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The psychophysics of inflation
Journal of Economic North-Holland
Psychology
THE PSYCHOPHYSICS
R.A. BATCHELOR TheCity University Business Received
December
269
7 (1986) 269-290
OF INFLATION
School, London,
3, 1985; accepted
*
UK
May 12, 1986
This study examines whether perceptions of and attitudes towards inflation obey standard psychological laws. Survey data on eight countries are used to generate estimates of the threshold for perceptions of changes in inflation, and the rate of inflation which consumers consider moderate. Weber’s Law, that the thresholds are constant, is firmly rejected. Fechner’s conjecture, that these thresholds, though variable, nonetheless always and everywhere represent changes which are regarded as of equal importance by consumers, is also rejected, but only narrowly. The popular theory of signal detection, that perceptual thresholds depend systematically on the level and noisiness of inflation, is, however, supported.
Introduction This study brings aggregate survey-based empirical evidence to bear on the following question: do the perceptions and feelings of individuals concerning changes in the macroeconomic environment, in particular changes in the general price level, obey the same ‘laws’ which experimental psychologists have observed in the reactions of individuals to changes in physical stimuli? The question is interesting because subjective variables, perceptions and sensations, play an increasing role in economic analysis. Normative economics has long recognised the distinction between the objective and the subjective magnitudes of economic stimuli - between the * The ,author is indebted to the Commission of the European Communities for financial support of the research programme on Expectations, Inflation and Growth, of which this study forms a part; to John Carlson and the Krannert School of Management at Purdue University for hospitality during its writing; and to participants in seminars at Birkbeck College, and the Universities of Hull and York, for their helpful comments. Mailing address: R.A. Batchelor, Centre for Banking and International Finance, The City University Business School, Frobisher Crescent, Barbican, London ECZY 8HB, UK.
0167-4870/86/%3.50
0 1986, Elsevier Science
Publishers
B.V. (North-Holland)
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amount of income, say, and the utility derived from that income. In recent years, positive economics has also come to be concerned as much with subjective as objective variables: with perceived and expected inflation as well as actual inflation; with permanent income as well as actual income. While such theories are potentially very insightful, their validation is difficult. The econometrician can most conveniently observe the objective magnitudes of variables such as inflation and income, whereas the individual can only act on the basis of their subjective magnitudes. In order to connect the subjective world of economic agents with the objective reality of official statistics, economists have been obliged to make strong and often not wholly explicit assumptions about how feelings arise, how perceptions are formed and how decisions are taken. In normative economics, for example, it has long been assumed that the subjective magnitude of an economic stimulus is related to its objective size through a particular law, of diminishing marginal utility, a law which finds support in empirical observation. In positive economics, it has become conventional to assume that in forming perceptions and expectations individuals employ optimal inferential and predictive techniques. Such assumptions are, for example, implicit in the ‘rational expectations’ approach to modelling price level expectations (Muth 1961), relative price perceptions (Lucas 1973), and permanent income (Nerlove 1967). These assumptions are less well grounded in empirical observation, and are made more from a desire to maintain consistency between what is assumed about decisions concerning the use of information, and the axioms of rationality normally applied to decisions concerning the use of resources. This central contention of economic theory, that individuals are good intuitive statisticians, has been strongly disputed by, for example, Simon (1959) and Kahneman et al. (1982). There are two reasons for thinking that the concepts and language of experimental psychology might help in establishing the validity of these assumptions about economic behaviour. One is that the establishment of law-like relations between the objective magnitudes of stimuli and their perceived, subjective, magnitudes has been a central concern of psychologists for over a century. In particular, early in the history of psychology, two ‘laws’ were proposed, which seemed to link objective with subjective magnitudes across a wide range of different physical stimuli. The first was due to Ernst Weber (1846) who suggested, on the basis of experimental evidence,
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that the subjectively just-noticeable-difference ( jnd) in the level of any stimulus was not constant, but rather proportionate to the initial objective level of the stimulus. The second was due to Gustav Fechner (1860) who suggested, on the basis of the more metaphysical speculation that jnds in stimuli must be considered equal in terms of subjective magnitude, that the subjective magnitude of any stimulus should be measured not in objective, physical, units, but in terms of the number of jnds it contained. These laws make a natural starting point for thinking about how perceptions of economic stimuli, and the sensations they produce, are formed. The second reason for supposing that such an approach might be productive is that both economics and psychology now employ the common language of statistical decision theory. Tanner and Swets (1954) showed persuasively that the jnd in perceptions should be regarded not as some physiologically determined parameter, but rather as a decision variable, the size of which is determined by the size and variability of the stimulus, the noisiness of the environment and the relative costs of failing to detect changes and incorrectly anticipating change. This theory of signal detection contains as a special case - the so-called ‘ideal receiver’ - the optimal forecasting assumptions currently popular in macroeconomics. The investigation of the analogy between economic and physical stimuli below falls into three sections. In the first section, a formal statement of Weber’s Law and Fechner’s Law is developed in the context of inflation perceptions. In the second section a technique is proposed by which aggregate qualitative survey data on perceptions and expectations in eight European Community countries in the period 1974-82 can be used to derive quantitative measures of jnds in the price level, and of rates of inflation which consumers in different countries consider ‘moderate’. The third section reports results from empirical tests of the predictions of Weber’s Law (strictly signal detection theory) and Fechner’s Law, based on this data. One key prediction of signal detection theory - that the jnd threshold should rise as uncertainty about the rate of inflation increases, is consistently borne out by the data. However, Fechner’s Law, which predicts that the moderate rate of inflation, while differing from country to country, and year to year, should nonetheless always represent the same number of just-noticeable-steps in inflation, is not strongly supported. To summarise, we cannot reject the hypothesis that on average over a large
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number of individuals, consumers process information about inflation in a statistically optimal way. We can, however, just reject the idea that the resulting perceptual thresholds can be used normatively, as measures of the seriousness of inflation. The psychophysical
laws
In order to focus the discussion, consider the following situation. A consumer starts his year subject to the ‘stimulus’ of a price level P. During the year, he visits a number of ‘markets’ or information sources which - together with any prior information on inflation - help him form an estimate Ap of the change in the price level stimulus during the year. At the end of the year he is asked to report _ whether any change has occurred; - how ‘important’ to him the change has seemed. Weber’s Law gives some guidance as to how the consumer might answer the first question. The empirical regularity which Weber noted in his experiments can be summarised as saying that the consumer will only report that a change has occurred if the mean observed change Ap exceeds the just-noticeable-difference in price Aj P, where AjP=GP.
0)
The law states that for any individual, and for any stimulus, the jnd in the stimulus is a fixed percentage of the level of the stimulus. The parameter 6 is to be interpreted as a measure of the individual’s sensitivity to the stimulus. It will vary across individuals according to their powers of discrimination, and will for any one individual vary according to the sensory mode involved. Fechner’s Law gives some guidance as to how the consumer might answer the second question, concerning the ‘sensation’ that the perceived -price change produces. It starts from the assumption that all just-noticeable-differences in perception represent equal chtnges in sensation, or subjective magnitude. An observed change of AP in the price level will therefore represent a change of AM units of subjective magnitude, where AM=b-AP/AjP,
(2)
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273
and djp is the average size of a jnd in price in the range P, P + A$. The constant b simply translates the number of jnds in this interval into a corresponding number of units of subjective magnitude, each just-noticeable-change in P resulting in a change of b units on the sensation scale. If both Weber’s Law and Fechner’s Law hold good, then for a small change in P, dP say, the corresponding change in sensation can be written dM=
b- dP/SP.
Integration M=i
(3)
of this yields
In P+c,
(4)
where c is a constant of integration. Thus the psychophysical laws predict that the subjective magnitude of any stimulus is proportionate to the logarithm of its objective magnitude. There is at least prima facie evidence that such psychophysical laws operate in economics. The just-noticeable-difference in price, for example, is indeed likely to be a function of the price level. In shopping for low value items, we assiduously inspect penny differences in the prices of different products. In purchasing high value items - motor cars, houses - we will happily bargain in steps of tens or hundreds of pounds, and sign large bills without inspecting the last few figures too closely. This commonplace observation received formal support in a survey by Pratt et al. (1979) of differences across neighbouring stores in the selling prices of a number of standard items. Their data show a strong and virtually proportionate relation between the standard deviation of prices, and the mean price for each item. The coexistence of such prices in competitive stores must mean that the price difference which buyers regard as effectively zero is, as Weber’s Law predicts, higher in the case of high-priced items than in the case of low-priced items. It is more difficult to adduce evidence on Fechner’s Law. However, the idea of diminishing marginal utility, a subjective measure of magnitude, from increased wealth is a standard assumption in microeconomits. It is a property of the semi-log function (4) that the marginal
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change in subjective magnitude will fall as the level of stimulus is increased; and, indeed, precisely such a function was proposed by Bernoulli (1738) as a description of the marginal utility of money, in his famous exposition of Cramer’s resolution of the St. Petersburg paradox. Weber’s Law has been substantially modified and reinterpreted as a result of experimental evidence and theoretical developments over the past century. The experimental evidence has generally pointed to the need to modify the functional form of the law at extreme stimulus values. Changes in the conceptual basis of the law, resulting from the development of the theory of signal detection by Birdsall (1955), and Tanner and Swets (1954), have, however, been of more profound significance. In most early theory, the perceptual threshold was regarded as the product of innate physiological factors in sight, hearing, length and colour judgment, sensitivity to pain, and so on. Tanner and Swets suggested that it would be better regarded as a decision criterion, when it should depend not only on random physiological factors, but also on such systematic considerations as the difficulty of disentangling the stimulus change (the signal) from other events (the noise), and indeed on the costs of failing to perceive such stimuli. The principles are those of statistical decision theory, and their application in psychology is fully discussed in Green and Swets (1974), and more succinctly in Coombs et al. (1970: ch. 6). The principles can be simply illustrated if we consider the conditions under which our representative consumer will find it worth reporting that the general price level has risen. It will help if we start by expressing actual and perceived changes in percentage terms, and give some stochastic structure to these inflation rates. Specifically, assume that the general rate of inflation p and the rates of inflation in individual markets i are distributed as p=p+v, pi=P
v+
wi,
wi -
iv@, Tu’),
(5)
N(0, Tw’).
(6)
Eq. (5) suggests inflation fluctuates randomly around a mean rate j. The consumer’s problem is to disentangle the inflationary ‘signal’ p from a sample of market inflation rates pi which differ from the general inflation rate by relative price disturbances wi. If the consumer
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215
visits m markets his observed average price change A? will by the Central Limit Theorem correspond to a mean perceived inflation rate of 1, where $=p+w,
w-
$0, rn’),
(7)
and rn2= 7:/m represents the ‘noise’ surrounding the consumer’s observation on p. When the consumer is asked the question - has any change occurred in the price level? - the theory of signal detection assumes that he responds as if conducting a formal test of the hypothesis p = 0. In other words, he tests whether i, has been drawn from the noise-only distribution fi = w against the alternative that it is drawn from the signal-plus-noise distribution $ =p + w. The distribution of p under the hypothesis that p = 0 is N(0, 7:); its distribution under the alternative that p = j, say, is N( j, r,,2+ 7,‘). These two distributions are shown on fig. 1. The general form of optimal decision rule for this type of problem is well known (see, for example, Green and Swets 1974: ch. 1). The consumer will choose a critical value 6, and the hypothesis p = 0 will be accepted or rejected according as j 2 8. The threshold 6 is chosen so as to minimise the expected disutility of Type I (‘false alarm’) and Type II (‘miss’) errors. The probabilities of making such errors are indicated by the shaded areas I and II in fig. 1. The threshold depends both on factors determining these probabilities and on the relative costs of the two types of error. To be precise,
s’s($, :,, i”, T),
(8)
where y is the ‘relative importance’ of Type II vis-a-vis Type I errors. The rationale for the signs of the various effects is as follows. For any given value of 6, a rise in the expected value of inflation p will reduce the probability of Type II errors; hence 6 will be raised to restore the original balance between Type I and Type II errors. Intuitively, if when inflation occurs it is very rapid, the consumer will need to observe a higher rate i before he is persuaded that inflation is indeed non-zero. A rise in the noise variance will increase the probabilities of both types of error but will have a relatively greater impact on Type I errors, and
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Fig. 1. Distributions of perceived inflation under Key: p = inflation; j = perceived inflation; p = mean inflation; 7,’ = variance of inflation; 7,” = variance of inflationary ‘noise’; S = critical value for testing p = 0 against IT = moderate rate of inflation.
of inflation
noise and signal-plus-noise
hypotheses.
p = p;
so require an increase in 6 to restore balance. In our scenario above, noise arises from relative price variations wi; the greater these are, the more evidence the consumer needs in the form of a high perceived rate j, before he is persuaded that he has not simply sampled an unusually inflationary set of goods, but that a general inflation is occurring. A rise in signal variance increases the probability of Type II errors, and so requires a downward revision in S; if the general rate if inflation itself is highly variable then the consumer will be quite ready to accept that whatever rate of inflation he perceives, even if it is rather low, is representative of what is happening to the price level in general. Finally, any increase in the costs of unanticipated inflation will penalise the making of Type II errors, and so cause the threshold S to be reduced. Eq. (8) defines necessary qualitative characteristics of the jnd for a
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consumer who is, in psychophysical parlance, an ‘ideal receiver’ of information about inflation, or in economic terms is forming ‘rational expectations’ about general price inflation on the basis of limited information about relative prices. Our test of the validity of this extension of Weber’s Law consists of establishing whether the predicted pattern of relationships between 6 and its determinants is observed in practice. Just as Weber’s Law has been subject to significant revision and extension in the past century, so also has the status of Fechner’s Law changed as a result of developments in psychological scaling techniques. Two types of technique have been developed to assess subjective magnitude. One is based on category scaling. The subject is presented with a stimulus, say a sound of a particular volume, and is asked to assign it to one of a ranked series of categories - very soft, soft, medium, loud, very loud. The other technique is based on magnitude estimation. The subject is asked to assign numbers to the perceived intensity of a series of stimuli with differing objective magnitudes. Both types of measure have been used to test whether the semi-logarithmic relation (4) exists between subjective magnitude (position on the scale) and stimulus size. Most studies based on category scaling show the predicted relation between subjective magnitude (position on the scale) and the logarithm of stimulus size. Studies based on the more convincing magnitude estimation technique, on the other hand, show very mixed results (see Stevens 1975), and the semi-logarithmic relationship (4) has been seriously undermined as a description of how subjective magnitude varies with stimulus size. However, there is a serious conceptual problem with these tests, viewed as tests of Fechner’s Law (2), since they depend on the joint validity of both Fechner’s Law and the original form (1) of Weber’s Law. As we have seen, the theory of signal detection suggests that perceptual thresholds are systematically variable, in which case the slope coefficient in (4) could not be expected to be constant, even if Fechner’s Law were true. A more direct test of (2) is simply to check whether the number of jnds necessary to produce a given change in sensation is indeed constant. Our procedure is to generate, for a number of countries k, and a number of years r, an. estimate of what rate of inflation is considered ‘moderate’ on average across consumers. If this is denoted rkr, then application of Fechner’s Law would suggest that this should
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everywhere and always represent sensory magnitude, so that T/J&~
= A M/b
Psychophysics
of inflation
a change of A@ units on the scale of
= 9,
where skt is the average jnd in inflation in the range (0, rkTTkr), and 0 is the fixed number of jnds in the moderate rate of inflation. Our test of Fechner’s Law is therefore a test of the invariance of 0 across countries and over time.
The measurement
of perceptions
In order to implement the tests described above, some means must be found of measuring the various subjective concepts involved. Normally, these measurements would be drawn directly from experiments in which individuals are presented with a range of stimuli and asked to discriminate between them, and to assign them some measure of subjective magnitude. In the present study of price level perceptions, the measures are instead made indirectly, on the basis of the aggregate responses of large numbers of individuals to a series of questionnaires about past and projected price changes. This section of the paper describes the quantification technique in some detail. The questionnaires involved have been distributed thrice-yearly to samples of between 1,000 and 5,000 consumers in each of eight member countries of the European Community, in the years 1974-82. These harmonised consumer surveys ask the following questions concerning perceptions of, and expectations about, the general price level: - Compared to what they were 12 months ago, do you think that prices in general are now: lower/about the same/a little higher/ moderately higher/ much higher/ don’t know? - By comparison with what is happening now, do you consider that in the next 12 months prices will: fall slightly/be stable/increase at a sloer rate/increase at the same rate/increase more rapidly/don’t know? The results of the questionnaire are summarised into five response the percentage of proportions (say, A, B, C, D, E) representing respondents (excluding don’t-knows) answering in each possible re-
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of inflation
sponse category. Since the two questions are similar in structure, we describe in detail only how the response proportions for the first question, on perceptions, can be used to generate measures of the jnd in price, and an estimate what is meant by a ‘moderately higher’ price level. The technique is a generalisation to a five-category tendency survey of the method developed by Carlson and Parkin (1975) to obtain quantitative measures of the mean and standard deviation of expectations from a simpler, three-category survey in which respondents could only indicate whether prices were rising, falling, or remaining unchanged. The technique relies on the following three assumptions: (1) The perceptions distribution. Individual uted across consumers as +( p, a*).
perceptions
jjj are distrib-
In the particular case of market sampling in an environment with relative price changes described by (5)-(7), + will be normal, with /.A=j, a2=7,2+ r2n . For simplicity, we have assumed 9 has a logistic, rather than a normal, cumulant. (2) Weber’s Law. Individuals will report that no change in level has occurred if jj is less than some threshold 8,. individuals will report that prices are ‘moderately higher’ within some threshold k&j around the rate of inflation they consider moderate. For simplicity, sumers. (3) Unbiasedness. are unbiased,
6, E and
7~ have been
In any survey, perceptions so that p =p.
assumed
equal
the price Similarly, if jj lies rj which
for all con-
of past rates of inflation
The perceptions distribution and the jnd bands + 6 and r + E are shown on fig. 2. Under the above assumptions, the sample response proportions A, B, C, D and E are maximum likelihood estimates of the areas under the distribution in the ranges (2 cc, -S), (-6, +S), (6, 7~- E), (7~ E, 7~+ E) and (7~ + E, cc), respectively. We denote by a, b, c and d the abscissae of the perceptions distribution corresponding to cumulative
R.A. Batchelor/ Psychophysics of infiaiion
280
-6
I
+6
u
n
--E
tl
n
4-E
Fig. 2. The distribution of mean inflation perceptions. Key: j = mean perceived inflation; p= average perceived inflation in sample survey; a = S. D. of perceived inflation in sample survey; n= moderate rate of inflation; 6 = jnd in inflation around zero; e = jnd in inflation around r.
probabilities of A, A + B, A + B + C and A + B + C + D. The benefit of assuming + has a logistic cumulant is that these abscissae are simply the log-odds statistics a = ln[A/(l - A)], b = ln[( A + B)/(l - A - B)] etc. It is a straightforward matter to show that p=7r-(a+b)/e,
S=7r*(a-b)/e, e=57-(c-d)/e,
00)
where e = (a + b) - (c + d). The system (10) defines the mean and standard deviation of the perceptions distribution, and the two thresholds, in terms of sample-based statistics a, b, c and d, together
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of inflation
with the as yet undefined moderate rate 7~. However, assumption requires that p = p, so that
the unbiasedness
m=p*e/(a+b), and this can be used to scale all of the expressions in (10). An estimate of the Fechner constant for the moderate inflation, the number of jnds it contains, can be obtained as
rate
of
e = T/( ihy2, =
e/[(a - b)(c - d)y2.
(12)
Note that the Fechner constant 8 is independent of 7~. We have used this procedure to generate estimates of mean perceived inflation, the perceptual thresholds S and E, and the Fechner constant 6 for each survey conducted in each of eight member countries of the European Community, starting with the May 1974 survey and ending in May 1982. An exactly analogous procedure can be used to generate estimates of mean expected inflation $ over the following year, the dispersion of these expectations across individuals, u’, and thresholds 6’ and E’ around a zero inflation rate, and around the perceived prior inflation rate p. Since the reference point for the expectations question is the ‘same rate’ of inflation, the perceived recent rate of inflation p takes over the role of the moderate rate in the system (lo), and this can be solved directly for $, u’, S’ and E’. To give a flavour of the large data set produced by this technique, tables l-3 present some summary statistics on the objective size and variability of inflation in the eight European countries surveyed, and on the subjective perceptions and expectations of consumers in these countries. Table 1 displays the average annual rate of consumer price inflation in each country, measured for the year ending in January, May and October, the data starting in May 1974 and ending in May 1982. Our set of eight countries can be seen to cover a wide range of inflationary experience , from as low as 5 percent in Germany to over 17 percent in Italy. The time period covered also contains marked fluctuations in inflation within each country with all countries experiencing peak rates in 1974-5 and 1979-80 associated with two major oil price hikes.
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Table 1 Mean and standard
deviation
of consumer
Country
price inflation
Inflation
Belgium Denmark France Germany Ireland Italy The Netherlands United Kingdom
of inflation
in eight European
economies,
1974-82.
in January,
May and
a
Mean
SD.
8.2 11.0 11.6 4.8 16.4 17.2 7.1 15.4
3.4 2.8 2.1 1.4 4.5 3.8 2.1 5.1
a Percentage changes in the consumer price index over twelve months October each year. Data starts in May 1974, ends in May 1982. Source: Eurostat.
ending
Table 2 lists average values for the mean and dispersion of inflation perceptions, the moderate rate of inflation, the perceptual thresholds 6 and E around zero and the moderate inflation rate, and thenumber 8 of just-noticeable-differences in price which go to make up the moderate inflation rate. Perceptions are by definition unbiased. The dispersion of perceptions across individuals, in our earlier theory a function of the sum of the variance of inflation over time and the variability of relative price changes, shows the expected correlation with the standard
Table 2 Perceived
consumer
Country
Belgium Denmark France Germany Ireland Italy The Netherlands United Kingdom
price inflation:
summary
statistics.
Mean value a of P
0
n
s
e
8
8.1 11.0 11.7 4.8 16.3 17.1 7.0 15.3
3.2 5.0 4.6 2.3 6.7 6.7 2.8 7.1
4.4 10.0 7.8 5.2 8.8 10.3 4.5 13.8
0.7 2.7 1.3 1.0 1.6 1.3 0.7 2.4
1.6 2.4 2.1 1.2 1.9 4.1 1.9 4.0
4.8 4.0 5.2 4.9 5.4 4.7 4.5 4.6
a Average value of each parameter, per annum.
over 25 surveys
in the period
May 1974-May
1982: percent
R.A. Batchelor/ Table 3 Expected
consumer
Country
price inflation:
Psychophysics
summary
283
of inflation
statistics.
Mean value a of
8’
a’
6’
E’
7.0 9.5 10.4 4.5 18.7 19.0 7.4 13.7
3.4 4.8 5.0 2.3 9.4 9.7 3.3 7.8
1.9 3.1 3.0 1.1 3.9 5.2 1.5 2.9
2.5 2.5 4.3 2.2 4.5 5.5 2.7 4.0
a Average value of each parameter, per annum.
over 25 surveys
Belgium Denmark France Germany Ireland Italy The Netherlands United Kingdom
in the period
May 1974-May
1982: percent
deviation of inflation. On the other hand, the ratio of the cross-sectional dispersion of perceptions to the variance of inflation - an index of the amount of noise relative to signal variation - is definitely higher in some countries (France, Denmark, Italy, Germany) than others (Belgium, The Netherlands, United Kingdom, Ireland). The moderate rate of inflation is clearly correlated across countries with the rate of inflation actually achieved, but it lies below actual rates for all countries except Germany, the least inflationary. The thresholds for reporting price differences are smaller when a zero price change is the standard than they are when the actual price change is taken to be this moderate rate of inflation. In other words, given the same evidence, consumers are more inclined to reject the hypothesis that inflation has been zero than they are to reject the hypotheses that it has been moderate. Finally, there seem to be roughly 4-S; perceptual thresholds contained in the moderate inflation rate in all countries, giving at least prima facie credence to Fechner’s Law. Table 3 lists similar statistics relating to inflation expectations. Although it is by no means guaranteed by the quantification method, expectations do lie on average quite close to outturns. The dispersion of expectations across individuals is correlated with the mean and variability of inflation; it is also correlated with, and larger than, the dispersion of perceptions of earlier inflation - consumers are, as we would hope, more at odds over what will happen than over what has happened. Finally, the thresholds 6’ and E’ also vary in size according
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of injiotion
to the inflationary history of the economy and, in general, again the threshold around zero is lower than the threshold around the perceived inflation rate. On the same evidence, consumers are more willing to reject the proposition that inflation will be zero than that inflation will continue at its previous rate. The thresholds for reporting expectations are also larger - almost double in most cases - than the thresholds for reporting changes in perceptions. Consumers are more inclined to reject hypotheses about past inflation than they are to reject hypotheses about the future. This is what one would expect, if the thresholds are indeed to be interpreted as critical values, given that consumers are inevitably more uncertain about the future than about the past. All of the measures of perceptions and expectations reported in tables 2 and 3 are conditional on the assumptions of the quantification procedure. Before using the data in formal tests, it is therefore worth considering the validity of these assumptions, and consequences of their relaxation. As a description of individual differences in perceptions, the logistic distribution is very similar to the theoretically indicated normal distribution; and both are for practical purposes not much different from that observed in reality by Carlson (1975). The assumption that all individuals use the same thresholds is clearly unrealistic. However, making these thresholds stochastic, with variance p*, say, can be shown to affect only the estimator for a* (deflating it by a*/( a* + p*)). Finally, the unbiasedness assumption in each survey is clearly very strong. An alternative is to make the weaker assumption that perceptions are unbiased only on average over a series of T surveys, so that &.J = C,p. This more plausible condition can be imposed by selecting an appropriate value for r which is, less plausibly, constant over time, and using this value to scale the thresholds in (10). To establish the sensitivity of our results to the unbiasedness assumption, we have carried out our statistical tests of Weber’s Law using both the data summarised on tables 2 and 3, and an alternative data set based on the assumption of a constant moderate rate. No significant qualitative differences emerged between results for the two data sets, so only the first group of results is reported below. Tests of Fechner’s Law are not affected by the validity of the unbiasedness assumption since, as we have seen from expression (12) the Fechner constant is independent of V.
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Testing the psychophysical
of inflation
285
laws
We look first at tests of the signal detection theory of perceptual thresholds based on estimates of the parameters of (8), then at tests of Fechner’s Law based on tests of the stability of the number 8 of jnds in the moderate rate of inflation. Given our data on perceptual thresholds, three steps are necessary to implement an empirical test of the model of these thresholds represented by eq. (8). First, a functional form must be chosen; second, empirical counterparts to the signal, noise variance, signal variance, and costs of inflation variables must be found; third, an appropriate estimation procedure must be selected. We have considered the linear model 6 = a, + alp + a,n, + a37ru+ a,y + u,
03)
where u is a normally distributed, zero-mean, serially uncorrelated error term. The dependent variable in (8) is the threshold of inflation perceptions around zero. Our data contain figures for three other thresholds E, S’ and E’ - and we have attempted to model all of these using variants of the above equation. The first determining variable in (13), is, is the signal size. A natural measure of this is the absolute difference between mean perceptions or expectations of inflation and the reference value around which the threshold under scrutiny operates. Hence, in the model for 6, the signal size is taken to be ( /A- 0 I; for E, the signal is 1p - T I; for a’, the signal is 1~’ 7 0 I; and for E’, ( p’ - p I. The second variable in (13), the noise standard deviation, can be measured by the dispersion of perceptions across individuals. The consumer surveys in effect sample from the sampling distribution of mean perceptions, so that, from (7), a* = r2 = 7,‘. The signal standard deviation r0 has been proxied for each survey date by the standard deviations of our various signal measures within a five-period window centred on that survey date. The final determinant of threshold size is the relative cost of missing the inflation signal. This has been proxied by the signal size, on the grounds that the costs of failing to adjust to inflationary changes are likely to be greater, the larger is the change. The problem with this
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of inflation
procedure is that the coefficient on signal size in (13) now becomes sign, since a, > 0, a4 < 0. A number a43 which is of indeterminate of other factors are likely to determine the costs of inflation in any particular country. For example, if a country has a highly indexed wage-bargaining or savings structure, perceptual thresholds might be expected to be high, since the costs of failing to notice inflation will be relatively small. It is, however, difficult to quantify the degree of indexation in different economies, and such variables do not enter our model explicitly. Threshold models based on (13) have been estimated in two ways. First, the time series data from all eight countries have been pooled, and the parameters a,, CI*, a3, and a4 initially estimated by applying ordinary least squares to this pooled data. However, all models showed serial correlation in residuals, and they have been re-estimated by the Cochrane-Orcutt procedure. In order to allow for the possibility that costs of inflation may differ across countries, the models include seven country dummies in place of the single constant term a,. Table 4 summarises the results of estimating models based on this pooled data. All of the models reveal a negative relationship between signal size and perceptual threshold. This is consistent with signal detection theory only if the costs of missing inflation rise as the rate of inflation increases (or departs further from moderate or past values). All of the models show a strong positive relationship between the thresholds and the noise variance, in conformity with the theoretical model. However, although three of the four thresholds show the predicted negative relationship with the signal variance, only in the case of E’ is this statistically significant; and the model for 8 yields an (insignificant) positive coefficient. Inspection of the country dummies shows considerable country-tocountry variations in thresholds not accounted for by signal and noise considerations. There does not, however, appear to be any systematic tendency for thresholds in highly indexed economies - Italy, Belgium to be higher than would otherwise by expected. Pooling the data does involve imposing the restriction that coefficients a,, u2, u3, and u4 are the same for consumers in all countries. These restrictions are just rejected by a formal F-test. The second step in our testing of signal detection theory is therefore to estimate (13) using time series data on each country separately. Rather than list all a1 +
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287
Table 4 Determinants of thresholds in inflation perceptions and expectations: pooled data. Coefficient a on Belgium Denmark France Germany Ireland Italy The Netherlands United Kingdom Signal Noise variance Signal variance Standard error P
Threshold 8
6’
E 0.05 (0.61) 0.39 (4.71) 0.10 (1.22) 0.07
-0.01 (0.10) - 0.05 (0.92) -0.10 (1.76) 0.00
E’
0.18 (1.88)
0.39 (4.08)
(14.54) 0.03 (0.74)
0.15 (2.41) 0.12 (2.29) 0.00 (- ) 0.06 (4.52) 0.50 (29.89) -0.06 (1.41)
0.33 (3.38) 0.0s (0.81) 0.09 (0.76) 0.40 (3.49) 0.09 (1.03) 0.00 (- ) -0.12 (4.94) 0.63 (11.42) - 0.02 (0.43)
0.34 0.46
0.23 0.65
0.42 0.43
(0.89) - 0.02
(0.23) 0.02 (0.18) 0.04 (0.55) 0.00 (- ) - 0.26 (9.90) 0.87
(0.08)
-0.51 (a.021
0.24 (2.90) 0.04 (0.53) 0.52 (6.09) 0.28 (3.50) 0.07 (0.76) 0.21 (2.22) 0.21 (3.28) 0.00 (- ) -0.12 (5.33) 0.51 (23.31) -0.13 (1.75) 0.36 0.77
a Regression by OLS with Cochrane-Orcutt correction for serial correlation. Figures in parentheses beneath estimated coefficients are t-statistics. p is the estimated first-order serial correlation coefficient.
32 (8 countries x 4 jnd thresholds) regressions, it may be sufficient to indicate the role played by each determining variable. In 19 out of the 32 models, signal size entered significantly, and with a negative sign; in only 2 cases was it significant and positive. In 31 out of the 32 models, the noise variance appeared to have a significant and positive effect on the jnd, a very striking result. However, in only 8 cases did the signal variance exercise a significant negative effect on threshold size, in all other cases being statistically insignificant. These results, taken as a whole, are not inconsistent with the signal detection variant of Weber’s Law. However, it is important to recognise the econometric limitations of our tests. The presence of serial corre-
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Table 5 Analysis of variance in the ‘Fechner constant’. Source of
Degrees of freedom
Sum of squares
Mean square
F
VariaIlCe
All countries country Year Residual
7 24 168
31.20 58.30 123.70
4.46 2.43 0.74
6.06 B 3.30 a
Total
199
213.19
2.45 2.79 0.79
3.08 b 3.52 b
Ail countries excluding Denmark Country 6 Year 24 Residual 144
14.67 67.06 114.31
Total
196.04
174
a 5% critical levels are 2.01 for country effects, 1.52 for year effects; b 5% critical levels are 2.10 for country effects, 1.52 for year effects.
lation in 10 out of the 32 country models point to the possibility that some important variable has been omitted from these models. The fact that the jnd measures, and the signal and noise measures, are all functions of the same set of sample response proportions through (10) means that spurious correlation may be introduced into (13) although detailed investigation indicated that the size of this should in practice be rather small. Finally, the relatively poor showing of the signal variance term may be more due to the poor quality of our proxy than to any failure in the underlying theory. The test of Fechner’s Law is more straightforward. We postulate the two-way analysis of variance model for the number of thresholds in the moderate rate of inflation in each country k and each year t:
Under the null hypothesis of Fechner’s Law, the country and year effects uk and u, have zero mean. The results of the analysis are set out on table 5. The calculated F-statistics strongly reject the notion that the number of jnds in the moderate rate is constant across countries, and also, to a lesser degree, that the number is constant over time. A major reason for the large country effect is the fact that the Fechner constants
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for Denmark are markedly lower than for all other countries in the sample (see table 2). Removing the observations on Denmark does reduce the relevant F-statistics, as shown in the table. However, Fechner’s Law is still formally rejected as a description of our data.
Conclusion The main conclusions of our investigation of the psychophysics of inflation can be simply stated. Weber’s Law, in its original form, is clearly refuted. Just-noticeable-differences in the price level vary over time and across countries. These variations are, however, broadly in line with what would be predicted by signal detection theory, if consumers acted as ideal receivers of information about inflation. Fechner’s Law, that these variable thresholds represent increments in inflation which are regarded by consumers with equal seriousness, is not supported when the thresholds are compared with our estimates of the rates of inflation which consumers regard as moderate. These findings have implications for economic theory and for econometric practice. They suggest that the assumption, often made in economic theory, that, on average, agents efficiently use information on the variability of aggregate and relative prices in decision-making, is not unreasonable. While individuals may not be good intuitive statisticians, our results suggest that their aggregate behaviour does exhibit a rational response to inflation uncertainty. Finally, our findings suggest that the assumption, sometimes made in translating qualitative survey data into quantitative measures of inflation expectations (Carlson and Parkin 1975), that the jnd in inflation is constant, is untenable. Since the jnd is systematically variable, such measures are liable to be systematically in error.
References Bernoulli, D., 1738. Exposition of a new theory of measurement of risk (tr. 1954 from Latin). Econometrica 22, 23-25. Birdsall, T.G., 1955. ‘The theory of signal detectability’. In: H. Quaster (ed.), Information theory in psychology. Glencoe, IL: Free Press. Carlson, J.A., 1975. Are price expectations normally distributed? Journal of the American Statistical Association 70. 749-754.
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Carlson, J. and M. Parkin, 1975. Inflation expectations. Economica 42, 123-138. Coombs, C.H., R.M. Dawes and A. Tversky, 1970. Mathematical psychology: an elementary introduction. Englewood Cliffs, NJ: Prentice Hall. Fechner, G.T., 1860. Elemente der Psychophysik. [Elements of psychophysics, 1966.1 New York: Rinehart and Winston. Green, D.M. and J.A. Swets, 1974. Signal detection theory and psychophysics. Huntington, NY: Krieger. Kahneman, D., P. Slavic and A. Tversky, 1982. Judgement under uncertainty: heuristics and biases. Cambridge: Cambridge University Press. Lucas, R.E., Jr., 1973. Some international evidence on output-inflation tradeoffs. American Economic Review 63, 326-334. Muth, J.R., 1961. Rational expectations and the theory of price movements. Econometrica 29, 315-335. Nerlove, M., 1967. ‘Distributed lags and unobserved components in economic time series’. In: W. Fellner (ed.), Ten economic studies in the tradition of Irving Fisher. New York: Wiley. Pratt, J.W., D.A. Wise and R. Zeckhauser, 1979. Price differences in almost competitive markets. Quarterly Journal of Economics 93, 189-211. Simon, H., 1959. Theories of decision-making in economics and behavioural sciences. American Economic Review 69,253-283. Stevens, S.S., 1975. Psychophysics: introduction to its perceptual, neural and social prospects. New York: Wiley. Tanner, W.P. and J.A. Swets, 1954. A decision-making theory of visual detection. Psychological Review 61, 401-409. Weber, E.M., 1846. ‘Der Tastsinn und das Gemeingefihl’. In: R. Wagner (ed.), Handwijrterbuch der Physiologie, Vol. 3. Braunschweig: Vieweg.
Psychology
THE PSYCHOPHYSICS
R.A. BATCHELOR TheCity University Business Received
December
269
7 (1986) 269-290
OF INFLATION
School, London,
3, 1985; accepted
*
UK
May 12, 1986
This study examines whether perceptions of and attitudes towards inflation obey standard psychological laws. Survey data on eight countries are used to generate estimates of the threshold for perceptions of changes in inflation, and the rate of inflation which consumers consider moderate. Weber’s Law, that the thresholds are constant, is firmly rejected. Fechner’s conjecture, that these thresholds, though variable, nonetheless always and everywhere represent changes which are regarded as of equal importance by consumers, is also rejected, but only narrowly. The popular theory of signal detection, that perceptual thresholds depend systematically on the level and noisiness of inflation, is, however, supported.
Introduction This study brings aggregate survey-based empirical evidence to bear on the following question: do the perceptions and feelings of individuals concerning changes in the macroeconomic environment, in particular changes in the general price level, obey the same ‘laws’ which experimental psychologists have observed in the reactions of individuals to changes in physical stimuli? The question is interesting because subjective variables, perceptions and sensations, play an increasing role in economic analysis. Normative economics has long recognised the distinction between the objective and the subjective magnitudes of economic stimuli - between the * The ,author is indebted to the Commission of the European Communities for financial support of the research programme on Expectations, Inflation and Growth, of which this study forms a part; to John Carlson and the Krannert School of Management at Purdue University for hospitality during its writing; and to participants in seminars at Birkbeck College, and the Universities of Hull and York, for their helpful comments. Mailing address: R.A. Batchelor, Centre for Banking and International Finance, The City University Business School, Frobisher Crescent, Barbican, London ECZY 8HB, UK.
0167-4870/86/%3.50
0 1986, Elsevier Science
Publishers
B.V. (North-Holland)
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amount of income, say, and the utility derived from that income. In recent years, positive economics has also come to be concerned as much with subjective as objective variables: with perceived and expected inflation as well as actual inflation; with permanent income as well as actual income. While such theories are potentially very insightful, their validation is difficult. The econometrician can most conveniently observe the objective magnitudes of variables such as inflation and income, whereas the individual can only act on the basis of their subjective magnitudes. In order to connect the subjective world of economic agents with the objective reality of official statistics, economists have been obliged to make strong and often not wholly explicit assumptions about how feelings arise, how perceptions are formed and how decisions are taken. In normative economics, for example, it has long been assumed that the subjective magnitude of an economic stimulus is related to its objective size through a particular law, of diminishing marginal utility, a law which finds support in empirical observation. In positive economics, it has become conventional to assume that in forming perceptions and expectations individuals employ optimal inferential and predictive techniques. Such assumptions are, for example, implicit in the ‘rational expectations’ approach to modelling price level expectations (Muth 1961), relative price perceptions (Lucas 1973), and permanent income (Nerlove 1967). These assumptions are less well grounded in empirical observation, and are made more from a desire to maintain consistency between what is assumed about decisions concerning the use of information, and the axioms of rationality normally applied to decisions concerning the use of resources. This central contention of economic theory, that individuals are good intuitive statisticians, has been strongly disputed by, for example, Simon (1959) and Kahneman et al. (1982). There are two reasons for thinking that the concepts and language of experimental psychology might help in establishing the validity of these assumptions about economic behaviour. One is that the establishment of law-like relations between the objective magnitudes of stimuli and their perceived, subjective, magnitudes has been a central concern of psychologists for over a century. In particular, early in the history of psychology, two ‘laws’ were proposed, which seemed to link objective with subjective magnitudes across a wide range of different physical stimuli. The first was due to Ernst Weber (1846) who suggested, on the basis of experimental evidence,
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that the subjectively just-noticeable-difference ( jnd) in the level of any stimulus was not constant, but rather proportionate to the initial objective level of the stimulus. The second was due to Gustav Fechner (1860) who suggested, on the basis of the more metaphysical speculation that jnds in stimuli must be considered equal in terms of subjective magnitude, that the subjective magnitude of any stimulus should be measured not in objective, physical, units, but in terms of the number of jnds it contained. These laws make a natural starting point for thinking about how perceptions of economic stimuli, and the sensations they produce, are formed. The second reason for supposing that such an approach might be productive is that both economics and psychology now employ the common language of statistical decision theory. Tanner and Swets (1954) showed persuasively that the jnd in perceptions should be regarded not as some physiologically determined parameter, but rather as a decision variable, the size of which is determined by the size and variability of the stimulus, the noisiness of the environment and the relative costs of failing to detect changes and incorrectly anticipating change. This theory of signal detection contains as a special case - the so-called ‘ideal receiver’ - the optimal forecasting assumptions currently popular in macroeconomics. The investigation of the analogy between economic and physical stimuli below falls into three sections. In the first section, a formal statement of Weber’s Law and Fechner’s Law is developed in the context of inflation perceptions. In the second section a technique is proposed by which aggregate qualitative survey data on perceptions and expectations in eight European Community countries in the period 1974-82 can be used to derive quantitative measures of jnds in the price level, and of rates of inflation which consumers in different countries consider ‘moderate’. The third section reports results from empirical tests of the predictions of Weber’s Law (strictly signal detection theory) and Fechner’s Law, based on this data. One key prediction of signal detection theory - that the jnd threshold should rise as uncertainty about the rate of inflation increases, is consistently borne out by the data. However, Fechner’s Law, which predicts that the moderate rate of inflation, while differing from country to country, and year to year, should nonetheless always represent the same number of just-noticeable-steps in inflation, is not strongly supported. To summarise, we cannot reject the hypothesis that on average over a large
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number of individuals, consumers process information about inflation in a statistically optimal way. We can, however, just reject the idea that the resulting perceptual thresholds can be used normatively, as measures of the seriousness of inflation. The psychophysical
laws
In order to focus the discussion, consider the following situation. A consumer starts his year subject to the ‘stimulus’ of a price level P. During the year, he visits a number of ‘markets’ or information sources which - together with any prior information on inflation - help him form an estimate Ap of the change in the price level stimulus during the year. At the end of the year he is asked to report _ whether any change has occurred; - how ‘important’ to him the change has seemed. Weber’s Law gives some guidance as to how the consumer might answer the first question. The empirical regularity which Weber noted in his experiments can be summarised as saying that the consumer will only report that a change has occurred if the mean observed change Ap exceeds the just-noticeable-difference in price Aj P, where AjP=GP.
0)
The law states that for any individual, and for any stimulus, the jnd in the stimulus is a fixed percentage of the level of the stimulus. The parameter 6 is to be interpreted as a measure of the individual’s sensitivity to the stimulus. It will vary across individuals according to their powers of discrimination, and will for any one individual vary according to the sensory mode involved. Fechner’s Law gives some guidance as to how the consumer might answer the second question, concerning the ‘sensation’ that the perceived -price change produces. It starts from the assumption that all just-noticeable-differences in perception represent equal chtnges in sensation, or subjective magnitude. An observed change of AP in the price level will therefore represent a change of AM units of subjective magnitude, where AM=b-AP/AjP,
(2)
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and djp is the average size of a jnd in price in the range P, P + A$. The constant b simply translates the number of jnds in this interval into a corresponding number of units of subjective magnitude, each just-noticeable-change in P resulting in a change of b units on the sensation scale. If both Weber’s Law and Fechner’s Law hold good, then for a small change in P, dP say, the corresponding change in sensation can be written dM=
b- dP/SP.
Integration M=i
(3)
of this yields
In P+c,
(4)
where c is a constant of integration. Thus the psychophysical laws predict that the subjective magnitude of any stimulus is proportionate to the logarithm of its objective magnitude. There is at least prima facie evidence that such psychophysical laws operate in economics. The just-noticeable-difference in price, for example, is indeed likely to be a function of the price level. In shopping for low value items, we assiduously inspect penny differences in the prices of different products. In purchasing high value items - motor cars, houses - we will happily bargain in steps of tens or hundreds of pounds, and sign large bills without inspecting the last few figures too closely. This commonplace observation received formal support in a survey by Pratt et al. (1979) of differences across neighbouring stores in the selling prices of a number of standard items. Their data show a strong and virtually proportionate relation between the standard deviation of prices, and the mean price for each item. The coexistence of such prices in competitive stores must mean that the price difference which buyers regard as effectively zero is, as Weber’s Law predicts, higher in the case of high-priced items than in the case of low-priced items. It is more difficult to adduce evidence on Fechner’s Law. However, the idea of diminishing marginal utility, a subjective measure of magnitude, from increased wealth is a standard assumption in microeconomits. It is a property of the semi-log function (4) that the marginal
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change in subjective magnitude will fall as the level of stimulus is increased; and, indeed, precisely such a function was proposed by Bernoulli (1738) as a description of the marginal utility of money, in his famous exposition of Cramer’s resolution of the St. Petersburg paradox. Weber’s Law has been substantially modified and reinterpreted as a result of experimental evidence and theoretical developments over the past century. The experimental evidence has generally pointed to the need to modify the functional form of the law at extreme stimulus values. Changes in the conceptual basis of the law, resulting from the development of the theory of signal detection by Birdsall (1955), and Tanner and Swets (1954), have, however, been of more profound significance. In most early theory, the perceptual threshold was regarded as the product of innate physiological factors in sight, hearing, length and colour judgment, sensitivity to pain, and so on. Tanner and Swets suggested that it would be better regarded as a decision criterion, when it should depend not only on random physiological factors, but also on such systematic considerations as the difficulty of disentangling the stimulus change (the signal) from other events (the noise), and indeed on the costs of failing to perceive such stimuli. The principles are those of statistical decision theory, and their application in psychology is fully discussed in Green and Swets (1974), and more succinctly in Coombs et al. (1970: ch. 6). The principles can be simply illustrated if we consider the conditions under which our representative consumer will find it worth reporting that the general price level has risen. It will help if we start by expressing actual and perceived changes in percentage terms, and give some stochastic structure to these inflation rates. Specifically, assume that the general rate of inflation p and the rates of inflation in individual markets i are distributed as p=p+v, pi=P
v+
wi,
wi -
iv@, Tu’),
(5)
N(0, Tw’).
(6)
Eq. (5) suggests inflation fluctuates randomly around a mean rate j. The consumer’s problem is to disentangle the inflationary ‘signal’ p from a sample of market inflation rates pi which differ from the general inflation rate by relative price disturbances wi. If the consumer
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215
visits m markets his observed average price change A? will by the Central Limit Theorem correspond to a mean perceived inflation rate of 1, where $=p+w,
w-
$0, rn’),
(7)
and rn2= 7:/m represents the ‘noise’ surrounding the consumer’s observation on p. When the consumer is asked the question - has any change occurred in the price level? - the theory of signal detection assumes that he responds as if conducting a formal test of the hypothesis p = 0. In other words, he tests whether i, has been drawn from the noise-only distribution fi = w against the alternative that it is drawn from the signal-plus-noise distribution $ =p + w. The distribution of p under the hypothesis that p = 0 is N(0, 7:); its distribution under the alternative that p = j, say, is N( j, r,,2+ 7,‘). These two distributions are shown on fig. 1. The general form of optimal decision rule for this type of problem is well known (see, for example, Green and Swets 1974: ch. 1). The consumer will choose a critical value 6, and the hypothesis p = 0 will be accepted or rejected according as j 2 8. The threshold 6 is chosen so as to minimise the expected disutility of Type I (‘false alarm’) and Type II (‘miss’) errors. The probabilities of making such errors are indicated by the shaded areas I and II in fig. 1. The threshold depends both on factors determining these probabilities and on the relative costs of the two types of error. To be precise,
s’s($, :,, i”, T),
(8)
where y is the ‘relative importance’ of Type II vis-a-vis Type I errors. The rationale for the signs of the various effects is as follows. For any given value of 6, a rise in the expected value of inflation p will reduce the probability of Type II errors; hence 6 will be raised to restore the original balance between Type I and Type II errors. Intuitively, if when inflation occurs it is very rapid, the consumer will need to observe a higher rate i before he is persuaded that inflation is indeed non-zero. A rise in the noise variance will increase the probabilities of both types of error but will have a relatively greater impact on Type I errors, and
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Fig. 1. Distributions of perceived inflation under Key: p = inflation; j = perceived inflation; p = mean inflation; 7,’ = variance of inflation; 7,” = variance of inflationary ‘noise’; S = critical value for testing p = 0 against IT = moderate rate of inflation.
of inflation
noise and signal-plus-noise
hypotheses.
p = p;
so require an increase in 6 to restore balance. In our scenario above, noise arises from relative price variations wi; the greater these are, the more evidence the consumer needs in the form of a high perceived rate j, before he is persuaded that he has not simply sampled an unusually inflationary set of goods, but that a general inflation is occurring. A rise in signal variance increases the probability of Type II errors, and so requires a downward revision in S; if the general rate if inflation itself is highly variable then the consumer will be quite ready to accept that whatever rate of inflation he perceives, even if it is rather low, is representative of what is happening to the price level in general. Finally, any increase in the costs of unanticipated inflation will penalise the making of Type II errors, and so cause the threshold S to be reduced. Eq. (8) defines necessary qualitative characteristics of the jnd for a
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consumer who is, in psychophysical parlance, an ‘ideal receiver’ of information about inflation, or in economic terms is forming ‘rational expectations’ about general price inflation on the basis of limited information about relative prices. Our test of the validity of this extension of Weber’s Law consists of establishing whether the predicted pattern of relationships between 6 and its determinants is observed in practice. Just as Weber’s Law has been subject to significant revision and extension in the past century, so also has the status of Fechner’s Law changed as a result of developments in psychological scaling techniques. Two types of technique have been developed to assess subjective magnitude. One is based on category scaling. The subject is presented with a stimulus, say a sound of a particular volume, and is asked to assign it to one of a ranked series of categories - very soft, soft, medium, loud, very loud. The other technique is based on magnitude estimation. The subject is asked to assign numbers to the perceived intensity of a series of stimuli with differing objective magnitudes. Both types of measure have been used to test whether the semi-logarithmic relation (4) exists between subjective magnitude (position on the scale) and stimulus size. Most studies based on category scaling show the predicted relation between subjective magnitude (position on the scale) and the logarithm of stimulus size. Studies based on the more convincing magnitude estimation technique, on the other hand, show very mixed results (see Stevens 1975), and the semi-logarithmic relationship (4) has been seriously undermined as a description of how subjective magnitude varies with stimulus size. However, there is a serious conceptual problem with these tests, viewed as tests of Fechner’s Law (2), since they depend on the joint validity of both Fechner’s Law and the original form (1) of Weber’s Law. As we have seen, the theory of signal detection suggests that perceptual thresholds are systematically variable, in which case the slope coefficient in (4) could not be expected to be constant, even if Fechner’s Law were true. A more direct test of (2) is simply to check whether the number of jnds necessary to produce a given change in sensation is indeed constant. Our procedure is to generate, for a number of countries k, and a number of years r, an. estimate of what rate of inflation is considered ‘moderate’ on average across consumers. If this is denoted rkr, then application of Fechner’s Law would suggest that this should
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278
everywhere and always represent sensory magnitude, so that T/J&~
= A M/b
Psychophysics
of inflation
a change of A@ units on the scale of
= 9,
where skt is the average jnd in inflation in the range (0, rkTTkr), and 0 is the fixed number of jnds in the moderate rate of inflation. Our test of Fechner’s Law is therefore a test of the invariance of 0 across countries and over time.
The measurement
of perceptions
In order to implement the tests described above, some means must be found of measuring the various subjective concepts involved. Normally, these measurements would be drawn directly from experiments in which individuals are presented with a range of stimuli and asked to discriminate between them, and to assign them some measure of subjective magnitude. In the present study of price level perceptions, the measures are instead made indirectly, on the basis of the aggregate responses of large numbers of individuals to a series of questionnaires about past and projected price changes. This section of the paper describes the quantification technique in some detail. The questionnaires involved have been distributed thrice-yearly to samples of between 1,000 and 5,000 consumers in each of eight member countries of the European Community, in the years 1974-82. These harmonised consumer surveys ask the following questions concerning perceptions of, and expectations about, the general price level: - Compared to what they were 12 months ago, do you think that prices in general are now: lower/about the same/a little higher/ moderately higher/ much higher/ don’t know? - By comparison with what is happening now, do you consider that in the next 12 months prices will: fall slightly/be stable/increase at a sloer rate/increase at the same rate/increase more rapidly/don’t know? The results of the questionnaire are summarised into five response the percentage of proportions (say, A, B, C, D, E) representing respondents (excluding don’t-knows) answering in each possible re-
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of inflation
sponse category. Since the two questions are similar in structure, we describe in detail only how the response proportions for the first question, on perceptions, can be used to generate measures of the jnd in price, and an estimate what is meant by a ‘moderately higher’ price level. The technique is a generalisation to a five-category tendency survey of the method developed by Carlson and Parkin (1975) to obtain quantitative measures of the mean and standard deviation of expectations from a simpler, three-category survey in which respondents could only indicate whether prices were rising, falling, or remaining unchanged. The technique relies on the following three assumptions: (1) The perceptions distribution. Individual uted across consumers as +( p, a*).
perceptions
jjj are distrib-
In the particular case of market sampling in an environment with relative price changes described by (5)-(7), + will be normal, with /.A=j, a2=7,2+ r2n . For simplicity, we have assumed 9 has a logistic, rather than a normal, cumulant. (2) Weber’s Law. Individuals will report that no change in level has occurred if jj is less than some threshold 8,. individuals will report that prices are ‘moderately higher’ within some threshold k&j around the rate of inflation they consider moderate. For simplicity, sumers. (3) Unbiasedness. are unbiased,
6, E and
7~ have been
In any survey, perceptions so that p =p.
assumed
equal
the price Similarly, if jj lies rj which
for all con-
of past rates of inflation
The perceptions distribution and the jnd bands + 6 and r + E are shown on fig. 2. Under the above assumptions, the sample response proportions A, B, C, D and E are maximum likelihood estimates of the areas under the distribution in the ranges (2 cc, -S), (-6, +S), (6, 7~- E), (7~ E, 7~+ E) and (7~ + E, cc), respectively. We denote by a, b, c and d the abscissae of the perceptions distribution corresponding to cumulative
R.A. Batchelor/ Psychophysics of infiaiion
280
-6
I
+6
u
n
--E
tl
n
4-E
Fig. 2. The distribution of mean inflation perceptions. Key: j = mean perceived inflation; p= average perceived inflation in sample survey; a = S. D. of perceived inflation in sample survey; n= moderate rate of inflation; 6 = jnd in inflation around zero; e = jnd in inflation around r.
probabilities of A, A + B, A + B + C and A + B + C + D. The benefit of assuming + has a logistic cumulant is that these abscissae are simply the log-odds statistics a = ln[A/(l - A)], b = ln[( A + B)/(l - A - B)] etc. It is a straightforward matter to show that p=7r-(a+b)/e,
S=7r*(a-b)/e, e=57-(c-d)/e,
00)
where e = (a + b) - (c + d). The system (10) defines the mean and standard deviation of the perceptions distribution, and the two thresholds, in terms of sample-based statistics a, b, c and d, together
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of inflation
with the as yet undefined moderate rate 7~. However, assumption requires that p = p, so that
the unbiasedness
m=p*e/(a+b), and this can be used to scale all of the expressions in (10). An estimate of the Fechner constant for the moderate inflation, the number of jnds it contains, can be obtained as
rate
of
e = T/( ihy2, =
e/[(a - b)(c - d)y2.
(12)
Note that the Fechner constant 8 is independent of 7~. We have used this procedure to generate estimates of mean perceived inflation, the perceptual thresholds S and E, and the Fechner constant 6 for each survey conducted in each of eight member countries of the European Community, starting with the May 1974 survey and ending in May 1982. An exactly analogous procedure can be used to generate estimates of mean expected inflation $ over the following year, the dispersion of these expectations across individuals, u’, and thresholds 6’ and E’ around a zero inflation rate, and around the perceived prior inflation rate p. Since the reference point for the expectations question is the ‘same rate’ of inflation, the perceived recent rate of inflation p takes over the role of the moderate rate in the system (lo), and this can be solved directly for $, u’, S’ and E’. To give a flavour of the large data set produced by this technique, tables l-3 present some summary statistics on the objective size and variability of inflation in the eight European countries surveyed, and on the subjective perceptions and expectations of consumers in these countries. Table 1 displays the average annual rate of consumer price inflation in each country, measured for the year ending in January, May and October, the data starting in May 1974 and ending in May 1982. Our set of eight countries can be seen to cover a wide range of inflationary experience , from as low as 5 percent in Germany to over 17 percent in Italy. The time period covered also contains marked fluctuations in inflation within each country with all countries experiencing peak rates in 1974-5 and 1979-80 associated with two major oil price hikes.
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Table 1 Mean and standard
deviation
of consumer
Country
price inflation
Inflation
Belgium Denmark France Germany Ireland Italy The Netherlands United Kingdom
of inflation
in eight European
economies,
1974-82.
in January,
May and
a
Mean
SD.
8.2 11.0 11.6 4.8 16.4 17.2 7.1 15.4
3.4 2.8 2.1 1.4 4.5 3.8 2.1 5.1
a Percentage changes in the consumer price index over twelve months October each year. Data starts in May 1974, ends in May 1982. Source: Eurostat.
ending
Table 2 lists average values for the mean and dispersion of inflation perceptions, the moderate rate of inflation, the perceptual thresholds 6 and E around zero and the moderate inflation rate, and thenumber 8 of just-noticeable-differences in price which go to make up the moderate inflation rate. Perceptions are by definition unbiased. The dispersion of perceptions across individuals, in our earlier theory a function of the sum of the variance of inflation over time and the variability of relative price changes, shows the expected correlation with the standard
Table 2 Perceived
consumer
Country
Belgium Denmark France Germany Ireland Italy The Netherlands United Kingdom
price inflation:
summary
statistics.
Mean value a of P
0
n
s
e
8
8.1 11.0 11.7 4.8 16.3 17.1 7.0 15.3
3.2 5.0 4.6 2.3 6.7 6.7 2.8 7.1
4.4 10.0 7.8 5.2 8.8 10.3 4.5 13.8
0.7 2.7 1.3 1.0 1.6 1.3 0.7 2.4
1.6 2.4 2.1 1.2 1.9 4.1 1.9 4.0
4.8 4.0 5.2 4.9 5.4 4.7 4.5 4.6
a Average value of each parameter, per annum.
over 25 surveys
in the period
May 1974-May
1982: percent
R.A. Batchelor/ Table 3 Expected
consumer
Country
price inflation:
Psychophysics
summary
283
of inflation
statistics.
Mean value a of
8’
a’
6’
E’
7.0 9.5 10.4 4.5 18.7 19.0 7.4 13.7
3.4 4.8 5.0 2.3 9.4 9.7 3.3 7.8
1.9 3.1 3.0 1.1 3.9 5.2 1.5 2.9
2.5 2.5 4.3 2.2 4.5 5.5 2.7 4.0
a Average value of each parameter, per annum.
over 25 surveys
Belgium Denmark France Germany Ireland Italy The Netherlands United Kingdom
in the period
May 1974-May
1982: percent
deviation of inflation. On the other hand, the ratio of the cross-sectional dispersion of perceptions to the variance of inflation - an index of the amount of noise relative to signal variation - is definitely higher in some countries (France, Denmark, Italy, Germany) than others (Belgium, The Netherlands, United Kingdom, Ireland). The moderate rate of inflation is clearly correlated across countries with the rate of inflation actually achieved, but it lies below actual rates for all countries except Germany, the least inflationary. The thresholds for reporting price differences are smaller when a zero price change is the standard than they are when the actual price change is taken to be this moderate rate of inflation. In other words, given the same evidence, consumers are more inclined to reject the hypothesis that inflation has been zero than they are to reject the hypotheses that it has been moderate. Finally, there seem to be roughly 4-S; perceptual thresholds contained in the moderate inflation rate in all countries, giving at least prima facie credence to Fechner’s Law. Table 3 lists similar statistics relating to inflation expectations. Although it is by no means guaranteed by the quantification method, expectations do lie on average quite close to outturns. The dispersion of expectations across individuals is correlated with the mean and variability of inflation; it is also correlated with, and larger than, the dispersion of perceptions of earlier inflation - consumers are, as we would hope, more at odds over what will happen than over what has happened. Finally, the thresholds 6’ and E’ also vary in size according
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of injiotion
to the inflationary history of the economy and, in general, again the threshold around zero is lower than the threshold around the perceived inflation rate. On the same evidence, consumers are more willing to reject the proposition that inflation will be zero than that inflation will continue at its previous rate. The thresholds for reporting expectations are also larger - almost double in most cases - than the thresholds for reporting changes in perceptions. Consumers are more inclined to reject hypotheses about past inflation than they are to reject hypotheses about the future. This is what one would expect, if the thresholds are indeed to be interpreted as critical values, given that consumers are inevitably more uncertain about the future than about the past. All of the measures of perceptions and expectations reported in tables 2 and 3 are conditional on the assumptions of the quantification procedure. Before using the data in formal tests, it is therefore worth considering the validity of these assumptions, and consequences of their relaxation. As a description of individual differences in perceptions, the logistic distribution is very similar to the theoretically indicated normal distribution; and both are for practical purposes not much different from that observed in reality by Carlson (1975). The assumption that all individuals use the same thresholds is clearly unrealistic. However, making these thresholds stochastic, with variance p*, say, can be shown to affect only the estimator for a* (deflating it by a*/( a* + p*)). Finally, the unbiasedness assumption in each survey is clearly very strong. An alternative is to make the weaker assumption that perceptions are unbiased only on average over a series of T surveys, so that &.J = C,p. This more plausible condition can be imposed by selecting an appropriate value for r which is, less plausibly, constant over time, and using this value to scale the thresholds in (10). To establish the sensitivity of our results to the unbiasedness assumption, we have carried out our statistical tests of Weber’s Law using both the data summarised on tables 2 and 3, and an alternative data set based on the assumption of a constant moderate rate. No significant qualitative differences emerged between results for the two data sets, so only the first group of results is reported below. Tests of Fechner’s Law are not affected by the validity of the unbiasedness assumption since, as we have seen from expression (12) the Fechner constant is independent of V.
R.A. Batchelor / Psychophysics
Testing the psychophysical
of inflation
285
laws
We look first at tests of the signal detection theory of perceptual thresholds based on estimates of the parameters of (8), then at tests of Fechner’s Law based on tests of the stability of the number 8 of jnds in the moderate rate of inflation. Given our data on perceptual thresholds, three steps are necessary to implement an empirical test of the model of these thresholds represented by eq. (8). First, a functional form must be chosen; second, empirical counterparts to the signal, noise variance, signal variance, and costs of inflation variables must be found; third, an appropriate estimation procedure must be selected. We have considered the linear model 6 = a, + alp + a,n, + a37ru+ a,y + u,
03)
where u is a normally distributed, zero-mean, serially uncorrelated error term. The dependent variable in (8) is the threshold of inflation perceptions around zero. Our data contain figures for three other thresholds E, S’ and E’ - and we have attempted to model all of these using variants of the above equation. The first determining variable in (13), is, is the signal size. A natural measure of this is the absolute difference between mean perceptions or expectations of inflation and the reference value around which the threshold under scrutiny operates. Hence, in the model for 6, the signal size is taken to be ( /A- 0 I; for E, the signal is 1p - T I; for a’, the signal is 1~’ 7 0 I; and for E’, ( p’ - p I. The second variable in (13), the noise standard deviation, can be measured by the dispersion of perceptions across individuals. The consumer surveys in effect sample from the sampling distribution of mean perceptions, so that, from (7), a* = r2 = 7,‘. The signal standard deviation r0 has been proxied for each survey date by the standard deviations of our various signal measures within a five-period window centred on that survey date. The final determinant of threshold size is the relative cost of missing the inflation signal. This has been proxied by the signal size, on the grounds that the costs of failing to adjust to inflationary changes are likely to be greater, the larger is the change. The problem with this
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Psychophysics
of inflation
procedure is that the coefficient on signal size in (13) now becomes sign, since a, > 0, a4 < 0. A number a43 which is of indeterminate of other factors are likely to determine the costs of inflation in any particular country. For example, if a country has a highly indexed wage-bargaining or savings structure, perceptual thresholds might be expected to be high, since the costs of failing to notice inflation will be relatively small. It is, however, difficult to quantify the degree of indexation in different economies, and such variables do not enter our model explicitly. Threshold models based on (13) have been estimated in two ways. First, the time series data from all eight countries have been pooled, and the parameters a,, CI*, a3, and a4 initially estimated by applying ordinary least squares to this pooled data. However, all models showed serial correlation in residuals, and they have been re-estimated by the Cochrane-Orcutt procedure. In order to allow for the possibility that costs of inflation may differ across countries, the models include seven country dummies in place of the single constant term a,. Table 4 summarises the results of estimating models based on this pooled data. All of the models reveal a negative relationship between signal size and perceptual threshold. This is consistent with signal detection theory only if the costs of missing inflation rise as the rate of inflation increases (or departs further from moderate or past values). All of the models show a strong positive relationship between the thresholds and the noise variance, in conformity with the theoretical model. However, although three of the four thresholds show the predicted negative relationship with the signal variance, only in the case of E’ is this statistically significant; and the model for 8 yields an (insignificant) positive coefficient. Inspection of the country dummies shows considerable country-tocountry variations in thresholds not accounted for by signal and noise considerations. There does not, however, appear to be any systematic tendency for thresholds in highly indexed economies - Italy, Belgium to be higher than would otherwise by expected. Pooling the data does involve imposing the restriction that coefficients a,, u2, u3, and u4 are the same for consumers in all countries. These restrictions are just rejected by a formal F-test. The second step in our testing of signal detection theory is therefore to estimate (13) using time series data on each country separately. Rather than list all a1 +
R.A. Batchelor/ Psychophysics of inflation
287
Table 4 Determinants of thresholds in inflation perceptions and expectations: pooled data. Coefficient a on Belgium Denmark France Germany Ireland Italy The Netherlands United Kingdom Signal Noise variance Signal variance Standard error P
Threshold 8
6’
E 0.05 (0.61) 0.39 (4.71) 0.10 (1.22) 0.07
-0.01 (0.10) - 0.05 (0.92) -0.10 (1.76) 0.00
E’
0.18 (1.88)
0.39 (4.08)
(14.54) 0.03 (0.74)
0.15 (2.41) 0.12 (2.29) 0.00 (- ) 0.06 (4.52) 0.50 (29.89) -0.06 (1.41)
0.33 (3.38) 0.0s (0.81) 0.09 (0.76) 0.40 (3.49) 0.09 (1.03) 0.00 (- ) -0.12 (4.94) 0.63 (11.42) - 0.02 (0.43)
0.34 0.46
0.23 0.65
0.42 0.43
(0.89) - 0.02
(0.23) 0.02 (0.18) 0.04 (0.55) 0.00 (- ) - 0.26 (9.90) 0.87
(0.08)
-0.51 (a.021
0.24 (2.90) 0.04 (0.53) 0.52 (6.09) 0.28 (3.50) 0.07 (0.76) 0.21 (2.22) 0.21 (3.28) 0.00 (- ) -0.12 (5.33) 0.51 (23.31) -0.13 (1.75) 0.36 0.77
a Regression by OLS with Cochrane-Orcutt correction for serial correlation. Figures in parentheses beneath estimated coefficients are t-statistics. p is the estimated first-order serial correlation coefficient.
32 (8 countries x 4 jnd thresholds) regressions, it may be sufficient to indicate the role played by each determining variable. In 19 out of the 32 models, signal size entered significantly, and with a negative sign; in only 2 cases was it significant and positive. In 31 out of the 32 models, the noise variance appeared to have a significant and positive effect on the jnd, a very striking result. However, in only 8 cases did the signal variance exercise a significant negative effect on threshold size, in all other cases being statistically insignificant. These results, taken as a whole, are not inconsistent with the signal detection variant of Weber’s Law. However, it is important to recognise the econometric limitations of our tests. The presence of serial corre-
288
R.A. Batchelor / Psychophysics
of inflation
Table 5 Analysis of variance in the ‘Fechner constant’. Source of
Degrees of freedom
Sum of squares
Mean square
F
VariaIlCe
All countries country Year Residual
7 24 168
31.20 58.30 123.70
4.46 2.43 0.74
6.06 B 3.30 a
Total
199
213.19
2.45 2.79 0.79
3.08 b 3.52 b
Ail countries excluding Denmark Country 6 Year 24 Residual 144
14.67 67.06 114.31
Total
196.04
174
a 5% critical levels are 2.01 for country effects, 1.52 for year effects; b 5% critical levels are 2.10 for country effects, 1.52 for year effects.
lation in 10 out of the 32 country models point to the possibility that some important variable has been omitted from these models. The fact that the jnd measures, and the signal and noise measures, are all functions of the same set of sample response proportions through (10) means that spurious correlation may be introduced into (13) although detailed investigation indicated that the size of this should in practice be rather small. Finally, the relatively poor showing of the signal variance term may be more due to the poor quality of our proxy than to any failure in the underlying theory. The test of Fechner’s Law is more straightforward. We postulate the two-way analysis of variance model for the number of thresholds in the moderate rate of inflation in each country k and each year t:
Under the null hypothesis of Fechner’s Law, the country and year effects uk and u, have zero mean. The results of the analysis are set out on table 5. The calculated F-statistics strongly reject the notion that the number of jnds in the moderate rate is constant across countries, and also, to a lesser degree, that the number is constant over time. A major reason for the large country effect is the fact that the Fechner constants
R.A. Batchelor/
Psychophysics
of inflation
289
for Denmark are markedly lower than for all other countries in the sample (see table 2). Removing the observations on Denmark does reduce the relevant F-statistics, as shown in the table. However, Fechner’s Law is still formally rejected as a description of our data.
Conclusion The main conclusions of our investigation of the psychophysics of inflation can be simply stated. Weber’s Law, in its original form, is clearly refuted. Just-noticeable-differences in the price level vary over time and across countries. These variations are, however, broadly in line with what would be predicted by signal detection theory, if consumers acted as ideal receivers of information about inflation. Fechner’s Law, that these variable thresholds represent increments in inflation which are regarded by consumers with equal seriousness, is not supported when the thresholds are compared with our estimates of the rates of inflation which consumers regard as moderate. These findings have implications for economic theory and for econometric practice. They suggest that the assumption, often made in economic theory, that, on average, agents efficiently use information on the variability of aggregate and relative prices in decision-making, is not unreasonable. While individuals may not be good intuitive statisticians, our results suggest that their aggregate behaviour does exhibit a rational response to inflation uncertainty. Finally, our findings suggest that the assumption, sometimes made in translating qualitative survey data into quantitative measures of inflation expectations (Carlson and Parkin 1975), that the jnd in inflation is constant, is untenable. Since the jnd is systematically variable, such measures are liable to be systematically in error.
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Carlson, J. and M. Parkin, 1975. Inflation expectations. Economica 42, 123-138. Coombs, C.H., R.M. Dawes and A. Tversky, 1970. Mathematical psychology: an elementary introduction. Englewood Cliffs, NJ: Prentice Hall. Fechner, G.T., 1860. Elemente der Psychophysik. [Elements of psychophysics, 1966.1 New York: Rinehart and Winston. Green, D.M. and J.A. Swets, 1974. Signal detection theory and psychophysics. Huntington, NY: Krieger. Kahneman, D., P. Slavic and A. Tversky, 1982. Judgement under uncertainty: heuristics and biases. Cambridge: Cambridge University Press. Lucas, R.E., Jr., 1973. Some international evidence on output-inflation tradeoffs. American Economic Review 63, 326-334. Muth, J.R., 1961. Rational expectations and the theory of price movements. Econometrica 29, 315-335. Nerlove, M., 1967. ‘Distributed lags and unobserved components in economic time series’. In: W. Fellner (ed.), Ten economic studies in the tradition of Irving Fisher. New York: Wiley. Pratt, J.W., D.A. Wise and R. Zeckhauser, 1979. Price differences in almost competitive markets. Quarterly Journal of Economics 93, 189-211. Simon, H., 1959. Theories of decision-making in economics and behavioural sciences. American Economic Review 69,253-283. Stevens, S.S., 1975. Psychophysics: introduction to its perceptual, neural and social prospects. New York: Wiley. Tanner, W.P. and J.A. Swets, 1954. A decision-making theory of visual detection. Psychological Review 61, 401-409. Weber, E.M., 1846. ‘Der Tastsinn und das Gemeingefihl’. In: R. Wagner (ed.), Handwijrterbuch der Physiologie, Vol. 3. Braunschweig: Vieweg.