Does tangency of indifference curve with the budget line always result in optimal solution for the consumer utility maximisation problem? Discuss.

 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).

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An indifference curve is a line showing all the combinations of two goods which give a consumer equal utility. In other words, the consumer would be indifferent to these different combinations. People cannot really put a numerical value on their level of satisfaction. However, they can, and do, identify what choices would give them more, or less, or the same amount of satisfaction. An indifference curve shows combinations of goods that provide an equal level of utility or satisfaction. a consumer’s choices are limited by the budget available. Total spending for goods and services can fall short of the budget constraint but may not exceed it.

Algebraically, we can write the budget constraint for two goods X and Y as:

Equation 7.7

P X Q X + P Y Q Y ≤B

where PX and PY are the prices of goods X and Y and QX and QY are the quantities of goods X and Y chosen. The total income available to spend on the two goods is B, the consumer’s budget. Equation 7.7 states that total expenditures on goods X and Y (the left-hand side of the equation) cannot exceed B.

Utility maximisation refers to the concept that individuals and firms seek to get the highest satisfaction from their economic decisions.

For example, when deciding how to spend a fixed some, individuals will purchase the combination of goods/services that give the most satisfaction.

Utility maximisation can also refer to other decisions – for example, the optimal number of hours for labour to supply their labour. Working more increases income, but reduces leisure time.

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