- Email: [email protected]
Demand curves and welfare
Anim. Behav., 1997, 53, 983–990
Demand curves and welfare ALASDAIR I. HOUSTON Behavioural Biology Group, School of Biological Sciences, University of Bristol (Received 4 October 1995; initial acceptance 10 January 1996; final acceptance 18 July 1996; MS. number: 5041)
Abstract. Dawkins (1990, Behav. Brain Sci., 13, 1–61) argued that ideas from economics can be used to assess animal suffering. In particular, she argued that the slope or elasticity of a demand curve generated by making an animal work for a commodity or the chance to perform an activity are suitable measures. Here a simple model is presented which suggests that neither the slope nor the elasticity is likely to be a generally valid measure. Welfare economists do not use the slope or the elasticity of demand curves as a measure of changes in welfare; they use the area under a demand curve. It is argued that this measure is also the appropriate one to use in the context of animal suffering. ?
1997 The Association for the Study of Animal Behaviour
A fundamental problem in animal welfare is to identify conditions in which animals suffer. (Throughout this paper, I use the term ‘animals’ to denote non-human animals.) Dawkins (1983, 1990) suggested that measures based on demand curves (as used in economics) may be useful in this context. Her argument can be summarized as follows.
with a review of the relevant aspects of economics and then look at its application to animal behaviour. Finally, I construct a simple model to explore the relationship between demand and fitness. To develop my argument, it is necessary to comment on the term ‘perceived costs’. Dawkins’ approach to suffering is an evolutionary one.
(1) Suffering is an unpleasant subjective feeling that is likely to occur when an animal is prevented from performing an activity that it is highly motivated to perform. (2) Animals will be strongly motivated to perform activities that are perceived by the animal to be important in terms of Darwinian fitness. From (1) and (2):
I have suggested that suffering may occur when captive animals are unable to do something they are highly motivated to do because it would normally reduce a risk to their fitness. (Dawkins 1990, page 4)
Suffering occurs when unpleasant subjective feelings are acute or continue for a long time because the animal is unable to carry out the actions that would normally reduce risks to life and reproduction in those circumstances. (Dawkins 1990, page 2) (3) Concepts of demand and elasticity from economics can be used to infer the perceived fitness costs and benefits associated with an activity. In this paper I am not concerned with points 1 and 2, but I try to evaluate the third point. I start Correspondence: A. I. Houston, School of Biological Sciences, University of Bristol, Woodland Road, Bristol BS8 1UG, U.K. (email: [email protected]). 0003–3472/97/050983+08 $25.00/0/ar960397
Thus, if an animal is studied in the habitat in which it has evolved (the ‘natural’ habitat), suffering can be expected when the animal is prevented from performing activities that make an important contribution to fitness. The situation is not so straightforward when the animal is not in its natural habitat. In such circumstances, the animal may continue to use rules that are appropriate for the natural habitat but are not appropriate in the current context (e.g. McNamara & Houston 1980; Dawkins 1983; Houston 1987). It is for this reason that Dawkins (1990) talked about ‘perceived costs’, which she took to be the fitness costs as perceived by the animal. The approach that I adopt in this paper is to assume that demand curves result from an animal behaving in such a way as to maximize some function. I then investigate how various
? 1997 The Association for the Study of Animal Behaviour
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aspects of the demand curves are related to changes in this function if the animal is prevented from performing an activity. To ensure that this approach is relevant to Dawkins’ claim, I can assume that either (1) the animal is in its natural habitat, so that perceived costs correspond to actual fitness costs or ‘canonical costs’ (McNamara & Houston 1986); or (2) the animal is behaving so as to maximize its total perceived net benefit (perceived gains minus perceived costs). For a related suggestion, see Cabanac (1992). In the section called ‘Animals follow rules’ I argue that abandoning these assumptions does not necessarily lead to a justification of demand curves as indicators of suffering. DEMAND CURVES In economics, a demand curve relates the quantity consumed, Q, to the price, p. Economists usually take Q to be the independent variable and p to be the dependent variable. When demand curves are used to study animal behaviour, an analogue of price is varied, so it is natural to take p to be the independent variable and Q to be the dependent variable. The slope of a demand curve is dQ/dp. In general, we should expect Q to decrease as p increases, so that dQ/dp is negative. To eliminate the effect of the absolute levels of Q and p, economists characterize demand curves in terms of relative slope or elasticity. Assume that the price is p and the demand is Q, and that the price is increased by a small amount Äp. If the demand changes by an amount ÄQ, then the relative change in price is Äp/p and the relative change in quantity is ÄQ/Q. The elasticity ç is approximately equal to "ÄQ/Q/Äp/p. The exact equation is
On this definition, if dQ/dp is negative then ç lies between 0 and £. If a consumer spends a constant amount of money A on a commodity, regardless of its price, then Q=A/p and hence ç=1.
10
(a)
9
Elasticity = 0.1
8 7 6 Elasticity = 1
5 Quantity (Q)
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4 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
1
0
2
3
4
5
6 (b)
Elasticity = 0.5
Elasticity = 1 Elasticity = 2
1
2
3
4 5 6 Price (p)
7
8
9
10
Figure 1. (a) A demand curve with constant slope dQ/dp= "1, where Q is quantity consumed and p is price. Elasticity ç= "dQ/dp p/Q, so when Q=10 and p=1, ç=0.1, whereas when Q=p=5.5, ç=1.0. (b) Demand curves with constant elasticity.
Demand is said to be inelastic if ç<1 and elastic if ç>1. It is clear from equation (1) that if a demand curve has a constant slope, then it will not have a constant elasticity. An illustrative example is given in Fig. 1a. The demand curve shown has a constant slope of "1. When Q=10 and p=1, ç=0.1, whereas when Q=5.5 and p=5.5, ç=1. Demand curves with constant elasticity are linear in log–log co-ordinates with slope equal to the elasticity (see, for example, Mathews & Ladewig 1994). When such curves are plotted as Q against p, the absolute magnitude of the slope decreases as p increases. Some examples are given in Fig. 1b. It can be seen from the figure that when demand is inelastic, Q may change significantly with p over some ranges of p, that is the slope is large. Furthermore, when demand is elastic, Q may not be very sensitive to p over some ranges of p, that is, the slope is small. Thus the elasticity of
Houston: Demand curves and welfare demand does not necessarily tell us about the slope of the demand curve.
BEHAVIOURAL ECONOMICS I follow Hursh (1984) in using ‘behavioural economics’ to refer to the application of economic concepts to the study of animal behaviour. Reviews of this approach can be found in Hursh (1980, 1984), Hursh & Bauman (1987); Lea (1978); Rachlin (1980); Rachlin et al. (1976, 1981) and Staddon (1983). In economics, the price of a commodity is the amount of money that must be paid for a unit amount. In behavioural economics, the price is the amount of time or energy that must be spent for a unit amount. To construct a demand curve, an animal is trained to perform an operant response (e.g. pressing a lever or pecking a key) in order to obtain a commodity such as food (see, for example, Hursh 1984). The price can then be varied by varying the number of responses that must be made for a unit of the commodity. The price can also be varied by varying the amount of the commodity that is given per response (e.g. Hursh 1980; Collier et al. 1986; Hursh et al. 1988). Matthews & Ladewig (1994) measured the demand curves of pigs, Sus scrofa, working for either (1) access to food or (2) access to the sight of another pig: the curves were more or less linear in log–log co-ordinates, indicating that elasticity was roughly constant. Hursh et al. (1988) investigated the demand of rats, Rattus norvegicus, for food: the demand curves were clearly non-linear on log–log co-ordinates, which means that elasticity was not constant. Various other studies (e.g. Foltin 1991; Hursh 1991) have found non-constant elasticities.
DEMAND CURVES AS INDICATORS OF SUFFERING Dawkins argued that demand curves can be used to establish an animal’s priorities: The well-adapted animal is expected to give highest priority (most inelastic demand) to doing behaviour which, if it did not do it, would be most likely to lead to its death or reproductive failure. It should give lowest priority (most elastic demand) to behaviour where failure to
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do it has the least effect on fitness . . . (Dawkins 1984, page 1261) As we have seen, this argument requires modification if the animal is not in its natural habitat. It can no longer be assumed that the animal is maximizing fitness, but the demand curve can be taken to represent the animal’s best response to its perceived gains and costs: The slope of a demand curve reflects the tradeoff an animal makes between different courses of action under different circumstances. (Dawkins 1990, page 51) Given Dawkins’ assumptions about suffering, she concluded: Suffering is most likely to occur if animals are prevented from performing the activities or deprived of the commodities whose demand curves have the flattest slopes (inelastic demand). Note that this measure is not the same as the absolute frequency with which the animal performs the behaviour under natural conditions, . . . The demand curve for behaviour that occurs rarely but is still performed when costs are imposed would have a flat slope (inelastic demand); the curve for commonly occurring behaviour that disappears in the presence of imposed costs has a steep slope (elastic demand). (Dawkins 1990, page 7) One problem with this argument is that slope and elasticity are treated as equivalent measures with a flat slope corresponding to inelastic demand and a steep slope corresponding to elastic demand. I have argued above that such a correspondence does not always hold. To allow for this, I consider both the slope and the elasticity as possible indicators of suffering. If the argument put forward by Dawkins is correct, then we should expect that either (1) preventing an activity with a steep slope results in a smaller loss in fitness than preventing an activity with a shallow slope, that is, across activities the magnitude of the slope is negatively correlated with the loss in fitness or perceived benefit; or (2) preventing an activity with a large elasticity (e.g. >1) results in a smaller loss in fitness than preventing an activity with a small elasticity (e.g. <1), that is, across activities the elasticity is negatively correlated with the loss in fitness or perceived benefit.
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F=Q k "mpQ,
3 2.5
where k = 0.5
2 Gain
(2)
m=m ˜ +a.
k = 0.25 1.5
The appendix gives the equation for the value Q* of Q that maximizes F. The appendix also shows that if, as the price p is varied, the animal always adopts the appropriate Q*, then the elasticity ç of the demand curve is constant and is given by
1 0.5 0
5
10
15
20 25 Time, t
30
35
40
Figure 2. Two examples of the gain (t/p)k with price p=5.
I now present a simple model which shows that neither (1) nor (2) need be true. A Simple Model In this section I analyse a simple model for the time to spend performing an activity. I phrase the analysis in terms of maximizing fitness but it is clear that exactly the same argument applies if the animal is maximizing perceived benefit. I assume that the animal has an amount of time T available to it. Let us start by considering a single activity to which the animal can devote a time t¦T. If t
It can be seen from equation (3) that elasticity is always greater than one but tends to one as k tends to zero. Let F* be the animal’s fitness if it adopts the optimal time allocation t*. If the animal is prevented from performing the activity, then F* is its lost fitness. Using equations (1) and (2) together with the equation for Q*, F* can be given by an equation that depends on ç, m and p (see the appendix). We are now in a position to evaluate the claim that the slope or the elasticity of a demand curve should be negatively correlated with the loss in fitness. Before doing so, it is necessary to say something about how to compare two or more activities. For simplicity, assume that we are interested in two activities, which are known as activity 1 and activity 2. Activity i is performed for a time ti (i=1,2). One possibility is that we focus on activity 1 in one context, and activity 2 in another context. This means that the above analysis can be applied directly, with each activity having its own value of k, and perhaps also its own value of m. Alternatively, we may consider the animal to perform both activities during the period that we are considering. Provided that t*1 +t*2¦T, the above analysis carries through. It can be seen from the two values of k, illustrated in Fig. 2, that the activity with the bigger elasticity is not necessarily associated with the smaller loss in fitness F*. The figure is based on a price p of 5. When k=0.5, ç=2 and F*=1, whereas when k=0.25, ç=131 and F *=0.75. t* and F* for these values of k and a range of values of p are given in Table I. In Table II, m and p are fixed and k is varied to vary ç. The table gives "dQ*/dp and F*.
Houston: Demand curves and welfare Table I. The effect of price p on the optimal allocation of time t* and the loss in fitness F* for two activities that differ in k and hence in elasticity (the cost parameter m=0.05 throughout) 1 3
ç=2 (k=0.5)
ç=1 (k=0.25)
p
t*
F*
t*
F*
1 2 3 4 5
100 50 33.33 25 20
5 2.5 1.67 1.25 1.0
8.55 6.79 5.93 5.39 5.00
1.28 1.02 0.89 0.81 0.75
As described above, from the argument of Dawkins (1990), we expect either (1) the magnitude of the slope is negatively correlated with F* or (2) the elasticity ç is negatively correlated with F*. It can be seen from Table II that neither (1) nor (2) need hold. Sometimes the magnitude of the slope is positively correlated with F* while ç is negatively correlated with F*; sometimes the magnitude of the slope is negatively correlated with F* while ç is positively correlated with F*.
ANIMALS FOLLOW RULES I have argued that my analysis is relevant to Dawkins’ position if either (1) the animal is in its natural habitat, so that perceived costs correspond to actual fitness costs, or (2) the animal is behaving so as to maximize perceived gains minus perceived costs.
Table II. The elasticity ç together with the loss in fitness F* and the magnitude "dQ*/dp of the slope when the price, p=5 m=0.05 ç 1.5 2.0 2.5 3.0
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I believe that it is perfectly reasonable to argue that animals in a laboratory setting are following rules that perform well in the natural habitat. As a result, animals will not necessarily be maximizing perceived gains minus perceived costs. Although this undercuts the application of my argument to laboratory experiments, I do not think that it necessarily provides support for the use of demand curves as a measure of suffering. To justify this view, I shall consider possible decision rules for an animal that has evolved in a very simple environment. In this environment, the animal has a fitness of zero if it consumes a quantity less than Q | of some vital substance in a day, and a fitness of one if it consumes at least Q | . The quantity Q obtained is related to the time t devoted to the relevant activity by the equation Q=t/p. I assume that in the animal’s environment, p always takes the value pˆ, so the animal might evolve a rule of the form: (R1) perform the activity for time tˆ =Q | pˆ. If the animal follows this fixed-time rule in a laboratory experiment in which p varies, then the quantity obtained will be Q=tˆ/p, from which it follows that demand has a constant elasticity of 1. The animal might evolve another sort of rule: (R2) perform the activity until the quantity consumed reaches Q | . If the animal follows this fixedamount rule in a laboratory experiment in which p varies, then provided the animal does not run out of time the amount consumed will be Q | and so demand will be inelastic. Thus the two rules R1 and R2 are equally good in the animal’s natural environment and yet have different demand curves. Furthermore, under a range of conditions that generate these different demand curves, the animal can adopt the behaviour required by each rule. Under these circumstances, there seems to be no straightforward way to identify one of these demand curves with more suffering than is associated with the other.
m=0.2
F*
"dQ*/dp
F*
"dQ*/dp
0.77 1.0 1.49 2.37
0.46 1.60 4.46 11.38
0.38 0.25 0.19 0.15
0.06 0.10 0.14 0.18
Results are given for two values of the parameter m.
DISCUSSION The argument presented by Dawkins is based on relating suffering to a fitness cost. I have measured the fitness cost as the loss in fitness if the animal is unable to perform the activity under consideration. It is important to realize that we cannot use the marginal fitness )F/)t under an optimal time
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allocation as a measure. If the optimum is not constrained, then )F/)t=0. Various models in behavioural economics (e.g. Rachlin & Burkhard 1978; Hursh & Bauman 1987) consider the maximization of a function (referred to in these cases as ‘utility’ rather than ‘fitness’) by the allocation of time to several activities, subject to the time constraint that the sum of time allocations is equal to the total time T. When this constraint is imposed, then at the optimal allocation )F/)ti is equal for each activity ti, so once again the marginal fitness does not provide a suitable measure. I have emphasized that the slope of a demand curve and its elasticity are distinct measures, and that neither may be correlated with lost fitness in the way suggested by Dawkins (1990). I chose a model that was simple to analyse; it is possible that in a more realistic model the correlations suggested by Dawkins would hold. This is an obvious topic for future research. Economists did not introduce elasticity to look at issues of human welfare, so it will not be surprising to them that elasticity does not necessarily provide a general measure of animal welfare. In the context of human welfare, economists have devised a series of measures of compensation for changes in price (see, for example, Ng 1979). These measures can be obtained from areas under demand curves. If we consider an animal’s demand curve in which the price is measured as the time to obtain a given quantity, then the area has the units of time. It gives an indication of the time that must be given to, or taken from, an animal to compensate for a change in price. If the price is reduced (i.e. food is easier to get) then time can typically be taken away in order to maintain the status quo. Ng (1990) argued that although the use of areas under demand curves has its problems, it is roughly right, whereas using the slope or the elasticity may not even give the qualitatively correct answer. In fact for fitness functions of the form F=G(Q)"mpQ, where G(Q) is the gain from obtaining quantity Q the following result holds. If the price is p0, then the area under the optimal demand curve from p=p0 to p=£ is proportional to the loss in fitness if the animal is denied the opportunity for any
consumption (i.e. cannot perform the relevant activity). Viewed in this light, the loss in fitness that I have used as a measure can be thought of as a fitness version of Marshall’s consumer surplus. This surplus is defined as the difference between the price a consumer would be willing to pay rather than go without the commodity and the amount that the consumer actually pays. See Ng (1979) and Mishan (1981) for further discussion. Ng’s argument in favour of using areas under demand curves was dismissed by Dawkins (1990, page 52) on the grounds that it is difficult to determine absolute values on the graphs presented by Ng. I take this to mean that if we have demand curves for different activities, it is not easy to compare the ‘quantity’ axis in each case. This is true, but as I have pointed out above, when price is measured in terms of quantity obtained per unit time, the area under the curve in each case has the unit of time, and hence the areas can be compared. Given the practical difficulties in measuring demand curves (Dawkins 1990, page 8) it is advisable to use a logically correct measure based on areas, rather than to use measures based on the slope or the elasticity which lack a logical justification.
APPENDIX The animal can devote a time t¦T to an activity. The fitness cost associated with t is m ˜ t, and the fitness gain from time T"t is a(T"t). The ‘price’ associated with the activity is p, so that the ‘quantity’ Q obtained is given by
The resulting net fitness is Qk +a(T"t)"m ˜t This net fitness is maximized by maximizing F=Q k "mpQ where m=m ˜ +a>0
(A2)
Houston: Demand curves and welfare and 0
and so the optimal quantity Q* is given by
From equations (A1) and (A3),
I assume that t*¦T. Let ç be the elasticity. By definition,
From equation (A3),
and so
Let F* be the fitness obtained by performing the activity for the optimal time t*. If the animal is deprived of the opportunity to perform the activity, then F* is the resulting loss in fitness. From the above equations.
ACKNOWLEDGMENTS My thanks to John McNamara for several discussions of the theoretical issues, to Ian
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Jewitt for discussions of demand theory and to Marian Dawkins, Georgia Mason and Christine Nicol for comments on early versions of the manuscript.
REFERENCES Cabanac, M. 1992. Pleasure: the common currency. J. theor. Biol., 155, 173–200. Collier, G. H., Johnson, D. F., Hill, W. L. & Kaufman, L. 1986. The economics of the law of effect. J. exp. anal. Behav., 46, 113–136. Dawkins, M. S. 1983. Battery hens name their price: consumer demand theory and the measurement of ethological ‘needs’. Anim. Behav., 31, 1195–1205. Dawkins, M. S. 1984. Reply to Hutson. Anim. Behav., 32, 1261–1262. Dawkins, M. S. 1990. From an animal’s point of view: motivation, fitness and animal welfare. Behav. Brain Sci., 13, 1–61. Foltin, R. W. 1991. An economic analysis of ‘demand’ for food in baboons. J. exp. anal. Behav., 56, 445–454. Houston, A. I. 1987. The control of foraging decisions. In: Quantitative Analyses of Behavior Vol. 6 Foraging (Ed. by M. L. Commons, A. Kacelnik & S. J. Shettleworth), pp. 41–61. Hillsdale, New Jersey: Lawrence Erlbaum. Hursh, S. R. 1980. Economic concepts for the analysis of behavior. J. exp. anal. Behav., 34, 219–238. Hursh, S. R. 1984. Behavioral economics. J. exp. anal. Behav., 42, 435–452. Hursh, S. R. 1991. Behavioural economics of drug self-administration and drug abuse policy. J. exp. anal. Behav., 56, 377–393. Hursh, S. R. & Bauman, R. A. 1987. The behavioral analysis of demand. In: Advances in Behavioral Economics. Vol. 1 (Ed. by L. Green & J. H. Kagel), pp. 117–165. Norwood, New Jersey: Ablex Pubishing Corporation. Hursh, S. R., Raslear, T. G., Shurtleff, D., Baumann, R. & Simmons, L. 1988. A cost benefit analysis of demand for food. J. exp. anal. Behav., 50, 419–440. Lea, S. E. G. 1978. The psychology and economics of demand. Psychol. Bull., 85, 441–466. McNamara, J. M. & Houston, A. I. 1980. The application of statistical decision theory to animal behaviour. J. theor. Biol., 85, 673–690. McNamara, J. M. & Houston, A. I. 1986. The common currency for behavioural decisions. Am. Nat., 127, 358–378. Matthews, L. R. & Ladewig, J. 1994. Environmental requirements of pigs measured by behavioural demand functions. Anim. Behav., 47, 713–719. Mishan, E. J. 1981. Introduction to Normative Economics. Oxford: Oxford University Press. Ng, Y.-K. 1979. Welfare Economics. London: Macmillan. Ng, Y.-K. 1990. The case for, and the difficulties in, using ‘demand areas’ to measure changes in welfare. Behav. Brain Sci., 13, 30–31.
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Rachlin, H. 1980. Economics and behavioral psychology. In: Limits to Action (Ed. by J. R. Staddon), pp. 205–236. New York: Academic Press. Rachlin, H. & Burkhard, B. 1978. The temporal triangle: response substitution in instrumental conditioning. Psychol. Rev., 85, 22–47. Rachlin, H., Green, L., Kagel, J. H. & Battalio, R. C. 1976. Economic demand theory and psychologi-
cal studies of choice. Psychol. Learn. Motiv., 10, 129–154. Rachlin, H., Battalio, R. C., Kagel, J. H. & Green, L. 1981. Maximization theory in behavioral psychology. Behav. Brain Sci., 4, 371–388. Staddon, J. E. R. 1983. Adaptive Behavior and Learning. Cambridge University Press, Cambridge.
Demand curves and welfare ALASDAIR I. HOUSTON Behavioural Biology Group, School of Biological Sciences, University of Bristol (Received 4 October 1995; initial acceptance 10 January 1996; final acceptance 18 July 1996; MS. number: 5041)
Abstract. Dawkins (1990, Behav. Brain Sci., 13, 1–61) argued that ideas from economics can be used to assess animal suffering. In particular, she argued that the slope or elasticity of a demand curve generated by making an animal work for a commodity or the chance to perform an activity are suitable measures. Here a simple model is presented which suggests that neither the slope nor the elasticity is likely to be a generally valid measure. Welfare economists do not use the slope or the elasticity of demand curves as a measure of changes in welfare; they use the area under a demand curve. It is argued that this measure is also the appropriate one to use in the context of animal suffering. ?
1997 The Association for the Study of Animal Behaviour
A fundamental problem in animal welfare is to identify conditions in which animals suffer. (Throughout this paper, I use the term ‘animals’ to denote non-human animals.) Dawkins (1983, 1990) suggested that measures based on demand curves (as used in economics) may be useful in this context. Her argument can be summarized as follows.
with a review of the relevant aspects of economics and then look at its application to animal behaviour. Finally, I construct a simple model to explore the relationship between demand and fitness. To develop my argument, it is necessary to comment on the term ‘perceived costs’. Dawkins’ approach to suffering is an evolutionary one.
(1) Suffering is an unpleasant subjective feeling that is likely to occur when an animal is prevented from performing an activity that it is highly motivated to perform. (2) Animals will be strongly motivated to perform activities that are perceived by the animal to be important in terms of Darwinian fitness. From (1) and (2):
I have suggested that suffering may occur when captive animals are unable to do something they are highly motivated to do because it would normally reduce a risk to their fitness. (Dawkins 1990, page 4)
Suffering occurs when unpleasant subjective feelings are acute or continue for a long time because the animal is unable to carry out the actions that would normally reduce risks to life and reproduction in those circumstances. (Dawkins 1990, page 2) (3) Concepts of demand and elasticity from economics can be used to infer the perceived fitness costs and benefits associated with an activity. In this paper I am not concerned with points 1 and 2, but I try to evaluate the third point. I start Correspondence: A. I. Houston, School of Biological Sciences, University of Bristol, Woodland Road, Bristol BS8 1UG, U.K. (email: [email protected]). 0003–3472/97/050983+08 $25.00/0/ar960397
Thus, if an animal is studied in the habitat in which it has evolved (the ‘natural’ habitat), suffering can be expected when the animal is prevented from performing activities that make an important contribution to fitness. The situation is not so straightforward when the animal is not in its natural habitat. In such circumstances, the animal may continue to use rules that are appropriate for the natural habitat but are not appropriate in the current context (e.g. McNamara & Houston 1980; Dawkins 1983; Houston 1987). It is for this reason that Dawkins (1990) talked about ‘perceived costs’, which she took to be the fitness costs as perceived by the animal. The approach that I adopt in this paper is to assume that demand curves result from an animal behaving in such a way as to maximize some function. I then investigate how various
? 1997 The Association for the Study of Animal Behaviour
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aspects of the demand curves are related to changes in this function if the animal is prevented from performing an activity. To ensure that this approach is relevant to Dawkins’ claim, I can assume that either (1) the animal is in its natural habitat, so that perceived costs correspond to actual fitness costs or ‘canonical costs’ (McNamara & Houston 1986); or (2) the animal is behaving so as to maximize its total perceived net benefit (perceived gains minus perceived costs). For a related suggestion, see Cabanac (1992). In the section called ‘Animals follow rules’ I argue that abandoning these assumptions does not necessarily lead to a justification of demand curves as indicators of suffering. DEMAND CURVES In economics, a demand curve relates the quantity consumed, Q, to the price, p. Economists usually take Q to be the independent variable and p to be the dependent variable. When demand curves are used to study animal behaviour, an analogue of price is varied, so it is natural to take p to be the independent variable and Q to be the dependent variable. The slope of a demand curve is dQ/dp. In general, we should expect Q to decrease as p increases, so that dQ/dp is negative. To eliminate the effect of the absolute levels of Q and p, economists characterize demand curves in terms of relative slope or elasticity. Assume that the price is p and the demand is Q, and that the price is increased by a small amount Äp. If the demand changes by an amount ÄQ, then the relative change in price is Äp/p and the relative change in quantity is ÄQ/Q. The elasticity ç is approximately equal to "ÄQ/Q/Äp/p. The exact equation is
On this definition, if dQ/dp is negative then ç lies between 0 and £. If a consumer spends a constant amount of money A on a commodity, regardless of its price, then Q=A/p and hence ç=1.
10
(a)
9
Elasticity = 0.1
8 7 6 Elasticity = 1
5 Quantity (Q)
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4 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
1
0
2
3
4
5
6 (b)
Elasticity = 0.5
Elasticity = 1 Elasticity = 2
1
2
3
4 5 6 Price (p)
7
8
9
10
Figure 1. (a) A demand curve with constant slope dQ/dp= "1, where Q is quantity consumed and p is price. Elasticity ç= "dQ/dp p/Q, so when Q=10 and p=1, ç=0.1, whereas when Q=p=5.5, ç=1.0. (b) Demand curves with constant elasticity.
Demand is said to be inelastic if ç<1 and elastic if ç>1. It is clear from equation (1) that if a demand curve has a constant slope, then it will not have a constant elasticity. An illustrative example is given in Fig. 1a. The demand curve shown has a constant slope of "1. When Q=10 and p=1, ç=0.1, whereas when Q=5.5 and p=5.5, ç=1. Demand curves with constant elasticity are linear in log–log co-ordinates with slope equal to the elasticity (see, for example, Mathews & Ladewig 1994). When such curves are plotted as Q against p, the absolute magnitude of the slope decreases as p increases. Some examples are given in Fig. 1b. It can be seen from the figure that when demand is inelastic, Q may change significantly with p over some ranges of p, that is the slope is large. Furthermore, when demand is elastic, Q may not be very sensitive to p over some ranges of p, that is, the slope is small. Thus the elasticity of
Houston: Demand curves and welfare demand does not necessarily tell us about the slope of the demand curve.
BEHAVIOURAL ECONOMICS I follow Hursh (1984) in using ‘behavioural economics’ to refer to the application of economic concepts to the study of animal behaviour. Reviews of this approach can be found in Hursh (1980, 1984), Hursh & Bauman (1987); Lea (1978); Rachlin (1980); Rachlin et al. (1976, 1981) and Staddon (1983). In economics, the price of a commodity is the amount of money that must be paid for a unit amount. In behavioural economics, the price is the amount of time or energy that must be spent for a unit amount. To construct a demand curve, an animal is trained to perform an operant response (e.g. pressing a lever or pecking a key) in order to obtain a commodity such as food (see, for example, Hursh 1984). The price can then be varied by varying the number of responses that must be made for a unit of the commodity. The price can also be varied by varying the amount of the commodity that is given per response (e.g. Hursh 1980; Collier et al. 1986; Hursh et al. 1988). Matthews & Ladewig (1994) measured the demand curves of pigs, Sus scrofa, working for either (1) access to food or (2) access to the sight of another pig: the curves were more or less linear in log–log co-ordinates, indicating that elasticity was roughly constant. Hursh et al. (1988) investigated the demand of rats, Rattus norvegicus, for food: the demand curves were clearly non-linear on log–log co-ordinates, which means that elasticity was not constant. Various other studies (e.g. Foltin 1991; Hursh 1991) have found non-constant elasticities.
DEMAND CURVES AS INDICATORS OF SUFFERING Dawkins argued that demand curves can be used to establish an animal’s priorities: The well-adapted animal is expected to give highest priority (most inelastic demand) to doing behaviour which, if it did not do it, would be most likely to lead to its death or reproductive failure. It should give lowest priority (most elastic demand) to behaviour where failure to
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do it has the least effect on fitness . . . (Dawkins 1984, page 1261) As we have seen, this argument requires modification if the animal is not in its natural habitat. It can no longer be assumed that the animal is maximizing fitness, but the demand curve can be taken to represent the animal’s best response to its perceived gains and costs: The slope of a demand curve reflects the tradeoff an animal makes between different courses of action under different circumstances. (Dawkins 1990, page 51) Given Dawkins’ assumptions about suffering, she concluded: Suffering is most likely to occur if animals are prevented from performing the activities or deprived of the commodities whose demand curves have the flattest slopes (inelastic demand). Note that this measure is not the same as the absolute frequency with which the animal performs the behaviour under natural conditions, . . . The demand curve for behaviour that occurs rarely but is still performed when costs are imposed would have a flat slope (inelastic demand); the curve for commonly occurring behaviour that disappears in the presence of imposed costs has a steep slope (elastic demand). (Dawkins 1990, page 7) One problem with this argument is that slope and elasticity are treated as equivalent measures with a flat slope corresponding to inelastic demand and a steep slope corresponding to elastic demand. I have argued above that such a correspondence does not always hold. To allow for this, I consider both the slope and the elasticity as possible indicators of suffering. If the argument put forward by Dawkins is correct, then we should expect that either (1) preventing an activity with a steep slope results in a smaller loss in fitness than preventing an activity with a shallow slope, that is, across activities the magnitude of the slope is negatively correlated with the loss in fitness or perceived benefit; or (2) preventing an activity with a large elasticity (e.g. >1) results in a smaller loss in fitness than preventing an activity with a small elasticity (e.g. <1), that is, across activities the elasticity is negatively correlated with the loss in fitness or perceived benefit.
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F=Q k "mpQ,
3 2.5
where k = 0.5
2 Gain
(2)
m=m ˜ +a.
k = 0.25 1.5
The appendix gives the equation for the value Q* of Q that maximizes F. The appendix also shows that if, as the price p is varied, the animal always adopts the appropriate Q*, then the elasticity ç of the demand curve is constant and is given by
1 0.5 0
5
10
15
20 25 Time, t
30
35
40
Figure 2. Two examples of the gain (t/p)k with price p=5.
I now present a simple model which shows that neither (1) nor (2) need be true. A Simple Model In this section I analyse a simple model for the time to spend performing an activity. I phrase the analysis in terms of maximizing fitness but it is clear that exactly the same argument applies if the animal is maximizing perceived benefit. I assume that the animal has an amount of time T available to it. Let us start by considering a single activity to which the animal can devote a time t¦T. If t
It can be seen from equation (3) that elasticity is always greater than one but tends to one as k tends to zero. Let F* be the animal’s fitness if it adopts the optimal time allocation t*. If the animal is prevented from performing the activity, then F* is its lost fitness. Using equations (1) and (2) together with the equation for Q*, F* can be given by an equation that depends on ç, m and p (see the appendix). We are now in a position to evaluate the claim that the slope or the elasticity of a demand curve should be negatively correlated with the loss in fitness. Before doing so, it is necessary to say something about how to compare two or more activities. For simplicity, assume that we are interested in two activities, which are known as activity 1 and activity 2. Activity i is performed for a time ti (i=1,2). One possibility is that we focus on activity 1 in one context, and activity 2 in another context. This means that the above analysis can be applied directly, with each activity having its own value of k, and perhaps also its own value of m. Alternatively, we may consider the animal to perform both activities during the period that we are considering. Provided that t*1 +t*2¦T, the above analysis carries through. It can be seen from the two values of k, illustrated in Fig. 2, that the activity with the bigger elasticity is not necessarily associated with the smaller loss in fitness F*. The figure is based on a price p of 5. When k=0.5, ç=2 and F*=1, whereas when k=0.25, ç=131 and F *=0.75. t* and F* for these values of k and a range of values of p are given in Table I. In Table II, m and p are fixed and k is varied to vary ç. The table gives "dQ*/dp and F*.
Houston: Demand curves and welfare Table I. The effect of price p on the optimal allocation of time t* and the loss in fitness F* for two activities that differ in k and hence in elasticity (the cost parameter m=0.05 throughout) 1 3
ç=2 (k=0.5)
ç=1 (k=0.25)
p
t*
F*
t*
F*
1 2 3 4 5
100 50 33.33 25 20
5 2.5 1.67 1.25 1.0
8.55 6.79 5.93 5.39 5.00
1.28 1.02 0.89 0.81 0.75
As described above, from the argument of Dawkins (1990), we expect either (1) the magnitude of the slope is negatively correlated with F* or (2) the elasticity ç is negatively correlated with F*. It can be seen from Table II that neither (1) nor (2) need hold. Sometimes the magnitude of the slope is positively correlated with F* while ç is negatively correlated with F*; sometimes the magnitude of the slope is negatively correlated with F* while ç is positively correlated with F*.
ANIMALS FOLLOW RULES I have argued that my analysis is relevant to Dawkins’ position if either (1) the animal is in its natural habitat, so that perceived costs correspond to actual fitness costs, or (2) the animal is behaving so as to maximize perceived gains minus perceived costs.
Table II. The elasticity ç together with the loss in fitness F* and the magnitude "dQ*/dp of the slope when the price, p=5 m=0.05 ç 1.5 2.0 2.5 3.0
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I believe that it is perfectly reasonable to argue that animals in a laboratory setting are following rules that perform well in the natural habitat. As a result, animals will not necessarily be maximizing perceived gains minus perceived costs. Although this undercuts the application of my argument to laboratory experiments, I do not think that it necessarily provides support for the use of demand curves as a measure of suffering. To justify this view, I shall consider possible decision rules for an animal that has evolved in a very simple environment. In this environment, the animal has a fitness of zero if it consumes a quantity less than Q | of some vital substance in a day, and a fitness of one if it consumes at least Q | . The quantity Q obtained is related to the time t devoted to the relevant activity by the equation Q=t/p. I assume that in the animal’s environment, p always takes the value pˆ, so the animal might evolve a rule of the form: (R1) perform the activity for time tˆ =Q | pˆ. If the animal follows this fixed-time rule in a laboratory experiment in which p varies, then the quantity obtained will be Q=tˆ/p, from which it follows that demand has a constant elasticity of 1. The animal might evolve another sort of rule: (R2) perform the activity until the quantity consumed reaches Q | . If the animal follows this fixedamount rule in a laboratory experiment in which p varies, then provided the animal does not run out of time the amount consumed will be Q | and so demand will be inelastic. Thus the two rules R1 and R2 are equally good in the animal’s natural environment and yet have different demand curves. Furthermore, under a range of conditions that generate these different demand curves, the animal can adopt the behaviour required by each rule. Under these circumstances, there seems to be no straightforward way to identify one of these demand curves with more suffering than is associated with the other.
m=0.2
F*
"dQ*/dp
F*
"dQ*/dp
0.77 1.0 1.49 2.37
0.46 1.60 4.46 11.38
0.38 0.25 0.19 0.15
0.06 0.10 0.14 0.18
Results are given for two values of the parameter m.
DISCUSSION The argument presented by Dawkins is based on relating suffering to a fitness cost. I have measured the fitness cost as the loss in fitness if the animal is unable to perform the activity under consideration. It is important to realize that we cannot use the marginal fitness )F/)t under an optimal time
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allocation as a measure. If the optimum is not constrained, then )F/)t=0. Various models in behavioural economics (e.g. Rachlin & Burkhard 1978; Hursh & Bauman 1987) consider the maximization of a function (referred to in these cases as ‘utility’ rather than ‘fitness’) by the allocation of time to several activities, subject to the time constraint that the sum of time allocations is equal to the total time T. When this constraint is imposed, then at the optimal allocation )F/)ti is equal for each activity ti, so once again the marginal fitness does not provide a suitable measure. I have emphasized that the slope of a demand curve and its elasticity are distinct measures, and that neither may be correlated with lost fitness in the way suggested by Dawkins (1990). I chose a model that was simple to analyse; it is possible that in a more realistic model the correlations suggested by Dawkins would hold. This is an obvious topic for future research. Economists did not introduce elasticity to look at issues of human welfare, so it will not be surprising to them that elasticity does not necessarily provide a general measure of animal welfare. In the context of human welfare, economists have devised a series of measures of compensation for changes in price (see, for example, Ng 1979). These measures can be obtained from areas under demand curves. If we consider an animal’s demand curve in which the price is measured as the time to obtain a given quantity, then the area has the units of time. It gives an indication of the time that must be given to, or taken from, an animal to compensate for a change in price. If the price is reduced (i.e. food is easier to get) then time can typically be taken away in order to maintain the status quo. Ng (1990) argued that although the use of areas under demand curves has its problems, it is roughly right, whereas using the slope or the elasticity may not even give the qualitatively correct answer. In fact for fitness functions of the form F=G(Q)"mpQ, where G(Q) is the gain from obtaining quantity Q the following result holds. If the price is p0, then the area under the optimal demand curve from p=p0 to p=£ is proportional to the loss in fitness if the animal is denied the opportunity for any
consumption (i.e. cannot perform the relevant activity). Viewed in this light, the loss in fitness that I have used as a measure can be thought of as a fitness version of Marshall’s consumer surplus. This surplus is defined as the difference between the price a consumer would be willing to pay rather than go without the commodity and the amount that the consumer actually pays. See Ng (1979) and Mishan (1981) for further discussion. Ng’s argument in favour of using areas under demand curves was dismissed by Dawkins (1990, page 52) on the grounds that it is difficult to determine absolute values on the graphs presented by Ng. I take this to mean that if we have demand curves for different activities, it is not easy to compare the ‘quantity’ axis in each case. This is true, but as I have pointed out above, when price is measured in terms of quantity obtained per unit time, the area under the curve in each case has the unit of time, and hence the areas can be compared. Given the practical difficulties in measuring demand curves (Dawkins 1990, page 8) it is advisable to use a logically correct measure based on areas, rather than to use measures based on the slope or the elasticity which lack a logical justification.
APPENDIX The animal can devote a time t¦T to an activity. The fitness cost associated with t is m ˜ t, and the fitness gain from time T"t is a(T"t). The ‘price’ associated with the activity is p, so that the ‘quantity’ Q obtained is given by
The resulting net fitness is Qk +a(T"t)"m ˜t This net fitness is maximized by maximizing F=Q k "mpQ where m=m ˜ +a>0
(A2)
Houston: Demand curves and welfare and 0
and so the optimal quantity Q* is given by
From equations (A1) and (A3),
I assume that t*¦T. Let ç be the elasticity. By definition,
From equation (A3),
and so
Let F* be the fitness obtained by performing the activity for the optimal time t*. If the animal is deprived of the opportunity to perform the activity, then F* is the resulting loss in fitness. From the above equations.
ACKNOWLEDGMENTS My thanks to John McNamara for several discussions of the theoretical issues, to Ian
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Jewitt for discussions of demand theory and to Marian Dawkins, Georgia Mason and Christine Nicol for comments on early versions of the manuscript.
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