Bargaining and the provision of health services

ORIGINAL PAPER

Bargaining and the provision of health services

Luigi Siciliani • Anderson Stanciole

Received: 22 July 2010 / Accepted: 8 February 2012 / Published online: 16 March 2012

� Springer-Verlag 2012

Abstract We model and compare the bargaining process

between a purchaser of health services, such as a health

authority, and a provider (the hospital) in three plausible

scenarios: (a) activity bargaining: the purchaser sets the

price and activity (number of patients treated) is bargained

between the purchaser and the provider; (b) price bar-

gaining: the price is bargained between the purchaser and

the provider, but activity is chosen unilaterally by the

provider; (c) efficient bargaining: price and activity are

simultaneously bargained between the purchaser and the

provider. We show that: (1) if the bargaining power of the

purchaser is high (low), efficient bargaining leads to higher

(lower) activity and purchaser’s utility, and lower (higher)

prices and provider’s utility compared to price bargaining.

(2) In activity bargaining, prices are lowest, the purchaser’s

utility is highest and the provider’s utility is lowest; activity

is generally lowest, but higher than in price bargaining for

high bargaining power of the purchaser. (3) If the purchaser

has higher bargaining power, this reduces prices and

activity in price bargaining, it reduces prices but increases

activity in activity bargaining, and it reduces prices but has

no effect on activity in efficient bargaining.

Keywords Bargaining � Negotiation � Purchasing

JEL Classification I11

Introduction

Prospective payment systems are used widely to remu-

nerate health care providers. In the hospital sector, they

usually take the form of Diagnosis Related Groups (DRGs)

pricing or similar methods, such as Healthcare Resource

Groups (HRGs) in the United Kingdom or Group Hom-

ogenes de Maladie (GMC) in France. Depending on the

institutional context, purchasers and providers (i.e. hospi-

tals) bargain on price, activity (i.e. numbers of patients

treated), or both. For example, in the US, Health care

Maintenance Organisations (HMOs) or private health

insurers bargain with hospitals on price, and seldom the

number of patients treated, i.e. activity [2, 6]. In the United

Kingdom, public purchasers (i.e. Health Authorities and

Primary Care Trusts) have been negotiating price and

number of patients treated (i.e. activity) with hospitals

(known in England as NHS Trusts) under ‘‘cost and vol-

ume’’ or ‘‘sophisticated’’ contracts. The government has in

more recent years implemented a policy known as ‘‘Pay-

ment by Results’’, where prices are regulated, but activity

is negotiated between the purchaser (i.e. the Primary Care

Trust) and the hospital (i.e. the NHS Trust). In the future,

Electronic supplementary material The online version of thisarticle (doi:10.1007/s10198-012-0383-x) contains supplementarymaterial, which is available to authorized users.

Anderson Stanciole: Work undertaken while at the University of

York.

L. Siciliani

Department of Economics and Related Studies,

and Centre for Health Economics,

University of York, York, UK

e-mail: [email protected]

L. Siciliani

Centre For Economic Policy Research (C.E.R.P.),

90–98 Goswell Street, London EC1V 7DB, UK

Present Address:A. Stanciole (&)

World Bank, Health, Nutrition, and Population,

1818 H St NW, Washington, DC 20433, USA

e-mail: [email protected]

123

Eur J Health Econ (2013) 14:391–406

DOI 10.1007/s10198-012-0383-x

http://dx.doi.org/10.1007/s10198-012-0383-x

prices may be allowed to vary again. Within the Medicare

Programme in the US, prices are nominally chosen by the

purchaser (the government), while activity is either chosen

or bargained with the provider: the price decided by the

government could be seen as the outcome of a bargaining

process between the government and the hospital associa-

tion. Similar arrangements exist throughout Europe

[15, pp. 243–245, 20, ch. 1].

Although we observe a substantial amount of bargaining

between purchasers and providers, the theoretical literature

on the relative merits of hospital prospective payment

systems normally assumes that payers are able to set the

prices unilaterally, while hospitals choose the amount of

quality and cost-containment effort, and in cases the

number of patients treated (see, for example, [8, 9, 11, 21,

25, 26, 34]). This implies that purchasers have all the

bargaining power, which is a simplifying assumption, as

the empirical evidence suggests that providers may hold at

least some of it. Propper [32] shows that, in England,

purchasers with higher bargaining power could secure

lower prices. Brooks et al. [6] estimate that US hospitals

hold on average 65% of the bargaining power when

negotiating with private insurers. Melnick et al. [24] find a

negative association between purchasers with greater

market shares and prices charged by the providers.

This study models the bargaining process between a

purchaser of health services (a public or private insurer)

and a provider (a hospital) in three plausible institutional

settings. Under these three settings we always refer, for

short, to ‘‘activity’’ as the total number of patients treated

by a representative hospital. We refer to the ‘‘price’’ as the

monetary converter which translates a DRG with a weight

equal to one into money (US dollars, British pounds, etc):

(a) the purchaser sets the price (stage 1), and the activity

is bargained between the purchaser and the provider

(stage 2): activity bargaining. This scenario corre-

sponds for example to the current arrangements under

‘‘Payment by Results’’ in England. The government

sets the price, but the activity is negotiated between

the Primary Care Trust and the hospital.

(b) The price is bargained between the purchaser and the

provider (stage 1), and the activity is chosen unilat-

erally by the provider (stage 2): price bargaining. This

scenario corresponds for example to the case of an

HMO, which bargains on the price with the hospital,

but the activity (i.e. patients treated) is chosen by the

hospital. This model could also be applied to Medicare

if we interpret the price set by the government each

year to pay hospitals as the result of the negotiations

between Medicare and a hospital association. In

England, the government has recently discussed the

possibility of moving to a setting where the prices is

negotiated between individual hospitals and local

purchasers, on the grounds that this may reflect more

accurately the costs of the hospital (though ultimately

it was not adopted).

(c) Price and activity are bargained simultaneously

between the purchaser and the provider: efficient

bargaining. This scenario corresponds for example to

the old arrangements in England under ‘‘cost and

volume’’ or ‘‘sophisticated’’ contracts’’ where Health

Authorities and Primary Care Trusts have been

negotiating price and number of patients treated (i.e.

activity) with hospitals.

The first two models (activity and price bargaining) are

two-stage models. For both models, prices are decided

before activity takes place. This is a reasonable assump-

tion. Prices are normally set at the beginning of each fiscal

year, before the hospitals start to treat the patients. In the

third model (efficient bargaining), both prices and activity

are decided at the beginning of the financial year, and the

model has then one stage only. Our main objective is to

compare prices, activity and the utility of provider and

purchaser in each of the three different institutional set-

tings. We obtain the following key results.

The key result from the analysis is that if purchasers can

set prices (activity bargaining), purchaser’ utility, which is

given by the net consumer welfare (patient benefit, net of

transfer to the provider), is highest. This result holds for

any level of bargaining power of the purchaser. The anal-

ysis therefore supports policies such as ‘‘payment by

results’’ in the UK or DRG type of payments across Eur-

ope, where prices are fixed by the purchaser or the regu-

lator. This result holds despite the fact that total surplus, i.e.

the sum of purchaser’s and provider’s utility, is not maxi-

mised (the surplus is instead highest under efficient

bargaining).

A second key analytical result is that if the bargaining

power of the purchaser is low, efficient bargaining leads to

higher prices and provider’s utility, and lower activity and

purchaser’s utility, compared to price bargaining. This

result seems surprising, as one would expect the purchaser

to be better off when she can bargain with more instru-

ments, i.e. both price and activity. However, this intuition

holds true only if the bargaining power of the purchaser is

high. If her bargaining power is low, having more instru-

ments is counterproductive. One policy implication is that

purchasers with low bargaining power may be better off if

restricted to bargaining on prices only, and not on price and

activity.

A third counter-intuitive result, which may be important

for policy, is that by shifting from efficient and price bar-

gaining (as in ‘‘cost and volume’’ or ‘‘sophisticated’’ con-

tracts) to activity bargaining (as in ‘‘payment by results’’),

392 L. Siciliani, A. Stanciole

123

the level of activity is likely to decrease. More precisely,

this study predicts that moving from efficient to activity

bargaining will certainly reduce activity. This is in contrast

to what is normally thought, i.e. that ‘‘payment by results’’

will encourage activity. When moving from price to

activity bargaining, activity will decrease (increase) if the

bargaining power of the purchaser is low (high). These

results are consistent with recent empirical evidence [14]

which shows that the introduction of ‘‘payment by results’’

in England did not lead to any significant increase in

activity.

This study contributes to the literature on purchaser-

provider bargaining in healthcare (see [2], for a survey).

Ellis and McGuire [13] develop a model in which patients

and doctors bargain about the intensity of treatment, and

derive the optimal combination of patient’s insurance and

reimbursement for the provider which maximises consumer

welfare.1 The focus is on deriving the intensity of treatment

under different demand and supply side arrangements.

There is no comparison with other types of bargaining.

Barros and Martinez-Giralt [3] show that, when bargaining

with providers, purchasers may prefer to bargain with a

professional association rather than a subset of more effi-

cient providers. They focus on price bargaining and ignore

the other types of bargaining considered in this study.

Barros and Martinez-Giralt [5] analyse a bargaining pro-

cess in which the purchaser can choose whether to nego-

tiate with each provider separately or jointly, or announce a

contract that any provider is free to sign (the ‘‘any willing

provider’’ clause). They show that if the total surplus is

high, the purchaser prefers the system of ‘‘any willing

provider’’, but if it is low she prefers either joint or separate

negotiations. Again, the focus is on price bargaining. Gal-

Or [16] shows that purchasers (private insurers) might be

willing to sign exclusive contracts with a subset of pro-

viders in order to secure more favourable terms during

bargaining. Gal-Or [17] studies whether vertical mergers

between hospitals and physician practices might enhance

their bargaining power with the insurers (see also [18]).

Barros and Martinez-Giralt [4] explore the implications of

the coexistence of a public and a private sector in the

provision of health services. They argue that the public

sector might choose to hold idle capacity in order to extract

more beneficial conditions when bargaining with the pri-

vate sector for the provision of services. There are other

applications of bargaining in the health economics litera-

ture. Clark [10] examines how to divide a budget between

two patients with different health conditions and capacity

to benefit. Pecorino [31] models the effects of drug

reimports from Canada on the profitability of US domestic

pharmaceutical companies.2 With the exception of the

model by Ellis and McGuire [13], most of the existing

studies focus on price bargaining. Our main departure and

contribution to the literature is to compare different types

of bargaining models. The solution under price bargaining

is qualitatively analogous to those obtained in the cited

papers. The added value of our analysis consists in con-

sidering within the same set-up other forms of bargaining

(activity and efficient) in addition to price bargaining, to

compare them and to link the different regimes to different

institutional arrangements.

Some of the bargaining models presented below can be

interpreted as reduced forms of more complex institutional

bargaining arrangements between (public or private) pur-

chasing entities and associations (or organisations) of pri-

vate providers. A more detailed analysis of such

arrangements would quickly become intractable within the

current set up and are therefore outside the scope of this

study.

The study is organised as follows. ‘‘The model’’ presents

the model. ‘‘Regime comparison’’ provides a comparison

of the different scenarios. ‘‘Adding quality and effort’’

extends the model by adding quality and cost-containment

effort. ‘‘Extension’’ further extends the basic model by

endogenising the bargaining power of the purchaser and

the provider. ‘‘Conclusions’’ offers concluding remarks and

policy implications.

The model

We model the bargaining process between a purchaser of

health services, such as a health authority, and a provider (a

hospital). Define y as the number of patients treated and

p as the price the provider receives for each patient treated.

The provider’s utility U is given by its surplus U(p, y) =

py - C(y), where C(y) is the cost function of the provider,

which satisfies Cy [ 0, Cyy [ 0 (increasing marginal cost).

The purchaser’s utility (or health authority utility) is

given by the difference between the benefit for the patients

B(y) and the transfer to the provider: V(p, y) = B(y) - py.

The benefit function satisfies By [ 0 and Byy B 0.

A more general objective function for the purchaser is

B yð Þ � ð1þ kÞpyþ dU, where k is the opportunity cost of

public funds and d is the weight attached to the utility of

the provider. The main results of the analysis with this

more general specification would be qualitatively similar as

long as either k[ 0 or d\ 1. This is because a positive

1 Dor and Watson [12] evaluate how different payment mechanisms

affect the incentives in the relationship between hospitals and

physicians.

2 See also Wright [35] for a model of price regulation in the

pharmaceutical sector where the regulator and the pharmaceutical

company bargain over a subsidy.

Bargaining and the provision of health services 393

123

opportunity cost of public funds k simply implies a higher

marginal cost for the purchaser. The weight assigned to the

utility of the purchaser needs to be strictly less than one

(when k = 0), otherwise paying a higher price to the

hospital leaves the purchaser’s utility unchanged: the loss

from a higher price paid is exactly offset by the higher

revenues and utility of the provider. If d\ 1 higher

transfers to the provider reduce the utility of the purchaser,

which generates a tension between the purchaser and the

provider. For expositional simplicity, we focus on the

special case where k = d = 0.

We analyse three plausible scenarios. (1) Activity bar-

gaining: the purchaser sets the price (stage 1), and activity

is bargained between the purchaser and the provider (stage

2). (2) Price bargaining: the price is bargained between the

purchaser and the provider (stage 1), but activity is chosen

by the provider (stage 2). (3) Efficient bargaining: price and

activity are bargained simultaneously between the pur-

chaser and the provider.

Note that under scenarios (1) and (2) prices are deter-

mined before activity. This assumption is in line with the

institutional setting of several countries (see ‘‘Introduc-

tion’’). Prices are determined or agreed before the activity

takes place, i.e. before the hospital begins to treat the

patients. For example, prices are determined at the begin-

ning of the financial year, while activity (i.e. the number of

patients treated) is determined during the financial year (i.e.

after the resources allocation mechanism has been deter-

mined). In this respect, prices is a longer term decision

compared to activity.

Define c, with 0 B c B 1, as the bargaining power of the

purchaser, (1 - c) as the bargaining power of the provider,

V and U as the outside options for the purchaser and the

provider respectively, and eV ¼ V � V and eU ¼ U � U.

For notational simplicity let Vi = V(pi, yi), Ui = U(pi, yi),

where i = a, p, e denotes respectively activity, price and

efficient bargaining. In all the sections below we use Nash

bargaining to solve for optimal conditions [19, 27–29].3

Activity bargaining

In the first scenario, we assume that first the purchaser

chooses the price (stage 1), then the purchaser and the

provider bargain on activity (stage 2).4 We solve by

backward induction. For a given price p, the bargained

activity can be determined by solving:

maxy

BðyÞ � py� V� �c

py� CðyÞ � U� �1�c ð1Þ

The first order condition (FOC) is:

ya :ceVðBy � pÞ ¼ 1� c

eUðCy � pÞ ð2Þ

(See section The model in the Appendix for proof). To

interpret the optimal condition on the bargained activity it

is useful to characterise the solution on the basis of the

price level. We distinguish three possible cases. The first

(special) case is such that the price is bp ¼ ByðyÞ ¼ CyðyÞ,i.e. the price corresponds to the level where the marginal

benefit crosses the marginal cost. In such a case, the opti-

mal activity desired by the purchaser ðbp ¼ ByðyÞÞ is equal

to the activity desired by the provider ðbp ¼ CyðyÞÞ. The

equilibrium level of activity is such that By(ya) = Cy(ya).

The second case arises when p\bp which we refer to as

the ‘‘low’’ price. In this case we have that By(ya) [ p and

Cy(ya) [ p, i.e. the desired activity for the purchaser is

higher than the desired activity for the provider. The bar-

gained activity lies somewhere between the desired activity

of the two parties. The LHS of Eq. 2 is the net marginal

benefit of activity for the purchaser, weighted by her utility

and her bargaining power. The RHS is the net marginal

cost for the provider, also weighted by his utility and his

bargaining power.

The third case arises when p [ bp which we refer to as

the ‘‘high’’ price. In this case we have that p [ By(ya)

and p [ Cy(ya), i.e. the desired activity for the purchaser

is lower than the desired activity for the provider.

The FOC can be rewritten as c

eVp� By

� �

¼ 1�c

eUp� Cy

� �

.

Again, the bargained activity lies between the desired

activity of the two parties. We impose an upper bound

on the price: p [ p ¼ Byð0Þ: If p [ p ¼ Byð0Þ, then the

price is so high that the desired activity for the purchaser

is zero. Therefore, the equilibrium activity is character-

ised for any level of price between zero and an upper

bound p.

Figure 1 illustrates different bargained activity levels

[ya(p)] for three different values of the bargaining power of

the purchaser, equal to 0.3, 0.5 and 0.7, respectively. In

equilibrium it is always the case that eU � 0 and eV � 0, so

that the equilibrium lies in the area between the average

and marginal benefit, and the area between the average and

marginal cost.

Finally if p = By(ya) = Cy(ya) (i.e. where the marginal

benefit curve crosses the marginal cost curve), there is no

disagreement between purchaser and provider, so that ya is

such that By = Cy.

3 The Nash bargaining solution has been used extensively in labour

economics to examine negotiations between trade unions and firms

with respect to wages and employment. See, for example, Oswald

[30] for a survey of the literature, and Manning [22], McDonald and

Solow [23], Sampson [33] and Bulkley and Myles [7].4 A different interpretation is that the Department of Health fixes the

price, then the Health Authority and provider bargain on activity. The

implicit assumption is that the Department of Health and the Health

Authority share the same objective function.


123

By differentiating Eq. 2 with respect to c we obtain

oya

oc ¼ðBy�pÞeU�ðp�CyÞeV

eV eU ð�CÞ. If the price is low, a higher bar-

gaining power of the purchaser increases activity ðoya

oc [ 0Þ.If the price is high it reduces activity ðoya

oc \0Þ.The effect of a change of price on activity is:

oya

op¼ 1

�Cð1� cÞCy � C=y

eU2� c

B=y� By

eV 2

� �

ð3Þ

which in general is indeterminate. According to our

assumptions, it is always the case that Cy [ C/y and

B/y [ By, since the marginal cost is higher than the average

cost, and the average benefit is higher than the marginal

benefit. For low levels of p the provider utility eU is low (and

the purchaser utility eV is high) so that oya

op [ 0. Similarly, for

high levels of p the purchaser utility eV is low (and the

provider utility eU is high) so that oya

op \0 for low p. This

result is consistent with the example shown in Fig. 1.

The above analysis holds for a given price. The pur-

chaser chooses the price to maximise:

maxp

BðyaðpÞÞ � pyaðpÞ ð4Þ

The FOC is:

pa : Byyp ¼ yþ pyp ð5Þ

The optimal price is determined such that the marginal

benefit of higher activity equals the marginal cost. The

SOC is: Byyyp2 ? Byypp - 2yp - pypp. Dividing both terms

of Eq. 5 by yp, straightforward manipulations lead to

pa : By ¼ p 1þ 1

�yp

� �

where �yp ¼ ypp=y is the elasticity of activity with respect to

price. The optimal price is such that the marginal benefit

from activity is equal to the price, weighted by the inverse

of the elasticity of activity with respect to price: a higher

elasticity implies a lower marginal cost from an increase in

price, as intuitive.

Price bargaining

In the second scenario, we assume that first the purchaser and

the provider bargain on price (stage 1), then activity is chosen

unilaterally by the provider (stage 2).5 By backward induction,

for a given price, the hospital chooses the level of activity

which maximises U = py - C(y), leading to the FOC:

yp : p ¼ Cy ð6Þ

with oyp

op ¼ 1Cyy

[ 0 and o2yp

op2 ¼ 0 (the SOC is -Cyy \ 0). The

bargained price can be determined by solving:

maxp

BðypðpÞÞ � pypðpÞ � V� �c

pypðpÞ � CðypðpÞÞ � U� �1�c

ð7Þ

Thanks to the envelope theorem, Up = yp(p). The FOC for

the bargained price is:

pp :ceV

Byyp þð1� cÞeU

y ¼ cyþ pyp

eVð8Þ

(See section The model in the Appendix for proof). The

LHS of Eq. 8 is the benefit from a marginal increase in

price, and includes the marginal benefit for the purchaser

from a higher activity (weighted by her bargaining power,

her utility and the responsiveness of supply), and the

marginal benefit for the provider from a higher surplus

(also weighted by his bargaining power and utility). The

RHS is the cost for the purchaser from a marginal increase

in price and an overall higher transfer (also weighted).

If the purchaser holds all the bargaining power (c = 1),

the optimal price is such that: Byyp = y ? pyp. If the

provider holds all the bargaining power (c = 0), the opti-

mal price is the highest possible compatible with the pur-

chaser having a non-negative utility. The bargained price is

an intermediate level between these two extremes.

Efficient bargaining

In the third scenario, purchaser and provider bargain simul-

taneously on activity and price. This setting is called efficient

bargaining, because it reduces the potential for unexplored

opportunities from mutual gain.6 The bargaining problem is:

=0.3 =0.5 =0.7

Cy

By

yyB )(

or V=0

yyC )(

or U=0

y - Activity

High

price

Low

price

)( pya

Fig. 1 Activity bargaining

5 This setup is analogous to the model of bargaining between a firm

and a union over wage and employment [22, 23], where the firm sets

the employment, but the wage is bargained with the union.6 The outcome achieved in price bargaining is not efficient. As

remarked by Aronsson et al. [1], ‘‘there are unexplored profits and/or

utility gains from bargaining’’.


123

maxp;y

BðyÞ � py� V� �c

py� CðyÞ � U� �1�c ð9Þ

After obtaining the FOCs and re-arranging, we obtain:

ye : By ¼ Cy ð10Þ

pe ¼ ð1� cÞBðyeÞ � V

yeþ c

CðyeÞ þ U

yeð11Þ

(See section The model in the Appendix for proof). The

negotiated level of activity maximises the sum of the

surplus for the purchaser and for the provider

U ? V = B(y) - C(y). In this respect the level of

activity is efficient. The optimal price is a weighted

average of the average cost of the provider and the average

benefit for the patients.7 If the purchaser holds all the

bargaining power (c = 1), the price is equal to the average

cost: the purchaser extracts all the surplus from the

provider. If the provider holds all the bargaining power

(c = 0), the price is equal to the average benefit: the

provider extracts all the surplus from the purchaser.

Regime comparison

Constant marginal benefit

To gain some insights into how the different scenarios

relate to each other, we consider the following functional

forms: (a) the benefit function is linear in activity:

B(y) = ay; (b) the cost function is quadratic: CðyÞ ¼ c2y2

with Cy = cy; (c) the outside options are normalised to

zero ðV ¼ U ¼ 0Þ.The equilibrium for the three scenarios is reported in

Table 1 (See section Constant marginal benefit in the

Appendix for proof).

The following proposition compares prices, activity and

utility under different regimes.

Proposition 1 (a) pe = pp C pa; (b) ye C {yp;ya},

yp C ya if c B 0.59; (c) Va C Ve C Vp; (d) Up C Ue C Ua

(e) Se C {Sp, Sa}; Sp [ Sa if c B 0.59.

(See section Constant marginal benefit in the Appendix

for proof). The price in efficient bargaining is equal to the

price in price bargaining, which is higher than or equal to

the price in activity bargaining. Compared to activity bar-

gaining, under both price and efficient bargaining, the

purchaser cannot set the price unilaterally but will have to

negotiate it with the provider. Since, in general, higher

price reduces the utility of the purchaser, the purchaser will

set a lower price when this can be decided unilaterally as

opposed to when it has to be negotiated (in which case the

provider will use some of the bargaining power to get a

better deal).

The activity in efficient bargaining is the highest. The

activity in price bargaining is higher than in activitybar-

gaining when the bargaining power of the purchaser is

below 0.59. Under efficient bargaining the activity is

chosen such that the marginal benefit is equal to the mar-

ginal cost, regardless of the bargained price. Under price

bargaining the activity is chosen unilaterally by the pro-

vider. Since the bargained price is strictly below the mar-

ginal (and average) benefit, and the activity is chosen such

that the price is equal to the marginal cost, it follows that

activity will be below the point where marginal benefit and

cost are equal. The result follows. Under activity bargain-

ing, the price chosen by the purchaser is set at relatively

low levels to keep payments to the provider low, which in

turn implies that the activity that the purchaser can nego-

tiate in return is lower compared to the efficient one (where

marginal benefit crosses the marginal cost).

The purchaser weakly prefers activity bargaining to

efficient bargaining, and efficient bargaining to price bar-

gaining. Since prices are lowest under activity bargaining,

the purchaser is better off under this scenario despite the

activity being lower compared to efficient bargaining.

Under price bargaining, activity is lower (which reduces

purchaser’s utility) compared to efficient bargaining while

prices are the same. Therefore, in this case it is the lower

activity that drives the lower purchaser’s payoff under

price as opposed to efficient bargaining. The results and

intuition are reversed from the provider’s perspective: the

provider weakly prefers price bargaining to efficient bar-

gaining, and prefers efficient bargaining to activity

bargaining.

Total surplus S, defined as the sum of provider’s and

purchaser’s utility (U ? V), is highest under efficient bar-

gaining. This is not surprising as, by definition, efficient

Table 1 Equilibrium with constant marginal benefit

Activity bargaining Price bargaining Efficient

bargaining

pa ¼ a2 pp ¼ að2�cÞ

2pe ¼ að2�cÞ

2

ya ¼ acð2�cÞ yp ¼ að2�cÞ

2cye ¼ a

c

Va ¼ a2

2cð2�cÞ Vp ¼ ca2ð2�cÞ4c

Ve ¼ ca2

2c

Ua ¼ a2ð1�cÞ2cð2�cÞ2 Up ¼ a2ð2�cÞ2

8cUe ¼ a2ð1�cÞ

2c

Sa ¼ a2ð3�2cÞ2cð2�cÞ2 Sp ¼ ð4�c2Þa2

8cSe ¼ a2

2c

7 This result is in line with the model of employment-wage

bargaining analysed by Manning [22] in the context of firm-union

negotiations. The level of employment does not depend on the payoffs

of firm and union. Consequently, they ‘‘can agree on this level and

then bargain about the distribution of the rents’’ ([22], p. 131).


123

bargaining is the procedure that maximises total surplus

(purchaser’s plus provider’s surplus). More interestingly,

both activity and price bargaining are inefficient, in the

sense that they do not maximise total surplus. Proposition 1

shows that the comparison between activity and price

bargaining is in general indeterminate: however, if the

bargaining power is below 0.59 then the surplus is higher

under price bargaining. The intuition for the result is the

following. Total surplus is maximised at ye = a/c where

the marginal benefit crosses the marginal cost. Activity is

always too low under activity and price bargaining because

ye C {yp;ya} so that the marginal benefit is strictly above

the marginal cost. Whether the surplus is highest under

activity or price bargaining depends on whether activity is

highest in either of the two regimes. But, as already dis-

cussed, activity is highest under price bargaining compared

to activity bargaining only for sufficiently high bargaining

power. The result follows.

One implication of our results is that if the provider

could choose between different bargaining regimes, the

provider would choose price bargaining, while the pur-

chaser would choose activity bargaining. Note that this is

in contrast to the result that the total surplus (the sum of the

surplus of provider and the purchaser) is highest under

efficient bargaining. Therefore, both the provider and the

purchaser would be willing to forego some of the total

surplus in favour of a more inefficient bargaining proce-

dure, which, however, maximises their own surplus. This

result may not arise if side payments between the parties

were costless and credible to make, in which case total

surplus would be maximised as under efficient bargaining,

and the surplus for provider and purchaser may be higher.

The results may help to explain some of the observed

institutional settings. In England, for example, the govern-

ment has moved from a system of ‘‘cost and volume’’ or

‘‘sophisticated’’ contracts’’ (described in the Introduction),

which is equivalent to efficient bargaining to a system of

‘‘payment by results’’, which is equivalent to activity bar-

gaining. As shown in proposition 1, although this move

increases the (public) purchaser’s payoff, it reduces the

provider’s one and total surplus. This may be explained by

the strong monopsony power of the government in England

(being the only buyer of public services from public hospi-

tals). The government has recently discussed the possibility

of moving to price bargaining where the prices are negotiated

between individual hospitals and local purchasers on the

ground that this may reflect more accurately the costs of the

hospital. Ultimately, price bargaining was not adopted in part

because of the fear of (local) purchasers paying too high

prices for care. Our model suggests that from the government

perspective this was the most rationale choice, since activity

bargaining under ‘‘payment by results’’ leads to a higher

payoff than under price bargaining.

Many European countries make use of DRG-type of

payments to remunerate hospitals (i.e. the equivalent of

payment by results in England), which in our framework

coincides with activity bargaining. This has become the

dominant payment system. Our analysis suggests that such

form of bargaining is, however, inefficient since it does not

maximise total surplus (though it does maximises pur-

chaser’s one) and generates welfare losses. Table 1 shows

that the welfare loss from having activity bargaining as

opposed to efficient bargaining is equal to: Se � Sa ¼a2

2cð1�cÞ2

ð2�cÞ2.8 A move from activity to efficient bargaining is not

interesting from the purchaser perspective. For the pur-

chaser to gain from a move from activity to efficient bar-

gaining would require hospitals to make side payments

(say a positive lump-sum transfer) from the hospitals to the

government. This is unlikely to happen as it is the pur-

chaser who ‘‘pays’’ for services and not the other way

around. The difficulty then of moving from activity to

efficient bargaining may therefore lie in the credibility of

making payments that go from the provider to the

purchaser.

To summarise, the purchaser is better off under activity

bargaining and the provider is better off in price bargain-

ing. Activity and surplus is highest in efficient bargaining

and prices are highest in efficient or price bargaining.

Figure 2 below displays the solution under different

regimes. An arrow indicates increasing bargaining power

of the purchaser. In efficient bargaining, a higher bargain-

ing power of the purchaser reduces prices but has no effect

on the level of activity. In activity bargaining, higher bar-

gaining power of the purchaser induces higher activity, but

has no effect on prices. In price bargaining, higher bar-

gaining power of the purchaser reduces both prices and

activity.

Interestingly, the solution in price bargaining, where the

purchaser holds all the bargaining power, coincides with

the solution in activity bargaining, where the provider has

all the bargaining power (point A). The solutions in price

and efficient bargaining coincide when the provider holds

all the bargaining power (point B). The solutions in activity

and efficient bargaining coincide when the purchaser holds

all the bargaining power (point C). Finally, the activity in

price bargaining is higher than in activity bargaining only

for low bargaining power of the purchaser.

Figure 2 also compares the solution when both parties

have the same bargaining power (c = 0.5). Prices are

higher in efficient and price bargaining (points Ec=0.5 and

8 Similarly, the welfare (total surplus) loss from having price

bargaining as opposed to efficient bargaining is equal to

Se � Sp ¼ a2

8cc2.


123

Pc=0.5 respectively). Activity is highest in efficient bar-

gaining and lowest in activity bargaining (point Ac=0.5).

Decreasing marginal benefit

We extend the previous analysis, and assume a more

general specification of the benefit function: BðyÞ ¼ay� b

2y2, with decreasing marginal benefit, while we

maintain the other assumptions: CðyÞ ¼ c2y2, V ¼ U ¼ 0.

Table 2 reports the solution in price and efficient bargain-

ing. Proofs are in the Appendix (Decreasing marginal

benefit). The solution for activity bargaining is more

involved, and is derived separately in ‘‘Decreasing mar-

ginal benefit and activity bargaining’’.

The following proposition compares the two regimes.

Proposition 2 If c[ bbþc; then (a) pp [ pe, (b)

ye [ yp, (c) Up [ Ue, (d) Ve [ Vp.

If the bargaining power of the purchaser is sufficiently

high ðc[ bbþcÞ prices are higher in price bargaining,

activity is lower, the provider is better off and the pur-

chaser is worse off than under efficient bargaining. If the

bargaining power of the purchaser is sufficiently low

ðc\ bbþcÞ all the results are reversed. The threshold b

bþc

increases with b and decreases with c. Note that if b = 0

we are back to the results of proposition 1. Therefore, if the

purchaser has low bargaining power, efficient bargaining

yields a lower utility for the purchaser than in price bar-

gaining. This is a surprising result: we would expect the

purchaser to be better off when she can bargain with more

instruments, i.e. both prices and activity. But this holds true

only if her bargaining power is high. If her bargaining

power is low, having more instruments is counterproduc-

tive. The purchaser is better off when she cannot bargain

on activity.

Figure 3 below displays the solution under the two

regimes. The solutions in efficient and price bargaining are

depicted by line BC and AD respectively. An arrow indi-

cates increasing bargaining power of the purchaser. As

before, in efficient bargaining activity is constant, irre-

spective of the distribution of bargaining power, and the

price decreases as the bargaining power of the purchaser

increases. In price bargaining, both prices and activity

decrease as the bargaining power of the purchaser

increases.

It is useful to compare these results with those obtained

in the previous section by assuming constant marginal

benefit. When the bargaining power of the purchaser is

low, the activity in efficient bargaining is lower than in

price bargaining but with constant marginal benefit it is

always higher.

If the marginal benefit is constant (and equal to the

average benefit), the purchaser is always better off

regardless of its bargaining power: this arises because

activity is always higher under efficient bargaining than

under price bargaining, while the price is the same.

Activity under efficient bargaining is always determined

such that the marginal benefit is equal to the marginal cost.

yC

By

yBBy

)(=

E =0.5P =0.5y

yC )(

ACA =0.5

y - Activity

Fig. 2 Comparison of scenarios with constant marginal benefit

Table 2 Equilibrium with decreasing marginal benefit

Price bargaining Efficient bargaining

pp ¼ acð2�cÞbþ2c pe ¼ aðð1�cÞbþð2�cÞcÞ

2ðbþcÞ

yp ¼ að2�cÞbþ2c

ye ¼ abþc

Vp ¼ ca2ð2�cÞ2ðbþ2cÞ Ve ¼ ca2

2ðbþcÞ

Up ¼ a2cð2�cÞ2

2ðbþ2cÞ2Ue ¼ ð1�cÞa2

2ðbþcÞ

CyB

D

yB(y) or V=0

E =0.5 P =0.5

yC(y) or U=0

A=0.5

ByC

A

y - Activity

Fig. 3 Comparison of scenarios with decreasing marginal benefit


123

If the marginal benefit is strictly decreasing and the pur-

chaser has low bargaining power, activity under price

bargaining can be such that the marginal benefit is below

the marginal cost (so that activity is higher than under

efficient bargaining) but the average benefit is above the

price, so that the purchaser’s utility is positive. This cannot

arise if the marginal (and average) benefit is constant: if the

marginal benefit is below the cost, then the average benefit

would also be below the price, which in turn would imply a

negative utility for the purchaser: however, this can never

arise under Nash Bargaining as both parties always have

positive utilities in equilibrium. Therefore, the activity

under price bargaining will be always lower than under

efficient bargaining if the marginal benefit is constant.

Decreasing marginal benefit and activity bargaining

In this section we derive the solution under activity bar-

gaining. For a given price, the optimal bargained activity

is:

See section Decreasing marginal benefit and activity

bargaining in the Appendix for the proof. The optimal price

is given by the price which maximises V ¼ ayaðpÞ�b2yaðpÞ2 � pyaðpÞ. Given the complexity of the solution, it is

not possible to derive manageable expressions for price and

activity. To compare the solutions for the three scenarios

we resort to numerical simulations. Our strategy is to

specify a grid of values for all the parameters of the model

(a, b, c and c), and compute the solution numerically. We

fix a = 1, and specify a grid for b 2 f0; 0:5; 1; 1:5; . . .; 30g,c = {0, 0.5, 1, 1.5, … , 30} and c = {0, 0.1, … , 0.9, 1}.

For example, supposing that a = b = c = 1 and

c = 0.5, then yaðpÞ ¼ 34�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðp� 1Þ2pþ 916

q

and

V ¼ ð1� pÞ 3

4�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðp� 1Þ2pþ 9

16

r !

� 1

2

3

4�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðp� 1Þ2pþ 9

16

r !2

the solution of which is pa = 0.29 and ya = 0.36. Table 3

reports the solution for a = b = c = 1 and c =

{0, 0.1, 0.25, 0.5, 0.75, 0.9, 1}. The tables for the other

values of b and c are omitted, but are available from the

authors.

Overall, the numerical simulations suggest that in

activity bargaining prices are lowest, the purchaser’s utility

is highest and the provider’s utility is lowest (note the

similarity with proposition 1). Activity is lower than in

efficient bargaining. It is lower than in price bargaining

when the bargaining power of the purchaser is below a

certain threshold, which is between 0.7 and 0.95 in our

simulations.

The solution in activity bargaining is displayed in Fig. 3

on the line AC, which was derived by plotting the

numerical solution 1,000 times. In contrast to the solution

with constant marginal benefit, in activity bargaining the

price is no longer fixed. As the bargaining power of the

purchaser increases, the price decreases and activity

increases.

As in the previous section, the solution in price bar-

gaining with c = 1 coincides with activity bargaining when

c = 0 (point A), and the solution in activity and efficient

bargaining coincide when c = 1 (point C). However, when

c = 0 (points B and D) efficient and price bargaining yield

different solutions. Finally, when both parties have the

same bargaining power, the solutions in efficient bargaining

and price bargaining coincide at the point where marginal

cost equals marginal benefit.

Finally, in price bargaining, an increase in the bar-

gaining power of the purchaser reduces prices and activity,

but in activity bargaining it reduces prices but increases

activity.

Adding quality and effort

In this section we extend the model by introducing quality

and cost containment effort, and we show that the results

using this more general specification are qualitatively

similar to the ones obtained above. We follow the approach

suggested by Ma [21] and Chalkley and Malcomson [9].

Define q as the quality generated by the provider and e as

the cost-containment effort. The cost function of the pro-

vider is Cðy; q; eÞ þ uðy; q; eÞ. C includes the monetary

cost, which increases with quality and activity but

decreases with effort: C(y, q, e), with Cy [ 0, Cq [ 0 and

yaðpÞ ¼2�c

2cða� pÞ þ bp1þc

2

� �

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2�c2

cða� pÞ þ bp1þc2

� �2�2bcpða� pÞq

bcð12Þ


123

Ce \ 0. u is the non-monetary cost, or disutility, which

increases with activity, quality and effort: uðy; q; eÞ, with

uy [ 0;uq [ 0 and ue [ 0.

We also assume that the demand for treatment depends

positively on quality so that y = y(q) with yq [ 0. This

assumption implies y = y(q), q = q(y), qy [ 0. Therefore

by contracting activity the purchaser can implicitly contract

the level of quality. The benefit function of the patients is

B = B(y, q) with By [ 0 and Bq [ 0. Since quality is a

positive function of activity, we can also write

B = B[y, q(y)] with oBoy ¼ oB

oy þ oBoq

oqoy [ 0. The provider’s

utility is given by the surplus: U ¼ py� Cðy; qðyÞ; eÞ�uðy; qðyÞ; eÞ. The purchaser’s utility is V = B[y, q(y)] -

py.

Activity bargaining

We assume that first the purchaser sets the price (stage 1);

second, purchaser and provider bargain on activity (stage

2); third, the provider chooses effort (stage 3). We solve by

backward induction. For a given price and activity (stage

3), the provider maximises the surplus U with respect to

effort so that:

Ueðe�Þ ¼ 0 : �Ceðy; qðyÞ; e�Þ ¼ ueðy; qðyÞ; e�Þ ð13Þ

The optimal effort for the provider e*(y) is such that the

marginal benefit of lower cost is equal to the marginal

disutility of effort. The indirect utility function of the

provider is Uðp; y; qðyÞ; e�ðyÞÞ ¼ py� Cðy; qðyÞ; e�ðyÞÞ�uðy; qðyÞ; e�ðyÞÞ.

For a given price (stage 2), the activity bargaining

problem between purchaser and provider is:

maxy

Vðp; y; qðyÞÞ � V� �c

Uðp; y; qðyÞ; e�ðyÞÞ � U� �1�c

ð14Þ

whose FOC is:

ya : cBy þ Bqqy � p

eV¼ ð1� cÞ

Cy þ uy þ ðCq þ uqÞqy � p

eU

ð15Þ

The volume of activity is such that the difference

between the marginal benefit and the price (weighted by

the relevant factors) equals the difference between the

marginal cost and the price (also weighted by the relevant

factors). The condition is analogous to Eq. 2. However, the

marginal benefit and marginal cost also include the

additional benefit and cost from higher quality. The

marginal cost includes both the monetary and non-

monetary cost.

In stage 1 the purchaser sets the price to maximise:

maxp

BðyaðpÞ; qðyaðpÞÞÞ � pyaðpÞ ð16Þ

The FOC is:

pa : Byyp þ Bqqyyp ¼ yþ pyp ð17Þ

The optimal price is such that the marginal benefit of

higher activity and quality induced by a higher price is

equal to the marginal cost.

Price bargaining

First the purchaser and the provider bargain on price (stage

1), and then the provider chooses the level of activity and

cost-containment effort (stage 2). We solve by backward

induction. For a given price (stage 2) the provider maxi-

mises the surplus U with respect to activity and effort, so

that:

Uyðy�; e�Þ ¼ 0 : p ¼ Cy þ uy þ ðCq þ uqÞqy ð18Þ

Ueðy�; e�Þ ¼ 0 : �Ce ¼ ue ð19Þ

Table 3 Numerical simulation

of equilibrium with decreasing

marginal benefit. Simulation

based on parameters

a = 1, b = 1, c = 1

c = 0 c = 0.1 c = 0.25 c = 0.5 c = 0.75 c = 0.9 c = 1

ya 0.33 0.33 0.34 0.36 0.40 0.44 0.50

ye 0.50 0.50 0.50 0.50 0.50 0.50 0.50

yp 0.67 0.63 0.58 0.50 0.42 0.37 0.33

pa 0.33 0.32 0.31 0.29 0.27 0.25 0.25

pe 0.75 0.70 0.63 0.50 0.38 0.30 0.25

pp 0.67 0.63 0.58 0.50 0.42 0.37 0.33

Va 0.17 0.17 0.18 0.19 0.21 0.23 0.25

Ve 0 0.03 0.06 0.13 0.19 0.23 0.25

Vp 0 0.03 0.07 0.13 0.16 0.17 0.17

Ua 0.06 0.05 0.05 0.04 0.03 0.02 0

Ue 0.25 0.23 0.19 0.13 0.06 0.03 0

Up 0.22 0.20 0.17 0.13 0.09 0.07 0.06


123

The provider chooses the level of activity, which

equates the price to the marginal monetary and non-

monetary cost. The marginal cost also takes into account

the indirect effect of activity caused by increased quality,

which is captured by the last term on the RHS. The optimal

effort is such that the marginal benefit of lower cost is

equal to the marginal disutility of effort. The indirect utility

function of the provider is U(p, y*(p), q(y*(p)), e*(p)).

Note that oy�

op ¼�Uee

UyyUee�U2ye[ 0, oe�

op ¼Uye

UyyUee�U2ye?0 and oU

op ¼y� (by the envelope theorem). The price bargaining

problem (stage 1) is given by:

maxp

Bðy�ðpÞ; qðy�ðpÞÞÞ�py�ðpÞ � V

c

py�ðpÞ � Cðy�ðpÞ; qðy�ðpÞÞ; e�ðpÞÞ�uðy�ðpÞ; qðy�ðpÞÞ; e�ðpÞÞ � U

1�c ð20Þ

The FOC is:

pp :ceVðBy þ BqqyÞyp þ

ð1� cÞeU

y ¼ ceVðyþ pypÞ ð21Þ

The optimal price is such that the weighted marginal

benefit for the purchaser of higher activity and quality, plus

the weighted marginal benefit for the provider in terms of

higher surplus, is equal to the weighted marginal cost for

the purchaser.

Efficient bargaining

First the purchaser and the provider bargain on price and

activity (stage 1), then the provider chooses the cost-con-

tainment effort (stage 2). By backward induction, for a

given activity and price (stage 2) the supplier maximises

the surplus U with respect to effort,

Ueðe�Þ ¼ 0 : �Ce ¼ ue ð22Þ

which provides e*(y). The bargaining problem is:

maxp;y

Bðy; qðyÞÞ � py� �V½ �c py� Cðy; qðyÞ; e�ðyÞÞ�uðy; qðyÞ; e�ðyÞÞ � U

1�c

ð23Þ

whose FOCs are:

ye : By þ Bqqy ¼ Cy þ uy þ qyðCq þ uqÞ ð24Þ

pe ¼ ð1� cÞB� V

yþ c

C þ uþ U

yð25Þ

The price equals the weighted sum of the average

benefit for the purchaser and the average cost of the

provider, which includes the non-monetary cost. The

optimal activity balances the purchaser’s marginal benefit

with the provider’s marginal cost.

Regime comparison

Suppose that the benefit and cost functions are separable in

activity, quality and effort, and that demand is linearly

increasing in quality: (a) Bðy; qÞ ¼ a1y� b1

2y2 þ a2q� b2

2q2;

(b) y = hq; Cðy; q; eÞ ¼ F þ c1

2y2 þ c2

2q2 � c3e, where F is a

fixed cost; (c) uðy; q; eÞ ¼ d1

2y2 þ d2

2q2 þ d3

2e2. ai, bi, di and h

are all positive parameters.

Define: bBðyÞ ¼ ða1 þ a2

h Þy� ðb1

2þ b2

2h2Þy2; bCðyÞ ¼ ðc1þd1

2

þc2þd2

2h2 Þy2; bV ðyÞ ¼ bBðyÞ � py� V ; bUðyÞ ¼ py� bCðyÞ �F� c3

2d3� U, where � c3

2d3¼ d3

2ðe�Þ2 � c3e� and e* is such

that �Ce ¼ ue.

Now, define: a ¼ ða1 þ a2

h Þ, b ¼ ðb1

2þ b2

2h2Þ, c ¼ ðc1þd1

2þ

c2þd2

2h2 Þ, and assume V ¼ 0 and U ¼ F þ c3

2d3.

Compare this formulation with ‘‘Decreasing marginal

benefit’’. It is straightforward to show that all results con-

tained in that section also hold for the more general for-

mulation developed in ‘‘Adding quality and effort’’.

Intuitively, since activity is an increasing function of

quality, by choosing or agreeing a certain level of activity,

the provider also determines the level of quality. Therefore,

adding quality adds complexity to the model but does not

alter the main incentives. The only difference is that the

marginal cost is now interpreted as the marginal cost of

activity and quality; similarly, the marginal benefit

includes the marginal benefit of activity and quality. For

what concerns effort, since the provider is residual claimant

in all the scenarios, effort is set such that marginal benefit

from lower cost is equal to the marginal disutility of effort,

regardless of the specific institutional setting. Therefore,

also adding this variable does not alter the main results of

the analysis.

Extension

In this section we assume that each party (i.e. the purchaser

and the provider) can exert costly investments that can

increase their bargaining power.9 We assume that the

bargaining power of the purchaser has the following linear

specification: c = c0 ? v - u, where v is the investment

by the purchaser to increase his bargaining power, and u is

the investment by the provider. Therefore, the bargaining

power of the provider is 1 - c = 1 - c0 - v ? u. Invest-

ment efforts are costly and are given by the function

K(v) for the purchaser and k(u) for the provider, respec-

tively. Investments are realised before the bargaining pro-

cess begins (simultaneously and non-cooperatively). We

9 We would like to thank an anonymous referee for suggesting this

extension.


123

also assume that such investments are wasteful: their only

purpose is to increase the bargaining power but have no

effect on patients’ health. The purchaser’s and provider’s

utility under each bargaining procedure i (and excluding

the cost of investment efforts) is respectively equal to

V(pi(c), yi(c)) = [B(pi(c)) - pi(c)yi(c)] and U(pi(c), yi(c))

= pi(c)yi(c) - C(yi(c)) where i = a, p, e.

The optimal level of investment for the purchaser is

such that it maximises the following function:

maxv

ViðpiðcÞ; yiðcÞÞ � KðvÞ;

which gives the optimality condition:

oVi

oc¼ K 0ðv�Þ:

The marginal benefit from a higher investment, in terms of

larger bargaining power (and a higher payoff in the bar-

gaining stage) is equal to its marginal cost.

Similarly, the maximisation problem for the provider is:

maxu

UiðpiðcÞ; yiðcÞÞ � kðuÞ;

from which we obtain:

�oUi

oc¼ k0ðu�Þ:

A higher investment reduces c and increases the bargaining

power of the provider and his utility, which is traded-off

with its marginal cost.

We now ask how the incentives to invest in such costly

(but wasteful) investments to increase the bargaining

power varies across the three regimes. We focus on the

special case of proposition 1, which assumes constant

marginal benefit. It is straightforward to obtain that

oVe

ov ¼ a2

2c,oVp

ov ¼a2ð1�cÞ

2c and oVa

ov ¼ a2

2c1

ð2�cÞ2. It follows that the

marginal benefit from increasing the bargaining power is

highest under efficient bargaining oVe

ov [ foVa

ov ;oVp

ov g� �

, and

lower in the other two cases. Therefore, the incentive to

invest is highest when total surplus S is highest (and per-

haps counter-intuitively it is not when the utility of the

purchaser is highest).

For the provider we obtain: oUe

ou ¼ a2

2c,oUp

ou ¼ a2

2cð1�c2Þ and

oUa

ou ¼ a2

2cc

ð2�cÞ3. It follows again that the marginal benefit

from increasing the bargaining power is highest under

efficient bargaining oVe

ou [ foVa

ou ;oVp

ou g� �

, and lower in the

other two cases. Therefore, the incentive to invest is

highest when total surplus S is highest.

Note also that the marginal benefit from increasing the

bargaining power is the same for the purchaser and pro-

vider under efficient bargaining, it is higher for the provider

under price bargaining and it higher for the purchaser

under activity bargaining. Suppose that the K(v) = k(v), ie

the cost of the investment is the same for the purchaser and

the provider. It follows that under efficient bargaining

v* = u* and c = c0. Under price bargaining we have

u* [ v* and c\ c0. Under activity bargaining we have

v* [ u* and c[ c0.

A key insight from this extension is that wasteful

investments will be highest when the efficient bargaining

procedure is used. We have shown above that total surplus

is highest when the bargaining power is exogenous. As we

show below this is not necessarily the case when the bar-

gaining power is endogenous: since wasteful investments

are higher under efficient bargaining, welfare defined as the

surplus S minus the costly investments may be lower.

To make this point simply, we focus on efficient versus

price bargaining. We normalise a = c = 1. We also

assume that the cost function of the investments is qua-

dratic: K(v) = zv2/2 and k(u) = zu2/2, and that c0 = 0.5.

The optimal levels of investments under efficient bar-

gaining are: ve ¼ ue ¼ 12z, which gives a total cost of

investment KðveÞ þ kðueÞ ¼ 14z . Welfare, which is defined

as the surplus minus the costly investments, is Se�KðveÞ � kðueÞ ¼ 1

2ð1� 1

2zÞ.The optimal levels of investments under price bargain-

ing are obtained solving the simultaneous equation system

v ¼ 1�ð0:5þv�uÞ2z , u ¼ 1�0:5ð0:5þv�uÞ

2z , which gives: up ¼ 3zþ12zþ8z2

and vp ¼ 14zþ2ð1þ 3zþ1

4z2þzÞ. Welfare is equal to Sp � KðvpÞ�

kðupÞ ¼ 4�ð0:5þvp�upÞ28

� z2ðvpÞ2 � z

2ðupÞ2.

Figure 4 illustrates the comparison between the two

welfare functions for different values of the marginal cost

of the investment z. The welfare under efficient and price

bargaining is depicted with a red and a black line,

respectively. When investments are costly, i.e. z is high, we

recover the results obtained in the previous section. Wel-

fare is higher under efficient bargaining: graphically, the

1 2 3 4 5 6 7 8 9 10

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

We

Wp

z

Fig. 4 Net surplus under efficient and price bargaining


123

red line is above the black one. When z is low, i.e.

investments are not very costly, then both parties will

engage in high levels of wasteful efforts that will increase

significantly wasteless expenditure (the red line lies below

the black one). Such expenditure is higher under efficient

bargaining, because as shown above the marginal benefit

from the investment is higher. For low levels of z, the latter

effect is so strong that overall welfare is higher under price

bargaining than under efficient bargaining. Finally, note

that when z is very low, wasteful expenditure is so high that

welfare is negative, and that efficient bargaining generates

a higher welfare for z [ 5.05.

Conclusions

Different countries have different institutional and bar-

gaining settings for purchasers and providers. They usually

follow one of three scenarios: the purchaser first sets the

price (stage 1), and activity is then bargained between

purchaser and provider (stage 2): activity bargaining; the

price is first bargained between purchaser and provider

(stage 1), but activity is then chosen unilaterally by the

provider (stage 2): price bargaining; and price and activity

are bargained simultaneously between purchaser and pro-

vider: efficient bargaining. This study has compared prices,

activity and the utility of provider and purchaser in each of

the three different institutional settings. We obtain three

main results:

First, if the bargaining power of the purchaser is higher

than a certain threshold and the marginal benefit of activity

is strictly decreasing, efficient bargaining leads to higher

activity and lower prices compared to price bargaining. As

a consequence, the purchaser’s utility is higher under effi-

cient bargaining than under price bargaining, while pro-

vider’s utility is lower. The results are reversed if the

bargaining power of the purchaser is below a certain

threshold: the activity is higher and the price are lower

under price bargaining rather than efficient bargaining.

Therefore, the purchaser’s utility is higher under price

bargaining than under efficient bargaining, while provider’s

utility is lower.

This result is surprising, as one would expect the pur-

chaser to be better off when she can bargain with both

instruments, price and activity. This intuition proves cor-

rect only when the bargaining power of the purchaser is

high. When it is low, the purchaser would be better off

contracting on prices only: having more instruments is not

useful, and actually is counter-productive. This is because

when the bargaining power of the purchaser is very low,

the provider will bargain a very high price, which under

price bargaining will be accompanied by a large volume of

activity. In contrast the level of activity under efficient

bargaining is always determined such that the marginal

benefit of quantity is equal to the marginal cost, regardless

of the price: therefore under efficient bargaining when the

purchaser is weaker, she will pay a higher price without

obtaining any extra activity.

Interestingly, the threshold level of bargaining power of

the purchaser over which the purchaser is better off,

depends critically on the shape of the marginal benefit

curve. The threshold is higher when the marginal benefit

function is steeper (i.e. the benefit function is more con-

cave) and when the marginal cost function is flatter (i.e. the

cost function is less convex).

Also, the threshold is strictly positive only when the

marginal benefit of activity is decreasing: if the marginal

benefit is constant (and equal to the average benefit), the

purchaser is always better off regardless of its bargaining

power (the threshold is zero in this case). Intuitively, this

arises because if the marginal benefit is constant, activity is

always higher under efficient bargaining than under price

bargaining, while the price is the same.

Activity under efficient bargaining is always determined

such that the marginal benefit is equal to the marginal cost.

If the marginal benefit is strictly decreasing and the pur-

chaser has low bargaining power, activity under price

bargaining can be such that the marginal benefit is below

the marginal cost but the average benefit is above the price,

so that the purchaser’s utility is positive. If the marginal

(and average) benefit is constant, having a level of activity

such that the marginal benefit is below the marginal cost

would imply that the average benefit is also below the

price. But this would imply a negative utility for the pur-

chaser, which can never arise as under Nash Bargaining

both parties always end up with positive utilities. Therefore

activity under price bargaining will be always lower than

under efficient bargaining if the marginal benefit is

constant.

Our second main result is that under activity bargaining,

price and provider’s utility are lowest and the purchaser’s

utility is highest. The level of activity in activity bargaining

is always lower than in efficient bargaining. It is also lower

than in price bargaining, but only if the bargaining power

of the purchaser is below a certain threshold (which,

according to our numerical simulations is above 50%). If

the bargaining power of the purchaser is high, then the

level of activity is higher under activity bargaining than

under price bargaining.

Even if activity is generally lower under activity bar-

gaining, the lower price more than compensates for the

reduction in the benefit for the patients from the lower

activity, so that the purchaser is overall better off. The

analysis therefore supports policies such as ‘‘payment by

results’’ in the UK, where prices are fixed by the purchaser

or the regulator.


123

One perhaps less intuitive implication of our results is

that by shifting from efficient and price bargaining (as in

‘‘cost and volume’’ or ‘‘sophisticated’’ contracts) to activity

bargaining (as in ‘‘payment by results’’), the level of

activity is likely to decrease. This is in contrast to what is

normally thought, i.e. that ‘‘payment by results’’ will

encourage activity. However, our results are consistent

with recent empirical evidence [14], which finds that the

introduction of ‘‘payment by results’’ in 2003–2005 gen-

erally did not lead to any subsequent significant increase in

the volume of activity in England.

Our third result is that under price bargaining, higher

bargaining power of the purchaser reduces prices and

activity; in activity bargaining it reduces prices, but

increases activity; and in efficient bargaining it reduces

prices but has no effect on activity. Therefore, when the

bargaining power of the purchaser increases, price and

activity move in the same direction under price bargaining

but in opposite directions under activity bargaining.

The intuition for these results is the following. Under

price bargaining, the optimal activity is chosen by the

provider such that the price is equal to the marginal cost.

Therefore, whenever the price increases, as a result of a

stronger purchaser, activity follows. Under efficient bar-

gaining, the optimal activity is such that it maximises the

sum of the purchaser and provider utility. Since purchaser’s

utility is given by the benefit minus the transfer to the

provider, while provider’s utility is given by the transfer

minus the cost, this is equivalent to maximise the differ-

ence between benefit and cost. The optimal activity is

chosen such that the marginal benefit is equal to the mar-

ginal cost of activity, regardless of the bargaining power.

Therefore, a stronger purchaser will obtain a lower price

but not a lower activity. Under activity bargaining, a

stronger purchaser is able to agree with the provider a

higher volume of activity for a given price; if the marginal

benefit of activity is decreasing, the higher agreed activity

reduces the marginal benefit for the purchaser from fixing a

higher price, so that the price is lower and activity is higher

when the purchaser is stronger.

The above results are derived in ‘‘The model’’ and

‘‘Regime comparison’’ and focus on price and activity

only. In Adding quality and effort we extended the analysis

by adding quality and cost-containment effort as choice

variables of the provider. This makes activity bargaining a

three-stage model, and efficient and price bargaining a two-

stage model. Cost-containment effort is always decided by

the provider in the last stage of the game as, realistically,

effort takes place after the negotiation stage, i.e. the

beginning of the financial year. We assume that demand

responds positively to quality. Therefore, the quality

decisions always happen when decisions on activity take

place. This is because by committing or deciding on a

certain level of activity, indirectly the provider commits as

well to a certain level of quality.

We show that under this more general setting, the main

results of the analysis in terms of regime comparison still

hold. The only difference is that the marginal cost is now

interpreted as the marginal cost and disutility of activity

and quality; similarly, the marginal benefit includes the

marginal benefit of activity and quality. For what concerns

cost-containment effort, since the provider is residual

claimant in the three different settings, effort is always set

such that marginal benefit from lower cost is equal to the

marginal disutility of effort, regardless of the institutional

setting. Therefore, adding effort to the analysis does not

alter the main results.

In ‘‘Extension’’ we show that, although welfare is higher

under efficient bargaining for a given bargaining power,

this may not be the case when bargaining power is

endogenised and the marginal cost of wasteful effort is

sufficiently low. This arises because the returns from

wasteful investments are highest when the efficient bar-

gaining procedure is used, so that welfare net of the

wasteful investment may now be lower under efficient

bargaining compared to the other bargaining procedures.

In terms of implications for future work, there is a need

for empirical work that quantifies the bargaining power of

the purchaser and the provider in health care markets. This

might help governments to decide whether to encourage

purchasers to bargain on prices only, or on price and

activity simultaneously. Moreover, most of the empirical

work focuses on the effect of bargaining power on prices

[2]. This study provides clear predictions of the effect of

the bargaining power on activity as well as price. More

precisely, under price bargaining, a higher bargaining

power of the purchaser reduces activity; under activity

bargaining it increases activity; and under efficient bar-

gaining it has no effect on activity. Future empirical work

might test such predictions. We have also shown that a

switch from efficient to price bargaining does not neces-

sarily lead to an increase in activity. Further empirical

work might test whether policies such as ‘‘payment by

results’’ are likely to increase or decrease activity com-

pared to previous policies. On the theory side, more work is

needed which integrates bargaining models with political-

economy ones.

Appendix

The model

Proof of Eq. 2. Activity bargaining The result is obtained

by differentiating c log½BðyÞ � py� V � þ ð1� cÞ log½py�CðyÞ � U� with respect to y. The second order condition


123

(SOC) is C ¼ cByyV�ðBy�pÞ2

eV 2� ð1� cÞCyy

eUþðp�CyÞ2

eU 2\0, which

is always satisfied. h

Proof of Eq. 8. Price bargaining By taking the log

and differentiating with respect to p we obtain

cByyp�yðpÞ�pyp

eVþ ð1� cÞyðpÞþpyp�Cyyp

eU¼ 0. From the FOC of

the provider we know that p = Cy. By simplifying, we

obtain: cByyp�yðpÞ�pyp

eVþ ð1� cÞyðpÞ

eU¼ 0. The SOC is

�ceU 2ððBy�pÞyp�yÞ2þð1�cÞeV 2y2

eV 2eU 2

�cð2�Byy

CyyÞeU�ð1�cÞeVeV eU

yp. h

Proof of Eq. 11. Efficient bargaining Define

X ¼ BðyÞ � py� V� �c

py� CðyÞ � U� �1�c

Then: o log Xop ¼ �

cy

BðyÞ�py�Vþ ð1�cÞy

py�CðyÞ�U¼ 0 and o log X

oy ¼cðBy�pÞ

BðyÞ�py�Vþ ð1�cÞðp�CyÞ

py�CðyÞ�U¼ 0. From the first equation we

obtain p ¼ c½CðyÞþU�þð1�cÞ½BðyÞ�V �y , which, substituted into

the second one, yields: By = Cy. The SOCs are:

o2 log Xop2 ¼ �y2ð c

~V2 þ 1�c~U2 Þ\0, o2 log X

oy2 ¼ cByy~V�ðBy�pÞ2

~V2 � ð1� cÞCyy

~Uþðp�CyÞ2~U2 \0, and o2 log X

op2

o2 log Xoy2 [ ðo

2 log Xopoy Þ

2. o2 log X

opoy ¼ �c~V

þ1�c~Uþ yðBy � pÞð c

~V2 þ 1�c~U2 Þ ¼ ð1�cÞBþcC�py

~V ~Uþ yðBy � pÞð c

~V2

þ1�c~U2 Þ ¼ yðBy � pÞð c

~V2 þ 1�c~U2 Þ, where the last simplification

follows from the FOC for price. o2 log Xop2

o2 log Xoy2 [ ðo

2 log Xopoy Þ

2 ¼

�y2ð c~V2 þ 1�c

~U2 Þ½cByy

~V~V2 � ð1� cÞCyy

~U~U2 � ðBy � pÞ2ð c

~V2 þ 1�c~U2 Þ��

y2ðBy � pÞ2ð c~V2 þ 1�c

~U2 Þ2¼ �cByy~V

~V2 þ ð1� cÞCyy~U ~U2 [ 0. All

three SOCs are always satisfied, since Byy B 0. h

Constant marginal benefit

Activity bargaining. pa ¼ a2, ya ¼ a

cð2�cÞ, Va ¼ a2

2cð2�cÞ,

Ua ¼ a2ð1�cÞ2cð2�cÞ2.

Proof The rule determining activity is, for a given price:

c a�pða�pÞyþ ð1� cÞ p�cy

ðp�c2yÞy ¼ 0, from which y ¼ 2p

cð2�cÞ. The

FOC for price is: 2acð2�cÞ �

4pcð2�cÞ ¼ 0, from which: pa ¼ a

2

(the SOC is � 4pcð2�cÞ\0). The bargained activity is there-

fore: ya ¼ acð2�cÞ. The utility of the purchaser and the pro-

vider are: Va ¼ ða� pÞy ¼ a2

2cð2�cÞ and Ua ¼ ðp� c2yÞy

¼ a2ð1�cÞ2cð2�cÞ2. h

Price bargaining. pp ¼ að2�cÞ2

, yp ¼ að2�cÞ2c , Vp ¼ ca2ð2�cÞ

4c ,

Up ¼ a2ð2�cÞ28c .

Proof Since y ¼ pc with yp ¼ 1

c, the FOC for the bargained

price is: cða�pÞ1c�pc

apc�

p2

c

þ ð1� cÞpc

p2

c �p2

2c

¼ 0, which gives: pp ¼að2�cÞ

2(the SOC is � 1

ða�pÞ2p2ðða� pÞ2 þ p2Þ � 2

p2ð1� cÞ\0).

Hence yp ¼ að2�cÞ2c , Vp ¼ ða� pÞy ¼ ca2ð2�cÞ

4c and Up ¼ ðp�c2yÞy ¼ a2ð2�cÞ2

8c . h

Efficient bargaining. pe ¼ að2�cÞ2

, ye ¼ ac, Ve ¼ ca2

2c , Ue ¼a2ð1�cÞ

2c .

Proof The FOC with respect to price implies:

p ¼ ð1� cÞaþ cc2y. The FOC with respect to activity

implies: ye ¼ ac. Therefore pe ¼ að2�cÞ

2and Ve ¼ ða� pÞy ¼

ca2

2c and Ue ¼ ðp� c2yÞy ¼ ð1� cÞa2

2c. h

Proof of Proposition 1 (a) pp ¼ að2�cÞ2� a

2¼ pa if c B 1.

(b) ya ¼ acð2�cÞ � ye ¼ a

c if acð2�cÞ � a

c or c B 1; yp ¼að2�cÞ

2c � ye ¼ ac if c C 0; yp ¼ að2�cÞ

2c � ya ¼ acð2�cÞ if (2 -

c)2 C 2 or c B 0.59. (c) Va ¼ a2

2cð2�cÞ �Ve ¼ ca2

2c if 2c -

c2 - 1 B 0 or -(c - 1)2 B 0; Ve ¼ ca2

2c �Vp ¼ ca2ð2�cÞ4c if c

C 0. (d) Up ¼ a2ð2�cÞ28c �Ue ¼ a2

2cð1� cÞ ifð2�cÞ2

4�ð1� cÞ

or 4 ? c2 - 4c C 4 - 4c, or if c2 [ 0; Ue ¼ a2

2cð1�

cÞ�Ua ¼ a2

2c1�cð2�cÞ2 if (2 - c)2 C 1, which is always the

case, since 0 B c B 1. h

Decreasing marginal benefit

Price bargaining. pp ¼ acð2�cÞbþ2c , yp ¼ að2�cÞ

bþ2c , Vp ¼ ca2ð2�cÞ2ðbþ2cÞ ,

Up ¼ a2cð2�cÞ2

2ðbþ2cÞ2 .

Proof Since y ¼ pc with yp ¼ 1

c, the FOC for the bargained

price is: cða�bpc�pÞ1c�

pc

ðapc�b

2p2

c2�p2

c Þþ ð1� cÞ

pc

p2

c �p2

2c

¼ 0, which simplifies

to cða�bpc�pÞ�p

ða�b2

pc�pÞ þ 2ð1� cÞ ¼ 0 or cða� bp

c � pÞ � cpþ 2

ð1� cÞða� b2cp� pÞ ¼ 0, giving: pp ¼ acð2�cÞ

bþ2c . Hence

yp ¼ að2�cÞbþ2c , Vp ¼ ða� b

2y� pÞy ¼ ca2ð2�cÞ

2ðbþ2cÞ and Up ¼

ðp� c2yÞy ¼ a2cð2�cÞ2

2ðbþ2cÞ2 . h

Efficient bargaining. pe ¼ aðð1�cÞbþð2�cÞcÞ2ðbþcÞ , ye ¼ a

bþc,

Ve ¼ ca2

2ðbþcÞ, Ue ¼ a2ð1�cÞ2ðbþcÞ .

Proof The FOC with respect to price implies:

pe ¼ ð1� cÞða� b2yÞ þ ccy

2. The FOC with respect to

activity implies: ye ¼ abþc. Therefore pe ¼ aðð1�cÞbþð2�cÞcÞ

2ðbþcÞ


123

and Ve ¼ ða� b2y� pÞy ¼ ca2

2ðbþcÞ and Ue ¼ ðp� c2yÞy ¼

a2ð1�cÞ2ðbþcÞ . h

Proof of Proposition 2 (a) pp [ pe ifacð2�cÞ

bþ2c [aðð1�cÞbþð2�cÞcÞ

2ðbþcÞ or b(cc ? bc - b) [ 0 or c [ bbþc.

(b) ye [ yp if abþc [ að2�cÞ

bþ2c or b ? 2c - (2 - c) (b ? c) [ 0

or c [ bbþc. (c) Up [ Ue if

a2cð2�cÞ2

2ðbþ2cÞ2 [ ð1�cÞa2

2ðbþcÞ or

b2c ? bcc2 ? c2c2 - b2 [ 0 or c ¼ f�bc;

bbþcg. (d) Ve [ Vp

if ca2

2ðbþcÞ[ca2ð2�cÞ2ðbþ2cÞ or (b ? 2c) - (b ? c) (2 - c) [ 0 or

c[ bbþc. h

Decreasing marginal benefit and activity bargaining

Proof From FOC with respect to y we have c ða�byÞ�p

ay�b2y2�py

¼�ð1� cÞ p�cy

py�c2y2 or cða� by� pÞðp� c

2yÞ þ ð1� cÞðp� cyÞ

ða� b2y� pÞ ¼ 0. Upon expanding, we obtain

bc2

y2 � yðc2�c2ða� pÞ þ bp1þc

2Þ þ pða� pÞ ¼ 0, with solu-

tion y ¼ ðc2�c

2ða�pÞþbp1þc

2Þ�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðc2�c2ða�pÞþbp1þc

2Þ2�4bc

2pða�pÞ

pbc . h

References

1. Aronsson, T., Lofgren, K., Wikstrom, M.: Monopoly union ver-

sus efficient bargaining: wage and employment determination in

the Swedish construction sector. Eur. J. Polit. Econ. 9, 357–370

(1993)

2. Barros, P.P., Martinez-Giralt, X.: Models of negotiation and

bargaining in health care. In: Jones, A. (ed.) The Elgar com-

panion to health economics, chapter 21. Elgar, Cheltenham

(2006)

3. Barros, P.P., Martinez-Giralt, X.: Negotiation advantages of

professional associations in health care. Int. J. Health Care

Finance Econ. 5(2), 191–204 (2005)

4. Barros, P.P., Martinez-Giralt, X.: Bargaining and idle public

sector capacity in health care. Econ. Bull. 9(5), 1–8 (2005)

5. Barros, P.P., Martinez-Giralt, X.: Selecting negotiation processes

with health care providers. Mimeo (2000)

6. Brooks, J., Dor, A., Wong, H.: Hospital-insurer bargaining: an

empirical investigation of appendectomy pricing. J. Health Econ.

16, 417–434 (1997)

7. Bulkley, G., Myles, G.: Bargaining over effort. Eur. J. Polit.

Econ. 13, 375–384 (1997)

8. Chalkley, M., Malcomson, J.: Contracting for health services

when patient demand does not reflect quality. J. Health Econ. 17,

1–19 (1998)

9. Chalkley, M., Malcomson, J.: Contracting for health services

with unmonitored quality. Econ. J. 108, 1093–1110 (1998)

10. Clark, D.: Priority setting in health care: an axiomatic bargaining

approach. J. Health Econ. 14, 345–360 (1995)

11. De Fraja, G.: Contracts for health care and asymmetric infor-

mation. J. Health Econ. 19(5), 663–677 (2000)

12. Dor, A., Watson, H.: The hospital-physician interaction in US

hospitals: evolving payment schemes and their incentives. Eur.

Econ. Rev. 39, 795–802 (1995)

13. Ellis, R., McGuire, T.: Optimal payment systems for health ser-

vices. J. Health Econ. 9, 375–396 (1990)

14. Farrar, S., Yi, D., Scott, T., Sutton, M., Sussex, J., Chalkley, M.,

Yuen, P.: National evaluation of payment by results interim

report: quantitative and qualitative analysis, October, Report to

the Department of Health; available at http://www.

aberdeen.ac.uk/heru/research/bpoc/projects/performance/

bpoc_ongo_po2.php (2006)

15. Figueras, J., McKee, M., Mossialos, E., Saltman, R.B.: Pur-

chasing to improve health systems performance. European

observatory on health systems and policies series. Open Uni-

versity Press, UK (2005)

16. Gal-Or, E.: Exclusionary equilibria in health care markets.

J. Econ. Manag. Strategy 6(1), 5–43 (1997)

17. Gal-Or, E.: The profitability of vertical mergers between hospitals

and physician practices. J. Health Econ. 18(5), 621–652 (1999)

18. Gal-Or, E.: Mergers and exclusionary practices in health care

markets. J. Econ. Manag. Strategy 8(3), 315–350 (1999)

19. Kalai, E.: Nonsymmetric nash solutions and replications of

2-person bargaining. Int. J. Game Theory 6, 129–133 (1977)

20. Le Grand, J., Mossialos, E.: Health care and cost containment in

the EU. Ashgate, Aldershot (1999)

21. Ma, A.: Health care payment systems: cost and quality incentives.

J. Econ. Manag. Strategy 3(1), 93–112 (1994)

22. Manning, A.: An integration of trade union models in a sequential

bargaining framework. Econ. J. 97, 121–139 (1987)

23. McDonald, I., Solow, R.: Wage bargaining and employment. Am.

Econ. Rev. 71(5), 896–908 (1981)

24. Melnick, G.A., Zwanziger, J., Bamezai, A., Pattison, R.: The

effects of market structure and bargaining position on hospital

prices. J. Health Econ. 11(3), 217–233 (1992)

25. Montefiori, M.: Spatial competition for quality in the market for

hospital care. Eur. J. Health Econ. 6, 131–135 (2005)

26. Mougeot, M., Naegelen, F.: Hospital price regulation and

expenditure cap policy. J. Health Econ. 24, 55–72 (2005)

27. Nash, J.: The bargaining problem. Econometrica 18, 155–162

(1950)

28. Nash, J.: Two-person cooperative games. Econometrica 21,

128–140 (1953)

29. Osborne, M., Rubinstein, A.: Bargaining and markets, 1st edn.

Academic, San Diego (1990)

30. Oswald, A.: The economic theory of trade unions: an introductory

survey Scandinavian. J. Econ. 87(2), 160–193 (1985)

31. Pecorino, P.: Should the US allow prescription drug reimports

from Canada? J. Health Econ. 21, 699–708 (2002)

32. Propper, C.: Market structure and prices: the responses of hos-

pitals in the UK national health service to competition. J. Public

Econ. 61, 307–335 (1996)

33. Sampson, A.: Bargaining over effort and the monitoring role of

unions. Eur. J. Polit. Econ. 9, 371–381 (1993)

34. Sanjo, Y.: Quality choice in a health care market: a mixed

duopoly approach. Eur. J. Health Econ. 10(2), 207–215 (2009)

35. Wright, D.J.: The drug bargaining game: pharmaceutical regu-

lation in Australia. J. Health Econ. 23(4), 785–813 (2004)


123

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