- Email: [email protected]
Economic implications of antibiotic resistance in a global economy
Journal of Health Economics 21 (2002) 1071–1083
Economic implications of antibiotic resistance in a global economy Niklas Rudholm∗ Department of Economics, Umeå University, SE-90187 Umeå, 90187 Umeå, Sweden Received 23 May 2000; accepted 20 December 2001
Abstract This paper concerns the economic implications of antibiotic resistance in a global economy. The global economy consists of several countries, where antibiotic consumption creates a stock of bacteria which is resistant to antibiotics. This stock affects the welfare in all countries because of the risk that resistant bacterial strains may be transmitted. The main purpose of the paper is to compare the socially optimal resource allocation with the allocation brought forward by the decentralized market economy. In addition, a dynamic Pigouvian tax designed to implement the globally optimal resource allocation is presented. © 2002 Elsevier Science B.V. All rights reserved. JEL classification: I18 Keywords: Antibiotic resistance; Nash equilibrium; Pigouvian taxes
1. Introduction Ever since the discovery of penicillin in 1928, antibiotic resistance among bacteria has been causing concern within the medical profession. The increase in antibiotic resistance is reported to be due to the extensive use of antibiotics worldwide (Finland, 1979; Levy, 1982; McGowan, 1983). The global implications of the development of resistant bacterial strains in any one country is one important aspect of the problem. In fact, Taxue et al. (1990) found that the best predictor of resistance in Shigella, a food-borne bacteria who causes diarrhea, was a history of foreign travel by the patient. The importance of the geographical transmission of resistant bacteria is also discussed in Cohen’s (1992) article on the epidemiology of drug resistance. Cohen fears that the problem of drug resistant bacteria might increase as ∗ Tel.: +46-90-786-99-40; fax: +46-90-77-23-02. E-mail address: [email protected] (N. Rudholm).
0167-6296/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 6 2 9 6 ( 0 2 ) 0 0 0 5 3 - X
1072
N. Rudholm / Journal of Health Economics 21 (2002) 1071–1083
international travel makes the transmission of such bacteria more frequent. As such, the use of antibiotics in any particular country affects the well-being of the population in other countries. This paper concentrates on two aspects of the resistance problem. Firstly, the use of antibiotics in one country is assumed to lead to the development of a resistant bacterial strain, which might cause disease in neighboring countries as well. Assuming that these countries play a non-cooperative differential game, taking the level of antibiotic use in other countries as given, this results in a suboptimal resource allocation at the global level. Secondly, the health of the consumer is assumed to affect output in each country. As the consumer does not recognize this relationship between health and output when choosing the level of antibiotic consumption, an externality in the production of goods arises. One purpose of the paper is to compare the socially optimal allocation of resources, at a global level, with the allocation in a non-cooperative differential game between countries. Another is to analyze the resource allocation brought forth by a decentralized market economy without any policies to control the use of antibiotics. A final concern is to design the Pigouvian tax/subsidy system required to make the market economy reach the globally optimal resource allocation. Most of the earlier studies concerning the economic impact of antibiotic resistance concentrate on the direct costs arising from prolonged hospitalization and increased mortality (Carmeli et al., 1999; Holmberg et al., 1987; Liss and Batchelor, 1987). Holmberg et al. (1987) report that the length of the hospital stay is at least twice as great for patients infected by resistant bacteria compared with patients infected by antibiotic susceptible ones, although they do not estimate the cost increases caused by prolonged hospitalization. Liss and Batchelor (1987) argue that, as policies to restrict the use of antibiotics are being adopted, the pharmaceutical industry loses part of its economic incentive to develop new antibiotic substances designed to combat resistant bacteria. Carmeli et al. (1999) report that the national cost of antibiotic resistance in the US lies somewhere between US$ 100 million and US$ 30 billion annually. However, despite presenting these estimates, Carmeli et al. (1999) admit that there is a lack of studies that thoroughly examine the costs arising from antibiotic resistance. Although there appear to be differing opinions about the actual size of the costs, it seems nevertheless to be a widespread view that antibiotic resistance increases treatment costs. Tisdell (1982) was the first to study antibiotic resistance as a utility maximization problem. He assumes that the drugs are effective only for a finite number of exposures, and that there is free access to this drug for a given cost. Tisdell then uses a two period model and shows that without intervention, there will be extensive antibiotic use in the first period, rendering them useless in the second period. In order to reach a first best equilibrium, he suggests that the government should either regulate first period consumption of antibiotics or grant monopoly rights to the sellers of antibiotics. Brown and Layton (1996) set up a dynamic model where a consumer maximizes the utility of antibiotic treatments and a farmer is maximizing the net benefits of antibiotic use in agricultural production. Both the consumer and the farmer take the effectiveness of antibiotic treatments as exogenously given, ignoring the impact of their antibiotic use on treatment efficiency. The social planner maximizes social welfare from antibiotic use, taking the resistance problem into account. A comparison of the different resource allocations shows
N. Rudholm / Journal of Health Economics 21 (2002) 1071–1083
1073
that both the consumer and the farmer choose a higher level of antibiotic consumption than the social planner. However, Brown and Layton do not suggest any corrective policy to be implemented. This paper contributes to the literature in at least three ways. First, this is (at least to my knowledge) the first attempt to analyze the problem of antibiotic resistance in a dynamic general equilibrium context. Second, the global aspects of antibiotic resistance are addressed. Finally, a dynamic Pigouvian tax is presented that is designed to internalize all external effects arising from antibiotic use at the global level. The type of model used in this paper has earlier been used to analyze different aspects of environmental problems. In a recent paper, Aronsson and Löfgren (2000) analyze the global implications of transboundry pollution in a Nash non-cooperative differential game, as well as in the uncontrolled market economy, and suggested a Pigouvian tax to reach a globally optimal resource allocation in a setting quite similar to mine. The paper is set out as follows. In Section 2, I describe the model. Section 3 presents a non-cooperative differential game between countries, while Section 4 analyzes the global social optimum. The difference between these two equilibria is that the Nash equilibrium does not internalize the part of the externalities which is due to the interaction between countries, whereas all externalities are internalized at a global level in the social optimum. In Section 5, I examine the resource allocation in the uncontrolled market economies. The policies required to make the agents in the decentralized economy choose the socially optimal resource allocation are discussed in Section 6. Section 7 concludes the paper.
2. The settings Assume that there are I countries in the global economy.1 In addition, to focus on the external effects of antibiotic use, I shall disregard international trade between the countries. The population in each country is assumed to be constant and can thus be normalized to one. Further, the consumers are assumed to have preferences over a composite commodity and health. The instantaneous utility function of the consumer in county i, at time t, can be written as ui (t) = ui (ci (t), hi (t))
(1)
for i = 1, . . . , I , where c is consumption of goods and h an indicator of health status.2 The utility function is assumed to be increasing in both arguments, twice continuously differentiable, and strictly concave. The health status is determined by the consumption of antibiotics, ai (t), and by the stocks of resistant bacteria created by all countries. We can write the health indicator function as hi (t) = hi (ai (t), x1 (t), . . . , xI (t))
(2)
1 Note that the analysis could be interpreted in terms of households, or hospital wards, within the same country instead of as separate countries. However, in order to focus on the global aspects of antibiotic resistance, the results presented below will be interpreted it terms of countries. 2 Time is considered to be continuous throughout the paper.
1074
N. Rudholm / Journal of Health Economics 21 (2002) 1071–1083
where the function hi (t) is assumed to be increasing in the consumption of the antibiotic, ai (t), decreasing in the stocks of resistant bacteria, x1 (t), . . . , xI (t), concave and twice differentiable. The assumption that only resistant bacteria affect the health indicator function should be interpreted such that the effects of non-resistant bacteria on health have been suppressed for notational convenience. This simplification is made in order to focus on the effects of resistant bacteria on health and production possibilities in the economy. Net output in country i is characterized by the production function: yi (t) = fi (ki (t), hi (t))
(3)
implying that net output is a function of the capital stock, ki (t), and the health status of the labor force, hi (t), measured per unit of labor. The effects of the health indicator should be interpreted as a direct effect reflecting the average health status per unit of labor, which is not internalized by the wage formation system. This production function is assumed to be increasing in hi (t), twice differentiable and strictly concave. Note that the production function measures net output, meaning that the depreciation of capital has been accounted for. Thus, if the capital stock becomes sufficiently large, the marginal product of capital may become negative. Now, substituting the expression for the health indicator into the utility and production functions yields: ui (t) = ui (ci (t), hi (ai (t), x1 (t), . . . , xI (t)))
(4)
yi (t) = fi (ki (t), hi (ai (t), x1 (t), . . . , xI (t)))
(5)
and
meaning that the utility of the consumer in country i is a function of his/her consumption of a composite good, ci (t), the consumption of antibiotics, ai (t), and the stocks of resistant bacteria generated by all countries. Net output in country i is, in turn, a function of the capital stock, ki (t), antibiotic treatments and the stocks of resistant bacteria. The net investments in country i is given by dki (t) = fi (ki (t), hi (ai (t), x1 (t), . . . , xI (t))) − ςi (ai (t)) − ci (t) dt
(6)
where ςi (ai (t)) is the amount of resources used by country i in the production of antibiotics. Finally, one key feature of the use of antibiotics is that it puts selective pressure upon bacteria to develop resistance against such treatments. Assume, along the lines of Finland (1979), that the accumulation of resistant bacteria in country i is an increasing function in the use of antibiotics, and that the stock of resistant bacteria evolves over time according to: dxi (t) = bi (ai (t)) dt where ∂bi (ai )/∂ai > 0 and ∂ 2 bi (ai )/∂ai2 ≤ 0.
(7)
N. Rudholm / Journal of Health Economics 21 (2002) 1071–1083
1075
3. The non-cooperative equilibrium 3.1. A description of the game between countries The problem of antibiotic resistance has been given increased attention by governments around the world. However, it seems as if individual countries are adopting their own programs to fight the problem by creating national organizations with the purpose of fighting antibiotic resistance.3 If national decision makers take the actions of other countries as given, this strategy might result in a suboptimal allocation of resources. To explore this idea further, assume that the resource allocation in each country is decided upon by a national social planner, who treats the creation of resistant bacteria in other countries as exogenous. This situation can be described by a differential game in open loop form between the countries. In this model, the strategies available to each planner are the choices of consumption of the composite commodity and antibiotic use at each instant. The payoff from the game is given by the utility level achieved from consuming the optimal amounts of goods and antibiotics chosen by the planner. The planner thus maximizes the utility of the consumer in country i, subject to the equations of motion for the capital stock, ki (t), and the stock of resistant bacteria, xi (t). The initial conditions ki (0) = ki0 > 0 and xi (0) = xi0 > 0, as well as the terminal condition limt→∞ ki (t) ≥ 0 are also imposed. Neglecting the time indicator for notational convenience, the present value Hamiltonian for country i can then be written Hi = ui [ci , hi (ai , x1 , . . . , xI )] e−θt + λi [fi (ki , hi (ai , x1 , . . . , xI ) − ς (ai ) − ci ] +µi [bi (ai )]
(8)
where λi and µi are the present value shadow prices, in terms of utility, of capital and the stock of resistant bacteria. 3.2. The Nash equilibrium It is well known that dynamic games of this type are difficult to solve analytically, and that a unique solution might not exist.4 However, since the purpose of this paper is to compare the welfare aspects of the Nash equilibrium to those of the other equilibrium concepts, I shall assume that a unique equilibrium does exist. The first-order conditions for country i, which gives the optimal choice of antibiotic- and goods consumption, can be written5 3 In Sweden there is an organization called STRAMA created with the purpose of promoting rational use of antibiotics in order to fight the emergence of resistant bacterial strains. Similar organizations can be found in Finland (MIKSTRA), Canada (CCAR), as well as in several other countries. During the last few years, some international organizations like the European antimicrobial resistance surveillance system (EARSS) and the alliance for prudent use of antibiotics (APUA), have been created with similar purposes. 4 The interested reader is referred to Hoel (1978) or Clarke (1980) who show that explicit solutions generally require simplifying assumptions. 5 The transversality conditions presuppose that certain growth conditions are fulfilled. In short, these growth conditions constitute upper bounds on the influence of the state variables on the functions involved. The reader is referred to Seierstad and Sydsaeter (1987, Theorem 16 of Chapter 3) for further details.
1076
N. Rudholm / Journal of Health Economics 21 (2002) 1071–1083
∂ui (ci , hi ) e−θt − λi = 0 ∂ci
(9)
∂ui (ci , hi ) e−θt ∂hi ∂fi (ki , hi ) ∂hi ∂bi (ai ) ∂ςi (ai ) + µi =0 + λi − ∂hi ∂ai ∂hi ∂ai ∂ai ∂ai λ˙ i = −λi
µ˙ i = −
∂fi (ki , hi ) ∂ki
(10)
(11)
∂fi (ki , hi ) ∂hi ∂ui (ci , hi ) e−θt ∂hi − λi ∂hi ∂xi ∂hi ∂xi
(12)
lim λi ≥ 0 (= 0 if lim ki > 0)
(13)
lim µi = 0
(14)
t→∞
t→∞
t→∞
where the time indicator has been neglected for notational convenience. Each planner is aware of the presence of the other planners, as well as how their choices of goods- and antibiotics-consumption affect his/her state equations. The planner thus faces the problem of choosing his/her control variables in such a way that the utility of the consumer in country i is maximized for every possible choice of goods- and antibiotics-consumption made by all other planners. When the control variables are chosen such that there are no incentives for the planner in any of the I countries to revise his/her choice, the economy is in Nash equilibrium. The first-order condition for antibiotic use in country i, Eq. (10), consists of three terms. The first term measures the consumer’s marginal discounted valuation (in utility terms) of antibiotic consumption. The second term represents the difference between the marginal product of antibiotics in goods production (which works through the health status of the labor force), and the marginal cost of producing antibiotics. Finally, the third term reflects the buildup of the stock of resistant bacteria in country i, where µi (t) measures the shadow price of additions to this stock. This shadow price is found by solving Eq. (12), subject to the transversality condition, limt→∞ µi = 0, and equals µni (t)
=
∞ t
∂ui (cin , hni ) e−θs ∂hni ∂fi (kin , hni ) ∂hni + λni ∂hi ∂xi ∂hi ∂xi
ds
(15)
where the superscript n denotes the Nash equilibrium. Note that the shadow price described in Eq. (15) only reflects the consequences for welfare and production capabilities arising in country i, when a stock of resistant bacteria builds up. It does not reflect the consequences for utility and production in other countries. This means that the Nash equilibrium is characterized by externalities, arising from interactions between countries. Thus, this allocation of resources can not be optimal, and a global view on the problem of antibiotic resistance has to be adopted in order to come to terms with the problem.
N. Rudholm / Journal of Health Economics 21 (2002) 1071–1083
1077
4. The globally optimal resource allocation I now turn to the task of describing the optimal resource allocation, were all externalities have become internalized at the global level. The results from this section will later be used when designing the Pigouvian taxes which makes the market economy reach the global optimum. The optimization problem facing a global utilitarian social planner can be written as a standard control problem with an infinite time horizon. To derive the globally optimal equilibrium, where all externalities have been internalized, assume that the planner maximizes the sum of the utilities for all countries, subject to the equations of motion for k1 (t), . . . , kI (t) and x1 (t), . . . , xI (t).6 That is ∞ I Max ui [ci (t), hi (t)] e−θt dt (16) ci (t),ai (t) 0
i=1
subject to dki (t) (17) = fi (ki (t), hi (t)) − ςi (ai (t)) − ci (t) dt dxi (t) (18) = bi (ai (t)) dt for i = 1, . . . , I , where hi (t) = hi (ai (t), x1 (t), . . . , xI (t)). I also impose the initial conditions ki (0) = ki0 > 0, xi (0) = xi0 > 0, and the terminal condition limt→∞ ki (t) ≥ 0. Neglecting the time indicator, the present value Hamiltonian can be written H =
n
[ui (ci , hi ) e−θt + λi (fi (ki , hi ) − ςi (ai ) − ci ) + µi (bi (ai )]
(19)
i=1
where λi and µi are present value shadow prices in terms of utility. In addition to the equations of motion for ki and xi , as well as the initial and terminal conditions, the first-order conditions for country i = 1, . . . , I can be written (neglecting the time indicator for notational convenience) ∂ui (ci , hi ) e−θt − λi = 0 ∂ci ∂ui (ci , hi ) e−θt ∂hi ∂fi (ki , hi ) ∂hi ∂bi (ai ) ∂ςi (ai ) + µi =0 + λi − ∂hi ∂ai ∂hi ∂ai ∂ai ∂ai λ˙ i = −λi µ˙ i = −
∂fi (ki , hi ) ∂ki
(21) (22)
I I ∂uj (cj , hj ) e−θt ∂hj ∂fj (kj , hj ) ∂hj − λj ∂hj ∂xi ∂hj ∂xi j =1
(20)
(23)
j =1
6 The assumption of an additive social welfare function is made to preserve simplicity and make comparisons between the different equilibria possible.
1078
N. Rudholm / Journal of Health Economics 21 (2002) 1071–1083
lim λi ≥ 0 (= 0 if lim ki > 0)
(24)
lim µi = 0
(25)
t→∞
t→∞
t→∞
where i = 1, . . . , I . Suppose that Λ∗i (t) = [ci∗ (t), ai∗ (t)], i = 1, . . . , I are the optimal control variables chosen by the global social planner. Let us briefly discuss the resource allocation presented above from the point of view of the use of antibiotics. Note that the first-order condition for antibiotic use has the same general form as in the Nash equilibrium analyzed in Section 3. The fundamental difference between the two equilibria refers to the way in which the shadow price of resistant bacteria are measured. By solving Eq. (23) subject to the transversality condition, limt→∞ µi = 0, the shadow price of resistance creation is found to equal ∞ I ∂u (c∗ , h∗ ) e−θs ∂h∗ I ∂fj (kj∗ , h∗j ) ∂h∗j j j j j ∗ ∗ ds µi (t) = (26) + λj ∂hj ∂xi ∂hj ∂xi t j =1
j =1
which clearly differs from the shadow price in the Nash equilibrium presented above. In this case µ∗i (t) takes into account how the creation of resistant bacteria in country i affects the production capabilities and welfare in all other countries as well. To see this, note that the first two terms in Eq. (26) reflect how the buildup of the stock of resistant bacteria in country i affects health, and thus utility, of the consumer in all countries. The last terms measure the impact of the creation of resistant bacteria on production possibilities in all countries. As such, this shadow price will reflect the effects on production possibilities and welfare in every country, caused by the creation of resistant bacteria in country i. Eq. (26) will play an important role when designing the dynamic Pigouvian taxes in later sections, as this shadow price is used in order to place the correct value on additions to the stocks of resistant bacteria. Comparing the two solutions presented so far, this means that the global planner acts to maximize utility globally, while the national planner ignores that the use of antibiotics in his/her country affects the welfare and production possibilities of other countries. As mentioned in footnote (2), some international organizations have been created in recent years to meet the increasing problem of resistant bacterial strains. The motive for creating such international organizations is most likely the increasing understanding that the problem of antibiotic resistant bacteria cannot be fought on the national level.
5. The uncontrolled market economy In most countries, the resource allocation is determined by some form of controlled market economy. In such economies, decision makers use economic policy, such as taxes and subsidies, in order to internalize external effects. In the following sections of the paper, I will show that the globally optimal solution can be implemented in the market economy.7 7 Aronsson and Löfgren (2000) show that the Nash equilibrium could be implemented in the market economy as well.
N. Rudholm / Journal of Health Economics 21 (2002) 1071–1083
1079
The next two sections thus concern two main issues. Firstly, in this section the equilibrium in the uncontrolled market economy will be derived and compared with the previous resource allocations. Secondly, in the next section a dynamic Pigouvian tax will be designed to show that the first best equilibrium can be implemented in the market economy, as long as the decision makers have enough information to be able to set the tax correctly. Each country consists of three agent types: one representative consumer, one firm producing a composite good and another firm producing and selling antibiotics. The consumer chooses the consumption of the composite good, ci (t), and the level of antibiotic use, ai (t), to maximize the present value of future utility, subject to the equation of motion for the capital stock. The representative consumer acts myopically and does not take into account the effect on the stock of resistant bacteria of his consumption of antibiotics. The stock of resistant bacteria is thus considered exogenous by the consumer. The utility maximization problem of the consumer in country i can then be written ∞ Max [ui (ci (t), hi (t))] e−θt dt (27) ci (t),ai (t) 0
subject to dki (t) = ri (t)ki (t) + πi (t) + wi (t) − pi (t)ai (t) − ci (t) dt
(28)
where ri (t) is the interest rate, πi (t) total profit income from both firms (as the consumer owns the firms), wi (t) labor income and pi (t) is the price of antibiotics, which are all considered exogenous by the consumer. Finally, impose the initial condition ki (0) = ki0 > t 0, as well as the No Ponzi Game condition limt→∞ ki (t) exp(− 0 ri (s) ds) ≥ 0, which implies that the present value of the capital stock has to be non-negative at the terminal point. Turning to the production side of the economy, both agent types act competitively. The producer of the composite good chooses capital, ki (t), in order to maximize πi1 (t) = fi (ki (t), hi (ai (t), x1 (t), . . . , xI ) − wi (t) − ri (t)ki (t)
(29)
To simplify the analysis, I disregard the use of capital in the production of antibiotics, and write the objective function of the firm producing antibiotics as follows: πi2 (t) = pi (t)ai (t) − ςi (ai (t))
(30)
where pi (t)ai (t) is the sales revenue for antibiotics and ςi (ai (t)) represents all the costs of producing antibiotics. Solving the utility and profit maximization problems of the different agents, and combining the first-order conditions, gives the following general equilibrium conditions −θt ∂ui (cim , hm i )e − λm i =0 ∂ci
m −θt ∂hm ∂ui (cim , hm i )e i m ∂ςi (ai ) =0 − λi ∂hi ∂ai ∂ai m λ˙ m i = −λi
∂fi (kim , hm i ) ∂ki
(31)
(32)
(33)
1080
N. Rudholm / Journal of Health Economics 21 (2002) 1071–1083
dkim m m = fi (kim , hm i ) − ςi (ai ) − ci dt
(34)
for i = 1, . . . , I , where the superindex m is used to denote the market equilibrium, and Eq. (34) is derived by combining Eqs. (28)–(30). It is apparent that the resource allocation in the uncontrolled market economy differs from the socially optimal allocation, as well as from the Nash equilibrium. In the uncontrolled market economy, the consumer chooses the level of antibiotic use ignoring the fact that the consumption of antibiotics affects the stock of resistant bacteria and, hence, the production capabilities of his/her own country as well as the production capabilities of the other countries. The equilibrium presented is characterized by externalities, since neither domestic nor transboundry external effects of resistant bacteria are internalized.
6. Implementing the cooperative equilibrium In order to implement the cooperative equilibrium a tax on antibiotic sales, τi (t)ai (t), is introduced. The objective function of the antibiotics producer changes to become: πi2 (t) = pi (t)ai (t) − ςi (ai (t)) − τi (t)ai (t)
(35)
The tax revenue is returned to the consumer as a lump sum transfer, Ti (t). The government in country i is assumed to balance the budget at each instant, that is τi (t)ai (t) − Ti (t) = 0 The representative consumer then solves: ∞ Max [ui (ci (t), hi (t))] e−θt dt ci (t),ai (t) 0
(36)
(37)
subject to dki (t) = ri (t)ki (t) + πi (t) + wi (t) − pi (t)ai (t) + Ti (t) − ci (t) dt
(38)
Solving the utility and profit maximization problems of the different agents, and combining the first-order conditions in the same fashion as above, gives the following general equilibrium conditions (neglecting the time indicator): ∂ui (ci , hi ) e−θt − λi = 0 ∂ci ∂ui (ci , hi ) e−θt ∂hi ∂ςi (ai ) =0 − λ i τi + ∂hi ∂ai ∂ai
(39) (40)
∂fi (ki , hi ) ∂ki
(41)
dki = fi (ki , hi ) − ςi (ai ) − ci dt
(42)
λ˙ i = −λi
N. Rudholm / Journal of Health Economics 21 (2002) 1071–1083
1081
Consider Proposition 1 as follows: Proposition 1. If the tax on antibiotic consumption is chosen such that ∂fi (ki∗ (t), h∗i (t)) ∂h∗i (t) µ∗i (t) ∂bi (ai∗ (t)) − ∗ τi∗ (t) = − ∂hi (t) ∂ai (t) λi (t) ∂ai (t)
(43)
for all t and i = 1, . . . n, where all entities are evaluated at the first-best equilibrium, the market economy will reach the globally optimal resource allocation. Substituting the expression for τi∗ (t) into Eq. (40), and evaluating Eqs. (39)–(42) in the first-best equilibrium, it is apparent that the first-order conditions of the two equilibria coincide. The first term in Eq. (43) represents the value of antibiotic consumption in the production of goods in country i. This term reflects the benefits for the firm of having a healthy labor force. Assuming for a moment that the second term equals zero, this would mean that the use of antibiotics should be subsidized by the planner. The second term captures the social value of additions to the stock of resistant bacteria, caused by antibiotic consumption in country i, implying that the use of antibiotics should be taxed. Taken together, this means that the tax presented in Eq. (43) could be either positive or negative, depending on the term which outweighs the other. In addition, note that the buildup of resistant bacteria affects the welfare in the other countries as well, meaning that this has to be accounted for in order to reach the optimal solution. In the Pigouvian tax presented in Eq. (43), this is accomplished through the shadow price, µ∗i (t), which has the form presented in Eq. (26). One can easily see that this shadow price takes into account how the creation of resistant bacteria in each country affects other countries as well. Introducing the tax presented above internalizes all external effects of antibiotic consumption and makes the market economy reach the globally optimal resource allocation. The tax/subsidy system presented above thus consists of two parts. The first part of Eq. (43) suggests that antibiotics should be subsidized, while the second part suggests that antibiotics should be taxed. Note that the second term in Eq. (43) is likely to increase over time if the stock of resistant bacteria is growing in the first-best equilibrium. However, it is not possible to determine whether the second term will eventually dominate the first. The question of whether antibiotics should be taxed or subsidized is thus a question that warrants empirical research. To implement the tax presented above, the government in each country would require large amounts of information. They would have to have access to information on exactly how antibiotic treatments affect the production of goods, as well as the value of additions to the stock of resistant bacteria, if they are to choose the tax correctly. Note also that without information on the size of the different effects of antibiotic consumption, the government in each country does not even know if antibiotic treatments should be taxed or subsidized.
7. Summary The purpose of this paper is to analyze the global and national economic aspects of antibiotic resistance in bacteria. The socially optimal resource allocation is compared
1082
N. Rudholm / Journal of Health Economics 21 (2002) 1071–1083
with the unrestricted market solution, as well as with a non-cooperative Nash equilibrium. It is assumed that antibiotic consumption creates a stock of resistant bacteria which affects the welfare of the consumers in all countries because of the risk of transmission of resistant bacterial strain between countries. Health, and thus the use of antibiotics, is also assumed to have a positive effect on the production of goods. However, the consumer/labor force in the market economy does not take these effects of antibiotic consumption into account when choosing the level of antibiotic consumption. The expression for the tax/subsidy system which makes the market economy reach the globally optimal solution consists of two parts; one part describing the positive effect of antibiotic use on production, and the other part describing the social value of additions to the stock of resistant bacterial strains. Whether the governments should tax or subsidize the use of antibiotics is dependent on which part outweighs the other. This means that a considerable amount of information is needed in order for them to set the tax or, as may very well be the case, the subsidy correctly. The problem of antibiotic resistant bacteria has received increased attention in recent years and several countries are adopting strategies to fight the creation of resistant bacterial strains. However, in this paper, it is shown that if decision makers in one country act in their own self-interest, and take the actions of neighboring countries as exogenously given, this results in a suboptimal allocation of resources. Thus, a global view of the problem of antibiotic resistance should be adopted in order to come to terms with the problem.
Acknowledgements The author would like to thank Thomas Aronsson, Runar Brännlund, Tomas Sjögren, Magnus Wikström and all participants in a seminar at UmeåUniversity for helpful comments and suggestions. Financial support from the Browaldh–Wallander–Hedelius foundation is gratefully acknowledged.
References Aronsson, T., Löfgren, K.-G., 2000. Green accounting and green taxes in the global economy. In: Folmer, A., Gabel, L., Rose, A. (Eds.), Frontiers in Environmental Economics. Cheltenham, Edward Elgar. Brown, G., Layton, D.F., 1996. Resistance economics: social costs and the evolution of antibiotic resistance. Environment and Development Economics 1, 349–355. Carmeli, Y., Troillet, N., Karchmer, A.W., Samore, M.H., 1999. Health and economic outcomes of antibiotic resistance in Pseudomonas aeruginosa. Archives of Internal Medicine 159, 1127–1132. Clarke, C.M., 1980. Restricted access to common property fishing resources: a game theoretic analysis. In: Liu, P.T. (Ed.), Dynamic Optimization and Mathematical Economics. New York, Plenum Press. Cohen, M.L., 1992. Epidemiology of drug resistance: implications for a post-antimicrobial era. Science 257, 1050–1055. Finland, M., 1979. Emergence of antibiotic resistance in hospitals, 1935–1975. Reviews of Infectious Diseases 1, 4–21. Hoel, M., 1978. Distribution and growth as a differential game between workers and capitalists. International Economic Review 19, 335–350.
N. Rudholm / Journal of Health Economics 21 (2002) 1071–1083
1083
Holmberg, S.D., Solomon, S.L., Blake, P.A., 1987. Health and economic impacts of antimicrobial resistance. Reviews of Infectious Diseases 9, 1065–1078. Levy, S.B., 1982. Microbial resistance to antibiotics: an evolving and persistent problem. Lancet 2, 83–88. Liss, R.H., Batchelor, F.R., 1987. Economic evaluations of antibiotic use and resistance—a perspective: report of task force 6. Reviews of Infectious Diseases 9, 297–312. McGowan, J.E., 1983. Antimicrobial resistance in hospital organisms and its relation to antibiotic use. Reviews of Infectious Diseases 5, 1033–1048. Seierstad, A., Sydsaeter, K., 1987. Optimal Control Theory with Economic Applications. North-Holland, New York. Taxue, R.V., Puhr, N.D., Wells, J.G., Hargrett-Bean, N., Blake, P.A., 1990. Antimicrobial resistance of Shigella isolates in the USA: the importance of International travelers. Journal of Infectious Diseases 162, 1107–1111. Tisdell, C., 1982. Exploitation of techniques that decline in effectiveness with use. Public Finance 37, 428–437.
Economic implications of antibiotic resistance in a global economy Niklas Rudholm∗ Department of Economics, Umeå University, SE-90187 Umeå, 90187 Umeå, Sweden Received 23 May 2000; accepted 20 December 2001
Abstract This paper concerns the economic implications of antibiotic resistance in a global economy. The global economy consists of several countries, where antibiotic consumption creates a stock of bacteria which is resistant to antibiotics. This stock affects the welfare in all countries because of the risk that resistant bacterial strains may be transmitted. The main purpose of the paper is to compare the socially optimal resource allocation with the allocation brought forward by the decentralized market economy. In addition, a dynamic Pigouvian tax designed to implement the globally optimal resource allocation is presented. © 2002 Elsevier Science B.V. All rights reserved. JEL classification: I18 Keywords: Antibiotic resistance; Nash equilibrium; Pigouvian taxes
1. Introduction Ever since the discovery of penicillin in 1928, antibiotic resistance among bacteria has been causing concern within the medical profession. The increase in antibiotic resistance is reported to be due to the extensive use of antibiotics worldwide (Finland, 1979; Levy, 1982; McGowan, 1983). The global implications of the development of resistant bacterial strains in any one country is one important aspect of the problem. In fact, Taxue et al. (1990) found that the best predictor of resistance in Shigella, a food-borne bacteria who causes diarrhea, was a history of foreign travel by the patient. The importance of the geographical transmission of resistant bacteria is also discussed in Cohen’s (1992) article on the epidemiology of drug resistance. Cohen fears that the problem of drug resistant bacteria might increase as ∗ Tel.: +46-90-786-99-40; fax: +46-90-77-23-02. E-mail address: [email protected] (N. Rudholm).
0167-6296/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 6 2 9 6 ( 0 2 ) 0 0 0 5 3 - X
1072
N. Rudholm / Journal of Health Economics 21 (2002) 1071–1083
international travel makes the transmission of such bacteria more frequent. As such, the use of antibiotics in any particular country affects the well-being of the population in other countries. This paper concentrates on two aspects of the resistance problem. Firstly, the use of antibiotics in one country is assumed to lead to the development of a resistant bacterial strain, which might cause disease in neighboring countries as well. Assuming that these countries play a non-cooperative differential game, taking the level of antibiotic use in other countries as given, this results in a suboptimal resource allocation at the global level. Secondly, the health of the consumer is assumed to affect output in each country. As the consumer does not recognize this relationship between health and output when choosing the level of antibiotic consumption, an externality in the production of goods arises. One purpose of the paper is to compare the socially optimal allocation of resources, at a global level, with the allocation in a non-cooperative differential game between countries. Another is to analyze the resource allocation brought forth by a decentralized market economy without any policies to control the use of antibiotics. A final concern is to design the Pigouvian tax/subsidy system required to make the market economy reach the globally optimal resource allocation. Most of the earlier studies concerning the economic impact of antibiotic resistance concentrate on the direct costs arising from prolonged hospitalization and increased mortality (Carmeli et al., 1999; Holmberg et al., 1987; Liss and Batchelor, 1987). Holmberg et al. (1987) report that the length of the hospital stay is at least twice as great for patients infected by resistant bacteria compared with patients infected by antibiotic susceptible ones, although they do not estimate the cost increases caused by prolonged hospitalization. Liss and Batchelor (1987) argue that, as policies to restrict the use of antibiotics are being adopted, the pharmaceutical industry loses part of its economic incentive to develop new antibiotic substances designed to combat resistant bacteria. Carmeli et al. (1999) report that the national cost of antibiotic resistance in the US lies somewhere between US$ 100 million and US$ 30 billion annually. However, despite presenting these estimates, Carmeli et al. (1999) admit that there is a lack of studies that thoroughly examine the costs arising from antibiotic resistance. Although there appear to be differing opinions about the actual size of the costs, it seems nevertheless to be a widespread view that antibiotic resistance increases treatment costs. Tisdell (1982) was the first to study antibiotic resistance as a utility maximization problem. He assumes that the drugs are effective only for a finite number of exposures, and that there is free access to this drug for a given cost. Tisdell then uses a two period model and shows that without intervention, there will be extensive antibiotic use in the first period, rendering them useless in the second period. In order to reach a first best equilibrium, he suggests that the government should either regulate first period consumption of antibiotics or grant monopoly rights to the sellers of antibiotics. Brown and Layton (1996) set up a dynamic model where a consumer maximizes the utility of antibiotic treatments and a farmer is maximizing the net benefits of antibiotic use in agricultural production. Both the consumer and the farmer take the effectiveness of antibiotic treatments as exogenously given, ignoring the impact of their antibiotic use on treatment efficiency. The social planner maximizes social welfare from antibiotic use, taking the resistance problem into account. A comparison of the different resource allocations shows
N. Rudholm / Journal of Health Economics 21 (2002) 1071–1083
1073
that both the consumer and the farmer choose a higher level of antibiotic consumption than the social planner. However, Brown and Layton do not suggest any corrective policy to be implemented. This paper contributes to the literature in at least three ways. First, this is (at least to my knowledge) the first attempt to analyze the problem of antibiotic resistance in a dynamic general equilibrium context. Second, the global aspects of antibiotic resistance are addressed. Finally, a dynamic Pigouvian tax is presented that is designed to internalize all external effects arising from antibiotic use at the global level. The type of model used in this paper has earlier been used to analyze different aspects of environmental problems. In a recent paper, Aronsson and Löfgren (2000) analyze the global implications of transboundry pollution in a Nash non-cooperative differential game, as well as in the uncontrolled market economy, and suggested a Pigouvian tax to reach a globally optimal resource allocation in a setting quite similar to mine. The paper is set out as follows. In Section 2, I describe the model. Section 3 presents a non-cooperative differential game between countries, while Section 4 analyzes the global social optimum. The difference between these two equilibria is that the Nash equilibrium does not internalize the part of the externalities which is due to the interaction between countries, whereas all externalities are internalized at a global level in the social optimum. In Section 5, I examine the resource allocation in the uncontrolled market economies. The policies required to make the agents in the decentralized economy choose the socially optimal resource allocation are discussed in Section 6. Section 7 concludes the paper.
2. The settings Assume that there are I countries in the global economy.1 In addition, to focus on the external effects of antibiotic use, I shall disregard international trade between the countries. The population in each country is assumed to be constant and can thus be normalized to one. Further, the consumers are assumed to have preferences over a composite commodity and health. The instantaneous utility function of the consumer in county i, at time t, can be written as ui (t) = ui (ci (t), hi (t))
(1)
for i = 1, . . . , I , where c is consumption of goods and h an indicator of health status.2 The utility function is assumed to be increasing in both arguments, twice continuously differentiable, and strictly concave. The health status is determined by the consumption of antibiotics, ai (t), and by the stocks of resistant bacteria created by all countries. We can write the health indicator function as hi (t) = hi (ai (t), x1 (t), . . . , xI (t))
(2)
1 Note that the analysis could be interpreted in terms of households, or hospital wards, within the same country instead of as separate countries. However, in order to focus on the global aspects of antibiotic resistance, the results presented below will be interpreted it terms of countries. 2 Time is considered to be continuous throughout the paper.
1074
N. Rudholm / Journal of Health Economics 21 (2002) 1071–1083
where the function hi (t) is assumed to be increasing in the consumption of the antibiotic, ai (t), decreasing in the stocks of resistant bacteria, x1 (t), . . . , xI (t), concave and twice differentiable. The assumption that only resistant bacteria affect the health indicator function should be interpreted such that the effects of non-resistant bacteria on health have been suppressed for notational convenience. This simplification is made in order to focus on the effects of resistant bacteria on health and production possibilities in the economy. Net output in country i is characterized by the production function: yi (t) = fi (ki (t), hi (t))
(3)
implying that net output is a function of the capital stock, ki (t), and the health status of the labor force, hi (t), measured per unit of labor. The effects of the health indicator should be interpreted as a direct effect reflecting the average health status per unit of labor, which is not internalized by the wage formation system. This production function is assumed to be increasing in hi (t), twice differentiable and strictly concave. Note that the production function measures net output, meaning that the depreciation of capital has been accounted for. Thus, if the capital stock becomes sufficiently large, the marginal product of capital may become negative. Now, substituting the expression for the health indicator into the utility and production functions yields: ui (t) = ui (ci (t), hi (ai (t), x1 (t), . . . , xI (t)))
(4)
yi (t) = fi (ki (t), hi (ai (t), x1 (t), . . . , xI (t)))
(5)
and
meaning that the utility of the consumer in country i is a function of his/her consumption of a composite good, ci (t), the consumption of antibiotics, ai (t), and the stocks of resistant bacteria generated by all countries. Net output in country i is, in turn, a function of the capital stock, ki (t), antibiotic treatments and the stocks of resistant bacteria. The net investments in country i is given by dki (t) = fi (ki (t), hi (ai (t), x1 (t), . . . , xI (t))) − ςi (ai (t)) − ci (t) dt
(6)
where ςi (ai (t)) is the amount of resources used by country i in the production of antibiotics. Finally, one key feature of the use of antibiotics is that it puts selective pressure upon bacteria to develop resistance against such treatments. Assume, along the lines of Finland (1979), that the accumulation of resistant bacteria in country i is an increasing function in the use of antibiotics, and that the stock of resistant bacteria evolves over time according to: dxi (t) = bi (ai (t)) dt where ∂bi (ai )/∂ai > 0 and ∂ 2 bi (ai )/∂ai2 ≤ 0.
(7)
N. Rudholm / Journal of Health Economics 21 (2002) 1071–1083
1075
3. The non-cooperative equilibrium 3.1. A description of the game between countries The problem of antibiotic resistance has been given increased attention by governments around the world. However, it seems as if individual countries are adopting their own programs to fight the problem by creating national organizations with the purpose of fighting antibiotic resistance.3 If national decision makers take the actions of other countries as given, this strategy might result in a suboptimal allocation of resources. To explore this idea further, assume that the resource allocation in each country is decided upon by a national social planner, who treats the creation of resistant bacteria in other countries as exogenous. This situation can be described by a differential game in open loop form between the countries. In this model, the strategies available to each planner are the choices of consumption of the composite commodity and antibiotic use at each instant. The payoff from the game is given by the utility level achieved from consuming the optimal amounts of goods and antibiotics chosen by the planner. The planner thus maximizes the utility of the consumer in country i, subject to the equations of motion for the capital stock, ki (t), and the stock of resistant bacteria, xi (t). The initial conditions ki (0) = ki0 > 0 and xi (0) = xi0 > 0, as well as the terminal condition limt→∞ ki (t) ≥ 0 are also imposed. Neglecting the time indicator for notational convenience, the present value Hamiltonian for country i can then be written Hi = ui [ci , hi (ai , x1 , . . . , xI )] e−θt + λi [fi (ki , hi (ai , x1 , . . . , xI ) − ς (ai ) − ci ] +µi [bi (ai )]
(8)
where λi and µi are the present value shadow prices, in terms of utility, of capital and the stock of resistant bacteria. 3.2. The Nash equilibrium It is well known that dynamic games of this type are difficult to solve analytically, and that a unique solution might not exist.4 However, since the purpose of this paper is to compare the welfare aspects of the Nash equilibrium to those of the other equilibrium concepts, I shall assume that a unique equilibrium does exist. The first-order conditions for country i, which gives the optimal choice of antibiotic- and goods consumption, can be written5 3 In Sweden there is an organization called STRAMA created with the purpose of promoting rational use of antibiotics in order to fight the emergence of resistant bacterial strains. Similar organizations can be found in Finland (MIKSTRA), Canada (CCAR), as well as in several other countries. During the last few years, some international organizations like the European antimicrobial resistance surveillance system (EARSS) and the alliance for prudent use of antibiotics (APUA), have been created with similar purposes. 4 The interested reader is referred to Hoel (1978) or Clarke (1980) who show that explicit solutions generally require simplifying assumptions. 5 The transversality conditions presuppose that certain growth conditions are fulfilled. In short, these growth conditions constitute upper bounds on the influence of the state variables on the functions involved. The reader is referred to Seierstad and Sydsaeter (1987, Theorem 16 of Chapter 3) for further details.
1076
N. Rudholm / Journal of Health Economics 21 (2002) 1071–1083
∂ui (ci , hi ) e−θt − λi = 0 ∂ci
(9)
∂ui (ci , hi ) e−θt ∂hi ∂fi (ki , hi ) ∂hi ∂bi (ai ) ∂ςi (ai ) + µi =0 + λi − ∂hi ∂ai ∂hi ∂ai ∂ai ∂ai λ˙ i = −λi
µ˙ i = −
∂fi (ki , hi ) ∂ki
(10)
(11)
∂fi (ki , hi ) ∂hi ∂ui (ci , hi ) e−θt ∂hi − λi ∂hi ∂xi ∂hi ∂xi
(12)
lim λi ≥ 0 (= 0 if lim ki > 0)
(13)
lim µi = 0
(14)
t→∞
t→∞
t→∞
where the time indicator has been neglected for notational convenience. Each planner is aware of the presence of the other planners, as well as how their choices of goods- and antibiotics-consumption affect his/her state equations. The planner thus faces the problem of choosing his/her control variables in such a way that the utility of the consumer in country i is maximized for every possible choice of goods- and antibiotics-consumption made by all other planners. When the control variables are chosen such that there are no incentives for the planner in any of the I countries to revise his/her choice, the economy is in Nash equilibrium. The first-order condition for antibiotic use in country i, Eq. (10), consists of three terms. The first term measures the consumer’s marginal discounted valuation (in utility terms) of antibiotic consumption. The second term represents the difference between the marginal product of antibiotics in goods production (which works through the health status of the labor force), and the marginal cost of producing antibiotics. Finally, the third term reflects the buildup of the stock of resistant bacteria in country i, where µi (t) measures the shadow price of additions to this stock. This shadow price is found by solving Eq. (12), subject to the transversality condition, limt→∞ µi = 0, and equals µni (t)
=
∞ t
∂ui (cin , hni ) e−θs ∂hni ∂fi (kin , hni ) ∂hni + λni ∂hi ∂xi ∂hi ∂xi
ds
(15)
where the superscript n denotes the Nash equilibrium. Note that the shadow price described in Eq. (15) only reflects the consequences for welfare and production capabilities arising in country i, when a stock of resistant bacteria builds up. It does not reflect the consequences for utility and production in other countries. This means that the Nash equilibrium is characterized by externalities, arising from interactions between countries. Thus, this allocation of resources can not be optimal, and a global view on the problem of antibiotic resistance has to be adopted in order to come to terms with the problem.
N. Rudholm / Journal of Health Economics 21 (2002) 1071–1083
1077
4. The globally optimal resource allocation I now turn to the task of describing the optimal resource allocation, were all externalities have become internalized at the global level. The results from this section will later be used when designing the Pigouvian taxes which makes the market economy reach the global optimum. The optimization problem facing a global utilitarian social planner can be written as a standard control problem with an infinite time horizon. To derive the globally optimal equilibrium, where all externalities have been internalized, assume that the planner maximizes the sum of the utilities for all countries, subject to the equations of motion for k1 (t), . . . , kI (t) and x1 (t), . . . , xI (t).6 That is ∞ I Max ui [ci (t), hi (t)] e−θt dt (16) ci (t),ai (t) 0
i=1
subject to dki (t) (17) = fi (ki (t), hi (t)) − ςi (ai (t)) − ci (t) dt dxi (t) (18) = bi (ai (t)) dt for i = 1, . . . , I , where hi (t) = hi (ai (t), x1 (t), . . . , xI (t)). I also impose the initial conditions ki (0) = ki0 > 0, xi (0) = xi0 > 0, and the terminal condition limt→∞ ki (t) ≥ 0. Neglecting the time indicator, the present value Hamiltonian can be written H =
n
[ui (ci , hi ) e−θt + λi (fi (ki , hi ) − ςi (ai ) − ci ) + µi (bi (ai )]
(19)
i=1
where λi and µi are present value shadow prices in terms of utility. In addition to the equations of motion for ki and xi , as well as the initial and terminal conditions, the first-order conditions for country i = 1, . . . , I can be written (neglecting the time indicator for notational convenience) ∂ui (ci , hi ) e−θt − λi = 0 ∂ci ∂ui (ci , hi ) e−θt ∂hi ∂fi (ki , hi ) ∂hi ∂bi (ai ) ∂ςi (ai ) + µi =0 + λi − ∂hi ∂ai ∂hi ∂ai ∂ai ∂ai λ˙ i = −λi µ˙ i = −
∂fi (ki , hi ) ∂ki
(21) (22)
I I ∂uj (cj , hj ) e−θt ∂hj ∂fj (kj , hj ) ∂hj − λj ∂hj ∂xi ∂hj ∂xi j =1
(20)
(23)
j =1
6 The assumption of an additive social welfare function is made to preserve simplicity and make comparisons between the different equilibria possible.
1078
N. Rudholm / Journal of Health Economics 21 (2002) 1071–1083
lim λi ≥ 0 (= 0 if lim ki > 0)
(24)
lim µi = 0
(25)
t→∞
t→∞
t→∞
where i = 1, . . . , I . Suppose that Λ∗i (t) = [ci∗ (t), ai∗ (t)], i = 1, . . . , I are the optimal control variables chosen by the global social planner. Let us briefly discuss the resource allocation presented above from the point of view of the use of antibiotics. Note that the first-order condition for antibiotic use has the same general form as in the Nash equilibrium analyzed in Section 3. The fundamental difference between the two equilibria refers to the way in which the shadow price of resistant bacteria are measured. By solving Eq. (23) subject to the transversality condition, limt→∞ µi = 0, the shadow price of resistance creation is found to equal ∞ I ∂u (c∗ , h∗ ) e−θs ∂h∗ I ∂fj (kj∗ , h∗j ) ∂h∗j j j j j ∗ ∗ ds µi (t) = (26) + λj ∂hj ∂xi ∂hj ∂xi t j =1
j =1
which clearly differs from the shadow price in the Nash equilibrium presented above. In this case µ∗i (t) takes into account how the creation of resistant bacteria in country i affects the production capabilities and welfare in all other countries as well. To see this, note that the first two terms in Eq. (26) reflect how the buildup of the stock of resistant bacteria in country i affects health, and thus utility, of the consumer in all countries. The last terms measure the impact of the creation of resistant bacteria on production possibilities in all countries. As such, this shadow price will reflect the effects on production possibilities and welfare in every country, caused by the creation of resistant bacteria in country i. Eq. (26) will play an important role when designing the dynamic Pigouvian taxes in later sections, as this shadow price is used in order to place the correct value on additions to the stocks of resistant bacteria. Comparing the two solutions presented so far, this means that the global planner acts to maximize utility globally, while the national planner ignores that the use of antibiotics in his/her country affects the welfare and production possibilities of other countries. As mentioned in footnote (2), some international organizations have been created in recent years to meet the increasing problem of resistant bacterial strains. The motive for creating such international organizations is most likely the increasing understanding that the problem of antibiotic resistant bacteria cannot be fought on the national level.
5. The uncontrolled market economy In most countries, the resource allocation is determined by some form of controlled market economy. In such economies, decision makers use economic policy, such as taxes and subsidies, in order to internalize external effects. In the following sections of the paper, I will show that the globally optimal solution can be implemented in the market economy.7 7 Aronsson and Löfgren (2000) show that the Nash equilibrium could be implemented in the market economy as well.
N. Rudholm / Journal of Health Economics 21 (2002) 1071–1083
1079
The next two sections thus concern two main issues. Firstly, in this section the equilibrium in the uncontrolled market economy will be derived and compared with the previous resource allocations. Secondly, in the next section a dynamic Pigouvian tax will be designed to show that the first best equilibrium can be implemented in the market economy, as long as the decision makers have enough information to be able to set the tax correctly. Each country consists of three agent types: one representative consumer, one firm producing a composite good and another firm producing and selling antibiotics. The consumer chooses the consumption of the composite good, ci (t), and the level of antibiotic use, ai (t), to maximize the present value of future utility, subject to the equation of motion for the capital stock. The representative consumer acts myopically and does not take into account the effect on the stock of resistant bacteria of his consumption of antibiotics. The stock of resistant bacteria is thus considered exogenous by the consumer. The utility maximization problem of the consumer in country i can then be written ∞ Max [ui (ci (t), hi (t))] e−θt dt (27) ci (t),ai (t) 0
subject to dki (t) = ri (t)ki (t) + πi (t) + wi (t) − pi (t)ai (t) − ci (t) dt
(28)
where ri (t) is the interest rate, πi (t) total profit income from both firms (as the consumer owns the firms), wi (t) labor income and pi (t) is the price of antibiotics, which are all considered exogenous by the consumer. Finally, impose the initial condition ki (0) = ki0 > t 0, as well as the No Ponzi Game condition limt→∞ ki (t) exp(− 0 ri (s) ds) ≥ 0, which implies that the present value of the capital stock has to be non-negative at the terminal point. Turning to the production side of the economy, both agent types act competitively. The producer of the composite good chooses capital, ki (t), in order to maximize πi1 (t) = fi (ki (t), hi (ai (t), x1 (t), . . . , xI ) − wi (t) − ri (t)ki (t)
(29)
To simplify the analysis, I disregard the use of capital in the production of antibiotics, and write the objective function of the firm producing antibiotics as follows: πi2 (t) = pi (t)ai (t) − ςi (ai (t))
(30)
where pi (t)ai (t) is the sales revenue for antibiotics and ςi (ai (t)) represents all the costs of producing antibiotics. Solving the utility and profit maximization problems of the different agents, and combining the first-order conditions, gives the following general equilibrium conditions −θt ∂ui (cim , hm i )e − λm i =0 ∂ci
m −θt ∂hm ∂ui (cim , hm i )e i m ∂ςi (ai ) =0 − λi ∂hi ∂ai ∂ai m λ˙ m i = −λi
∂fi (kim , hm i ) ∂ki
(31)
(32)
(33)
1080
N. Rudholm / Journal of Health Economics 21 (2002) 1071–1083
dkim m m = fi (kim , hm i ) − ςi (ai ) − ci dt
(34)
for i = 1, . . . , I , where the superindex m is used to denote the market equilibrium, and Eq. (34) is derived by combining Eqs. (28)–(30). It is apparent that the resource allocation in the uncontrolled market economy differs from the socially optimal allocation, as well as from the Nash equilibrium. In the uncontrolled market economy, the consumer chooses the level of antibiotic use ignoring the fact that the consumption of antibiotics affects the stock of resistant bacteria and, hence, the production capabilities of his/her own country as well as the production capabilities of the other countries. The equilibrium presented is characterized by externalities, since neither domestic nor transboundry external effects of resistant bacteria are internalized.
6. Implementing the cooperative equilibrium In order to implement the cooperative equilibrium a tax on antibiotic sales, τi (t)ai (t), is introduced. The objective function of the antibiotics producer changes to become: πi2 (t) = pi (t)ai (t) − ςi (ai (t)) − τi (t)ai (t)
(35)
The tax revenue is returned to the consumer as a lump sum transfer, Ti (t). The government in country i is assumed to balance the budget at each instant, that is τi (t)ai (t) − Ti (t) = 0 The representative consumer then solves: ∞ Max [ui (ci (t), hi (t))] e−θt dt ci (t),ai (t) 0
(36)
(37)
subject to dki (t) = ri (t)ki (t) + πi (t) + wi (t) − pi (t)ai (t) + Ti (t) − ci (t) dt
(38)
Solving the utility and profit maximization problems of the different agents, and combining the first-order conditions in the same fashion as above, gives the following general equilibrium conditions (neglecting the time indicator): ∂ui (ci , hi ) e−θt − λi = 0 ∂ci ∂ui (ci , hi ) e−θt ∂hi ∂ςi (ai ) =0 − λ i τi + ∂hi ∂ai ∂ai
(39) (40)
∂fi (ki , hi ) ∂ki
(41)
dki = fi (ki , hi ) − ςi (ai ) − ci dt
(42)
λ˙ i = −λi
N. Rudholm / Journal of Health Economics 21 (2002) 1071–1083
1081
Consider Proposition 1 as follows: Proposition 1. If the tax on antibiotic consumption is chosen such that ∂fi (ki∗ (t), h∗i (t)) ∂h∗i (t) µ∗i (t) ∂bi (ai∗ (t)) − ∗ τi∗ (t) = − ∂hi (t) ∂ai (t) λi (t) ∂ai (t)
(43)
for all t and i = 1, . . . n, where all entities are evaluated at the first-best equilibrium, the market economy will reach the globally optimal resource allocation. Substituting the expression for τi∗ (t) into Eq. (40), and evaluating Eqs. (39)–(42) in the first-best equilibrium, it is apparent that the first-order conditions of the two equilibria coincide. The first term in Eq. (43) represents the value of antibiotic consumption in the production of goods in country i. This term reflects the benefits for the firm of having a healthy labor force. Assuming for a moment that the second term equals zero, this would mean that the use of antibiotics should be subsidized by the planner. The second term captures the social value of additions to the stock of resistant bacteria, caused by antibiotic consumption in country i, implying that the use of antibiotics should be taxed. Taken together, this means that the tax presented in Eq. (43) could be either positive or negative, depending on the term which outweighs the other. In addition, note that the buildup of resistant bacteria affects the welfare in the other countries as well, meaning that this has to be accounted for in order to reach the optimal solution. In the Pigouvian tax presented in Eq. (43), this is accomplished through the shadow price, µ∗i (t), which has the form presented in Eq. (26). One can easily see that this shadow price takes into account how the creation of resistant bacteria in each country affects other countries as well. Introducing the tax presented above internalizes all external effects of antibiotic consumption and makes the market economy reach the globally optimal resource allocation. The tax/subsidy system presented above thus consists of two parts. The first part of Eq. (43) suggests that antibiotics should be subsidized, while the second part suggests that antibiotics should be taxed. Note that the second term in Eq. (43) is likely to increase over time if the stock of resistant bacteria is growing in the first-best equilibrium. However, it is not possible to determine whether the second term will eventually dominate the first. The question of whether antibiotics should be taxed or subsidized is thus a question that warrants empirical research. To implement the tax presented above, the government in each country would require large amounts of information. They would have to have access to information on exactly how antibiotic treatments affect the production of goods, as well as the value of additions to the stock of resistant bacteria, if they are to choose the tax correctly. Note also that without information on the size of the different effects of antibiotic consumption, the government in each country does not even know if antibiotic treatments should be taxed or subsidized.
7. Summary The purpose of this paper is to analyze the global and national economic aspects of antibiotic resistance in bacteria. The socially optimal resource allocation is compared
1082
N. Rudholm / Journal of Health Economics 21 (2002) 1071–1083
with the unrestricted market solution, as well as with a non-cooperative Nash equilibrium. It is assumed that antibiotic consumption creates a stock of resistant bacteria which affects the welfare of the consumers in all countries because of the risk of transmission of resistant bacterial strain between countries. Health, and thus the use of antibiotics, is also assumed to have a positive effect on the production of goods. However, the consumer/labor force in the market economy does not take these effects of antibiotic consumption into account when choosing the level of antibiotic consumption. The expression for the tax/subsidy system which makes the market economy reach the globally optimal solution consists of two parts; one part describing the positive effect of antibiotic use on production, and the other part describing the social value of additions to the stock of resistant bacterial strains. Whether the governments should tax or subsidize the use of antibiotics is dependent on which part outweighs the other. This means that a considerable amount of information is needed in order for them to set the tax or, as may very well be the case, the subsidy correctly. The problem of antibiotic resistant bacteria has received increased attention in recent years and several countries are adopting strategies to fight the creation of resistant bacterial strains. However, in this paper, it is shown that if decision makers in one country act in their own self-interest, and take the actions of neighboring countries as exogenously given, this results in a suboptimal allocation of resources. Thus, a global view of the problem of antibiotic resistance should be adopted in order to come to terms with the problem.
Acknowledgements The author would like to thank Thomas Aronsson, Runar Brännlund, Tomas Sjögren, Magnus Wikström and all participants in a seminar at UmeåUniversity for helpful comments and suggestions. Financial support from the Browaldh–Wallander–Hedelius foundation is gratefully acknowledged.
References Aronsson, T., Löfgren, K.-G., 2000. Green accounting and green taxes in the global economy. In: Folmer, A., Gabel, L., Rose, A. (Eds.), Frontiers in Environmental Economics. Cheltenham, Edward Elgar. Brown, G., Layton, D.F., 1996. Resistance economics: social costs and the evolution of antibiotic resistance. Environment and Development Economics 1, 349–355. Carmeli, Y., Troillet, N., Karchmer, A.W., Samore, M.H., 1999. Health and economic outcomes of antibiotic resistance in Pseudomonas aeruginosa. Archives of Internal Medicine 159, 1127–1132. Clarke, C.M., 1980. Restricted access to common property fishing resources: a game theoretic analysis. In: Liu, P.T. (Ed.), Dynamic Optimization and Mathematical Economics. New York, Plenum Press. Cohen, M.L., 1992. Epidemiology of drug resistance: implications for a post-antimicrobial era. Science 257, 1050–1055. Finland, M., 1979. Emergence of antibiotic resistance in hospitals, 1935–1975. Reviews of Infectious Diseases 1, 4–21. Hoel, M., 1978. Distribution and growth as a differential game between workers and capitalists. International Economic Review 19, 335–350.
N. Rudholm / Journal of Health Economics 21 (2002) 1071–1083
1083
Holmberg, S.D., Solomon, S.L., Blake, P.A., 1987. Health and economic impacts of antimicrobial resistance. Reviews of Infectious Diseases 9, 1065–1078. Levy, S.B., 1982. Microbial resistance to antibiotics: an evolving and persistent problem. Lancet 2, 83–88. Liss, R.H., Batchelor, F.R., 1987. Economic evaluations of antibiotic use and resistance—a perspective: report of task force 6. Reviews of Infectious Diseases 9, 297–312. McGowan, J.E., 1983. Antimicrobial resistance in hospital organisms and its relation to antibiotic use. Reviews of Infectious Diseases 5, 1033–1048. Seierstad, A., Sydsaeter, K., 1987. Optimal Control Theory with Economic Applications. North-Holland, New York. Taxue, R.V., Puhr, N.D., Wells, J.G., Hargrett-Bean, N., Blake, P.A., 1990. Antimicrobial resistance of Shigella isolates in the USA: the importance of International travelers. Journal of Infectious Diseases 162, 1107–1111. Tisdell, C., 1982. Exploitation of techniques that decline in effectiveness with use. Public Finance 37, 428–437.