Nursing home choice, family bargaining, and optimal policy in a Hotelling economy

• Author: Gregory Ponthiere,Marie‐Louise Leroux
• Date: 01 August 2020
• Published date: 01 August 2020
• DOI: http://doi.org/10.1111/jpet.12327

Received: 31 August 2017 Accepted: 1 June 2018

DOI: 10.1111/jpet.12327

ARTICLE

Nursing home choice, family bargaining, and

optimal policy in a Hotelling economy

Marie-Louise Leroux1,2,3 Gregory Ponthiere4

1Départementdes Sciences Économiques, ESG

-Université du Québec à Montréal, Montréal,

Canada

2CORE,Université catholique de Louvain,

Louvain-la-Neuve,Belgium

3CESifo,Munich, Germany

4UniversityParis Est (ERUDITE), Paris School of

Economicsand Institut universitaire de France,

Paris,France

Correspondence

Marie-LouiseLeroux, Département des Sciences

Économiques,ESG - Université du Québec à

Montréal,Montréal, Canada.

Email:leroux.marie-louise@uqam.ca

We develop a model of family bargaining to study the impact of the

distribution of bargaining power within the family on the choice of

nursing homes by families, and on the locations and prices chosen

by nursing homes in a Hotelling economy. In the baseline (static)

model, where the dependent parent cares only about the location

of the nursing home, the markup of nursing homes is increasing in

the bargaining power of the dependent parent, and nursing homes

are located at the extreme periphery. Wecompare the laissez-faire

with the social optimum (which involves more central locations of

nursing homes), and examine its decentralization in first-best and

second-best settings. We explore the robustness of our results to

introducing a bequest motive in a dynamic overlapping generations

model, which allows us to study the joint dynamics of wealth accu-

mulation and nursing home prices. If the bequest motive is strong,

the markup is decreasing in the bargaining power of the dependent.

However, wealth accumulation, by reducing interest rates, raises

markup rates and nursing homes prices.

1INTRODUCTION

Due to the aging process, the provision of long-term care (LTC)to the dependent elderly has become a major chal-

lenge for advanced economies. According to the European Commission (2015), the number of old dependent persons

in the Euro area is expected to grow from about 27 million in 2013 to about 35 million bythe year 2060. Although that

forecast depends on scenarios concerning mortality and disability trends, it is nonetheless widely acknowledged that,

whatever the scenarios are, there will be a substantial rise in LTC needs in the next decades.1

Nursing homes are important agents in the provision of LTC,especially when serious degrees of dependency are

reached. Brown and Finkelstein (2009) show that the probability to enter a nursing home at one point in one's life is

large, and lies between 35% and 50%. Moreover,because of the increase in the prevalence of serious old-age patholo-

gies, the demand for institutionalized LTCservices is expected to grow sharply in the years to come.

At the empirical level,the determinants of nursing home choices were studied by Schmitz and Stroka (2014), on the

basis of data from the German sickness fund TechnikerKrankenkasse on 2,534 people above age 65 who newly moved

1See Norton (2000) and Cremer, Pestieau, and Ponthiere(2012). Challenges raised by LTC for insurance markets are examined in Brown and Finkelstein

(2011).

Journal of Public Economic Theory.2018;1–34. wileyonlinelibrary.com/journal/jpet c

2018 Wiley Periodicals,Inc. 1

ORIGINAL ARTICLE

markup rates and nursing homes prices.

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J Public Econ Theory. 2020;22:899–932. wileyonlinelibrary.com/journal/jpet © 2018 Wiley Periodicals, Inc.

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to a nursing home in 2010. Matching those individual data (on chosen nursing home, zip code before moving, and care

level)with institutional data (on prices and reported quality of nursing homes), Schmitz and Stroka (2014) showed that

two dimensions drive the choice of nursing homes: location and price. The probability of choosing a nursing home is

decreasing in the distance (with respect to the previous household) and in the price.2On the contrary, the reported

quality has no significant impact on the choice of nursing homes.

From the perspective of microeconomic theory,the choice of nursing homes raises specific difficulties. A major dif-

ficulty lies in the fact that the main beneficiary of the nursing home is dependent, and faces cognitive and/orfunctional

limitations. Because of his limited autonomy,the dependent can hardly be regarded as a “sovereign” consumer select-

ing the best alternative among the set of affordable nursing homes. Thus, one cannot apply standard consumer theory

to the choice of nursing home. On the contrary,it is more relevant to consider a model where the family of the depen-

dent parent—including, to an extent varying with the degree of dependency, the dependent himself—will collectively

choose the nursing home the dependent will be sent to.

The goal of this paper is to try to open the black boxof the choice of nursing homes, by developing a microeconomic

model where the selected nursing home is the outcome of bargaining between the dependent parent and his family.We

study a model of cooperative decision making where the selected option—that is, the selected nursing home (defined

in terms of price and location)—maximizes a weighted sum of the utilities of family members, the weights reflecting

the bargaining power of each family member. As such, our model is in line with other models of cooperativedecision

making concerning LTCarrangements, such as Hoerger, Picone, and Sloan (1996), Sloan, Picone, and Hoerger (1997),

and Pezzin, Pollak, and Schone (2007).3

Models of family bargaining point to an important determinant of social outcomes: the distribution of bargaining

power within the family.4As stressed by Sloan et al. (1997), the dependent parent and his children can disagree on

the kind of supply of LTC(e.g., formal vs. informal care), because they do not have the same preferences. Hence, the

option that will emerge depends on the distribution of bargaining power within the family.Sloan et al. (1997) stressed

that the bargaining power of the parent depends on three main features: first, his degree of cognitiveawareness (which

could limit his capacity to take part to the decision); second, his number of children (which can favor competition for

gifts); and third, his wealth (the strategic bequest motive).

This paper proposes to explore further the consequences of the distribution of bargaining power within the family

on LTCoutcomes, by considering its impact on the choice of nursing home, and, also, on the characteristics (price and

location) of nursing homes that drive that choice. The underlying intuition for considering the impact of family bar-

gaining on nursing homes'characteristics goes as follows. When facing the choice of a nursing home, families take the

characteristics of available nursing homes as given. However,those characteristics cannot be taken as parameters, but

are variables chosen strategically bynursing homes. Thus, we need also to explain how family bargaining affects, at the

equilibrium, nursing homes price and location.

Forthat purpose, this paper considers an economy àlaHotelling(1929), where a continuum of families, composed of

a dependent parent and a child, choose between two nursing homes located along a line, those nursing homes choosing

their prices and locations. Two variants of the Hotelling economy will be studied. First, we will consider a baseline

static model, where the dependent parent, who has limited capacity to enjoy consumption, wants the nursing home to

be as close as possible to his family,whereas his child, being the bearer of LTC spending, cares both about location and

price. Second, we will extend this frameworkto a dynamic overlapping generations model (OLG), in order to introduce

a parental bequest motive. This extended model allows us to study the joint dynamics of wealth accumulation and

nursing home prices.

2Note that there existssignificant heterogeneity in the factors driving nursing home choices (in particular, concerning the role of nursing home location), as

shownby Ramos-Gorand (2016) for the case of France.

3However,our model differs from models of noncooperative decision making applied to LTC, such as Hiedemann and Stern (1999), Stern and Engers (2002),

Konrad,Kunemund, Lommerud, and Robledo (2002), Kureishi and Wakabayashi (2007), and Pezzin, Pollak, and Schone (2009).

4Onthe impact of the distribution of bargaining power on time allocation, see Konrad and Lommerud (2000). de la Croix and Vander Donkt (2010) and Leker

andPonthiere (2015) studied the impact of bargaining on education outcomes.

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There are several reasons why the Hotelling model is relevant for the issue at stake. First, the nursing homes

sector exhibits entry barriers and is thus imperfectly competitive. For instance, the opening of a nursing home in

France can only be made provided that there is an official call for tenders from the French State. The opening of a

nursing home also requires a specific authorization, to be signed by national and local authorities. Thus, the num-

ber of nursing homes is limited, as in the Hotelling model. Second, nursing homes are generally left free regard-

ing their geographical location. In the French case, official calls for tenders do not, in general, specify extremely

precise locations, so that nursing homes have some room for location choices. Third, nursing homes exhibit pos-

itive markup rates. Martin (2014) estimated that the average markup rate for French nursing homes ranges in

2011 from 4.2 % to 11.8 %.5Those features make the Hotelling model a natural candidate to study nursing

homes.

Anticipating our results, we find, in the baseline model (i.e., without bequest motive), that nursing homes locate at

the extremes of the Hotelling line (following the principle of maximum differentiation), whereas the markup of nurs-

ing homes is increasing in the bargaining power of the dependent parent. The laissez-faire equilibrium is contrasted

with the utilitarian social optimum, where nursing homes locate more centrally. If the governmentcan force nursing

home locations, the social optimum can be decentralized by merely subsidizing nursing homes to achieve pricing at

the marginal cost. However,if locations cannot be forced, the decentralization requires in addition a nonlinear tax on

location.

Turning now to the dynamic OLG model with wealth accumulation, it is shown that if the parental bequest

motive is strong, the markup can be decreasing in the bargaining power of the dependent parent, unlike in the

baseline model. Moreover, the markup is shown to be decreasing with the interest rate, because a higher interest

rate raises the opportunity cost of LTC expenditures. The existence, uniqueness, and stability of stationary equilib-

ria is then studied, and it is shown that, starting from low wealth levels, the convergence towards the stationary

equilibrium leads to a rise in the price of nursing homes through higher markup rates induced by lower interest

rates.

Our paper is related to several aspects of the literature on LTC. First, it is related to models of family bargaining,

such as Hoerger et al. (1996) and Sloan et al. (1997), which studied how family bargaining affects the choice of formal

versus informal LTCprovision, as well as the choice of living arrangement. Our contribution is to study how nursing

homes react strategically and set prices according to the distribution of bargaining power within the family.Our paper

is also related to the literature on location games in the context of LTC, such as Konrad et al. (2002) and Kureishi and

Wakabayashi (2007). Although those papers studied the strategic location of children with respect to a given nurs-

ing home location, we do the opposite and study the strategic location of nursing homes with respect to a given loca-

tion of children. We also complement IO papers applying Hotelling's model to health issues, such as Brekke, Siciliani,

and Straume (2014), who studied competition in prices and quality among hospitals. Our paper complements this IO

approach by considering interactions between family bargaining, pricesand location outcomes. 6Wealso complement

the literature on optimal public policies under LTC,such as Jousten, Lipszyc, Marchand, and Pestieau (2005), Pestieau

and Sato (2008), and Cremer,Lozachmeur, and Pestieau(2016) by exploring the optimal public intervention when nurs-

ing homes choose prices and locations strategically. Finally,our paper is also related to studies of long-run dynamics

using OLGmodels with LTC, such as Canta, Pestieau, and Thibault (2016) and Pestieau and Ponthiere (2016).

The rest of the paper is organized as follows. Section 2 presents the main assumptions of the baseline model. Sec-

tion 3 characterizes the laissez-faire, and exploresthe links between the distribution of bargaining power in the family

and the markup rate of nursing homes. The social optimum and its decentralization are studied in Section 4. Section 5

extends our previous baseline static model to dynamic OLGframework, and examines the robustness of our results to

the introduction of a parental bequest motive. Conclusions are drawnin Section 6.

5Althoughthose figures may seem surprising given that the price of LTC formal services in nursing home is regulated in France,it should be reminded thatthe

Frenchauthorities do not regulate the price of accommodation services, which can thus give rise to a significant markup.

6Interactionsbetween bargaining and spatial competition are also studied by Bester (1989) in a Hotelling model where prices are the outcome of bargaining

betweenconsumers and firms. In our model, the bargaining occurs on the consumer side only.

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