Hospital Differentiation and Network Exclusivity⇤ Peichun Wang† April 11, 2017

[Preliminary draft. Do not cite.] Abstract Hospitals in the US differ significantly in their quality. These differences are also increasing over time due to hospitals quality competition. This paper develops a model of network formation where contracts between hospitals and insurers are determined by Nash bargaining. Model simulations show that hospital differentiation leads to more exclusive networks, high insurance premiums and low enrollment rates, although the magnitudes depend on the relative quality and cost of the hospitals as well as the degree of insurer differentiation. Unlike previous studies of insurer-provider networks, this paper also shows that the ability to renegotiate contracts in the model strengthens the exclusivity of networks. I estimate hospital and insurance demand and insurer-hospital contracting decisions on Massachusetts’ State Inpatient Database over 2002-2011. Structural estimates suggest that the exclusive networks observed in the data can be partially attributed to hospital quality differentiation. Further descriptive evidence from New Hampshire’s commercial claims data over 20052014 also lends support. JEL Classification: Keywords:

⇤I

am grateful to Ulrich Doraszelski, Katja Seim, and Michael Sinkinson for their guidance. I especially thank Robert Town for accessing the HCUP State Inpatient Database at NBER. I also thank Peter Blair, Mark Duggan, Hanming Fang, Jin Soo Han, JF Houde, Jon Kolstad, Robin Lee, Eli Liebman, Volker Nocke, Dan Sacks, Amanda Starc, Boris Vabson, Xingtan Zhang, and seminar participants at Wharton, London Business School TADC, and CES for helpful comments. Lynn Hua provided excellent research assistance. This research was funded in part by the Leonard Davis Institute of Health Economics. All remaining errors are my own. † The Wharton School, University of Pennsylvania, 3620 Locust Walk, Philadelphia, PA 19104. E-mail: [email protected].

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1

Introduction

In many markets contracts and/or relationships are developed either between upstream suppliers and downstream retailers or other forms of bilateral oligopolies. These contracts are usually bilaterally negotiated over the contract terms and often lead to complex buyer-seller networks. The complexity of these networks creates various contracting externalities to the market participants as the formation and destruction of any link would have an impact on many if not all other parties through the network. The network complexity also presents challenges to evaluate the impacts of market structure changes on either side of the market. For example, an otherwise undesirable merger could now be beneficial to downstream consumers despite the increased market power if it eliminates exclusive contracts, or vice versa. Thus it is important to understand how networks are formed and how bilateral transfers are determined in these markets in theory as well as in empirical settings in order to conduct counterfactual analysis and evaluate mergers. However, the complexity of these network formation processes via bargaining, together with other issues such as the presence of multiple equilibria, prohibitive computational costs, and crucially the lack of detailed data, has prevented us from having a tractable empirical framework. In this paper, I develop a model of bilateral contracting where both networks and transfer prices arise endogenously in equilibrium. I estimate the model with an application to the U.S. medical care market, which is both theoretically suitable and empirically important. However, this approach can be applied in any other bilateral oligopoly settings to understand the implications of competitions and market structures. Accounting for about 18% of the U.S. economy, the health care industry deserves a close scrutiny. In particular, this paper speaks to the anti-trust issues that have arisen in this market through the lens of the negotiation process between health providers and health insurers. Figure 1 illustrates the basic market structure of the U.S. health care industry. In the U.S., health insurers only allow their enrollees to access a particular set of hospitals conditional on enrollment into a plan. With each hospital on the plan’s network, the insurer negotiates a base reimbursement price that will be used to calculate the actual bill once incurred by the plan’s enrollees1 . These transfers account for over 40% of total health care spending or about $1 trillion annually. However, very little is understood about how these payments are negotiated and especially how these negotiations might be affected in light of market structure changes. Both the hospital industry and the health insurance industry have become much more concentrated over the last two decades. The hospital industry has had thirteen federal horizontal merger litigation trials over the past 25 years, the most out of any industry in the U.S.. On 1 The

actual reimbursement payment from the insurers is the base rate multiplied by a severity index of the disease being treated.

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Figure 1: U.S. Medical Care Market Structure

the other hand, the health insurance industry has already become “highly concentrated2 ,” as defined by the Department of Justice (DoJ), in 94% of the markets across the U.S. in 2008. Either industry’s growing concentration seems to have fueled even more incentives for the other to consolidate, pointing to the possibility of an underlying bargaining process between hospitals and insurers that exacerbates the potential anti-trust issues. Research on insurer-provider networks and bargaining, however, is not new. In particular, Ho (2009) identifies important determinants of hospital-insurer network formations through bargaining. Gowrisankaran, Nevo and Town (2013) takes network configuration as given and models negotiated prices between hospitals and insurers to evaluate potential impacts of hospital mergers. This paper, on the other hand, models both network formations and equilibrium contract terms to answer two important welfare questions: 1) how is health care accessibility affected by the potential mergers, or in other words, how does the network configuration change in response to the more concentrated market structure, and 2) given changes in the hospitalinsurer networks, how do prices (hospital base prices and plan premiums) change and affect firm profitability and consumer welfare. My model of network formations and competitions proceeds in five stages. The focus is on the first two stages where bilateral contracting takes place, but I also carefully model the downstream retail market in the last three stages to correctly specify the bargaining values. I solve the model with a numerical example and show that contracting externalities lead to 2A

Market is highly concentrated if it has an Herfindahl-Hirschman Index (HHI) above 1800. HHI is defined as the sum of squares of all market shares multiplied by 10000.

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large market inefficiencies when taking into account network formations. I then conduct a stylized counterfactual merger experiment with the example and show that ignoring endogenous network formations produces downward bias in estimated welfare loss. Although I plan to use a unique set of detailed data later3 , part of the advantage of this model is that I can estimate it with only data on network configurations, and hospital and insurance plan characteristics4 . Particularly, I estimate the model without either negotiated base prices or insurance plan premiums. This paper uses the HCUP State Inpatient Database (SID) of Massachusetts from 2002 to 2011. The SID is a 100% sample of hospital visits. Each observation is on the individual visit level and contains hospital and payer identifiers, from which I infer insurer-provider network structures, and hospital and insurance plan demand. To estimate the model, I use a method of simulated moments estimator. With only the limited data from SID, my estimator yields generally reasonable estimates and the model fits the moments of network outcomes very well. Together with some preliminary empirical evidence, I show that vastly different hospitals in a market and more concentrated hospital systems often sign more exclusive contracts with insurers, resulting in more restrictive networks, market inefficiencies, and welfare loss.

Literature review This paper is closely related to several literatures spanning game theory, industrial organization, and the empirical literature on health care. Among the large literature on strategic network formations, two papers are of particular relevance. Uetake (2014) estimates an empirical model based on Jackson and Wolinsky (1996) using data from the venture capital industry. Partial identification with moment inequalities is used in Uetake (2014) by taking advantage of the concept pairwise stability of the networks to get around the problems of multiple equilibria and high computational costs. My paper, on the other hand, solves for the full information solution and specifies a more general Nash bargaining game to determine network transfers. Lee and Fong (2013) provides an applied framework for estimating dynamic network formation games with transfers determined by Nash bargaining solutions. My model is similar to theirs but differs in several ways. I specify a static model for the empirical application but allow for a sequential contracting game for equilibrium purification. Due to the structure of the sequential game, I am also able to adopt a more robust bargaining model at no additional computational cost. The use of sequential games to achieve unique equilibrium for counterfactual analysis was initially due to Berry (1992) and has been widely used in the literature on entry games in industrial organization. 3 See

Section ?? for details. is similar to Ho (2009) on this front, whereas Gowrisankaran et al. (2013) requires observed negotiated prices to estimate its model. 4 This

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This paper also draws heavily from both the theoretical and empirical literature on bargaining. Much of the empirical applications on bilateral bargaining builds on the model from Horn and Wolinsky (1988) (HW hereafter), where contracting participants do not immediately renegotiate following a negotiation breakdown and thus prices for each contract solve the Nash bargaining problem for that contract, conditional on all other contract prices under the current configuration. Capps, Dranove and Satterthwaite (2003), Gowrisankaran, Nevo and Town (2014), and Lee and Fong (2013) use the HW protocol to model the bargaining game between health care providers and insurers. Crawford and Yurukoglu (2012) applies the same bargaining model to the contracting process between upstream content providers and downstream broadcasters in the television market. However, Stole and Zwiebel (1996) (SZ hereafter) specifies a slightly different bargaining game that is robust to immediate contract renegotiations and direct price externalities, at the cost of large computational burden in empirical estimations. Dranove, Satterthwaite and Sfekas (2013) develops and estimates a simplified model of SZ to investigate the level of rationality in these contracting environments. In this paper, I fully adopt the SZ bargaining model to allow for the full extent of contracting externalities. Two other papers also highlight the importance of bargaining in the health care market. Ho (2009) points out the important determinants of the surplus division rules between insurers and providers. Lewis and Pflum (2015) teases out the effect of hospital systems on insurer reimbursement rates due to increased bargaining power from local market concentration by specifying a bargaining model. There is also a booming literature on the impacts of hospital and insurer consolidations. Dafny (2010) and Dafny, Duggan and Ramanarayanan (2012) both identify causal effect of insurer market concentration on plan premiums. Gowrisankaran, Nevo and Town (2014) provides a structural model to evaluate the price effects of hospital mergers. Ho and Lee (2013) finds that increased insurer competition reduces negotiated hospital prices on average but the effect is the opposite for more attractive hospitals. This paper fits in the goals of this literature to evaluate effects of mergers but provides a comprehensive model to conduct counterfactual analysis for anti-trust agencies. The final strand of related literature is the standard discrete choice literature in differentiated product markets applied in the health industry to estimate demand and evaluate market power. Using random-coefficient logit models as in Berry, Levinsohn and Pakes (1995), Capps, Dranove and Satterthwaite (2003), Gowrisankaran, Nevo and Town (2014), and Ho (2009) all calculate the consumer willingness-to-pay to hospital networks as the logit inclusive values. Starc (2014) also uses similar demand estimation techniques but models adverse selection into different plans as a way to mitigate the effects of market powers. In this paper, I temporarily only specify a simple logit model but allow insurers to optimally play a Bertrand pricing game. The rest of the paper is as follows. Section 2 presents the SID data and preliminary empirical analysis. Section 3 introduces the model. Section 4 discusses simulation results of the model 5

Table 1: Sample market Year Market 2011 011 2011 011 2011 011 2011 011

Payer Hospital Health New England 25051 Health New England 25160 HealthNet 25051 HealthNet 25160

Visits 770 266 2857 839

Payer Total 1125 1125 3819 3819

Hospital Total 5907 1921 5907 1921

and their implications. Section 5 and 6 goes over the empirical strategy and estimates from the model. In section 7, I conclude and discuss future work for this project.

2

Data and descriptive evidence

The data used in this paper comes from the HCUP State Inpatient Database (SID). Although I have access to data from multiple states with various periods of time in SID, I restrict my attention to Massachusetts for consistency of insurer identifiers as well as for future analysis with the MA All-Payer Claims Database (APCD). Advantages of using SID include that it is a 100% sample of all hospital visits in a year and that it includes specific payer (insurance plan) identifiers for Massachusetts5 . The SID data observations are on each hospital visit level and span ten years from 2002 to 2011, with almost 1 million observations each year. With current data limitations, I estimate hospital demand with a simple logit model and thus collapse the data onto year-markethospital-payer level6 . I define markets as each 3-digit ZIP code area within MA, which gives me 18 markets each year7 and 180 markets in total. Therefore, I drop any hospital visit observations from outside these markets. I also focus on the privately insured market in this paper and drop all government sponsored insurance plans (e.g. Medicare, Medicaid, Medigap, and etc.). For reasonable computational cost in the estimation and to capture the first-order effects, I also only focus on major hospitals and insurance plans that have at least 10% share of either side of the market. I thus effectively group the uninsured and patients insured with small plans together as the outside option to choosing major insurance plans and group all small hospitals as out-of-network options conditional on being enrolled in a plan. Finally, for the remaining market participants, I infer the network structure by only counting a hospital-payer pair as having a link between them if the hospital accounts for at least 10% of that payer’s total visits. Otherwise, the hospital is counted as outside the payer’s network in that market. Table 1 presents the data structure of a sample market. 5 For

example, Aetna PPO would have a different ID than CIGNA PPO or Aetna HMO. the future, with location data for hospitals and patients as well as other hospital characteristics, I can form a likelihood function for each hospital choice observation and estimate demand with maximum likelihood. 7 The 3-digit ZIP codes range from 010 to 027 in MA. 6 In

6

Figure 2: Number of players

Figure 3: Insurance and hospital market concentration

2.1

Summary Statistics

This section presents various summary statistics of the SID data. In Figure 2, I show the number of major players (hospitals and insurers) in each market across years in the frequency plot. Across the 180 markets in my sample, most markets are highly concentrated, with two to three players on each side of the market. There is also some variation in the number of players and more in the concentration measures in either side of the market, as shown in Figure 3. Even after I exclude all smaller players from the sample, both markets still exhibit reasonably high concentration. However, even with so few players in the market, we see in Figure 4 that nearly half of all markets do not have full-contract networks. I define the contract saturation rate as the total number of actual contracts divided by the total number of possible contracts in a market. And 7

Figure 4: Network saturation rates across markets

Figure 5: Within-payer network concentration

we see that the network can miss as much as half of all possible contracts in certain markets. Thus it is important to understand the reason and impacts of these missing links. Combining the links with their relative shares, there is even more variation in the withinpayer HHI computed from conditional hospital shares across payers and markets. Finally, Table 2 summarizes key statistics in the collapsed SID. Because for the current estimation I assume a representative consumer in all markets and thus a common risk pool, I can take the total number of patients as my ex-post market size and the hospital visits as demand. With more detailed data on plan enrollment and patient characteristics, I can relax this assumption in the future. The private insurance market, which I focus on in this paper, is a substantial portion (average 40%) of the total health care market in Massachusetts. In the hospital characteristics part, I separately construct the number of actual contracts, contract saturation ratio, and the within-hospital payer HHI for each hospital to investigate the relationships of hospital quality and hospital contracting behaviors in the preliminary analysis. Finally, in the insurer characteristics section, note that the portion of enrollees going out-of-network is substantial because I defined hospitals with small conditional shares as outside the network and grouped 8

Table 2: Summary statistics Mean Market Characteristics Obs.: Year-Market # of Patients 43076 # of Uninsured 1400 # of Privately Insured 17045 # of Insurers 2.294 # of Hospitals 2.467 # of Contracts Possible 5.639 # of Actual Contracts 4.994 Overall Uninsurance Rate 0.031 Private Uninsurance Rate 0.073 Private Insurance Rate 0.398 Total Population 352724 Hospital Characteristics Obs.: Year-Market-Hospital # of Hospital Visits 3514 # of Contracts 2.025 Contract Saturation Ratio 0.898 Within-Hospital HHI 0.11 Insurer Characteristics Obs.: Year-Market-Insurer # of Hospital Visits 2726 # of Out-of-Network Visits 1160 Within-Insurer HHI 0.289

Std

Min

Max

N

(35784) (1903) (13672) (0.810) (0.999) (2.978) (2.511) (0.013) (0.031) (0.062) (277945)

985 31 427 1 1 1 1 0.009 0.023 0.272 22801

(2530) (0.79) (0.195) (0.075)

48 1 0.333 0.004

12134 5 1 0.572

444 444 444 444

(2378) (1485) (0.206)

50 5 0.015

12083 7613 0.965

413 413 413

166948 180 13168 180 63420 180 5 180 5 180 15 180 13 180 0.084 180 0.177 180 0.51 180 1247268 180

them all together.

2.2

Preliminary Analysis

In this section, I first estimate conditional hospital demand to construct a consistent index for the perceived hospital qualities. I then present some empirical evidence on the relationship between hospitals’ qualities and their contracting behavior with insurers. To estimate demand for hospitals conditional on being enrolled in a plan, I specify the following simple logit model for the utility of individual i, enrolled in plan m, choosing hospital h, in year t, uimht = bh HOSPITALh + lt YEARt + eimht

(1)

where HOSPITALh and YEARt are dummy variables for hospital h and year t, and eimht is distributed i.i.d. type I extreme value. I define the outside option as the consumer going out-of-network (to other major hospitals 9

who do not have contracts with her plan, other non-major hospitals, or hospitals outside the market), while still getting reimbursed by her insurer, and normalize its characteristics to zero, uim0t = eim0t

(2)

Note that the data observations are on the payer-hospital pair level but I do not include a zero observation for insurers and hospitals who do not have a contract as I group them into the out-of-network category. As the group of out-of-network hospitals is relatively large, the outside option remains reasonably consistent across payers and markets with additions of a few excluded major hospitals. In the demand specification, I include hospital and year fixed effects as demand shifters but do not allow for hospital quality to vary individually across time for better identification. In Figure 6, I scatter plot the log of share ratio (conditional hospital share over outside share) over time by individual hospitals to show that hospitals generally do not exhibit individual heterogenous time trends and thus hospital and time fixed effects should explain most of the demand variations.

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Figure 6: Hospital-specific time trends

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Figure 7: Hospital quality and network concentration

Most hospitals exhibit positive perceived qualities (hospital fixed effect plus year fixed effect) despite the large set of out-of-network hospitals defined in my sample, indicating consumers’ strong preference towards high quality hospitals. The year fixed effects are mostly significant and display an upward trend, revealing the increase in major hospital quality (relative to non-major out-of-network hospitals) over time. Now with these estimates, I construct a hospital quality index, QUALITYht = bˆ h + lˆ t

(3)

which represents the perceived quality of hospital h in year t. To investigate the relationships between the network outcomes in a market and the market’s primitives, I explore the empirical link between hospital quality and its network diversity, measured by the within-hospital payer HHI. In Figure 7, I scatter plot the logarithm of withinhospital HHI over hospital qualities and see a negative relationship between hospital quality and its network diversity - since higher HHI means less diversity. However, this relationship is obviously plagued with many identification issues, as hospital quality is correlated with location, time, and etc. In what follows, I try to control for these unobserved heterogeneities to more formally test this relationship. In Table 3, I adopt three specifications to test the relationship between hospital quality and its contracting behaviors. In column 1, I regress the logarithm of within-hospital HHI on hospital quality, while controlling for year, market, and hospital fixed effects8 . The relationship is still positive and significant, suggesting that a one unit increase in the quality index (one log point in demand measures) leads to 11.5% increase in the within-hospital HHI. In column 2, I simply use the number of 8A

data observation is on the year-market-hospital level and the same hospital appears in different markets and years and thus the hospital fixed effects are identified.

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contracts a hospital has in a market as the dependent variable and find that an unit increase in the quality index leads to about 0.8 fewer contracts for the hospital, which, again, suggests a negative relationship between hospital quality and its network diversity. Finally, in column 3, to control for differences in market sizes, I define a hospital’s contract saturation rate as the number of contracts it has divided by the total number of possible contracts available in the market (number of payers) and use it as the dependent variable. The estimate is marginally not significant but also suggests the same relationship as before - that higher quality hospitals have less network diversity. Finally, in the fourth column, I explore the effects of hospital quality and competitions on the network diversity outcome in the market. I define the market saturation rate analogously as the total number of contracts in the market divided by total possible contracts. Controlling for market and year fixed effects, I find that one unit increase in the variance of hospital qualities leads to more than 30 percentage points decrease in the market contract saturation rate, which is substantial. Table 3: Effects of Quality on Hospital Network Diversity Log(HHI)

# Contracts

Hospital Saturation

0.115*** (0.033)

-0.786* (0.436)

-0.150 (0.108)

Year FEs Market FEs Hospital FEs

Yes Yes Yes

Yes Yes Yes

Yes Yes Yes

-0.313*** (0.064) Yes Yes No

Adjusted R2 : Observations:

0.650 444

0.461 444

0.451 444

0.335 180

Quality Variance

Market Saturation

p-values in parentheses * p < 0.10, ** p < 0.05, *** p < 0.01

This relationship could be attributed to high quality hospitals being more selective in contracting with insurers, or that high quality hospitals are associated with higher costs and are thus less affordable to many small payers. It also suggests that heterogeneity among hospital competitions likely leads to exclusive contracts and less network diversity. However, due to the complexity and endogeneity of the bilateral contracting process, and to understand the mechanisms through which the market primitives affect the network outcomes, I resort to a structural model that explicitly specifies the contracting process. I now turn to illustrate the model.

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3

Model

I model the game played by hospitals and insurers in the following five stages. I. Insurers and hospitals sequentially send out contract negotiation requests to potentially all other players, bearing a negotiation cost per each request sent. II. Negotiation takes place when it is mutually requested by an insurer-hospital pair. Insurer payment base prices are determined through bilateral bargaining and are conditional on all other contracts in the market. III. Insurers simultaneously determine their plan premiums through a Bertrand pricing game. IV. Consumers choose plans based on their preferences over networks available and plan premiums. V. Consumers incur claims at a hospital of her choice, potentially reimbursed by her insurer. This paper intends to investigate stages I and II with respect to market structure changes, but I also model stages III - V to carefully evaluate the impacts of said changes. In the rest of the section, I first introduce key notations and assumptions and then present the model backwards as I solve it.

3.1

Notations & Assumptions

• Notations – i - individual – m - insurer – h - hospital – Nm - set of hospitals contracted with insurer m – Nh - set of insurers contracted with hospital h – N - set of all contracts between all hospital-insurer pairs • Assumptions 1. Individual hospitals bargain with individual insurers (no hospital systems) 2. I treat each insurance plan as a separate entity 3. Zero co-insurance rate: consumers are fully insured after plan purchases 4. All insurers and hospitals share one homogenous risk pool with full information

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3.2

Stage V: Patient Hospital Choice

This is the same conditional hospital demand model I estimate in Section 2. I will briefly recap the model and then derive quantities that will be necessary for the rest of the model. Individual i’s utility in choosing hospital h when enrolled in plan m is, uimh = bh xh + eimh

(4)

where xh are the hospital-year dummies and bh the uniform (across consumers) marginal utility of the hospital fixed effects (which, in this case, correspond to the estimates in Table ??). Then the mean utility for hospital h is, dh = bh xh

(5)

And the ex-ante expected utility for choosing insurer m’s hospital network is the logit inclusive value of all the hospitals in m’s network, Wm = log

Â

h20,Nm

exp(dh )

!

(6)

Finally, hospital h’s conditional share among all of m’s enrollees is, smh =

3.3

exp(dh ) Âk20,Nm exp(dk )

(7)

Stage IV: Insurance Plan Choice

When making insurance plan decisions, consumers consider insurer characteristics xm with uniform marginal utility bm , her expected utility from the insurer’s hospital network Wm , and plan premium pm , vim = bm xm + Wm + am pm + eim (8) where the outside option is to remain uninsured. Integrating out the logit error, I obtain the insurance demand exp( bm xm + Wm + am pm ) sm = (9) 1 + Âl exp( bm xm + Wl + am pl )

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3.4

Stage III: Insurance Plan Pricing

Given the negotiated reimbursement rates pmh (determined in stage II), insurers first calculate their expected marginal costs, MCm =

Â

h20,Nm

pmh smh

Following Gowrisankaran et al. (2013), I define pm0 , the reimbursement rate of insurer m for out-of-network hospitals, is the unweighted mean of p~m , m’s vector of negotiated prices. Insurers then simultaneously set plan prices that solve the Bertrand pricing game, p⇤m = max( pm

MCm )sm

pm

3.5

Stage II: Payment Base Negotiation

I now formulate the Nash bargaining game between each hospital-payer pair that has mutually agreed to negotiate in stage I. The Nash bargaining problem consists of bargaining values of the bargaining partner, defined as the difference between their agreement values and disagreement values, which are in turn the profits they make with or without the contract being negotiated9 . I first calculate the bargaining value of hospital h when facing insurer m as the difference in profits under the current contract configuration and the configuration if the m h negotiation breaks down. Denote the whole vector of negotiated prices pmh ’s as ~p, then h’s bargaining value is as follows, Vh ( N, ~p) = ph ( N, ~p)

ph ({ Nh \m, N

h },

~p0 )

(10)

To decompose equation (10), let MCmh be the insurer specific hospital marginal cost, representing heterogeneous administrative cost associated with patients from different insurers, and M the market size, then the hospital profits in equation (10) are, ph ( N, ~p) = ph ({ Nh \m, N

h },

~p0 ) =

 ( p jh

j2 Nh

Â

j2 Nh ,j6=m

MCjh )s jh s j M

( p0jh

MCjh )s0jh s0j M

(11) (12)

where ~p0 = ~p({ Nh \m, N h }), s0jh = s jh ({ Nh \m, N h }, ~p0 ), and s0j = s j ({ Nh \m, N h }, ~p0 ) are all re-computed under the contract configuration after dropping the m h contract. And this is 9 Although

the majority of hospitals in Massachusetts are not-for-profit, the large literature on hospital ownership has suggested that these hospitals usually act similarly to their for-profit counterparts, often maximizing a linear combination of profits and quantity. In this paper, I make the standard simplifying assumption in this literature that not-for-profit hospitals maximize profits as their objective.

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precisely the difference between the SZ model and the HW Model. In the HW model, they do not re-compute the entire vector ~p to calculate the disagreement values but only set pmh to zero and keep ~p mh constant. Whereas in the SZ model, a new vector of ~p0 mh is computed under the absence of the contract and used to specify the disagreement value10 . In other words, after a negotiation pair breaks down, all other players immediately renegotiate their contracts, and in expectation, the m h pair also takes that into account. As a result, to compute the network transfers under any configuration with the SZ protocol, we need to solve the model recursively until the degenerate case where all contracts are null. This leads to high computational costs, which I will defer the discussion until the next section where I solve the model. Analogously, insurer m’s bargaining value when facing hospital h is, Vm ( N, ~p) = pm ( N, ~p)

pm ({ Nm \h, N

m },

~p0 )

(13)

Note that { Nm \h, N m } is the same contract configuration as { Nh \m, N h }, but just from different perspectives. Insurers’ profits under the two scenarios are similarly, pm ( N, ~p) = ( p⇤m

MCm )sm M

(14)

~p0 ) = ( p⇤0 m

0 MCm )s0m M

(15)

m },

pm ({ Nm \h, N

0 0 where ~p0 , p⇤0 h contract. m , MCm , and sm are all re-computed without the m Finally, I formulate the Nash bargaining problem using the bargaining values of insurers and hospitals and solve for prices that maximize the joint surplus of both parties, given their respective bargaining weights, and conditional on prices for the other contracts, b

NBh,m ( pmh | p ~mh ) = Vh h Vmbm

p⇤mh = max NBh,m ( pmh | p ~mh ) pmh

(16) (17)

Stage II through V essentially determine each player’s playoff in the network formation game under different configurations. I now specify the structure of the network formation game to solve for the network equilibrium where each player potentially contracts with a different set of other players to maximize their payoffs.

3.6

Stage I: Network Formation

In this stage, insurers and hospitals strategically form negotiation partnerships before they start bargaining in stage II. All players know, in expectation, their payoffs under every possible network configuration, derived from stages II to V. I specify the game such that each insurer 10 Lee

and Fong (2013) provides a nice example to illustrate this difference as well.

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Figure 8: A simple example: monopolists

first sends out negotiation requests to all hospitals, and then each hospital responds to all the requests, all in sequence11 . Insurers’ strategies are specified as a vector of zeros (no request) and ones (request) with length NH , the number of hospitals in that market. Thus, each insurer has 2 NH strategies and similarly, each hospital has 2 NM strategies to respond. Only mutual requests result in a negotiation pair in stage II. Therefore, the game is described by a 22NH NM by ( NH + NM ) matrix although there are only 2 NH NM possible network configurations due to multiple strategy combinations resulting in the same network configuration. In execution, I solve for payoffs under 2 NH NM scenarios and map them back into the game matrix of 22NH NM by ( NH + NM ). In addition to the payoffs computed from stages II to V, each player also pays a fixed cost for every request (or positive request response) she sends, representing negotiation administrative costs. I also do not explicitly model hospital capacity constraints but only lump the potential penalty of exceeding capacity into the hospital fixed cost of agreeing to negotiate. Finally, I use backward induction to solve for the unique subgame perfect equilibrium of this game. To illustrate how this game is set up and solved, I provide a simple example below. For simplicity, assume that there are one hospital and one insurer in a market. With a contract, the hospital provides $2 million worth of services, and the insurer retails for $10 million of profits. Without a contract, they both make zero profits. With equal bargaining power, they evenly divide the contract surplus, resulting in a transfer of $6 million from the insurer to the hospital. The bargaining process (stages II - V) is illustrated on the left in Figure ??. On the right, both the insurer and the hospital have two strategies and the strategy of one incurs a fixed cost of $1 million, thus the payoffs displayed in the game tree. In this case, the subgame perfect equilibrium outcome is (1, 1), thus having a contract. I now turn to solving the model with a numerical example and exploring the properties and implications of the simulation results. 11 The

particular sequence order used will be discussed in Section ??.

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4

Implications

4.1

Curse of Dimensionality

In this section, I discuss the computational costs of this model. Solving the model is very computation intensive, as noted in other empirical network papers12 as well. The model suffers from the curse of dimensionality as the computation cost increases exponentially in the number of players. Specifically, the main computation burden lies in the 2 NH NM times that I solve the bargaining game to compute payoffs under each network configuration. The reason I have to solve all 2 NH NM possibilities of network transfers is twofold. First, due to the use of the SZ model, to compute payoffs for any network configuration, I need to solve all configurations that are less saturated (having less contracts in the network). Second, to solve the network formation game, I need to compute payoffs under each network outcome to map back into each strategy combinations. Therefore, the use of the SZ model does not incur any additional computation costs in the presence of the network formation game. To mitigate the computation burden, I employ parallel processors to break up the computation13 . Specifically, I use each individual thread to compute a single game (market) before I group them back together. Note that I cannot break the loop within a game due to data dependency due to the recursive nature of the SZ model, whereas in HW, I would be able to divide up the task even further. However, the additional CPU time gain is minimal since the number of markets usually exceeds the number of threads I have. The parallel computation results in large reduction in CPU time cost.

4.2

Simulated Competitions

To understand the implications of the model and to further investigate the relationships between the market primitives and its outcomes, I simulate the model with a numerical example. Specifically, in a hypothetical market with two hospitals and two insurers, I hold the cost and quality of one hospital constant, while varying the cost and quality of the other hospital. I solve the model on the grid of cost-quality pairs and compare various market outcomes. I now specify the parameters used and present the results. Parameters. Market size is 1. Insurers have a single characteristics index, both of which are set as 0.5, with consumers’ marginal utility of 1. Consumers’ price elasticity for insurance is -1. Insurers’ contracting fixed cost is constant at 0.05. One hospital’s cost and quality are held at 1 for both. The other hospital’s cost and quality vary between 0.5 and 1.5 with 0.1 increments for both. The latter hospital also moves first in the network formation game. Consumers’ 12 See

Lee and Fong (2013) and Uetake (2012). also exists algorithmic improvement methods that involve approximations to the equilibrium solution such as state space interpolation methods, which are not discussed in this paper. 13 There

19

marginal utility for hospital quality is also 1. Hospitals’ contracting fixed cost is 0.075. Finally, the bargaining weights for hospitals and insurers are both 0.5. Results. Figure 9 shows the simulated results. In the top right panel, I show the total number of contracts in the market on the grid of cost and quality of one hospital. Blue blocks indicate two contracts, green three, and maroon four (full network). We see that the network is most saturated in the middle when cost and quality are comparable across hospitals. Or in other words, more intense competition leads to more diverse networks. In Figure 10, I decompose the contracts into individual hospitals, with blue indicating zero contracts, green one, and maroon 2. On the left, the graph of hospital one shows that naturally the hospital gets more contracts with higher quality and lower costs. Combined with the empirical evidence found in Section ??, it suggests that, for hospitals who already have high qualities, higher quality leads to fewer contacts because their costs rise faster than qualities. On the other hand, it is likely that for low quality hospitals, costs rise slower than qualities but they also demand more surplus with increased quality. On the right of Figure 10, the change in cost and quality of the competitor has an asymmetric impact on its network diversity, especially as a second mover in the network formation game. Coming back to Figure 9, the top middle panel shows the share-weighted hospital industry profits. The wave shape exactly resembles that of the top left panel due to the dependency of profits on networks. We see that the industry profits maximize at where contracts are fully saturated (except the very corner of lowest cost and highest quality), indicating that a hospital system, which internalizes the contracting externalities, would do better. It also suggests that from a policy perspective, it might be more beneficial to help the low quality and inefficient hospital than to facilitate a more differentiated market. The top right panel, displaying the per-enrollee insurance plan premiums, also exhibits the wave shape of the networks and shows positive correlation between contract counts and plan premiums. Although premium increases seem to hurt consumers, in this case, they come with higher health care quality and network accessibility, instead of market powers. The welfare benefits of network diversity are also evidenced in the bottom left panel. With mediocre hospital costs and qualities in the middle of the grid, full network achieves almost the max insurance rate, comparable to the corner of highest quality and lowest cost. Analogous to the top middle panel, the bottom middle graph shows that the insurer industry also enjoys high level of profits with more saturated networks but fails to do so due to externalities. Finally, in the bottom right panel, I plot the total welfare, sum of consumer surplus and industry profits, over the grid. The rough surface is due to conflicts of interests of different parties in the market. The total welfare again peaks in the middle where contracts are full, instead of the corner. This again suggests that from the policymakers’ perspectives, it is more important to promote competition among hospitals than to blindly improve quality or reduce 20

cost.

Figure 9: Simulation Results: A Numerical Example

Figure 10: Individual Hospital Contract Counts I now propose a stylized merger model to understand potential impacts of hospital consolidations with the network formation framework. In the following example, I simulate a merger between the two hospitals above exogenously. Following the merger, the merged hospitals form a hospital system that restricts insurers’ strategies to include or exclude the whole system 21

together. The hospital system internalizes total profits of the system in stage I but individual hospitals in the system still bargains separately in stage II. I keep all the parameters the same as in the example above. Figure 11 presents the same six graphs post-merger. Now the hospital system always signs an exclusive contract with one insurer, resulting in two contracts in the market. Therefore, the consolidation in the upstream industry automatically forecloses the downstream market for the other insurer, resulting in a monopoly retail market. Total welfare decreases, while industry profits go up after the market foreclosure. We see that in this example, not only consumers are hurt with increased prices, but also fewer product choices due to network exclusions. In this case, a counterfactual hospital merger analysis without considering endogenous network formations would underestimate the negative effects of the merger.

Figure 11: A Stylized Counterfactual Merger Example

5

Estimation Strategy

I now turn to the estimation of this model. Due to the limitations of the SID data, I will make several simplifying assumptions in the estimation strategy. The model is estimated in two steps. In Section ??, I estimate the perceived hospital qualities using conditional hospital demand data as the first step. I then jointly estimate insurance demand, plan premiums, bargaining, and network formation using the method of simulated moments (MSM). Specifically, I estimate the following five structural parameters, which will be further explained later in the estimation procedure. 22

• b - hospital cost coefficient on hospital quality • g - time trend in hospital costs beyond quality shifts • µ - intercept for hospital marginal costs • a - consumer price elasticity for insurance plans • f - constant contracting fixed cost for both hospitals and insurers The model is identified with two sets of moment conditions. I first match moments on the individual insurance demand, which is on the year-market-payer level. The second set of moments is the set of individual contracts between each hospital-payer pair, which is on the year-marketpayer-hospital level and each moment takes on a value of one or zero. I now formally describe the estimation procedure, along which assumptions are made where appropriate. Given each evaluation of the parameter set q = { b, g, µ, a, f }, iterate through all markets in the data in the outer loop, and in the inner loop, • Solve a Nash bargaining problem under each possible network configuration (2 NH NM iterations, going from the least saturated networks to the most due to the data dependency of the SZ model) – In consumers’ hospital choices, uimh = bh xh + eimh , substitute in estimates for bˆ h from Section ?? – In insurance demand, I simplify the utility equation to vim = Wm + am pm + eim by suppressing insurers’ characteristics14 – In plan premiums setting, solve the standard Bertrand game with first order conditions – In formulating the Nash bargaining problem, specify hospitals’ marginal costs as MCh = bQUALITYht + gt + µ

(18)

where QUALITYht are again substituted in from the quality index from Section ??, t is a time trend that captures any cost shifts across time that are beyond quality improvements, and µ captures the average of the residual marginal costs 14 This

is assuming that insurance plans, as pure financial instruments, only carry values to the extent of the networks they offer and at what prices they are offered. I acknowledge that the literature has indicated that consumers potentially care about other plan characteristics, such as brand names, billing services, and etc., and thus I will explore these other specifications as I get access to more comprehensive data.

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– To solve for each contract price in the Nash bargaining problem, use a Gauss-Siedel procedure that maximizes the joint surplus conditional on all other prices and updates the current price to solve for the next price15 • Compute each player’s profit under all 2 NH Nm networks, and map them into a game tree of dimensions 22NH NM by ( NH + NM ) based on network configurations • Subtract the appropriate amount of contracting fees ( f ⇥ # of requests) off of each player’s profits to get their game payoffs • Solve the game by backward induction to get the equilibrium network structure • Compute the corresponding contracts between each hospital-payer pair under the equilibrium network and each payer’s shares • Evaluate the objective function value

~ˆ (W, q ) J = (m

~ (W ))0 (m ~ˆ (W, q ) m

~ (W )) m

(19)

~ (W ) are the data moments and m ~ˆ (W, q ) are the where W represents the data and thus m predicted moments from the model at the parameter guess q Finally, I solve for qˆ 2 argmin J (W, q ) and now I turn to the estimated results. q

6

Results

Table ?? presents coefficient estimates for the structural parameters of the model.16 All of the estimates exhibit the expected signs without constraints in the estimation. The hospital cost coefficient on hospital quality (b) is estimated to be greater than 1, suggesting that, as hypothesized in the preliminary analysis, hospital costs rise faster than quality improvements. There is also an upward trend in hospital marginal costs beyond quality changes over time, as indicated by the positive coefficient g. Note that with the SID data, it is hard to interpret the magnitude of the estimates since I do not observe any prices to pin down the units of any dollar amount. Therefore, I will only calibrate the magnitudes from the mean costs in past literature with my 15 Note

that I currently hold the bargaining weights constant at 0.5 for both hospitals and insurers and do not estimate them. This is fairly standard in the bargaining literature because: 1) the bargaining weights are interpreted as relative discount factors and thus it is reasonable to set them as 0.5, and 2) the bargaining weights are usually very difficult to be identified properly. See both Lee and Fong (2013) and Gowrisankaran et al. (2013) for discussions. 16 Note that I restrict attention to only 10 out of the 180 markets in my sample. Specifically, I focus on the market that has the 3-digit ZIP code 013, which covers Franklin County, Worcester County, and Berkshire County and exploit time series variation in the market structures in this market for 10 years. I also only report the point estimates from the estimator as standard errors are not yet available.

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estimates17 , shown in column 3. Hospital marginal costs on average rise $83.5 per year, with a base of $7980 in 2001 and $2053 per quality unit. The average fixed contracting cost across hospitals and insurers is estimated to be about $100 per contract, base on the unit calibration. To interpret the price sensitivity coefficient a, the average implied insurer own price elasticity is -0.103, with standard deviation 0.022, while the average implied insurer cross price elasticity is 0.015, with standard deviation 0.003. The relatively inelastic demand could potentially be attributed to the fact that with network contracting considerations, insurers have increased incentives to maintain high volume of demand in order to negotiate for lower reimbursement rates with hospitals18 . Other estimates are consistent with the past literature and also shed light on previously undiscovered quantities such as the contracting costs. I now turn to present other predicted equilibrium quantities that were also previously lacking in the literature. Table 4: Estimates: Structural Parameters Definitions

Estimates

Calibrated Coef.

b

∂MCh ∂QUALITY

1.23

$2053.4

g

∂MCh ∂t

0.05

$83.5

µ

MCh intercept

4.78

$7980.0

a

price elasticity

-0.88

NA

f

fixed cost

0.06

$100.2

N

50

As mentioned above, I adopt two sets of moments in the estimation - network outcomes and insurance plan shares, resulting in 50 moments in contracting outcomes and 25 in shares. My model successfully predicts 48 out of the 50 contracts in all markets with the estimates in Table ??, with an objective function value of 2.465. Figure 12 shows the predicted hospital marginal costs, insurance plan premiums, and negotiated transfers. Hospital marginal costs are calibrated at the mean but range from $8757 to $11302, with standard deviation $713. The predicted insurance plan premiums average $983.5, ranging from $915 to $1037, with standard deviation $37.219 . FInally, the average insurance claim after bargaining is $11160, ranging from $10116 to $12050, with standard deviation $518. 17 Specifically, I assume that the mean marginal cost of hospitals is about $10,000.

With a mean estimated hospital marginal costs of 5.99, I calibrate the magnitudes with a multiplier of 1669.45. 18 Other factors such as the logit structure and data inadequacy could also be the reasons. 19 The plan premiums have to be calibrated again due to the assumption that everyone gets sick and incurs insurance claims, i.e. only patients buy insurance. I follow Lee and Fong (2013) and calibrate the probability of getting sick at 0.075.

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Figure 12: Predicted Costs and Equilibrium Prices

7

Conclusions

In this paper, I specify and estimate a model of bilateral contracting that exists in many markets, including but not limited to health care, television, and manufacturing. Through the contracting, a certain network usually arises while the network transfers are determined. I show that ignoring the potential network structures leads to biased results. I apply this model to an application of the U.S. medical care market where insurers establish networks with hospitals for their enrollees to access while a reimbursement payment rate is determined between each insurer-hospital pair. Using limited data from the SID, I estimate the model with MSM that only requires information on the network outcomes and hospital and insurer demand. I find that large variations in hospital qualities within markets often lead to less diverse networks and lower welfare levels. On the other hand, closer and more intense competition among hospitals generate larger networks. Finally, hospital mergers lead to not only increased bargaining power in the newly established hospital system but also more exclusive contracts with insurers, resulting in greater welfare loss. Much work, however, still remains to be done in the future. I am expanding the data set by acquiring the All-Payer Claims Database (APCD) from Massachusetts, which will allow me to better estimate the model including the Nash bargaining weights. I will also supplement these data sets with hospital and insurance plan characteristics, such as hospital and patient locations and insurance plan premiums, for a more careful demand estimation. Moreover, I will complement my current estimation strategy with other econometric tools such as generalized method of moments and moment inequality estimators to compare results.

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Starc, Amanda. 2014. “Insurer pricing and consumer welfare: Evidence from medigap.” RAND Journal of Economics . Stole, Lars A and Jeffrey Zwiebel. 1996. “Intra-firm bargaining under non-binding contracts.” Review of Economic Studies . Uetake, Kosuke. 2014. “Estimating a model of strategic network formation and its effects on performance: An application to the US venture capital markets.” Unpublished manuscript, Yale School of Management .

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