Indifference Curves and Budget Lines

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Introduction A basic tool in economics is the mathematical representation of consumer behaviour. It is an abstract (i.e. theoretical) model, based on assumptions. Here, the model is presented in its simplest version, based on simplifying assumptions about the behaviour of an individual who wants to spend his/her money on a variety of goods available on the market. Needless to say, the simplified model is not always suitable for the analysis of complex situations. Advanced versions of the model, based on more realistic assumptions, have been designed to cope with non-standard real world situations.

The simplified model ultimately aims at representing the consumer's choice as the solution of an optimisation problem (so that algebra can be used to analyse his/her behaviour). In few words, the theory describes in mathematical language the behaviour of a consumer who selects the best available bundle on the market. The context should be taken as given and is made up of three 'exogenous' pieces of information: (i) an amount of money y (usually, but somehow improperly, called 'income') that can be entirely spent on consumption goods; (ii) the prices of all consumption goods; (iii) the consumer's preferences. The term 'exogenous' means that income, prices and preferences are not explained within (i.e. by) the model and are taken as given, preliminar information. The model in fact explains what amount of each good is bought by a consumer, given her/his income, the market prices and his/her preferences.

Indifference Curves is the curve that represents the bundle of goods which give consumer the same level of satisfaction, hence the word 'indifference' because consumer do not gain or lose utility or satisfaction if they move along the curve from one point to another. In simple model this is usually represented by two goods. However, it is possible to have more than two goods by writing it in vector and indifference curve will be all the combination of goods that give the same satisfaction to consumer.

Budget Lines Whilst the indifference curves are the mathematical representation of preferences, the budget set is the mathematical representation of all the bundles available to the consumer (because their cost does not exceed her/his income). It will turn out that, if a group of simplifying assumptions are met, the best choice for the consumer can be represented as a bundle in the budget set that also belongs to the highest attainable indifference curve. At this stage, this sounds obscure: in what follows, the attention will be focused at lenght on the definition and meaning of budget sets, lines and of indifference curves. Before, however, it is better to point out that, even at this very preliminar stage, some simplifying assumptions have indeed been introduced.

Optimizing behaviour The first assumption is that every consumer makes optimal choices, in other words she/he never selects intentionally a bundle A when a better bundle B is also available at the same (or lower) cost. This sounds quite reasonable and realistic. However, it is not entirely without problems. The main drawback concerns empirical research: if one assumes a priori that the consumer always makes optimal choices, then any bundle actually purchased should necessarily be considered by the researcher as the best available to her/him. This implies that the model does not permit any empirical test on the quality of consumption choices: mistakes are simply ruled out and, by assumption, the consumer never regrets about his/her choices. This may not be a severe limitation for the model: one has only to remind that it is not very well suited to analyse, for instance, those changes in the consumer's behavior that occur after a sequence of trials and errors (in real world situations a consumer may find the most preferred bundle after having proved and discarded some alternative choices).

Price-taking The second assumption underlying the simplified model maintains that a single consumer takes prices 'as they are' and does not bargain for a better price. Of course, price-makers will take account of the possible changes in the aggregate demand for their good when fixing the price. However, by assumption, a single consumer ignores this and acts accordingly 'as if' his/her consumption choices have no effect on market prices. Thus, the simplified model is not suited to analyze those situations when an individual buyer has in fact market power (and knows it).

Useful assumptions (conventions) Some ancillary assumptions are made just to ease the mathematical representation and to focus on the choice of the best available bundle. First of all, in the simplified model the number of goods available on the market is not infinite. Goods are distinguished according to their characteristics: for example, red apples are different from yellow apples. As in common language, a bundle is a list of quantities (one for each available good, including the zero-amount case). Only non-negative amounts of goods are considered (a more refined analysis would represent the goods possibly sold on the market by an individual as negative quantities, a possibility that is ruled out in the simplified model). Second, quantities are measured on a continuous scale: even the smallest amount of good can be further divided into smaller pieces (i.e. the goods are infinitely divisible). Such a convention permits to avoid 'kinks and holes' in the mathematical functions that represent the consumers' preferences and budget: a very nice property that allows the beginners to focus on economics and forget for a while about complex algebra. Obviously, it is a departure from many real world situations: one cannot easily buy a little bit of a car, nor it is customary to purchase two seconds of a movie... Third, in the simplified model the consumer is assumed to manage her/his budget in a sequence: at a preliminary stage, he/she has already decided how much work to sell on the labour market (that is, how much to earn) and also how much to save. In a last stage, he/she decides about consumption: the amount of money y left from the previous stages is thus given and can be spent entirely.

Mathematical symbols To save ink (digits), words are often replaced by mathematical symbols. A bundle is usually represented as a N-dimensional vector q, where N is the number of goods. Two bundles are distinguished by using superscripts:

  \mathbf{q^A}          \mathbf{q^B}

The budget set All the bundles that can be purchased with the given level of income y are elements of the 'budget set', the set of bundles available to the consumer (when his/her income is y). The obvious common property of the bundles in the budget set is that their cost is not greater than y. In mathematical terms, being p the N-vector of given market prices, a bundle q belongs to the budget set if:

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