Rationing in restaurants

International Journal of Industrial Organization 9 (1991) 291-301, North-Holland

Rationing in restaurants T h o m a s von Ungern-Sternberg University of Lausanne, Lausanne, Switzerland

Final version received February 1990

This paper provides an explanation why consumers are frequently rationed at certain types of restaurants, bars, cafes, etc. The argument is as follows: The more expensive the restaurant is, the lower will be the consumers' consumption per unit of time spent in the restaurant, Restaurants have a fixed seating capacity. When the capacity constraint becomes binding they will set their prices so as to maximise their profits per unit of time the customers spend in the restaurant, It may well be the case, that there is excess demand at this profit maximisingprice.

1. Introduction By and large we teach our students that in market economies prices adjust to clear the market. Exceptions are, of course, possible. Unforeseen spurts of demand may lead to rationing. At a more sophisticated level of analysis we also explain that informational asymmetries may create problems of moral hazard and adverse selection, and these put some constraints on the extent to which prices can be adjusted to demand conditions. Having worked hard to absorb these insights our students set off on holidays. There is a good chance that they will be unable to get a drink at an outdoor bar at the C6te d'Azur, find a seat at a Gelateria on the Piazza Navona, get a piece of cake in a caf6 on the Champs Elys6es or even a simple beer on the B~irenplatz in Bern (to cite just a few examples). Our students must find these experiences disagreeable (I do) and be disappointed with their teachers for not having prepared them for these hardships of life. To avoid such unnecessary disillusionment with economics it might help to teach this paper near the end of summer term. It proposes a plausible theory to explain the rationing phenomena described above. Indeed, I even believe that the theory is substantially correct, i.e. that it gives the principle explanation for the rationing actually taking places. Starting point of the analysis is an observation once made by the manager of an open-air tea-room in Lausanne. He claimed that perhaps one quarter of his customers actually came because they needed a drink, the remaining three quarters just wanted to rest their feet. This suggests that the tea-room 0167-7187/91/$03.50 © 1991--ElsevierSciencePublishersB.V. (North-Holland)

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is faced with the following problem. It supplies two commodities to the customers, drinks and seats (with a view), but it charges a price only for the drinks. The proportions in which the commodities are consumed are not fixed. Consumers may spend more or less time over their drinks. As long as there is excess seating capacity this causes no problems. Complications arise when seating capacity becomes scarce relatively to total demand. Roughly speaking the tea.room must then set its prices so as to maximise its profits per minute o f seating capacity (i.e. make cake sufficiently cheap so that a customer who wants to stay for 20 minutes has coffee plus cake and not just a coffee.) (When seating capacity is fully occupied, maximising profits per minute of seating capacity is, of course, equivalent to maximising total profits.) Suppose the tea-room has fixed its optimal prices in this fashion and total demand (i.e. the number of potential customers) increases even further. If the tea-room were to increase its price, it would decrease its profits per unit of seating capacity and thus total profits. The tea-room has no choice but to leave its price level constant and ration its customers. I would like to cite some casual empirical evidence to support this theory: - The story told here is, of course, valid only if there is no new entry into the market, i.e. no new tea-room can open in the same location. All the examples cited above have just this property. The owners have an important locational advantage and entry at the same or similar locations is impossible. - T h e story suggests that the problem is likely to be most prominent in those locations, where there is a large number of customers with a lot of time to spend. Tourists are an obvious example, and indeed the rationing tends to be severest in tourist areas. Many tourists like to spend their time outdoors. It is thus not surprising that the examples which come to mind most easily are about well situated outdoor locations. It is sometimes argued that these outdoor places have to ration their customers because of problems of implementing peak load pricing. I find this line of reasoning unconvincing. First, I know bars and restaurants that do practice extensive peak load pricing. Secondly the observed rationing is so extreme that one would expect the introduction of peak load pricing to be very profitable. At the worst it would probably be profitable to charge the peak-price during the entire period. At the very least, however, the peak load pricing story (or its absence) and the one presented here should be considered as complements. - T h e argument suggests that similar rationing phenomena should also be observable in establishments which are not dependant on climatic conditions or tourists, where peak load pricing is obviously no issue, and where market entry is difficult. High quality restaurants or bars with a

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293

particular atmosphere are obvious examples. The evidence seems to confirm the theory. It is well known that one usually has to book long in advance if one wishes to have a table at a top restaurant (Girardet in Crissier for example). When thinking about the problem from the perspective outlined above, one soon stumbles across one piece of evidence that would seem to directly contradict the theory. The theory suggest that the tea-room should charge more or less similar mark-ups on all the items it offers. Instead one observes that certain 'standardised' commodities (e.g. tea, coffee, beer) are much cheaper than the other items available. The usual argument to explain this is that these prices often serve as an indicator of the tea-room's average price level and are therefore downward biased. A place that is rationing a substantial part of its potential customers has only a limited interest in creating the impression of being low-price. The signalling theory does not work for this type of situation. Furthermore, the tea-room has little incentive to care for the customers with particularly low reservation prices (elastic demand). An alternative explanation has to be found. To understand the persistence of these low price articles one must take explicit account of consumer heterogeneity on the market. Suppose there are two kinds of customers. Business customers (who are in a hurry) and tourists (whose consumption is quite time intensive). It may well be that the business customers' demand (in the outdoor places we are talking about) is more price elastic than the tourists demand. After all the business men attach little or no value to the beauty of the scenery, If the tea-room increases its price too much it will drive out all the business customers. It does not want to do this. Business customers do not take up much of its spare seating-time and are thus (as their name suggests) good business. The tea-room will try to keep prices low on those items consumed by the business customers and increase the prices of those goods associated with time-intensive (tourist) consumption. This would go part of the way of explaining the high price of ice-creams and cocktails relatively to coffee or mineral water. There is, of course, some literature on alternative explanations for quantity rationing by price setting firms: B6hm et al. (1983) consider rationing as a particular case of (second degree) price discrimination. In their model quantity rationing can occur only if there are at least two types of consumers. In the model presented here rationing can occur even with homogeneous consumers. Basu (1987) motivates quantity rationing by introducing 'judging quality by excess demand' considerations. In his (very elegant) model it might pay a producer to ration customers even if it still has spare production capacities available. In our model rationing becomes profitable only when there is no spare capacity left. The rest of this paper is organised as follows: Section 2 analyses the basic

J.i,o. -

G

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T. yon Ungern-Sternberg, Rationing in restaurants

model with identical consumers, section 3 introduces consumer heterogeneity, and section 4 contains some concluding remarks,

2. The basic model When trying to model the problem set out in the introduction two basic approaches immediately come to mind: Either one builds a 'restaurant type' model: In certain expensive restaurants each seat is occupied only once every evening. In this type of model the amount of seating capacity (measured in seating hours) taken up by a customer is totally independent of the 'quantity' he consumes, or the amount of time he actually stays. (The seat is 'lost' for the evening.) Alternatively one builds a 'tea-room type' model. Here there is a certain amount of consumer turnover, i.e., if each customer spends less time in the tea-room, the number of customers the tea-room can serve on any given day increases. I shall adopt the second approach here, because it is somewhat more interesting. The assumptions of the basic model are as follows:

A, Consumers There are G identical consumers potentially willing to enter a tea-room. To keep matters simple it is assumed that the number of consumers wishing to enter the tea-room in any given time-interval of the day is the same. This allows us to put aside problems of fluctuating demand, peak load pricing etc. To further simplify matters we shall ignore 'integer number' problems due to the fact that the tea-room can fill up only slowly in the morning and that one cannot let customers enter just before closing time. The reader may wish to think in terms of a tea-room that stays open 24 hours a day and 7 days a week. The total time a customer spends in the tea-room t, and the quantity he consumes x, are simultaneously determined by a utility maximising procedure. We make the following assumption about consumer preferences. (1) Demand functions are downward sloping:

x=x(p),

x'<0.

(1)

(Note that if one wished to explicitly take into account the fact that the tea-room sells a number of different items, one could treat x and p as vectors instead of scalars). (2) Consumption and total time spent in the tea-room are (from the consumers' point of view) complements, i.e. an increase in price will lead to a decrease in x and also a decrease in total time t. We thus have

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295

t' < O.

(2)

t = t(p),

Some people I have discussed this model with have suggested that a consumer who has spent more on a given item will feel entitled to spend more time over it. Note that there is no contradiction between this notion and the assumption t ' < 0 . As the argument put forward suggests, we shall indeed assume (under 3) that the average time spent per unit of consumption increases with p. Note also that even if one assumed that total time spent in the tea-room increased with the price level, the argument presented here would go through afortiori. (3) The average time spent per unit of consumption t increases with the price level, t/x = t'(p),

f' > 0.

(3)

The more expensive a tea is, the more time customers will spend over it (on average). This is a plausible assumption in all those cases where consumers attach an independent value to sitting in the tea-room (e.g., for the rest or for the view). Substituting (I) into (3), we can write t(p) = x ( p ) . f(p).

The assumption that total time spent in the tea-room decreases with price i.e. t ' < 0 can thus be rewritten x'(p)I(p) + (p)f'(p) < 0 ~ [ex I > er"

(4)

The price elasticity of quantity consumed is necessarily greater than the price elasticity of the average time per unit of consumption. (If we had assumed t ' > 0 we would instead have [ex[ 0, can similarly be rewritten: dt

p_ dx.p,

(5)

i.e., the elasticity of total time with respect to price is greater than the elasticity of quantity with respect to price. (Note that (5) would always be fulfilled if we had t ' > 0.)

T. yon Ungern-Sternberg,Rationing in restaurants

296 B. Producers

The tea-room can 'produce' x at a constant marginal cost c. It has a capacity constraint due to the fact that it has only N seats available for T* (opening) hours, i.e. total seating capacity (measured in seating hours per day) is equal to: K=NT*,

(6)

Denote by small g (as opposed to capital G) the number of consumers who actually do enter the tea-room on any given day. The tea-rooms maximisation problem is then to choose a price level p so as to: m a x zr = g . x ( p ) [ p - - c ] p

(7)

s . t . g . ~(p)x(p) < K,

(8)

g
(9)

g = min G, K/f(p)x(p)],

(lo)

Eq. (8) tells us that the total time all customers actually spend in the restaurant cannot exceed the total seating capacity (measured in seating hours per day). Eq. (9) simply states that the total number of customers who actually do enter the restaurant (g) cannot exceed the number of potential customers (G). Eq. (9) states that in our model (with identical customers) all customers will actually enter the restaurant (i,e. we have g = G) provided that there is no shortage of seating capacity. When G is sufficiently low, only (9) will be binding. When G is sufficiently high, only (8) will be binding. For intermediate values of G both (8) and (9) will be binding. C. Profit maximising prices

When only (9) is binding (i.e. we have G < G in fig. 1) we obtain the standard first order condition for the profit-maximising price pn p.-e 1 Pn = ~ "

(11)

Note that when the capacity constraint just becomes binding (i,e. we have G = G_in fig. 1 below) we have f(Pn) "x(pn) = K/G.

(12)

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T. yon Ungern-Sternberg, Rationing in restaurants P

Pk

Pn

I i I I I I I

I I I

G Fig. 1 When the number of potential customers G is sufficiently larger so that only the capacity constraint is binding (i.e. we have G>(~ in fig. 1), we can substitute g = K/~p)" x(p)

in Eq. (7). This yields us the first order condition. pr-c

PK

1

~:r"

(13)

From (4) we know that lex)> el, i.e. the price when the capacity constraint is binding is always higher than the price p, when total demand is tow. (If one had assumed instead t ' > 0 one would obtain the curious result that as demand increases the tea-room decrease its prices to make the customers stay less long. One would, however, still observe rationing). When demand is just sufficiently high so that only the capacity constraint is binding (i.e, we have G = (3 in fig. 1), we have ~(Pr)" x(Pr) = K/G.

(14)

For intermediate values of G, where both (8) and (9) are binding (i.e. we have

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T. yon Ungern-Sternberg, Rationing in restaurants

G < G < G in fig. 1) the optimal price is (implicitly) determined by the two constraints to equal: f(p~)x(pG) = K / G .

(15)

As both the L.H.S. and the R.H.S. of (15) are decreasing functions of G, the optimal price Pc increases with the number of potential customers G. The situation is graphically depicted in fig. 1. For all values of G>C, the tea-room is completely bound by its capacity constraint and can no longer react to increases in demand by increasing its prices. For intermediate values of G on the other hand, i.e. G~ [G_,(~] the capacity constraint is also binding, but nevertheless an increase in G will lead to an increase in the price-level.

3. Heterogeneous consumers

The analysis of the previous section explains why tea-rooms may have no incentive to increase their prices when the number of potential customers increases. The reason is basically that once prices have reached a certain (high) level, the only effect of further price increases is to raise the timeintensity of consumption and thus to decrease total profits. As explained in the introduction, one observation which does not fit in with the above theory is the fact that the prices of certain items (such as coffee or tea) tend to remain relatively low. To explain this observation we shall now introduce two different types of consumers into our model, 'business customers' and 'tourists'. We assume that these two groups differ in both their price elasticity of demand and their price elasticity of time spent per unit of consumption. The rationing mechanism we analyse has the following simple structure: There is a continuous flow of potential customers passing in front of the tearoom, A fraction ~ are businessmen, a fraction ( l - a ) are tourists, Thus whenever a seat becomes available there is a chance ~ that it will be taken by a businessmen. To minimize on computational effort and to highlight as clearly as possible the incentives to price discriminate we assume that all business customers consume o n l y good X and all tourists consume o n l y good Y This assumption implies that practicing (second degree) price discrimination is particularly simple. (No matter how large the price differential, consumers always stay with 'their' product.) The potential for practicing price discrimination would, of course, be (considerably) reduced if one allowed the customers of the high price good to switch to consuming the low price good. The marginal cost of producing both X and Y is equal to c. All the

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299

variables and functions concerning business customers are indexed B, all the variables and functions concerning tourists are indexed S. The tea-room profit maximisation problem is now to choose two prices pX and pv so as to max rc = g[o~xn(p ~) [p~, - c] + (1 - a)Ys(P') [PY -

c]]

(16)

pX py

s.t.

g[exntB+(1-ct)y~ts] ~gT<__K,

g < G,

(17)

and

g = min [G, K/y]. By computations exactly analogous to these of the previous section we obtain: If the capacity constraint is not binding: p.~-c

i

p.~-c

I =-~,

(18)

i.e., the profit margin is inversely proportional to the price elasticity of demand. Business customers will be charged a lower price, if their price elasticity of demand is higher. If the number of potential customers is sufficiently high so that only the capacity constraint is binding, we have

p~:-c 1 p~ =e-~

and

p~:-c= 1 P,~ e~,

(19)

i.e., the profit margin is inversely proportional to the price elasticity of the time spent per unit of consumption. Business customers will be charged a lower price if their price elasticity of time spent per drink is higher (i.e. they don't come if Px is high and stay for a quick drink when Px is low). B B From eq. (4) we know that we must always have: lex)>ef and I~ffl>es, i.e. as the capacity constraint becomes binding the prices charged to both the businessmen and the tourists must increase. It may be the case, however, that the price elasticity of demand of the businesssmen is lower, i.e. I~[ < le~l while at the same time the price elasticity of the time spent per drink is higher, i,e. eeD> ~r, s The relative price of the good consumed by the businessmen will then fall as the capacity constraint becomes binding, Such a situation is depicted in fig. 2. The analysis of this section also sheds some light on the question, why the

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T. yon Ungern-Sternberg, Rationing in restaurants

py I

.x P

,,

Py

I

G

G

G

Fig. 2

use of non-linear prices may not be particularly attractive for the tea-room when the capacity constraint becomes binding. For example charging a fixed 'cover charge' and fairly low per unit prices may have the beneficial effect to extracting some of the tourists surplus. It may also have the disadvantage of driving out the business customers who just come in for a quick drink and are thus very attractive from the tea-room's point of view.

4. Conclusion The purpose of this paper was to present a simple and (I hope) stimulating model of rationing. The market studied was that of restaurants, bars, cafes, tea-rooms, etc. I believe the problem analysed to be of considerable economic importance. After all, in 1985 178.200 persons (5.6% of Switzerlands working population) were active in the hotel and restaurant business. This is more than the 149.000 persons working in Swiss banks and insurances taken together. It would seem that the restaurants have not as yet received their fair share of theoretical papers. Slightly more seriously, it should be noted that the service industries in general have been widely neglected by scholars in industrial organisation. An indication of this may be found by looking at the cover of this journal. Maybe a signal to promote research in a different direction could be set

T. yon Ungern-Sternberg, Rationing in restaurants

replacing the old steel mill by a picture of a waiter with a white napkin and a good bottle of wine. Finally I would like to point out, that many of the industries ! have managed to study in any empirical detail exhibit a degree of price rigidity which is not easily explained by our simple profit maximisation model. Considerable insight into the working of real world markets can be gained. I believe, if one tries to find good theoretical explanation not for price rigidities as such, but for the price rigidities in any specific market whose institutional setting one knows in some detail. On the restaurant business the key to understanding rationing is, I believe, the fact that the producers supply their customers with two commodities, refreshments and seating capacities, but charge a price for only one of them (refreshments). The interesting point is that the commodity for which no price is charged is also the one where capacity constraints are most likely to appear. Since it is also true that the two may be consumed in different proportions rationing may sometimes be the most profitable strategy. References Basu, K., 1987, Monopoly, quality uncertainty and 'status' goods, International Journal of Industrial Organisation 5, 435-446. B6hm, V., E. Maskin, H. Polemarchakis and A. Postlewaite, 1983, Monopolistic quantity rationing, Quarterly Journal of Economics 98, 189-197.