26 19,01 DHDHP = 30 Not converged - - 1,59 7,05 1,18 7,75 DHDHP = 60 All - - - - 1,39 25,40 DHDHP = 60 Not converged - - - - 1,37 17,10

Table 5: Model 1: λ_δ and difference in starting prices for different combinations of and D_HDHP.

indifferent probability can be found (for any λ, δ and ). It can be concluded that when the HDHP sets a high deductible, the convergence rate is generally lower. Table 5 shows the ratio λ_δ and the difference in starting prices for the same simulations. The average ratio λ_δ differs substantially. The runs that did not converge have a much lower λ_δ ratio. We find that the ratio λ_δ affects the difference in starting prices, that affects the convergence rate. However, a low ratio λ_δ negatively affects the convergence rate directly as well.

Deductible and welfare in the U.S. health insurance market : a two-sided duopoly approach