Contents

1. Overview

In neoclassical economics, the material world is reduced only to prices and quantities. Empty boxes like "good X" and "good Y" are exchanged on the market at certain prices in certain quantities, being the equilibrium quantities and prices at which demand is equal to supply.

If you take a real-world promotional leaflet of a retailer chain, you will be given a significantly larger number of elements on which to make your choice (to visit the shop or not, possibly in search for one or more goods you want to buy before the promotional period ends).

In this paper, we shall explore mismatches between 1-year ECON101 textbook theory of economics and empirical evidence from the real world that you can collect by yourself, so to replicate (confirming, widening or rejecting) our analyses, as a scientific approach would authorise and ask for.

In particular, we shall claim that "quality" is a missing element in the description of goods, that goods are inserted in broader categories (in a sort of tree structure), that quantities can have several measurement units at the same time while being prevalently discrete (and not continuous), that prices - asymmetrically distributed with odd endings - can be expressed not only in absolute but also in percentage terms, that time is crucial to understand the setting in which actual purchasing acts occur. Not only the triangle price-quantity-quality is considered by the consumer but one has to recognize that there is no one-to-one correspondence between any couple of this trilateral relation.

We shall raise issues that are all too obvious in marketing science but remain at the margin in microeconomics. In so doing, we shall offer to students and researchers a further justification for paying attention to a number of alternative economic models and approaches to consumer that we and our colleagues have being developing for years.

2. A real-world example of grocery retailers' weekly sales ads

Special buys can be advertised in written materials delivered to all houses in a certain geographical range from the point of sale. These home-drop advertising materials report a careful selection of products in promotion which should convince the potential customer to go to the shop until the offer is valid.

Click on this image to download the full leaflet

We use these written materials as a static, replicable, limited source for information flows from retailer to consumers, containing the minimum amount of information needed to attract customers. By contrast, within the shop, the consumer will actively move and act in the venues, zooming in and out from broad categories to specific products, will visually evaluate alternatives under her/his eyes, will look at what others are buying and saying. An analysis of the shop environment is much more complex. The leaflet is an easier material for analysis, while retaining the key elements that attract to POS.

You can find many more promotion leaflets at the end of the paper, while we remain sure that a more extensive comparison across leaflets will enrich the analysis we are going to present, so we invite you to collect leaflets in your area and reflect upon them. Ours is more a method than a conclusion.

But let's start. In the image above, representing the front page of a real leaflet, you shall see the brand of an Italian retail hard-discount chain (MD), the claim of the week ("Offerte strepitose" = "Amazing offers"), three products with their features.

On the left you see a pack of milk and 8 pieces of information: 1. the image of the product package, 2. the name of the product, 3. the qualifier(s) of the product, 4. the normal price, 5. the discounted price, 6. the percentage of price reduction, 7. the quantity as volume, 8. a text underlining the exceptionality of the offer (in other images you'll find also 9. the brand and further elements):

The image of the package in turn represent a whole set of images, including the brand ("Malga paradiso"), flowers, a container, and texts. In other words, in order to identify and qualify the product X of neoclassical approach, in real life it's necessary to highlight several elements, both textual and visual, some of which replicated and given twice, of which we distiguish:

1.1. "Product name" (here: Latte = Milk);

1.2. "Product qualifiers" (here: "parzialmente scremato" = partially screamed = low fat), which are a sort of sub-categories of the product, which distinguish it from other variants;

1.2. "Quality hints" (here: U.H.T., which is a process to obtain safety and longer shelf life), aimed at shortly push up the consumer's assessment of quality (as they were bullet points);

2. visual elements, which can refer e.g. to the "exhibited product" (when transparent glass or plastics reveal the exact colour and texture of the good) or to the "exalted product" (when a photo has been manipulated so as to show a particularly tempting product), to its "origin" (when the environment in which the product was produced is suggested), to its "use" and its "ultimate effects" (e.g. on happiness of the people).

In neoclassical theory, consumer is given preferences and a utility function. In real world, his/her assessment is endogenously changing, by on going support of hints and suggestions.

3. Price information

Price information is split in three elements: the normal price, the discounted price and the percentage of discount. A fourth element, the absolute difference between normal and discounted price, is not given, although the viewer can calculate it (which can be easier or more difficult, depending on the actual numbers).

The customer is given several pieces of information, from which it is expected that he/she will realize that has the opportunity of buying a good at a particularly interesting conditions (while probably retaining that also the normal price is good, in comparison what other retailers charge). At the same time, it has been created a tension between the different price elements and induce at least some activity from the consumer (look, read, calculate), hopefully concurring to the further step of a mental inner order to go to the shop before the deadline for the offer is elapsed.

It is not just the price P that let the consumer take decisions. It's the comparison with other reference prices (present in the leaflet and in the memory) that pushes action. As with prospect theory, action is the result of a comparison with a reference point and the shop itself provides "food for thought". Giving stimuli and suggesting a frame for interpreting them: both are relevant for marketing activities and sale promotion.

The unit of measure are the currency (the Euro sign) for the absolute prices and the percentage sign for the discount. All too often in models you keep out both, but the real consumer need such support to allocate the pure number into a useful information.

In particular, it's interesting to note the extensive use of percentages (whose actual formula of computation is not always immediately performed by real consumer), which are used here as well as for expressing macroeconomic dynamics. Percentages are used because of two main reasons: they make comparisons easy and they establish a loose relation with logarithmic sensibility to stimulus (some psychological research has shown that the minimal threshold for perception reaction rises with the increase of the total value at hand).

To provoke consumer reaction of positive suprise and of taking the inner decision of visiting the shop, it's useful to show high percentages of discount in a large enough number of items. An hyper-rational consumer would sum up the absolute value saved and compare with the cost to travel to the shop (for time spent and gasoline used). A bounded rational consumer would simply have routines about where to shop and might take note to go to that shop before the promotion ends (with different consumers having different rules of behaviour: some might totally ignore leaflets; others might study them carefully and stick to them when shopping; in the middle most would give a look, include the shop in their shopping routes and hop in shopping both discounted and non-discounted items).

As for the price numbers, they are integer Euro plus two decimals (a head with a tail). The theoretical minimum price is thus 0.01 Euro. All price numbers will be multiples of this minimum non-divisible unit, which is also the smallest value of a currency coin. It's impossible to have prices like 1.2342342342342 Euros in a grocery shop. In mathematical terms, prices are always integers with truncated decimals, not "rational numbers" or "real numbers" (with infinite number of decimals).

Outside grocery, it has been noticed that many prices are multi-dimensional, consisting of numerous price dimensions, as they relate to e.g. monthly payments for a certain number of months. As the customer has already difficulties in coping with simple arithmetics with one-dimentional prices, this paper shows the difficulties which multiplies when the price is multi-dimensional, arriving at the conclusion that "results from a laboratory study indicate that under conditions typical of the market place, consumers do not evaluate multi-dimensional prices rationally. Instead they utilize a simplified model, resulting in inaccuracies in their price perceptions".

Our empirical insights have far reaching consequences for economic theory and modelling. Integers with truncated decimals are key to computability using numerical methods, i.e. computer algorithms, instead of the smooth functions assumed by the neoclassical theory. It is fully possible that the equilibrium point there is for a price of 1.34234932090459056333333 etc., so the existence of the the equilibrium requires an excessive number of decimals. Conversely, by assuming, in econmic models, that prices are integers with truncated decimals allows for exhaustiveness in exploration (e.g. of the price that maximise profits, under certain information and knowledge of consumers).

4. Quantity information

Quantity can be expressed not only in weight (e.g. kilo) or volume (e.g. litre) but as the number of items inside the same pack.

In this example, in a pack you find 10 items ("Croissant"), which cumulatively weight 360 grams. Please note that the price is for the pack, not for a full kilo or for each croissant, as you cannot purchase the individual item, you have to buy the pack. This information is useful for many reasons.

Each item will likely represent a "consumption dose" during the actual consumption over time: the consumer will eat one croissant. In ten days of actual consumption she/he will have exhausted the pack, which meanwhile would have entered into her/his cumulative bundle. This gives a hint for consumer about when to come back to the shop and repurchase the product to avoid inventory breaks.

Moreover, this information is useful to judge the relative convenience with other competing substitute goods. Conversely, by bundling several items in one package, the supplier can modify the terms of comparison, increase the sold quantity per purchase occasion, and generate prices per item that can have a large, even infinite, number of decimals. However, these "prices per item" are non-transactionary, the consumer cannot purchase the items separatedly, they are just notional values, results of a mental computation.

Meanwhile, the retailer can choose to supply several package sizes of exactly the same product, as we widely analysed in this paper. But it cannot provide a continuum of sizes.

Coming back to the leaflet: the transactionary price is referred to the number of packs you buy, not the weight of them or the number of items included in each pack. The quantity purchased is the number of packs (1, 2, 3,...). This cannot be a number with decimals (1.234 packs).

All this is at odds with the smooths movements of the equilibrium point in the classical demand and supply diagram from courses in microeconomics:

Nothing here prevents the equilibrium to be reached at a quantity of 1.234; it can even occur that the quantity will be irrational (such as the square root of 2). Quantities can assume any value, they are totally flexible. Simmetrically, prices here can have any number of decimals. In equilibrium, the actual price and quantity purchased can have any number of decimals. Microeconomics prices are said to be flexible as quantities, with no restrictions about the ending number (e.g. an ending in 3 like with 1.233 is perfectly possible and equally likely as any other ending).

On the contrary, in real life, packaged goods are necessarily sold in "natural numbers" (discrete quantities). And prices never end in 3.

What about meat and other fresh goods that are sold on weight? The price is here given by weight.

In this example, the meat is sold at 7.99 euros per kilo. If in the shop you'll find it packed, they have already weight it and each pack will have a slightly different price.

If you can ask (or take directly by yourself - as it happens with fruits and vegetables in open spaces) the butcher (or you, respectively) will use a balance. The balance has a certain precision (e.g. 1 gram) under which the quantity cannot be measured. The price will be calculated by multiplication. Although the quantities are now many more than before, they continue to be discrete numbers (natural numbers of the precision-level of the balance). Conversely, given the price of the good, the minimal precision of the balance might lead to an actual price lower than the minimum currency threshold (1 cent in Euro countries). A price per kilo of less than 10 euro makes the cost per gram lower than 1 cent, to the effect that it is ignored in the final bill the consumer has to pay.

In sum, also goods purchased by weight have quantities and prices that are multiples of non-further-divisibile units (e.g. 1 g and 1 cent). The smooth, continuous, twice differentiable curves of demand and supply in neoclassical textbooks ignore this real-world texture on which real consumers make their choices.

Over time, the total consumption of a grocery good will be equal to the "consumption dose" times the number of consumption acts (consumption seized occasions). We define the consumption dose as the typical quantity used in every occasion, with containers (e.g. a cup of milk) and packaging (e.g. a croissant unit) offering a discretization actually used by consumers. For instance yogurt consumption usually occurs with single doses in a container, even when the product package links two containers:

The consumer usually finishes the full dose. When consumer can more finely choose the quantites, a consumption dose is the result of consumer practice having no (increasing or decreasing) trend (as with doses of shampoo used to wash hair, which are widely different each time but are without a trend: the sum of a sufficiently large number of doses (e.g. 10) remains constant over time).

The quantity in the cumulative bundle is reduced each act by one dose (on average), sending useful alerts for consumer routines to prevent inventory break and to assess urgency of repurchasing: the signal to go to buy might be sent when the quantity in the cumulative bundle approaches the level of inventory break before the second next visit to POS is forecast (so it's necessary to buy in the next visit to POS)

In choosing the quantity, one could look at the forecasted time to come back to POS and posit that the consumer will purchase a quantity that safely allows for the consumption acts x consumption doses that will be made over that period.

One could take into account how much is still in the cumulative bundle at home (so that the sum of it and the purchased quantity safely) as well as the number of components of the household: many people do not live alone as single but purchase for their family and other people living in the same house, which de-cumulate the inventory together.

In a model, POS could be visited according to a schedule plus variation of the schedule introduced for emergency reasons (e.g. forthcoming exhaustion of inventory in the cumulative bundle of special "necessity" items).

Duplication in purchases could happen in order to remove the possibility of emergencies or to take advantage of a special promotion or to take advantage of the POS competitiveness in price and quality - to avoid coming back.

In other terms, purchased quantity is not a simple function of price: it depends on the categorization of the good by the consumer, on the routines establishing the number of consumption occasions and which occasions are actually seized, on the consumption dose, and on the time of repurchase - all this cumulated for all the people living together at the household.

For instance milk purchases depends on milk being categorized as "standard necessity" with routines dictating that a cup of milk is eaten at breakfast each day (the cup being the "consumption dose"), thus over time the quantity of milk consumed (in terms of bottle of milk) will be equal to the number of days times the dose divided by the number of doses in each milk bottle. The quantity purchased will be allocated across different shops, possibly because of their relative price convenience (but also because of purchase trip decision), with large price discounts motivating more-than-usual quantity purchased. Price is important in categorization, in the habit formation, in the allocation across shops and specific shopping occasions but there is no one-to-one price-quantity relation. There is no curve of demand except as a possible, but not necessary, "emerging property" over time of consumer behaviour, each act being inserted in routines.

Other food goods are more surprises that are consumed only in special occasions or as special reward and happiness moment. They are not permanently in the comulative bundle at home: they appear and disappear. This can be modelled as a random process or being related to a flow of life (minor or major) events. In a single purchase event, the quantity purchased is often "one" or what it is used in one proximate future occasion of consumption. In total yearly quantities, they tend to be stable or be related to the expansion (or reduction) of income and the correspoding budget relaxation or budget tightening rules. Every person may have different tastes and categorisation of goods. However, from a supply perspective, these goods tend to be "delicacy" ("delikatessen"), rare food items, which are considered as highly desirable, sophisticated, or peculiarly distinctive in a given region or food culture. A delicacy may have an unusual flavour or be expensive compared to everyday foods. The terms for these goods can somehow be "specialty food" in opposition to "staple food", which indicates good that people routinely purchase without interruption in the cumulative bundle. Please note that these objective attribution from the supply need to be crossed with what subjectively the consumer believes and categories.

In neoclassical theory, demanded quantity is a function of price only, often a linear one or one deriving from indifference curves. In the real world, price is just one component of a much more structured set of routines and information.

In non-grocery items, like consumer electronics or other durables, quantity is often a size, with one or more dimensions.

In this example of image, the product (Fornello a gas) is qualified with "2 fires" and with "Color Black". The dimensions are in centimetres (width and length).

This information is useful for the consumer to imagine how the good fits the cumulative bundle of containers at home and represent a feature of the good having a potential impact on performance (e.g. easeness of manipulation).

Needless to say, the consumer cannot change the size of the product, can only choose not to buy it and buy another one instead (which is different from the first because of dimensions, but sometimes also because of other features, leading to the need of balancing the changes).

The quantity purchased by the consumer is usually one (or zero if not buying!). You usually have one copy of a book, not many (unless you are purchasing them for others). In certain cases, successive acts over the years lead to have more item than one in the cumulative bundle (possibly in relationship to the number of persons in the households). In other instances, a house might require a certain number of items, so that a first collection of that number is purchased when the house is used for the first time, then - depending on breaks and replacement policies - over time new items are purchased.

However the purchase in the same shopping occasion of more than one (perfectly identical) item of durable goods is unlikely. The attention of the consumer is on "which one to buy", more than "how many to buy". Think at your visits in a bookshop.

***

Some durable goods (such as garments) undergo a "wearing and tearing" process, for which, after a certain number of events of utilisation, some degradation of quality occurs. This in part may be due to unproper use but, in part, it is completely unavoidable. To produce this degradation may be the use itself or other concurrent actions, such as cleaning after use. Quality degradation can be prevented, reduced, or reversed by ordinary and extraordinary maintenance, including reparation. When quality degradation is advanced and the consumer considers it unbearable, he or she will decide to put an end to the life of the good and, if the good is considered as a necessity, to substitute it with a new one. Putting an end to the good may lead to disposal and rubbish or, for certain objects and in certain countries and historical periods, to recycling, upcycling or other processes. The circular economy starts in thiis simple act of the consumer but requires an industrial and territorial organisation. In a similar way, it's not a ongoing "wearing and tearing" process that puts an end to the good's life: it's an abrupt accident that destroys it in full or in part. Again, repair is an option here. But there are situations in which the good is not repaird and put in the process of disposal, recycle, upcycle or other alternatives.

In a sustainable economy, goods are designed to last, to have a reduced "wearing and tearing"; accidents are structurally reduced (e.g. with rules about maximum limits of speeds for vehicles); people have a right to repair; industries and territories mandate and facilitate recycling and upcycling, the utilisation in new products of such used parts and raw materials (which then become "secondary raw material") closing the circle.

***

Some relatively expensive and important goods are purchased once in a lifetime, possibly with the financial help of others (e.g. the family). They are the house (or flat), the car (or other vehicle), large pieces of furniture, solar panels, etc. They may be characterised as "investment" and extensive search process can inform "which one" to buy, with major life events being responsible to set "when" to buy. Financial conditions, such as loans parametres, may exert an influence (which is highly unlikely in the case of food and grocery items).

***

Under poverty constraints, surprise goods are almost avoided (or being relegated to annual events with the support of a solidarity network, typically within the family), staples are selected among the cheapest, with possible interruption of their supply (in isolation or in total), thus leading to a degradation of the quality and quantity of the diet, with episodes of hunger and malnutrition. Lack of durables expose the body and the material conditions to shocks and vulnerabilities. These events can be highly dramatic and expensive, in a chain of events in which the poor pays more than the rich.

5. Other pieces of information on the ad

If you browse entire full leaflet examples, freely downloadable here, you'll notice that many items are collected under broad categories (e.g. Fresh goods, Canned goods, Consumer electronics, etc.). In some offer seasons, innovative broad categories are temporarily introduced (e.g. Christmas presents, Easter eggs, etc.). The whole device of "category" is to provide an "introduction" to produts, to help localize them, offering a logical succession which does not disorient the customer and allow for some quick visual browsing, as well as to offer a "family" of relatively similar goods for comparison purposes.

Inside the shop, categories and the product tree are particularly important to cut the time to localize the good you were looking for. Too long a search would discourage the consumer - if it happens too often, the consumer would remember the awful shopping experience and would avoid to come back all together.

Needless to say, a key piece of information contained in the leaflet is the time of validity of the offer in promotion. In a real-world sample of ours, we found that most are valid for two weeks. However, marketing campaigns try and propose many other terms of validity.

Furthermore, the address of the shop, its available parking facilities, payment methods, opening hours and other services are included in the leaflet.

6. Price signals or quantity signals? What else?

Let's now explore the informational processes of consumers and retailers in terms of inputs and outputs. In standard textbooks, supply and demand are expressed as equations in which price and quantity are reversibly and symmetrically linked, to the effect that consumers and producers both respond to price and quantity "signals".

In our real-world supermarket, the consumer looks at the price (and the other elements we mentioned earlier) and decide whether to buy or not, how many units to buy of which one of the many goods in offer. The consumer "processes" pieces of information on prices and provides an output as a quantity (zero, one, two,...). He/she does not negotiate the price, nor proposes a price at which she/he promise to buy a certain quantity. The informational flow from consumer to producer (retailer) is a quantity response to a price (and the other elements) signal.

Conversely, the retailer, whose main task is to fix the price, will take into consideration a number of elements (its overall positioning in the competitive retailing landscape, its costs, the advantages in attracting new customer, etc.), will estimate how much it can sell at certain level of prices (an expected demand in function of the price) then the outcome of its information process is a price. From (expected) quantity to prices is, in all too simplified analysis, the direction taken by retailer. Furthermore, each day it receives a quantity signal (how many units are sold, how many remain on the shelves), to the effect that, looking at "unusual" or "unwanted" quantity outcomes (e.g. too little sales of a certain item) may decide to change the price (e.g. by reducing it) and to undertake a number of other promotional efforts (e.g. to put the product on an adv leaflet).

In sum, real-world consumers and retailers are asymetrically sending and receiving price and quantity signals to each other, while also using further types of signals:

* time signals - the retailer highlight the duration and the end of the promotional offer, the consumer elaborates on whether she/he has time to go to the POS; when there will expectedly be next promotional seasonM on how much time elapsed since last purchase of the item; the current state of his/her cumulative bundle, his/her experience of the good, etc.

* quality signals - consumers can protest about failures in quality of the good, the retailers gives quality hints about the products in order to differentiate it from others;

* popularity signals - consumer look in the streets, in others' houses, in television, etc. how many and who has bought the good, so to follow (or snobistically not follow) a fashion; retailers screen other competing POS to highlight if new products are in their supply and may decide to include them (or an imitation) so that their own customers will not be disappointed while searching for them;

* information-rich tools - e.g. retailers use advertising formats with some (or even) a lot of informations (including parking availability, payment methods, etc. - photo).

Meanwhile, both are immersed in a flow of information about how the general macroeconomy is going, how the specific location is developing, etc. and in broad social trends and influences.

Finally, it is clear that different groups of consumers and different retailing formats would be characterised by a varying degree of attention to a number of signal sources and contents. For instance, Turkish bazaars might allow for a wide negotiation of prices between the customer and the seller.

This means that a general theory of consumption should allow a wide variety of Point of Sales where to shop, spatially distributed and with a range of channels of communication with the perspective customer (as well as possibly providing post-sale assistance) and interaction settings.

7. Changing consumer behaviour: price instruments vs. quantity instruments - what else?

In normative economics, we judge what is good for society - or a collection of individuals - and suggest to policymakers what can be effective in changing behaviours. For instance you might judge that people should smoke less, eat less junk food or pollute less. As normative neoclassical economist, you would suggest either to raise prices (e.g. through a tax on the final good prices, VAT, etc.) or to establish quantity quotas (max production, max sales, etc.). Your expectation is that higher prices will lower consumption (but you can forecast by how much only if you know the "real" demand curve) and that quotas will raise prices so that only the most dependent consumer will continue to buy (or the richest of them!).

Because of the symmetry between price and quantity in signals, you probably will suggest that it's irrelevant whether the policymaker will use a price or a quantity policy.

In other cases, you may add some realistical detail about specific administrative costs or on the side of informational imperfections, so as to prefer prices or quantity instruments.

A large part of the debate in climate change economics between carbon taxes and cap-and-trade can be seen as a discussion on whether to employ price or quantity restrictions, possibly involving quite elaborated evaluations like this.

Our - very basic and sketchy - previous analysis would, however, suggest that you can (and should) suggest a much wider number of policies targeted at quality, time, product substitution, retail chain landscape, etc. to really move the system. This is, in much deeper and wider development, what we proposed in a long book on "Innovative economic policies for climate change mitigation".

8. Conclusions

Real people use a much wider number of signals and elements for choosing than just prices and quantities. Real-world retailers know it and subtly force the communication with consumers along lines that put their supply in an attractive light, so to have consumer shopping in their point of sales.

Neoclassical economics disregard these aspects as irrelevant details. Evolutionary agent-based models of consumer behaviour can instead include these empirical observations into simulations allowing for aggregate properties to emerge, including multi-stage simulation games in the distribution landscape during the different phases of the business cycles.

Now it's up to you

If you receive at home some ad leaflets or you find them at the shop, then please carry out an analysis of what is shown, which elements are present (prices, quantities, and what?), how prices and discounts are given, how are quantities expressed, and how our analysis is confirmed or not. What else can be added? What about odd pricing (i.e. the prevalence of price ending with odd numbers like 9)? What about the absence of the "3" as ending? If you find something interesting, let us know.

If you do not have examples, look at these examples of leaflets or these other examples of special offers.

In this text, we summarise arguments made in this paper and in other articles of mine. We address how much is purchased of a certain good in one purchase session by a individual consumer / purchaser, typically a member of an household. This is not about shops purchasing the large quantities they want to expose to final consumers or professional buyers in companies that stockpile pieces to be used in production processes.

Depending on the good, the purchased quantity can assume certain values. For durable goods like books, the typical quantity is one. You do not need two copies of the same book. The second copy is of no use in most situations. One can imagine certain situation where a family would like to have more copies of the same book but this is an effort of imagination: if you go to a random house and check whether they have two copies, you will not typically find any. Moreover, we are interested in the purchased quantity in the same purchase occasion. So if somebody forget to have at home a certain book and unintentially repurchase one, still the seller will have sold one copy, in that occasion.

Accordingly, for certain goods the normally purchased quantity, when different from zero, is one.

This is true for books, electric appliances, cars, etc. To repeat, this is about the purchase act (it can happen that a family has two cars) of the same identical good (not good category). In a family with two cars, the cars will not be identical, but rather e.g. one large for longer trips and one small for everyday city trips.

Some of those goods are highly differentiated and it is customary for certain consumers to build up over time wider or narrower collections of similar yet not identical goods (e.g. in a bookshelf you can have several dozens of books). This is part of the cumulative bundle.

A second group of goods, such as packed milk, is purchased in a certain package and then uses in standard consumption doses (and their variance and randomness in single future uses). Over time the content of a package wil be exhausted and, if there is a routine to avoid breaks in home inventories, repurchased. The previous group of goods had no such progressive elimination (e.g. of pages in a book). Here the choice is about the size of the package (e.g. half litre, one litre, two litres) and the integer number of packages. For these goods, it is meaningful to buy two, three or higher numbers of units, since over time they will be depleted (so a large number of units allows to postpone future purchases). They also enter into the cumulative bundle.

A third group of goods, such as hand-cut pieces of meat, are purchased in approximate quantities (e.g. half kilo) but the actual piece can be a (troncated) integer with two decimals, reflecting a randomness of the original piece in a living body and/or of the cut itself. The exposed price for such goods is a number with is then multiplied for the actual quantity, including its decimals, to get the final price actually paid by the customer (which in turn will be the sum of all goods purchased in that occasion and will have a cut, depending on the currency, e.g. 2 decimals in steps of 5 cents for Euro). They also enter into the cumulative bundle.

Further groups can exist and would deserve to be inserted here.

Once we have the functional domain of the purchase act (0, 1, an integer, a truncated integer - depending on the group), you can explore the routines that lead to the determination of which one of the legitimate values is actually taken in a (real or simulated) purchase act.