175

MODELLING ASYMMETRIC CONSUMER RESPONSES, WITH AN EXAMPLE*

Trevor Young University of Manchester

This paper attempts to provide a method by which differing consumer responses to up ward and down ward price movements may be accommodated in applied work. Three reoso~~s for asymmetry in demand are dis-ed-addiction to the product, addiction to competing products and limited information- and appropriate forms of the estimating equation are considered. An empirical euercire provides evidence of asymmetry in response of UK consumers of instant coffee.

The notion that the demand curve for a product may be irreversible dates at least to Marshall but empirical investigation of the idea did not begin before the work of Farrell (1952). Since then several models of dynamic consumer behaviour have been formulated (e.g. Houthakker and Taylor (1970). Stone and Rowe (1957)) but most of these have concentrated on the modelling of stock adjustment by the consumer: few have attempted to isolate directly the nature of the asymmetry of consumer response to changes in market forces. This note presents the results of some work on demand utilising recent techniques of price decomposition which accommodate the possibility of differing responses to upward and downward price movements.

Marshall (1927, p. 807) in his discussion of the limitations of static demand assumptions introduced asymmetry in the following way:

“. . . this theory is out of touch with real conditions of life, in so far as it assumes that, if the normal production of a commodity increases and afterwards again diminishes to its old amount, the demand price and the supply price will return to their old positions for that amount”.

“. . . habits which have once grown up around the use of a commodity while its price is low are not quickly abandoned when its price rises again”.

More recently the influence of habit formation on the form of the demand curve has been discussed by Scitovsky (1976,1978). Terming the phenomenon “addiction-asymmetry”, he argued that, although it would be undoubtedly a feature of the consumption of products such as drugs, cigarettes and alcohol, for which both a physiological and psychological dependency may arise, such asymmetry might be “pretty nearly universal”. A consumer may become attached to any aspect of a higher standard of living once he has experienced it. Graphically the demand curve would appear as in Figure la. Given that the consumer fmds himself at point a, he would move up the arm ab if the price of the produce rises, but down the more elastic segment uc if price falls. * The author is indebted to Bruce Trail1 and his other coileagucs at Manchcsta, as wdl

He went on to argue that adaptation to a change in price is gradual and that

as to an anonymous referee. for helpful comments.

176 TREVOR YOUNG

la LQ l b

Fipm 1 Types of Asymmetric Consumer R a p o a ~ c ~

The other form of asymmetry which may be relevant in particular instances is illustrated in Figure lb. Here the kink in the demand curve is in the opposite direction to that of the previous example, i.e. the consumer’s reaction to price falls is relatively more inelastic than his response to price rises. The demand curve might take this form for one of two reasons. Firstly, the argument could go as follows: as the price of a product rises, some customers, switching to the consumption of substitute goods, will leave the market altogether. To the extent that these customers establish new consumption patterns centred on the substitute products, they may not be wooed back easily to the original market, if the price in the latter were to fall again. Hence the asymmetry of the demand curve would be rationalised as addiction to the competing products. An eFample might be provided by the substitution of non-meat protein sources for meat products during the period of high meat prices in the mid-1970’s; the continued widespread presence of novel protein foods such as “textured protein” and “meat extenders” at the retail level suggests that some portion of this market might have been irredeemably lost to meat producers.

Scitovsky (1978) provides an alternative explanation. The market for a particular product may be thought to comprise two types of customers: those who already purchase the goods and those who do not purchase because the price is currently too high but who would enter the market at a lower price. However, whereas established customers gain information on market conditions whenever they go to make their routine purchases, potential customers will only be aware of price changes if the producer advertises them widely. So whereas a price fall will induce existing customers to buy more of the good, new customers will only enter the market if through advertising, the producer has made a special effort to solicit their patronage. A price fall which goes unadvertised will induce a smaller expansion in demand than a price decrease of which all consumers are fully informed. Hence the kd in the demand curve in Figure lb could be attributed to limited information in the marketplace*. For example a consumer who does not consume fish because it has become too expensive, no longer visits specialist fsh shops and may remain unaware of subsequent price changes.

Given these three reasons for asymmetric consumer response to price changes-addiction to the product, addiction to competing products, and

ScitOvsky (1978) terms this phenomenon “infonnational-asymmary”.

MODELLING ASYMMETRIC CONSUMER RESPONSES. WITH A N EXAMPLE 177

limited information-there remains the problem of modelling asymmetric demand functions in applied research. This is tackled in the next section.

Modelling Asymmetric Responses When attempting to model asymmetric demand functions, the crucial question will arise: how will the demand curve appear, if the consumer faces a series of price changes? One possibility is that the curve will be sawtoothed (Figures 2a and 2b). This is the type of consumer reaction implied by the method of price decomposition suggested by Wolffram (1971) and it may be described briefly as fo1lows:f

Figure 2 Wolffrnm-type Demand Curves

A simple symmetric demand curve, ignoring random disturbances might appear as:

(1) where Q is the quantity purchased at time t and PI is the price paid. The equivalent Wolffram-type irreversible demand curve would be:

(2)

where WR, is the sum of all period-to-period rises in 4 i.e.

QI = a, + b , PI

QI = a2 + b1 WR, + b, WF,

t

i = 1 WR, = Z A PRi, given

4 PRi = Pi - 4-1 ifpi > Pi-1 = o otherwise

and WF, is the sum of all period-to-period falls in P, i.e.

t

i = 1 WF, = Z APF;:, given

Wolffram’s method of price decomposition was developed for use in the analysis o f supply. Here it is simply translated into a demand context. In Trill el uf. (1978) the Wol f fmp approach is discussed and a modification of the method is suggested, again for use in supply analysis.

178 TREVOR YOUNG

A PFi = 4- Pi-1 ifP;< 4-1 = o otherwise

A convenient reformulation of equation (2), suggested by Gollnick (1972) is as follows*:

(3) QI = a , + bl Pr + i$ WF,

where bb is (b, - b,) and PI and WF, are defined as above. An advantage of this form is that the usual t test statistic on coefficient bbalso provides an appropriate test on the equality of the slope parameters (i.e. Ho : b, = b3) .

Irreversability of the Wolffram type produces a demand curve which will always be kinked at the prevailing price, no matter what the history of price changes has been. This implies that in Fig. 2a (where bb < 0) price variation expands the market ips0 facto, while Fig. 2b (where bd > 0) represents a market which contracts with variation. If this property seems undesirable, it can easily be eliminated by an appropriate transformation of the Wolffram method of price decomposition. That is to say, it is possible to generate a set of estimating equations depicting demand curves which are kinked only when certain criteria, specified in terms of price peaks and troughs, are met. In particular, three transformations of the Wolffram approach are suggested below and an attempt is made to give each a behavioural interpretation in line with the introductory discussion of the reasons for asymmetry in demand.

(i) addiction to the product-The behavioural hypothesis here is that at record low prices, new ‘addicts’ are captured and consumption habits built up around the product will persist if prices subsequently rise. To model this phenomenon, we transform the Wolffram price decomposition by adding to the falling series only that portion of the price decrease which takes us below the previous minimum price paid. That is to say, the estimating equation becomes (4) Qc = a, + b, P, + bb TWF,

t where W F t = 5 A P M I 4 with P M I 4 denoting the minimum price at time i and @MI4 = PMIN - PMIN.+. If the consumption of the product in uestion does exhibit aspects of “addiction-asymmetryy” then, bb < 0. The

in which case bdrepresents the expansion in ‘normal’ demand attributable to the new addicts.

The demand curve implied by this type of price decomposition is kinked when prices are at unprecedentedly low levels and it is illustrated in Figure 3a.

(ii) oddiction to competing products--If at unprecedentedly high prices, the consumer switches to a substitute product and it is difficult for the seller to win back his custom, a different specification of the demand curve is required. The estimating equation would become:

(5) Qc = a5 + b5 Pr + bh TwRt t

where TWRt = z APMAXi with PMAX i denoting the maximum price i = 1

toe ? ficient 6 , may be interpreted as the “normal” response to a price change,

~- ~ ~ _ _ _ _ _ _ ~~ ~~ ’ This qua& foUows directly by noting that Pf = PO + WR, + WF, , with PO being the price in the iniF doe period, and by substituting for WR,. Van Dijk (1978) uses the Gollnick specification in h a work on the demand for meat.

MODELLING ASYMMETRIC CONSUMER RESPONSES. WITH A N EXAMPLE 179

paid to date, APMAX, = PMA& - PMAXi - 1. If addiction to competing products were a feature of the market, then bb < 0.

In this case the demand curve is kinked when prices rise above previously recorded levels and it is illustrated in Figure 3b. It will be noted that this diagram is simply the demand curve of Figure 3a, rotated through 180”.

(iii) limited information-If potential customers of the product remain unaware of price changes, and of price reductions in particular, the market for the product would be smaller than if information were perfectly distributed. This type of market can be encompassed by the general form of equation (4) but with the expectation that bh > 0. That is to say, record low prices diminish “normal” demand, because potential customers are uninformed of the new price levels. This case is depicted* in Figure 3c.

P

Q 3a

‘ I

Y

3b

1 Q 3c

Figure 3 Transformed Wolffrnm Models

Before presenting an empirical example in which the demand for coffee is investigated, some remaining considerations should be examined.

Firstly, in a wider context the price of substitutes should appear in the demand equation. If the decomposition is performed on a ratio of prices

~ ~~ ~~ ~ ~

Graphically there exists a fourth possibility i.e. a diagram depicting quation (5) but with M > 0. This would appear as a rotation of Figure 3c. However, there scans to be no obvious rationalisation for such a demand m e .

180 TREVOR YOUNG

the implication would be that the consumption of the substitute commodity exhibits the same type of asymmetry as the commodity in question and this may be a justifiable assumption. However, if the prices of the dependent variable and of competing products are decomposed separately, this restriction need not apply. This proposition is examined in a little more detail in the Appendix.

Since it is also customary to include income as a dependent variable in the demand equation, one may wish to allow consumer response to income changes to be asymmetric too. For example the formation of habits may be encouraged as much by income rises as by price falls and so an appropriate demand curve might be estimated with the following equation:

QI = a6 + b6 PI + MTWF, + c6 Y, + cbTWRYI

t z i = l

where Y, is consumer income, TWRY, = AYMAXi with YMAX i

denoting the maximum income level attained at time i and AYMAXi = YMA4 - YMAX.-, . Asymmetry in consumer response caused by addiction to the product would be confirmed if M < 0 and c6 > 0

Furthermore, it should be noted that the Wolffram and transformed Wolffram methods of price decomposition have the property that past experience, no matter how distant, is still taken into account in present consumption., While habits may well die hard, one might wish to discount early market behaviour in some way. This is not easily accomplished within the Wolffram framework but one approach, albeit an imperfect one, would be to permit both addiction to the product in question and to competing products to influence the form of the demand equation. The argument would be that, whereas habits created whilst the product price was low would not be abandoned immediately in response to a price rise, the pattern of consumption may be broken if the price were to rise to unprecedented heights and the consumer were to adopt the competing good. For example the demand for coffee may be characterised by “addiction-asymmetry” but world coffee exporters often express concern about future consumption levels when its price reaches record heights. A simple way to enter the two phenomena into the equation would be to combine equations (4) and (5) by use of dummy variables: the ‘addiction’ variable of equation (4) appearing when PMIN, < Pr < PMAX, but being replaced by TWR, at record high price levels.

Finally, it is recognised that the durability of many consumption goods is a main source of lags in adjustment to new market conditions and that the models of consumer behaviour discussed above could be made more realistic by the explicit incorporation of stock-holding activities.

An Empirical Example For the purposes of this paper, attention is focused on testing the presence of addiction-asymmetry in the demand for coffee. It seems probable that elements of physiological and psychological addiction influence the behaviour of the coffee drinker. Not only might the consumer be attached to the custom of drinking coffee at certain times of the day but moreover coffee is an addictive drug with physiological withdrawal symptoms. Hence it seems reasonable to .suspect a priori that of the three causes of asymmetry considered, addiction to the product is the most likely to occur.

The empirical work is based on quarterly household budget data, collected and published by the National Food Survey Committee (NFS), for the

MODELLING ASYMMETRICCONSUMER RESPONSES. WITH AN EXAMPLE 181

period 1960 (1) to 1977 (4). The “symmetric” demand function was specified as follows-

QC, = a + bPCR, + cPCDYR, + dD1 + e D 2 + fD3

QC, = purchases of instant coffee-oz/person/week where

PCR, = average price paid (P per lb.), deflated by a Retail Price Index (1958 = 1-00).

PCDYR = per capita disposable income*, deflated by a Retail Price Index (1958 = 1.00).

Dl,D2,D3 = seasonal dummy variables. The base quarter (DO) is January - March.

The asymmetric form may be written as QC, = a + bPCR, + bh WF, + cPCDYR, + dD1 + eD2 + fD3

TWF, would replace WF, in the transformed Wolffram version of the model. An apparent misspecification of these equations is the absence of substitutes. Of the other beverages, tea, and to a lesser extent, cocoa seem likely candidates for inclusion as independent variables but attempts to include the price of each product in turn, either in a ratio with the price of coffee or as a separate explanatory variable proved less successful than the results presented below. The explanatory power of the equation was reduced and in some instances the estimated price coefficient had an incorrect sign or was statistically insignificant?. Furthermore, given that the price of coffee is the only price variable to appear and that the price at the beginning of the data period was not exceeded in real terms until the third quarter of 1977, it was not possible to test whether both addiction to the product and addiction to competing products were influencing simultaneously the demand for coffee.

Table 1 presents the results of the empirical exercise. Equation 1.1 of this table, which provides the symmetric form of the demand curve, gives a reasonable fit to the data, although the Durbin-Watson statistic is very low. The coefficients on the price and income variables are statistically significant with the correct signs and the seasonal pattern indicates that consumption would be lowest in the Spring and Summer quarters.

The Wolffram type demand curve is depicted in equation 1.2 and the hypothesis of equality of the rising price and falling price coefficients is rejected at the 99 per cent confidence level. Moreover the relative magnitude of the price variables conforms with Q priori reasoning; translating the slope coefficients into elasticities, the responses are -69 with respect to the price rise variable and -1.17 with respect to the price fall variable. Equation 1-3 presents the transformed Wolffram version of the demand curve and again the evidence of asymmetry is quite clear. However, the results are very similar to those of equation 1-2; the rising price and falling price elasticities are -67 and -1.33 respectively* Both irreversible functions provide a better fit to the

* The NFS does not provide income data in its annual reports. We have used the population and disposable income figures published by the Monthly Digest of Statistics for this pu’pose.

f Another study of coffee using similar data (Solis-Fallas (1978)) docs include the price of tea as an explanatory variable but it fails consistently to provide statistidy significant results. * Other studies of the UK market for coffee suggest a somewhat more inelastic response to price change: Solis-Fallas (1978) estimates a short run price elasticity of -a49 and the NF!3 (1978) calculate an dasticity of -32.

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MODELLING ASYMMETRIC CONSUMER RESPONSES. WITH AN EXAMPLE 183

data than the symmetric model. The final equation of Table 1 presents the results of fitting a demand curve for which both the price and income variables have been decomposed into their rising and falling components. Once again asymmetry of consumer response to price changes is clearly established. However, whereas there is evidence that the response to income changes is also irreversible, the null hypothesis is rejected only at the 90% confidence level and in any case the relative magnitude of the coefficients on the rising and falling income variables is inconsistent with the hypothesis of addiction to the product.

The final phase of the exercise is to investigate whether the asymmetry which has been revealed in the short run functions persists into the long run. This has been performed by estimating a demand equation with an A h o n lag* on each price series and the results, for both the Wolffram and transformed Wolffram versions, appear in Table 2. The degree of polynomial and lengths of lag were determined by experimentation, the choice being restricted to polynomials of the second and third degree and to lags to the maximum of ten periods. The equations presented here are those which provide the best results in terms of R2 value, the significance of individual lag coefficients and the shape of the lag distributions. From equation 2.1 the response to a price fall below the previous minimum level takes about two years; in short, habits seem to be slow to form. The response to price changes above the minimum price is more immediate i.e. about nine months. It is also interesting to note that, judging from the relative magnitudes of the immediate responses to rising and falling price changes, there may be some evidence of asymmetry due to limited information in the short run. Equation 2.2. provides very similar results for the Wolffram function, although in this case the response to price rises is completed in the first quarter. It may also be concluded from Table 2 that there is a more elastic response to both upward and downward price movements in the long run but that the difference in the slope coefficients is of the same order of magnitude as in the short run functions.

Conclusion There seems to be a stong case for believing that the demand curve in a dynamic setting is asymmetrical. In this paper the principal reasons for asymmetry in consumer response have been presented and methods of incorporating these concepts into an estimating equation have been examined. For the purpose of an empirical illustration the demand for coffee has been investigated using two alternative specifications of the asymmetric function. The evidence indicates that the market for coffee is characterised by the form of asymmetry consistent with addiction of consumers to the product (the short run response to a rise in the price of coffee is about half the size of the response to a price fall). However, at least over the data period chosen, the alternative estimating equations provide very similar results and neither is clearly superior.

APPENDIX On the use of price ratios in asymmetric functions.

Suppose the symmetric form of the demand equation were simply

W t = a + b (3 t ~ ~~ ~ ~

* Sec for example J . Johnston, EconometricMethods McGraw-Hill1972, Chapter 10.

184 TREVOR YOUNG

I 0

MODELLING ASYMMETRIC CONSUMER RESPONSES. WITH A N EXAMPLE 185

where QC is the quantity of coffee consumed, PC its price and P T the price of the substitute product, taken here to be tea.

The Wolffram-type demand curve could then be written either as QC, = a , + b , WR, + b, W h

where WR, is the sum of period to period rises in the price ratio @$) and WF, is the sum of period to period falls in the price ratio, or

pc + bA Wt , as in the Gollnick version. QC, = a2 + b , PT t

From the asymmetric model,

BQc -- --'lPC if PTrises. -b/PC , if P T falls; Moreover 'ac = - but -

6PT P T 2 BPT PT'

Thus if the consumer were addicted to coffee consumption

the response to falling tea prices must be less elastic than the response to an increase in the price of tea

If instead of using a price ratio, the prices of coffee and tea were to enter the equation separately, this restriction need not apply.

References Farrdl, M. J. (1952). IrreversibleDemand Functions, Econometrica, 20, 171 - 186. Gollnick, H. (1972). Zur statistixhen Schatzung und Priifung irreversibler Nachfragefunktionen

Houthakker. H. and Taylor, L. D. (1970). C o m e r Demand in the United Statts I929 - 1970,

Johnston, J. EconometricMethodr. 1972 McGraw-Hill Chapter 10. Marshall. A. (1927). Principles of Economics, Macmillan, 8th Edition. National Food Survey Committee. Houselrold Food Connrmprion and Expenditure : 1977.

Scitovsky. T. (1976). The J o y k Economy, Oxford University Press. Scitovsky, T. (1978). Asymmetrics in Economics, Scot. J. Pol. Econ., 25,227 - 237. Solis-Fallas, 0. (1978). The Demand for Coffee in the United Kingdom. An enquiry into the

Houthakks and Taylor State Variables Model. Unpublished M. A. dissertation. University of Manchater.

Stone, J. R. N. and Rowe. D. A. (1957). The Market Demand for Durable Goods. Econometrica, 25,423 - 443.

Traill, B.. Colman. D. and Young. T. (1978). Estimating Irreversible Supply Functions, Am. J. agric. Econ., 60,528 - 531.

van Dijk G. (1978). Prim Formation and Margin Behaviour of Meat in the Netherlands and West Germany. Unpublished P&D. thesis. University of A b a d a n .

wolffram. R. (1971). Some New Approaches-Some Critical Notes, Am. J. agric. Econ.. 53,356 - 59.

Agrarwir&h&t. 21,227 - 231.

Harvard University Press.

HMSO(I978).

Positivistic Measures of Aggregate Supply Elasticities:

186 TREVOR YOUNG

RbUIllC

MfiTHODES POSSIBLES DE MESURAGE DES aACTIONS A S m T R I Q U E S DES CONSOMMATEURS A L’AIDE D’UN EXEMPLE

Cet article cherche ci ktablir une mkthode servant li mesurer un ensemble de rdactions d$firentes du consommateur aux mouvements de prix en hausse et en bake. I1 examine trois des raisons du phknomine d’asymktrie dans la dernande- attachement au produit’ attachement ri des produits concurrents et in formation limitke-et envisage des formes d’kquation appropriipes pour prockder d l’estimation. Un exemple d’application empirique permet de dkmontrer la prkence d’une arymdtrie dans la rkaction des consommateurn de cafe instantanb en Grande-Bretagne.

Zusammenfassung

DIE AUFSTELLUNG ASYMMETRISCHER KONSUMENTEN- REAKTIONEN MIT EINEM BEISPIEL

Dieser Artikel versucht eine Methode zu liefern, mit deren Hive Konsumentenreaktionen au f Preisbewegungen nach oben und unten in der praktischen Tlitigkeit eingefugt werden kdnnen. Drei Griinde ftir eine &mmetrie der Nacllfage weden diskutierf: der Hang zu einem Produkt, der Hang zu Konkur- renzprodukten und begrenzte IMormation, und angemessene Formen der Schlytzungsgleichung werden untersuch t. Eine empirische AwfUhmng liefert den Eeweis far eine Asymmetrie in der Reaktion der Konsumenten im Vereinigten Kdnigreich auf Sofort ldslichen Kaffee.