Consumption I: Indifference curves. Budget constraint. Utility maximisation

 

Indifference curves

Indifference curves are lines in a coordinate system for which each of its points express a particular combination of a number of goods or bundles of goods that the consumer is indifferent to consume. This is, the consumer will have no preference between two bundles located in the same indifference curve, since they all provide the same degree of utility. The indifference curves, as we move away from the origin of coordinates, imply higher consumption and, therefore, increasing levels of utility.

An indifference map is a combination of indifference curves, which allows understanding how changes in the quantity or the type of goods may change consumption patterns.

Francis Y. Edgeworth, developed the mathematics concerning the drawing of indifference curves in his book “Mathematical Psychics: an Essay on the Application of Mathematics to the Moral Sciences”, 1881, from earlier works by William Stanley Jevons. However, Vilfredo Pareto was the first economist to draw indifference maps as we know them nowadays, in his book “Manual of Political Economy”, published in 1906.

The first example of indifference map showed in the adjacent graph is the most common representation. It shows four convex indifference curves (red), showing each curve what amount of a good or bundle of goods x1 the consumer has to give up in order to be able to consume more goods, or bundles of goods, x2. This relation gives us the marginal rate of substitution (MRS) between these goods, which is the slope of the curve in each of its points.

Our second example is an indifference map with four parallel lines (green). This is the case for goods or bundles of goods, y1 and y2, which are perfect substitutes, since the lines are parallel and MRS = 1, that is the slope has an angle of 45º with each axis. It can also be the case for goods or bundles of goods that are perfect substitutes but in different proportions. In that case, the slope will be different and the MRS can be defined as a fraction, such as 1/2 ,1/3 , and so on. For perfect substitutes, the MRS will remain constant.

Our third example shows an indifference map with four indifference curves (blue) that represent perfect complementary goods, z1 and z2. This is, there will not be an increase on the consumer’s utility unless both goods increase in the required proportion. The best example of complementary goods are shoes, since the consumer’s utility will not increase when he or she gets a new right shoe without getting a new left shoe. Notice that the elbows are collinear, and the line crossing them defines the proportion in which each good needs to increase in order to have an increase in the utility. In this case the horizontal fragment of each indifference curve has a MRS = 0 and the vertical fractions a MRS = ∞.

These explanations of indifference curves can also be applied to production. In that case, the MRS turns into marginal rate of technical substitution and marginal rate of transformation.
Video – Indifference curves:

Budget constraint

Consumer behaviour is a maximisation problem. It means making the most of our limited resources to maximise our utility. As consumers are insatiable, and utility functions grow with quantity, the only thing that limits our consumption is our own budget (assuming, of course, we are dealing with normal goods, not negative or harmful goods which consumption we want to limit).

A budget constraint (green line in the adjacent figure) provides the second half of the maximisation problem. We need to balance the utility we derive from consumption with the budget we have.

Supposing we have a choice of two goods, 1 and 2, then our restriction is as follows:

which simply means that our budget must be at least as much as the price of the two goods times their respective price.

This simply shows that our consumption is capped and that the more we spend on one good, the less we can on the other.
Video – Budget constraint:

Utility maximisation

Utility maximisation must be seen as an optimisation problem regarding the utility function and the budget constraint. These two sides of the problem, define Marshallian demand curves.

An individual is therefore faced with the following problem: faced with a set of choices, or baskets of goods, and a fixed budget, how to choose the basket which maximises their utility?

If we know an individual’s utility function, and we know their budget, we have the two restrictions necessary to maximise their utility. This can be done graphically, with the point where budget and utility function meet defining an optimum, as shown in the adjacent figure.

It can be also done mathematically, through a Lagrangian, where the first derivatives determine a system of equations that can be resolved by submitting our utility function to the restriction presented by the budget:

Video – Utility maximisation:

We have started by learning about the very basics of consumer theory. How much we like (or need) goods configure utility functions representing our preferences. This utility functions, when contrasted with our budget constraint, lead us to resolve our maximisation problem: get the most utility with a given budget.

However, we could ask ourselves: what if I wanted to get a given utility for the lowest possible cost? How price changes affect our wellbeing? Is there some way to actually draw these utility functions?

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