Medical Tourism 10

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International trade in medical services

Pierre-Yves Geoffard Thierry Verdier

Preliminary version

August 25, 2011

Abstract

This paper develops a theoretical analysis of international tradein health services, considering explicitly the interaction between themarket for health care services and the market for health care profes-sionals in a "general" equilibrium model. After analyzing the autar-kic" equilibrium in two separate countries (a poor, and a rich one), weconsider trade in medical services when travel costs may substantiallydiffer across patients according to their health conditions. This featureinduces a selection bias in patients movement, which has importantconsequences on the patterns of international trade. For intermediatetravel costs, only patients who would not have been treated in theirorigin country travel for treatment. There is no effect in the rich coun-try while in the poor country, the positive demand shock may increaseequilibrium medical wages and health care prices, which affects doc-

tors positively but southern patients negatively. In contrast, if travelcosts are low, the health sector in the rich country is also affected. Atthe margin, patients in a better condition than those who stay at hometo get treated abroad, worsening the case-mix of patients treated inthe origin country. This has conflicting effects on medical care prices,but unambiguously puts a downward pressure on medical wages.


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Keywords: International trade, health care.JEL codes:


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1 Introduction

To this date, there is no consensus about the overall net benefits of interna-tional trade in health care services in general, and of patients movements inparticular. Some WTO agreements are relevant to health, in what the GATS

defi

nes as four diff

erent modes: mode 1 refers to cross-border service tradesuch as telemedicine, medical education, and e-health; mode 2 refers to pa-tients seeking treatment abroad; mode 3 refers to foreign direct investmentsin health care services; mode 4 refers to international movements of healthcare professionals. However, only a limited number of countries (mostly de-veloping countries) have committed to full liberalisation of health servicesunder GATS1. In contrast, many organisations seem to be express strongreserves about international trade in health care. That position, which ishighly prevalent among the medical profession, is exemplified by the Amer-ican College of Surgeons, which recently issued a statement about medicaltourism, stressing the numerous risks for patients2.

Indeed, the effects of international trade in health services may sharecommon features with international trade in other sectors: overall efficiencygains may be obtained by greater specialisation, a more systematic exploita-tion of increasing returns to scale, and increased competitive pressures on

local producers, but these gains also lead to significant redistribution withinsocieties, among producers and consumers alike. 3.

The current paper develops a theoretical analysis of mode 2 interna-tional trade in health services, a phenomenon also referred to as medicaltourism, or patients movements.4 Our objective is to provide a compre-hensive theoretical model upon which a welfare analysis may be may beconstructed, and to guide future empirical analysis by pointing out what

should be measured to evaluate the costs and benefits of patients movementsfor different components of the society, in origin as well as in destinationcountries. Our general framework is, to our knowledge, the first which triesto blend two branches of economics : an international trade approach (ageneral equilibrium with two countries), together with elements specific tohealth and health care which the health economics literature has identified


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Cost (in 1000 USD)USA Singapore Thailand India

Heart ByPass 130 18.5 11 10Angioplasty 57 13 13 11Hip Replacement 43 12 12 9

Hysterectomy 20 6 4.5 3Table 1:

and analysed.We focus on patients movements across countries which significantly dif-

fer in terms of per capita income. Such movements may take two main forms.First, as resources devoted to health care are limited in poor countries, healthcare is usually of poor overall quality. However, some rich customers fromthese poor countries may afford to, and be willing to, pay for a better qualityof treatment. This drives a first type of patients movement, which is associ-ated to two features of poor countries : poor quality of health care services,high degree of inequality. In this type of medical trade, rich countries ex-port health care to poor ones. Second, in contrast with this situation, richcountries devote a substantial share of their income to the health system.

Nevertheless, the growth in health expenditures puts health financing underpressure. Households finance health care expenditures through a combina-tion of taxes which subsidise public insurance organisms, private insurancepremiums, and out of pocket expenditures. In most rich countries, public orprivate insurance is the main source of health care financing; but in somecountries, in particular the US, a substantial share of the population remainsuncovered for non emergency care, or some medical services are poorly cov-

ered by existing insurance schemes. Faced with increasing health care costs,all countries have attempted to contain this increase, through various formsof health care demand and/or health care supply regulation. Demand sideregulation usually attempts to decrease the demand for health care services,and may take the form of co-payments (deductible, caps on insurance reim-bursement ) or waiting lists These trends make patients in rich countries


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Based on similar figures, Mattoo and Rathindran (2002) estimate thatif 10% of US patients underwent elective surgery abroad for 6 procedures 5,more than 1 billion US dollars could be saved. These figures illustrate thatthe potential gains from trade are considerable. Many health insurers limitcoverage of health care treatments received abroad; but uninsured patients,

cost savings which may result from receiving treatment abroad may leadthem to seriously consider this possibility. We focus in this paper on theanalysis of the second kind of patients movements, where patients originatefrom rich countries.

At this stage, our analysis is theoretical. Our analysis aims to identify themain economic issues affected by international travel of patients. Two mainmarkets may be affected: the market for health care services, and the marketfor health care professionals. The interaction between these two markets liesat the heart of our analysis, and we investigate the equilibrium on thesetwo markets simultaneously, in a general equilibrium model. In a verystandard way, we first look at the situation in two separate countries (a poor,and a rich one), and study the autarky equilibrium in each country. Therich country enjoys an overall higher productivity of unskilled labour, whichaffects overall income, but also unit costs of medical care; the rich countryalso has a higher number of health care professionals (per capita); in addition,

it prohibits health care price discrimination across patients of different healthconditions. These ingredients induce potential gains from trade. Such gainsfrom trade may substantially differ across patients as they are related to eachpatients health condition, which affects both willingness to pay for healthcare, and treatment costs. In addition, travel costs may be substantiallylarger for patients in very bad condition, and may be less willing to travelabroad for treatment. This induces a selection bias in patients movement,

which has important consequences when we investigate the open economycase.

If travel costs are large but not so large as no trade would occur, onlypatients who would not have been treated in their origin country travel southfor treatment. There is no effect in the rich country; in the poor country, thepositive demand shock may increase equilibrium medical wages and health


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health sector in the rich country is also affected: some patients who wouldhave been treated at home now travel for treatment. At the margin, thesepatients are in a better condition than those who stay at home to get treated:in addition to the negative demand shock in the rich country, the case-mixof patients treated there gets worse. This has conflicting effects on medical

care prices, but unambiguously puts a downward pressure on medical wages.

2 International trade theory and health eco-

nomics

2.1 International trade in services

The main message drawn from international economics is that the increase incompetition due to international trade puts local producers under more pres-sure, and factor prices for input factors to tradable goods show a tendencyto equalise across countries. Overall, in the rich country this is beneficial toconsumers, but may hurt workers, especially those with low qualifications,which are employed in sectors for which the good may be imported ratherthan locally produced. Symmetrically, in the poor country the increase in ex-

ports may be beneficial to employment and wages, but international demandfor locally produced goods may lead to an increase in prices, which may hurtlocal consumers. Conversely in the North, supply of health care from lowerwage countries may put northern health care producers under pressure, andincreased competition may lead to a decrease in production costs, which aremostly labour costs.

2.2 Specificity of health care

Health care is often thought of as a very special kind of service.First, at an individual level, health is a risk. This risk induces an income

risk of having to pay for potentially very important treatment costs, eventhough the likelihood of being a health condition which requires extensive


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to health care. Together with potential market failures due to asymmetricinformation6, this explains why in most countries, public health insurance isvery prevalent, and markets for private health insurance are strongly regu-lated.

Second, patients characteristics are heterogeneous. Even for a precisely

defi

ned medical condition, the degree of illness severity may show substantialvariation, which leads to differences in treatment costs; this is reflected, forinstance, by the residual variance in treatment costs within a given diagnosisrelated group7. Some of these differences may be anticipated when patientsenter the health production facility, and some may not. Even when treatmentcosts differences can be anticipated by the health care production unit, thisdoes not automatically translates into differences in payments received by thisunit, insofar as hospital payment schemes are usually, at least to some extent,prospective. This risk pooling form of payment induces strong incentivesfor hospitals to select patients: try to attract less costly patients (cream-skimming), try to reject more costly ones (dumping), and try to reduce thequantity or quality of services provided, especially to the more costly patients(stinting). However, as reported by Newhouse (2002), there is very limitedempirical evidence of risk selection by hospitals. What is true at the hospitallevel is also true for insurance plans. Whether it is due to regulation or

to technical limitations of insurers, segmentation is always limited, and atypical health insurance contract covers, at least to some extent, insureeswith different levels of expected costs. In contrast with the absence of riskselection by hospitals, there is some empirical evidence of risk selection byhealth insurance schemes8.

Finally, insofar as patients movement is concerned, travel costs may alsodepend on patients health condition. Patients in very poor condition may

require some medical car to travel to and back from airports. Medical bodiessuch as the British Medical Association also claim that flying soon aftersurgery can cause complications because of the stress travelling puts on thebody, which may be particularly strong for patients in very bad health. And

6Arrow, in what is known recognised as the founding contribution to health economics


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of course, some patients are simply too sick to travel on long distance flights,as their health condition puts them at risk of needing some rapid medicalintervention.

However, other aspects of health care make it similar to other services inother sectors, in which demand depends on prices and quality.

As said in the introduction, one of the main drivers of patients move-ment from rich to poorer countries is the very large differences in treatmentcosts across countries. Do these huge cost differences result in importantexport flows? The available data is of very poor quality, but it seems thatthe potential benefits from trade are far from being all exploited. In a recentpaper, Herman (2009) reports actual known figures on cross border trade inhealth care services in the four modes defined by WTO, with an emphasison European countries. Within the EU, consumption of health care servicesabroad is still of a very small magnitude with respect to total health carespending. Outside Europe, it seems that no reliable data exists about USimport flows (US patients travelling abroad), or about export flows of vis-ible destination countries such as India, Thailand, Cuba, or Singapore (toname only a few). Yet, actual trade flows between the US and the countriesmentioned in Table 1 seem to be very small as compared to the potentialbenefits. Mattoo and Rathindran (2006) claim that the main reason for this

discrepancy between potential and actual gains from trade lies in the natureof existing health insurance plans, which discriminate explicitly or implicitlyagainst treatment abroad.

Indeed, if not organised by health insurers themselves, patient demandmay respond to price differences only if the share of out of pocket paymentsis sufficiently large. This is the case for uninsured patients, as well as formedical treatments poorly covered by existing insurance schemes, like dental

care. In that case, the health economics empirical literature has shown thatdemand does react to prices, especially for ambulatory care (including dentalcare) and non emergency hospital care9.

Quality differences may also be an important driver of patients movement.Indeed, this is the main driver for traditional patients movements frompoor to rich countries which stems from rich individuals from poor countries


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However, recent trends have shown significant improvements of hospital carequality in many developing countries, many of local hospitals now beingaccredited by international certification bodies such as the Joint CommissionInternational (US) or Trent (UK)10. Moreover, not only do some hospitalsin developing countries provide top quality services, but in many cases there

are no waiting times, in contrast of what can be observed in some countries,especially the UK, for some elective care procedures. In short, not only priceis lower, but overall quality may even be higher in destination countries.As the current paper is devoted to the analysis of patients movement fromdeveloped to developing countries, we abstract from quality differences acrosscountries by simply assuming that the health care service under considerationis of homogenous quality. Thus, the only driver remains price differences.

Important differences in treatment prices may provide incentives on in-surance funds to promote, or even organise, medical travel for their insurees.Indeed, some private US Health Maintenance Organisations (HMO) do covernon emergency care received in neighbourhood or even distant countries, butthis is not the case for public insurance schemes such as Medicare or Medicaidin the US, or other similar schemes in other countries 11. In other words, formost individuals in rich countries, insurance coverage distorts prices, and outof pocket costs paid by patients who could consider to travel for medical care

do not reflect differences in complete treatment costs. We nevertheless inves-tigate a model in which the patient faces the true differences in costs acrosscountries: this is obviously the case for uninsured patients, which constitutethe bulk of potential demand for cross border health care services, but mayalso be interpreted as a situation in which an insurer sets its reimbursementpolicy to make its insurees internalises these cost differences12.

2.3 Trying to combine both insights

The arguments above lead us to point out some elements which need to betaken into account in a formal model. First, health care costs should beendogenous. Since production of health care mostly relies on labour, themarket for medical labour should be explicitly modelled in interaction with


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the market for medical care. Second, individual health is a key determinantof demand for medical care, but may also affect treatment costs, as well astravelling costs. Third, regulation in rich countries imposes cross subsidies,in the sense that patients in bad (resp., good) health are charged a pricebelow (resp., above) their treatment cost. In the sequel, we develop a simple

framework which tries to capture these elements. To start with, we present amodel of the health sector that integrates both a market for health servicesand a market for health care professionals. An important institutional ele-ment of the model relates to how health prices are determined on the supplyside of the sector. Two alternative regimes are discussed. The first one (iden-tified to be the situation of the Northern country) considers that regulationprevents health care suppliers to base their price on patients health state.This implies therefore some pooling and average cost pricing of health careservices. The second possibility (corresponding to the case of the Southerncountry) assumes in contrast that price discrimination across health statesis fully possible, implying marginal cost pricing, under perfect competition.We derive then for each regime the health sector equilibrium under autarkyin each country, assuming as well that Northern patients are richer than theirSouthern counterparts.

3 A simple model of the health care industry

Consider an economy with two goods: a non health care consumption goodc and health care services H. We assume that the consumption good isproduced under competitive conditions with unskilled labour only, underconstant returns to scale: one unit of unskilled labour produces g units of the

consumption good. For simplicity we assume an infi

nitely elastic unskilledlabour supply. This implies that the equilibrium wage rate w for unskilledlabour must be equal to pg, where p is the price of the consumption good.Finally, we normalise the price of the consumption good to 1, so that theunskilled labour equilibrium wage w = g.


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country. We translate this fact into the assumption that their cost functionis identical. Naturally, unit cost of inputs may be different, in particularlabour costs, which make an important share of hospital costs. We assumethat health care production requires two kinds of input: skilled labour L,and unskilled labour l. To translate the fact that treatment of less healthy

patients may require more labour time, we assume that the treatment cost ofa patient of health level h is given by C(h; w, m) = x(h)c(w, m). c(w, m) is astandard cost function homogenous of degree 1 and strictly concave in w andm (respectively the wage rate of unskilled labour and skilled labour); x(.) is adecreasing function of health h and captures the fact that patients in betterhealth conditions are less costly to treat13. The cost C(h; w, m) thereforerepresents the marginal cost to treat one patient in state h. If patients areranked according to their health h and if patients treated are the sickest,then the average cost to treat all patients in a health condition s lower thana given value h is given by:

AC(h,w,m) =

Rh0

C(h; w, m)f(h)dh

F(h)=

Rh0

x(h)f(h)dh

F(h)c(w, m). (1)

with F(.) and f(.) respectively the cumulative and density function of thedistribution of health levels in the population. Under the assumption thatthe marginal cost decreases with h (ie. x(h) is decreasing in h), then theaverage cost AC(h,w,m) also decreases with h, but always remains abovethe marginal cost curve (ie. AC(h,w,m) C(h,w,m) for all w, m).

The supply of medical doctors (skilled labour) is fixed and denoted byM. By Shepards lemma, the demand for skilled labour to treat a patient of

health h is given by L(h; w, m) =

C

m (h; w, m) = x(h)

c

m

14

13 In the special case of a Cobb Douglas technology of health treatment, we would getC(h; w, m) = x(h)[aa(1 a)(1a)]wam1a with 0 < a < 1.

Alternatively, when there is no possibility of substitution between skilled and un-skilled labour in health treatment, the cost function would take the following linear form:C(h; w, m) = x(h)[w + m], with , > 0.


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3.2 Demand for health care

We assume that demand for health care is induced from utility of health. In agiven country, agents have identical preferences over consumption and health,represented by a utility function u(c, H). We assume that u is increasing inboth arguments, with decreasing marginal utility of consumption or health

(ie. u0c > 0; u0H > 0, ucc < 0 and uHH < 0). We also assume that thecross derivative ucH 0: a better health makes consumption goods moreenjoyable. We also assume for simplicity that in a given country all agentshave the same income Y.

Agents however differ in terms of health h, which is distributed among thepopulation according to a distribution function denoted by f, with cumula-tive distribution denoted by F(.). Medical treatment of someone of health h

improves health with after-treatment health h0

given by h0

= Max (H(h), h)with H(h) an increasing function of pre-treatment health h, and H(0) > 0and H0h < 1. These assumptions imply that there exists a level of healthhsup such that for all h hsup, medical treatment does not improve healthconditions (ie. after-treatment health h0 is just equal to pre-treatment healthh); This will imply that all individuals with health conditions better thanhsup will never demand for health care services. Indeed, consider now the

willingness to pay z for health care treatment. For a given individual withhealth h, it is given by the following condition:

u(Y z, h0) = u(Y, h). (2)

The willingness to pay z is a function z(h, Y) of income Y and initialhealth state h. Differentiation of (2) for h < hsup provides:

u0c(Yz, h0)dz = [u0c(Y, h)u

0

c(Yz, h0)]dY + [u0h(Y, h)u

0

h(Yz, h0)H0h]dh

(3)With the above assumptions on u(.) and H(.) and the fact that h0 h, simpleinspection of (3) gives immediately that z(h, Y) decreases with initial healthh and increases with income Y. Moreover it is easy to see that z(hsup, Y) =0. Given that all individuals with health conditions larger than hsup never


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Uniform pricing regime

We first investigate the case in which a uniform price p for health care ischarged to the patient. In that case, since willingness to pay decreases withhealth, demand is directly determined by the distribution of health withinthe population. Let us denote by h(p, Y) the largest level of health state forwhich the patient demands one unit of health care at price p. Formally, itis determined by u(Yp, H(h(p, Y)) = u(Y, h(p, Y)). Equivalently, we havethat h h(p, Y) if and only if z(h, Y) p. Note that h(p, Y) < hsup for allp > 0 and h(0, Y) = hsup.

Demand for health care treatment is then given by:

D(p, Y) = Zh(p,Y)

0 f(h)d

h = F(h(p, Y)) (4)

In short, at a given price, only patients in bad health (below the thresholdlevel h(p, Y)) demand health care. The threshold value h(p, Y) is decreasingwith p. Since the income level also determines willingness to pay, a higherincome Y leads to a larger health demand.

Marginal pricing regime z(h, Y) C(h; w, m)

In that regime there is perfect discrimination across patients and medicalcare to a patient in state h is just priced to its marginal cost C(h; w, m).Thedemand for health care stems then from patients for whom the willingnessto pay is larger than that production cost. Note however that a marginalimprovement in patients health h negatively affects the willingness to pay

for health care z(h, Y) as well as the marginal cost C(h; w, m) = x(h)c(w, m)to treat this patient. In the sequel, we assume that the first effect dominatesin absolute terms the second one. More precisely we assume that

Assumption H1 :

[ sup]z0h(h, Y ) x

0(h)


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Lemma 1: Suppose that assumption H1 holds, then there exists a uniquethreshold value h [0, hsup] such that all patients with health h h demand health treatment while all those with health h > h do notdemand health treatment. When h ]0, hsup[ , the health level h isgiven by the relationship z(h, Y) = C(h; w, m), or alternatively the

condition u(Y

C(h; w, m), H(

h)) = u(Y,

h).

Proof: See the appendix.

It follows from the above discussion that in the marginal pricing regime,the demand for health treatment is equal to F(h). Notice as well that incontrast to the uniform pricing regime, now there is no single price for medical

care, but a competitive price p(h) = C(h; w, m) for each value of h.

3.3 Equilibrium

Before allowing the possibility of medical travel, we first look at autarkicequilibrium in each country: the North and the South. The main purpose ofour theoretical analysis is to identify how each of these reference situations is

aff

ected by international trade in health care. In order to do so, we assumethat patient income in each country (YN and YS respectively for the Northand the South) is not affected by international trade in health services. Thiskeeps each equilibrium analysis tractable, and seems a reasonable assump-tion.

3.4 Northern country

As said above, we assume that in the Northern country, health care is suppliedat a uniform (pooling) price p, independent of patients health and undera balanced health sector budget. The total cost to treat all patients withhealth h lower than h(p, YN) is equal to:

h( Y ) h( Y )


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average pricing rule:p = AC(h(p, Y), w , m) = c(w,m)F(h(p,Y))

Rh(p,Y)0

x(h)f(h)dh

p =c(w, m)

F(h(p, YN))

Zh(p,YN)0

x(h)f(h)dh = AC(h(p, YN), w , m) (5)

with h(p, YN) determined by the condition

z(h, YN) = p (6)

It will be convenient to define

Ax(h) =1

F(h)

Zh0

x(h)f(h)dh (7)

such that the average cost AC(h,w,m) = Ax(h).c(w, m). Notice that, sincethe marginal cost C(h; w, m) = x(h)c(w, m) decreases with health h, thenthe average cost AC(h,w,m)) = Ax(h).c(w, m) is also decreasing in h;Also the average cost AC(h,w,m) remains above the marginal cost curveC(h; w, m)15.

Joining (5), (6) and (7), we get therefore the first equilibrium condition

z(h, YN) = AC(h,w,m) = Ax(h)c(w, m)

which is represented on Figure 1.In analogy with the marginal pricing regime and assumption H1, to ensure

a well defined equilibrium we make the following assumption

Assumption HA :

For all relevant income level Y and h [0, hsup] ,z0h(h, Y)

z(h, Y) hN do not demand health treatment. When hN ]0, hsup[ , the health level hN is given by the relationship z(hN, YN) =

AC(hN; w, m), or alternatively the condition u(YN

AC(hN; w, m), H(

hN)) =u(YN, hN).


For a given Y and w, this first equilibrium condition links the medicaldoctor wage m and hN, the threshold value for health which characterises

which patients get treated (ie. those with health h below hN). An increase inm shifts the average cost curve AC(h,w,m) upwards, and therefore increasesthe pooling health price p = AC(h,w,m) ; This in turn decreases hN: lesspatients get treated. When m is very low, average cost is small, and theequilibrium corresponds to a small value of z(h, YN), i.e. a high value ofh; inthe opposite case, when m is very large, average cost is high, and only thosein very poor health are willing to pay for health care.

The second market is the one for medical doctors (or for their labourtime). The market clearing condition writes as:ZhN

0

C

m(h; w, m)f(h)dh = MN.

The right hand side term MN is simply thefi

xed supply of medical doctorsin the North. The left hand side term LT(hN; w, m) =

RhN0

Cm

(h; w, m)f(h)dhis the demand for medical labour to treat all patients with health below hN.Due to substitution between skilled and unskilled labour, the cost functionis quasi-concave in (w, m), the marginal cost is decreasing with m and thedemand for labour also decreases with m for each value of h Given that


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fixed supply: the corresponding value ofhA should be small. When m is verylarge, the opposite holds: mostly unskilled labour is used in the productionfunction, and therefore many patients may be treated. Notice that this pos-itive relationship between m and hN depends crucially on the possibility tosubstitute between skilled and unskilled labour in the health care production

function. In the extreme case of a Leontieff

production function with perfectcomplement inputs, demand for skilled labour does not respond to changesin medical wages, and the total number of patients treated is determined bythe available supply of doctors MN, independently of their wage.

Given that the consumption good is produced with unskilled labour onlyand under constant returns to scale, the equilibrium wage rate is equal tog. However, the model is not yet fully specified: the demand for healthcare depends on income. Labour income is made of three parts: unskilled

labour used in consumption good production or in the health care sector, atthe equilibrium wage rate w = g; skilled labour, supplied at the equilibriumwage m. A complete general equilibrium approach would require to specifyunskilled labour supply, but such an extension is beyond the scope of thecurrent paper. Instead, we assume that countrys income YN is exogenouslygiven. Under this assumption, and given that the equilibrium wage w is equalto g for unskilled labour, the set of equations (8) determines the equilibrium

skilled wage m and the threshold value hN:z(hN, YN) = AC(hN; g, m)

MN = LT(hN; g, m)

(8)

These can be rewritten as :

(z(hN,YN)

Ax(hN)= c(g, m)

MN = Ax(hN)F(hN)c

0

m(g, m)As said above, each of these conditions links m and hA; the first gives anegative relationship, the second a positive one. Under reasonable boundaryconditions, a unique equilibrium (m, h

N) can be ensured. More precisely weget the following proposition:


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The complete equilibrium may be analysed on the following figure. Thedownward sloping curve reflects the health care services market and describesthe health threshold hN(m) below which get treated for a given medicaldoctor wage m. The upward sloping curve describes the equilibrium medical

doctor wage em(hN) coming from medical skilled labour market clearing. Theintersection of the two curves provides the health sector equilibrium (m, h

N)under autarky in the North. Notice that the upward sloping curve whichcorresponds to the market for medical labour is vertical when substitutionbetween inputs is not possible.

3.5 Southern country

In the poor country, the situation is different in two main respects. Firstly,population income is lower (YS < YN), which leads to a lower willingnessto pay for health care, and thus a lower demand. Secondly, skilled laboursupply is lower, in particular for medical doctors MS < MN. Thirdly, theoverall productivity of the economy is lower, which may be represented bythe assumption that gS is lower than g = gN. Finally, we assume that health

care is sold at competitive price, which may depend on patients health stateh. In other words, the market for medical care is perfectly segmented. Theremay be also differences in terms of population health distribution F(.), butwe abstract from that difference in the analysis.

The cost to provide one unit of medical care to a patient in state h is equalto C(h; w, m), but here wages refer to southern wages. Given our discussionof the marginal cost pricing regime, under assumption H1, applying lemma 1to the South context tells us that there exists a threshold value h

S [0, hsup],

defined by:

u(YS C(hS; w, m), H(hS, 1)) = u(YS, hS),

such that health treatment demand in this country is equal to F(hS).Th b diti l i d th t illi t (h Y ) i l


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than treatment cost C(h; w, m) if and only if health h is smaller than hS.Therefore the equilibrium condition on the market for health care is nowgiven by:

z(hS, YS) = C(hS; w, m).

The second equilibrium condition on the market for skilled labour is un-changed, and may be written in a compact format as:

MS = LT(hS; w, m).

Hence finally, the following health sector equilibrium conditions of theSouth

z(hS, YS) = C(hS; gS, m)MS = LT(hS; gS, m)

(9)

which can be rewritten as(z(hS ,YS)

x(hS)= c(gS, m)

MS = Ax(hS)F(hS)c0

m(gS, m)

The equilibrium analysis proceeds is the same than in the richer country.In particular we have the following proposition ensuring the existence andunicity of the autarkic equilibrium (m, h

S) in the South.

Proposition 2: Suppose that assumption H1 holds and that limm0 c(g, m) =0, limm0 c

0

m(g, m) = + and limm c0

m(g, m) = 017 , then there ex-

ists a unique health sector equilibrium (m, h

S) satisfying (9).


17 These boundary conditions are equivalent to Inada conditions and are satisfied forinstance for cobb-douglas treatment technologies of the health sector. Note that it requiressome degree of substituability between the two factors skilled and unskilled labour. In


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3.6 Comparison of equilibria

As said above, four elements are different between the rich and the poorcountry: income level Y, medical supply M, unskilled wage rate (productiv-ity) g, and pricing regulation. We study in turn the effect of each of thesedifferences. The reference country is the Southern (poor) one.

A higher income increases the willingness to pay for health care, and shiftsthe demand curve upwards. This implies that for a given m, more patientswill demand health care. This shifts the threshold health - medical wagecurve 1 on Figure 2 upwards, which increases medical wages and, to a degreewhich depends on substituability of inputs, increases the total number ofpatients treated.

A higher medical supply (per population) has different effects. For a

given m, more patients may be treated with the available supply of doctors,and this shifts the threshold health - medical wage curve 2 to the right.Equilibrium medical wages decrease, and the total number of patients treatedincreases.

A lower overall productivity g directly affects the cost of unskilled labourin the health production function. A higher unskilled wage increases produc-tion costs. For a given m, less patients will be treated, and the threshold

health - medical wage curve 1 shifts downards. But unskilled labour wagemay also affect the demand for medical labour: a higher value of w = gincreases demand for medical labour, which shifts the THMW curve 2 to theleft. The overall effect on medical wage is unclear; the effect on the totalnumber of patients treated is negative.

The last difference comes from pricing regulation. As said above, thecurve function to be considered on the market for health care is the marginalcost curve C(h) in the poor country, whereas it is the average cost curve

AC(h) in the rich one. All other things being equal, the average cost curve isabove the marginal cost curve. Regulation implies that less patients will betreated in the rich country. However, notice that in the rich country patientsin very bad health pay the pooling price, which is lower than their treatmentcost; patients in better health pay more than their treatment costs; and


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patients are treated, even when we consider treatments which are not orpoorly covered by health insurance, for which patients pay a substantialshare of the price. Overall, it seems that the income effect dominates all theothers.

The two situations may be represented on the same figure 3, where thethreshold health value is on the horizontal axis, price of medical labour (i.e.,medical wage) is on the upper vertical axis, and price of medical care is onthe lower vertical axis.

In the sequel we will note the autarkic values in the North (mN, h

N) and

in the South (mS, h

S) with a superscript A and respectively (mAN, h

A

N) and

in the South (mAS, hA

S).

4 Open trade

International trade in a good may occur when transportation costs are suffi-ciently low, so that the total cost of the good produced in an origin country,including production, transportation to the destination country, and deliverycosts, is competitive with respect to a similar good produced in the destina-tion country, which bears lower transportation costs. In terms of services,

the definition of transportation is less clear. Indeed in terms of medicalservices, for mode 2 international trade, the consumer has to travel to thepoint of service. The transportation cost is nothing but the costs associatedto this travel: they include pure transportation cost (airflight ticket) andcosts of stay (housing costs, and opportunity cost of time), which may alsohave to be paid for other persons accompanying the patient.

We denote this total transportation cost by t(h). These costs may depend

on patients health condition. The opportunity cost of time may be lowerfor individuals in poor health, but bad health also increases travel costs,especially for very poor health conditions: this may include the need to usean ambulance to travel to and from airports, or simply the pain or hassle totravel when ill. Overall, it seems that the association between health andtransportation cost may depend on the specific illness for which a treatment


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the variable part r(h) is a decreasing convex function of h (ie. r0(h) < 0and r(h) > 0) and reflects the intrinsic cost to trade health services inter-nationally. As discussed, it depends negatively on the patients individualhealth level. We further assume that these transportation costs to medicalservices are decreasing in health levels at a decreasing rate, so that the r(h)function is convex. We also impose for convenience the boundary conditionsthat r(0) = + and limh r(h) = 0 and r

0(0) = and limh r0(h) = 0

so that it is prohibitively costly to move the least healthy individuals to theSouth, while on the opposite, there are no moving costs for the most healthypatients.

We will denote all variables which refer to the Northern (resp., Southern)country with a subscript N (resp., S). Also, we restrict ourselves to themain case of northern individuals going to the south in order to benefit from

cheaper medical treatments18. The homogeneous good is freely traded acrossthe two regions and we consider trade regimes with non specialization andpositive production of that numeraire good in both regions. We assume thatunskilled labour productivity in the North g is normalized to 1 and thatunskilled labour productivity in the South is gS < 1. Competitive pricing inthat sector therefore pins down the wages of unskilled labor in both regionsas wN = 1 and wS = gS < 1.

Getting now to medical services, for a northern patient with health h, thefull cost of treatment in the South writes as x(h)c(gS, mS) + r(h) + . Thatpatient will opt for treatment when

z(h, YN) Min (pN, x(h)c(gS, mS) + r(h) + )

where pN is the price of medical services in the North and given by averagecosts in that region.

Formally, a North-South medical service trade equilibrium can then bedescribed by a non-discriminating northern medical price pN, medical doctorwages mN, mS in the North and South, a health threshold hS below which

18 C t ll th ti t ld l t t g t th N th t b fit


23/51

southern patients get treated in the South and sets of northern patients N1 ,N2 satisfying :

N1 (pN, mS, ) =

h [0, hsup] such that

z(h, YN) Min(pN, x(h)c(gS, mS) + r(h) + ) (10)

N2 (pN, mS, ) =

h [0, hsup] such that

pN > x(h)c(gS, mS)) + r(h) +

(11)

pN =

RhN1

Nc2

x(h)f(h)dhRhN1

Nc2

f(h)dh. c(1, mN) (12)

z(hS, YS) = x(

hS)c(gS, mS) (13)"Z

hN1Nc

2

x(h)f(h)dh

#c

m(1, mN) = MN (14)

LT(hS; gS, mS) +

"ZhN1 N2

x(h)f(h)dh

#c

m(gS, mS) = MS (15)

(10) describes the set N1 of northern agents with health levels such thatthey prefer to be treated in the North or in the South. Conversely, ifNc2denotes the complementary ofN2 , then

N1

Nc2 reflects the set of north-

ern patients that are treated in the North while N1 N2 denotes the set

of northern patients treated in the South. (12) simply restates that there isaverage cost pricing in the North while and (13) characterizes the threshold

hS below which southern patients get treated in the South, given that thatthere is marginal cost pricing. Finally, (??) and (??) describes the medicaldoctors labor market clearing conditions in the North (respectively in theSouth). In the South, this includes the labor demand LT(hS; gS, mS) ema-nating from treatment of southern patients plus the demand that comes from

di l t i f th ti t t th S th


24/51

r(h) to be convex enough so that for all m, (h, m) is a concave function of

h (ie. 2h2

< 0) (see the appendix). We also show in the appendix that for all

m the function (h, m) reaches a maximum at some value of health bh(m).Moreover, this maximum (m) = Maxh(h, m) is clearly decreasing in mand negative when m is large enough.

We then make the following assumption:

Assumption E: (mAS) = z(bh(mAS), YN) x(bh(mAS))c(gS, mAS) r(bh(mAS)) > 0

and bh(mAS) > hANThis assumption means that there is at least one individual in the North

that is not treated in autarky (as bh(mAS) > h

AN) and is ready to pay for med-

ical services in the South when labor wages there are the autarkic ones (gSand mAS) and there is no trade friction ( = 0). Without such an assump-tion, there would be no possibility of international trade in medical servicesbetween the two regions.

We then get the following proposition characterizing the flow of interna-tional medical services from the South to the North (and associated flows ofnorthern patients to the South) as a function of trade barriers and medical

doctor wages mS in the South:

Proposition 3: Under assumption E, we get the following:

i) There is a value of prohibitive trade frictions such that for ,there is no international trade in medical services.

ii) For 0 < , there exists a wage threshold mS(), such that forall mS mS(), then, again there is no international trade in medical

services. The threshold mS() is decreasing in and mS() = mAS.

iii) For 0 < , and mS < mS(), then there exists an interval[h1(mS, ); h2(mS, )] such that all individuals from the North withhealth h [h1(mS, ); h2(mS, )] are ready to pay medical services inthe South. The size of this interval is decreasing with trade frictions


25/51

Part i) of proposition 3 obviously says that trade barriers need to besufficiently small for international trade in heath services to occur. Partii) indicates moreover that the cost of medical treatment in the South (asindexed by medical doctor wages mS) should also be low enough for suchtransaction to be attractive to northern patients. More interestingly, partiii) indicates that when individual transport costs r(h) are decreasing convexenough, then only northern patients with intermediate values of health (iesuch that h [h1(mS, ); h2(mS, )]) are likely to travel to and get treatedin the South. The reason for this is the following. Individuals with very badhealth conditions and low values of h face prohibitive transports costs r(h)and therefore cannot be treated efficiently in the South. Conversely, individ-uals with good health and high values ofh, have a low willingness to pay formedical treatment,implying tha they are also unlikely to pay the transaction

costs r(h) + of international trade to be treated in the South. It followsthat only northern individuals with intermediate values of h may find it at-tractive to get medical treatment in the South. Indeed such individuals facemoderate personal costs r(h) of moving abroad while still having a relativelylarge willingness to pay z(h, YN) to pay for medical care.

Given the precedent discussion we are now ready to describe the nature

the international trade equilibrium in medical services. Two basic cases canbe differentiated.In the first one, all northern patients that get treated in the South were

not treated in the North under autarky. Trade in medical services betweenthe two regions comes from a pure northern demand expansion effect ofmedical services to the South without changes in the demand for medicalservices treated in the North itself. Consequently, all variables in the Northremain unaffected at their autarkic values.

The second case corresponds a situation where additionally there is adisplacement of the demand for medical services in the North induced byinternational trade with the South. In that situation, some patients that aretreated under autarky in the North, shift to be treated in the South. In sucha situation obviously the equilibrium variables of the northern region are


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Consider first the situation where all trade in medical services betweenNorth and South trade comes from a pure expansion of the demand in medicalservices from northern patients stimulated to choose to be treated in theSouth. In such a equilibrium the set of treated individuals in the North

N1 Nc2 remains the interval

h0, h

A

N

iand the equilibrium conditions in the

North imply pN = pAN, mN = m

AN and the conditions in the South write as:

z(hS, YS) = x(hS)c(gS, mS) (16)

LT(hS; gS, mS) +

"Zh2(mS ,)h1(mS ,)

x(h)f(h)dh

#c

m(gS, mS) = MS (17)

h1(mS, ) hA

N (18)

The first equation (16) determines the health threshold hS = hS(mS)below which southern individuals want to consume medical services in theirregion, given that there is competitive pricing there. Under the assumptionH1,this threshold is a decreasing function of mS. The second condition (17)is the labor market clearing condition for medical doctors in the South. Total

demand is made of two terms. The first term LT

(hS; gS, mS) is simply thedemand emanating from local southern consumers. The second term is theadditional demand coming from new northern patients going to the South inthe interval [h1(mS, ); h2(mS, )]. Finally, the last condition (18) ensuresthat only northern patients who were not consuming medical services under

autarky (ie with health levels h hA

N) are willing to consume medical servicesin the South.

Looking at equation (17), one may denote

(mS, ) the aggregate demandfor medical doctors in the South :

(mS, ) = LT(hS; gS, mS) +

"Zh2(mS ,)h1(mS ,)

x(h)f(h)dh

#c

m(gS, mS)


27/51


Substituting in (16) provides then the equilibrium southern health thresh-old heS = hS(m

e()) that is increasing in . Indeed, reduced trade frictionstend to increase the demand for medical services in the South from northernpatients. This creates a larger demand for medical doctors in the South and

increased pressures on their wage mS. As the price of medical services in theSouth increases, less southern patients get treated and heS goes down.

To be in an "pure demand expanding" equilibrium, finally condition (18)should be satisfied ie. h1(me(), ) hAN. The following proposition providesconditions under which this is going to happen.

Proposition 4: When the autarkic price of medical service in North pANis high enough, there exists a threshold [0, [ such that a "PureNorth demand expanding" health trade equilibrium exists for all valueof [, ].

Proof: see the appendix.

2) A "North demand shifting" trade equilibrium:When direct trade costs are below the threshold , trade in medical

services between the two regions leads to a situation where additionally thereis a displacement of the demand for medical services in the North. In thatsituation, some patients that are treated under autarky in the North, shiftto be treated in the South. It follows that aggregate demand for medicalservices in North is reduced, affecting therefore as well the northern medicalcosts.

One can first note that because of assumption HA and the fact thatz(h, YN), x(h), r(h) are all decreasing functions of h, the set of treated indi-viduals treated in the North N1

Nc2

N1

Nc2 =

h [0, hsup] p

z(h, YN) pN and

(h) ( ) (h)

(19)


28/51

and the trade equilibrium conditions in the North write as:

AC(hN; mN) = x(hN)c(gS, mS) + r(hN) + (20)

LT(hN; mN) = MNhN < h

A

N

The first equation characterizes the northern individual hN indifferentbetween consuming medical services in the North (and paying the averagecost pN = AC(hN; mN)) and getting treated in the South (and paying totalcompetitive pricing medical costs x(hN)c(gS, mS) plus the transaction costsr(hN) + ).

The second equation is the medical doctor market clearing condition inthe North. Finally, the last equation says that this marginal patient hN is

actually below the threshold level of autarky hA

N, ensuring that we are in an"North demand shifting" equilibrium.

Similarly, the equations for the South are described as follows:

z(hS, YS) = x(hS)c(gS, mS) (21)

LT(hS; gS, mS) +

"Zh2(mS ,)hN

x(h)f(h)dh

#c

m(gS, mS) = MS (22)

(21) shows again the level of health hS of the marginal patient in theSouth while (22) is again the market clearing condition for medical doctorsin the South. Note now that in the aggregate demand for medical doctorsin the South, the additional demand emanating from norther patients writesas: "Zh2(mS ,)

hN

x(h)f(h)dh

#c

m(gS, mS)


29/51

decreasing in and increasing in mN such that (mN, mS, ) T if and onlyifmS > fmS( , mN)19.

In the appendix we show that under reasonable assumptions on the func-tions x(h), r (h) and Ax(h), for all mS h do not. If there is no such h ]0, hsup[ with(h) = 0, then for all h ]0, hsup[ the function (h) has a constant sign.

If(h) > 0 then all patients with h < hsup are ready to pay for medicaltreatment. Two case can occur. The first one is (hsup) = 0 and h = hsup.Or (hsup) > 0 and for all h hsup patients are ready to pay for medicaltreatment. By convention one may again pose h = hsup.

If(h) < 0, then no patient h > 0 is ready to pay for medical treatment.

Then or (0) = 0 and h = 0.Or (0) < 0 and for all h > 0, patients are notready to pay for medical treatment. Again by convention we can pose h = 0.From the definition ofz(h, Y), note that when h ]0, hsup[ , an alternative wayto characterize h is also through the relationship u(YC(h; w, m), H(h)) =u(Y, h). QED

Proof of lemma 2: Identical to that of lemma 1, substituting Ax(h)for x(h) amd assumption HA for assumption H1.

Proof of proposition 1:


37/51

for all m [0, m0] with m0 given by the condition c(g, m0) =z(0,YN)Ax(0)

(ie

such that hN(m0) = 0). Moreover hN(m) is decreasing in m and becauselimm0 c(g, m) = 0 and z(hsup, YN) = 0, one also get hN(0) = hsup. Similarlythe second condition MN = LT(hN; g, m) can be written as

ZhN0 x(h)f(h)dh = MNc0m(g, m)Recalling that c0m(g, m) is decreasing in m and that limm0 c

0

m(g, m) = +and limm c

0

m(g, m) = 0, this relationship defines a unique value em(hN) forall hN [0, h

sup]. Moreover em(hN) is increasing in hN with em(0) = 0 andem(hsup) > 0.

Now define the function (m) =

em(hN(m)) m for all m [0, m0].

From the previous discussion, this function (m) is decreasing in m (ie.

0(m) = em0(hN)h0N(m) < 0); Moreover it is such that (0) = em(hN(0)) =em(hsup) > 0 and (m0) = em(hN(m0)) m0 = em(0) m0 = m0 < 0.Hence there exists a unique m ]0, m0[ such that (m) = 0. Consider thenh

N = hN(m). Then by construction (m, h

N) is the unique point satisfyingcondition (8) and therefore the unique equilibrium of the health sector. QED

Proof of proposition 2:

Given assumption H1 and lemma 1, there exists a unique hS(m) [0, hsup]

satisfying the condition

z(hS, YS)

x(hS)= c(gS, m)

for all m [0, m0] with m0 given by the condition c(gS, m0) =z(0,YS)x(0)

(ie

such that hS(m0) = 0). Moreover hS(m) is decreasing in m and becauselimm0 c(g, m) = 0 and z(hsup, YS) = 0, one also get hS(0) = hsup. Similarlythe second condition MS = LT(hS; g, m) can be written as


38/51

em(hS) for all hS [0, hsup]. Moreover em(hS) is increasing in hS with em(0) = 0and em(hsup) > 0.

Now define the function (m) = em(hS(m))m for all m [0, m0]. Fromthe previous discussion, this function (m) is decreasing in m (ie. 0(m) =

em0(hS)h

0

S(m) < 0); Moreover it is such that (0) =

em(hS(0)) =

em(hsup) > 0

and (m0) = em(hS(m0))m0 = em(0)m0 = m0 < 0. Hence there existsa unique m ]0, m0[ such that (m) = 0. Consider then hS = hS(m).Then by construction (m, h

S) is the unique point satisfying condition (8)and therefore the unique equilibrium of the health sector. QED

Properties of the function (h, m) :

i) Concavity of (h, m) in h..We easily get the relationships:

h=

z

h

x

hc(gS, m) r

0(h)

2

h2=

2z

h2

2x

h2c(gS, m) r(h)

2

hm=

x

h

c(gS, m)

m> 0

So 2h2

< 0 when r(h) is convex enough.QED.

.ii) The function (h, m) reaches a maximum at some value of healthbh(m)Using the fact that r0(0) = and limhhsup r

0(h) = 0, one can concludethat:

h

h=0

> 0

Given that 2h2

< 0 when r(h) is convex enough, .h

is decreasing in h.


39/51

Furthermore, it is easy to see that when it is an interior solution, bh(m) isincreasing in m.(as

2h2

< 0 and 2

hm> 0).

Finally note as well that limh0 (h, m) < 0 (as r(0) = +) andlimhhsup (h, m) < 0 (as z(h, YN) C(h; gS, m) is negative at h = hsup).QED.

iii) (m) = Maxh(h, m) is decreasing in m and negative when m islarge enough:

(m) = Maxh(h, m) = z(bh(m), YN) x(bh(m))c(gS, m) r(bh(m)). andsimple differentiation gives :

0(m) =

c(gS, m)

mx(

bh(m)) < 0 and

(m) = 2c(gS, m)

m2 x(bh(m)) c(gS, m)m x(bh(m))bh0(m) > 0Also it is easy to see that for m large enough, bh(m) gest closer to hsup.

Hence z(bh(m), YN) 0 and (m) < 0: QED. Proof of proposition 3::

i) Pose = (mAS) > 0 under assumption E. Then clearly for ,nobody in the North is ready to pay for medical services in South given thatthe medical doctor wage in the South are the autarkic one mAS). Thereforethere is no international trade in medical services.

ii) For 0 < , the function (m) is decreasing in m and such that(mAS) = > and(m) < 0 for m large enoug. Hence there exists a Southwage threshold mS() > m

AS such that(mS()) = . When the doctor wage

mS in the South is larger than mS(), then,(mS) < (mS()) = . Thisimplies again that there is no possibility of international trade in medicalservices, given the level of trade barriers . Moreover the fact that (m) isdecreasing in m implies obviously that the threshold mS() is decreasing in Also at the prohibitive trade friction one has mS() = m

A Finally


40/51

the fact that (h, mS) is a concave function ofh taking its maximum at pointbh(mS) with (bh(mS), mS) > 0, implies that for all mS mS(), thereexists an interval [h1(mS, ); h2(mS, )] with h1(mS, ) bh(mS) h2(mS, )and such that for all h [h1(mS, ); h2(mS, )], we have (h, mS) > .Henceall northern individuals with health h [h1(mS, ); h2(mS, )], are ready topay medical services in the South, given their individual moving costs r(h)

and the trade friction level .Notice that the bounds h1(mS, ) and h2(mS, ) are determined by the

condition (h1(mS, ), mS) = (h2(mS, ), mS) = .Note also that h1(mS, )is increasing in and h2(mS, ) is decreasing in , as (h, mS) is increasing

(resp. decreasing) in h for h bh(mS) (resp. h bh(mS)). Hence the size ofthis interval is decreasing with the level of trade frictions (ie. h1(mS, ) isincreasing in and h2(mS, ) is decreasing in ).

Finally, h1(mS(), ) = h2(mS(), ) = bh(mS(). Thus at the medicaldoctor threshold mS(), the size of this interval is zero (ie. h1(mS(), ) =h2(mS(), )).QED.

Proof of lemma 3:

i) Simple diff

erentiation of

(mS, )gives indeed:

(mS, )

mS=

LT(hS; gS, mS)

hS

hSmS

+LT(hS; gS, mS)

mS+

Zh2(mS ,)h1(mS ,)

2C

m2(h; gS, mS)f(h)dh

+C

m(h2(mS, ); gS, mS)f(h2(mS, ))

h2

mS

C

m(h1(mS, ); gS, mS)f(h1(mS, ))

h1mS

all the terms are negative and so (mS ,)mS

< 0.


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iii) Simple differentiation gives as well that (mS ,)

< 0. Because of this,it follows immediately that

dmeSd

= (me,)

(me,)mS

< 0

and therefore m

e

S() is a decreasing function of . QED.

Proof of proposition 4

One may first notice that h1(meS(), ) is an increasing function of .

Indeed at the medical doctor labor market equilibrium in the South, one has

LT(hS; gS, meS()) + "Z

h2(meS(),)

h1(meS(),)

x(h)f(h)dh# cm (gS, meS()) = MSHence "Zh2(meS(),)

h1(meS(),)

x(h)f(h)dh

#=

MS LT(hS; gS, m

eS())

cm

(gS, meS())

Given that meS() is a decreasing function of, LT(hS; gS, mS) and

cm

(gS, mS)are decreasing functions of mS, the RHS of this equation is decreasing in .

Therefore the LHS should also decrease with . Given that h1(m, ) andh2(m, ) move in opposite direction when changes, this can be the caseonly when h1(m

eS(), ) increases with . Note also that h1(m

eS(), ) =

h1(mAS, ) =

bh(mAS) > hAN.Two cases can then happen. First, suppose that h1(m

(0), 0) < hAN. This

implies that there is a unique > 0 such that h1(me(), ) = h

A

N. And for

all [, ] , h1(me(), ) > h1(me(), ) = hA

N,.condition (18) is satisfiedand there exists a "Pure North demand expanding" health trade equilibrium.

Suppose otherwise that h1(m(0), 0) hAN . Then in that case, for all

[0, ] , h1(me(), ) > h1(me(0), 0) hA

N and again condition (18) issatisfied and there exists a "Pure North demand expanding" health trade

ilib i O h i th lt f iti 4 ti b t i


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Proof of the existence of a threshold ehN(mN, mS, ) < hAN for allmS < fmS(, mN).

i) Denote (h, mN, mS) = Ax(h)c(1, mN) x(h)c(gS, mS) r(h). Thenwhen r(h) is sufficiently convex, for all mN mAN, and mS > m

AS, the

function (h) is concave. Consider then the following assumption:

Assumption HT :

Max

x0(h)

x(h);

r0(h)

r(h)

>

Ax0(h)

Ax(h)

Then, it is first easy to see that there is at most one value

ehN(mN, mS, )

satisfying (

ehN, mN, mS) = . Indeed suppose that there exists one value e

hN

such that (ehN, mN, mS) = or Ax(ehN)c(1, mN)x(ehN)c(gS, mS)r(ehN) =. Then

0h(ehN, mN, mS) = Ax0(ehN)c(1, mN) x0(ehN)c(gS, mS) r0(ehN)

> Ax(ehN)Max"x0(ehN)x(

ehN)

;r0(ehN)r(

ehN)

#c(1, mN) x

0(ehN)c(gS, mS) r0(ehN)= hx(ehN)c(gS, mS) + r(ehN) + iMax"x0(ehN)

x(ehN) ;r0

(ehN)r(ehN) #

x0(ehN)c(gS, mS) r0(ehN)>

hx(ehN)c(gS, mS) + r(ehN)iMax"x0(ehN)

x(

ehN)

;r0(ehN)r(

ehN)

#x0(ehN)c(gS, mS) r

0(ehN)or

0h(ehN, mN, mS) > x(ehN)c(gS, mS)x0(ehN)

x(ehN)!

+ r(ehN)r0(ehN)r(ehN)

!


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has 0h(ehN, mN, mS) > 0. This implies that there is at most one such valueehN.

Now for all mS < fmS( , mN), (hAN, mN, mS) = Ax(h)c(1, mN)x(hAN)c(gS, mS)r(hAN) > . Moreover, (0, mN, mS)) < 0 as limh0 r(h) = +.Therefore

by continuity, there exists a value

ehN such that (

ehN, mN, mS) = andit is

therefore unique under assumption HT. QED.

Proof of the existence of hN(mS, ) < hAN and mN(mS, ) < m

AN

for all mS < fmS( , mAN),Indeed the marginal threshold function (CC) hN =

ehN(mN, mS, ) de-

scribes a downard sloping curve between hN and mN while the medical doc-

tor labor market clearing condition (MM) : LT

(hN; 1, mN) = MN de

fines anupward sloping curve between hN and mN defined as

hN = mNNow, for a fixed value mS, define then bmN(mS, ); the value of mN such

that mS = fmS( , mN). Then mS < fmS( , mN) is equivalent to mN >bmN(mS, ). Therefore mS < fmS( , mAN) implies that bmN(mS, ) < mAN.Also note that

ehN(

bmN(mS, ), mS, ) = hAN while

ehN(mAN, mS, ) < h

AN.

Hence, along the (CC) curve, hN = hAN at mN =

bmN(mS, ) and hN =

ehN(mAN, mS, ) < hAN at mN = mAN.On the other hand, at mN = bmN(mS, ) < mAN, one has LT(hAN; 1, bmN(mS, )) >MN, Hence along the (MM) curve, one should necessarily have hN < h

AN at

mN = bmN(mS, ). Also, at mN = mAN , hN = hAN along the (MM) curve.Now, in the interval

bmN(mS, ), mAN, consider the function (mN) =(

ehN(mN, mS, ))mN. (mN) is a decreasing function of mN with

(bmN(mS, )) = hAN bmN(mS, ) = mAN bmN(mS, ) > 0and

(mAN) = ehN(mAN, mS, )mAN < (hAN)mAN = 0

Therefore there exists a unique mN = mN(mS, ) bmN(mS, ), mAN such


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Simple inspection also shows that mN(mS, ) and hN(mS, ) are increasingfunctions of mS, . QED.

Proof that

e(mS, ) is decreasing in mS

e(mS, )mS

=LT(hS; gS, mS)

hS

hSmS

+LT(hS; gS, mS)

mS+

"Zh2(mS ,)hN(mS ,)

x(h)f(h)dh

#2c

m2(gS, mS)

+

c

m (gS, mS)x(h2(mS, ))f(h2(mS, ))

h2

mS

c

m(gS, mS)x(hN(mS, ))f(hN(mS, )))

hNmS

with all terms negative. Hence e(mS, ) is a decreasing function of mSand (as well). QED.

Proof that for , there exists an equilibrium wage emeS() >mAS such that e(emeS(), ) = MS.

Note first that at = , C(hAN; 1, mAN) = z(h

AN, YN) = x(h

AN)c(gS, m

eS(

))+r(hAN) +

and therefore

fmS(

, mAN) = meS(

), h1(me(), ) = hAN and

h1(meS(

), ) = hA

N =

ehN(m

AN, m

eS(

), ) = hN(meS(

), )

Now consider for <

the South doctor wage m

e

S() > m

A

S which bydefinition is a solution of

(meS(), ) = LT(hS; gS, m

eS) +

"Zh2(meS ,)h1(meS ,)

x(h)f(h)dh

#c

m(gS, mS) = MS


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and by definition

(h1(meS(), ), m

eS()) =

Now in the "demand shifting" equilibrium, the northern patient with thresh-old health hN(m

eS(), ) that is indifferent between treatment in the North

and treatment in the South strictly prefers treatment to no treatment. Hence

peN = Ax(hN(meS(), ))c(1, mN(m

eS(), )) < z(hN(m

eS(), YN)

Also by definition of hN(meS(), ) one has

Ax(hN(meS(), ))c(1, mN(m

eS(), )) = x(hN(m

eS(), ))c(gS, m

eS())

+r(hN(meS(), )) +

Hence

(hN(meS(), ), m

eS()) = z(hN(m

eS(), YN)

x(hN(meS(), ))c(gS, m

eS())

r(hN(meS(), ))

= z(hN(meS(), YN)p

eN +

>

Moreover under assumption E, hN(meS(), ) < h

AN h1(m

eS(), ). From

this one concludes that

e(meS(), ) = LT(hS; gS, meS()) + "Zh2(meS(),)

hN(me

S(),) x(

h)f(

h)d

h# cm (gS, meS())

= MS

"ZhN(meS(),)h1(meS ,)

x(h)f(h)dh

#c

m(gS, m

eS())

< MS


46/51

willingness to pay

Average cost

Marginal cost

hsup Health h

PoolingPrice

Competitiveprice

Figure 1)

zh,Y

Ch;w,m xhcw,m

Ch,w,m Axhcw,m


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Threshold value h

wage m

hsup

m

h N

mhNMedical labour market Eqm curve

hNmHealth care market Eqm curve

Figure 2)

wage m Medical labour market South


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h

Price

Medical labour market North

Health care market North

Health care market South

hsup

mNA

h NA

mS

A

h SA

CSh

CNh

PN

PS

Figure 3)


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hsup Health h

Figure 4a): No trade in medical services

zh,Y

h NA

PNA

CNh

CSh th

CSh


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hsup Health h

Figure 4b): Pure Demand Expanding Equilibrium

zh,Y

h NA

PNA

CNh

CSh thCSh

h1mS,h2mS,

Demand expanding effect


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hsup Health h

Figure 4c): Demand Shifting Equilibrium

zh,Y

h NA

PNA

CNh

CSh th

CShhNmN,mS,

Demand expanding effect

Demand shifting effect

h2mS,

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