Utility is the ‘satisfaction’ we get from using, owning or doing something. It is what allows us to choose between options. This can be plotted on a chart.

A preference function therefore assigns values to the ranking of a set of choices. This is useful as it allows us to see consumer behaviour as a maximisation problem: faced with a set of options and a budget constraint, we will choose what satisfies us most. Utility functions are often expressed as U(x₁,x₂,x₃…) which means that U, our utility, is a function of the quantities of x₁, x₂ and so on. If A is a basket of goods, and , then U(A)>U(B). That is, if we prefer A to B it is because we derive greater utility from it.

Utility functions follow the same code of conduct, the same axioms, as preferences, because they are simply numerical representations of them. That is, they are transitive, complete, continuous and convex, for the same reasons. Being continuous allows us to differentiate them, and being insatiable allows us to say that:

This means that the more, the better, which is the same as saying that utility functions grow with quantity.

The most important thing to point out is perhaps the fact that utility functions do not assign a numerical value to our preferences. They simply indicate order and magnitude of preference, that is, what we like more and by how much.

Video – Utility function:

The foundation for Economics is rationality. Rationality implies that people will act in ways that best suit their particular set of circumstances, including, but not limited to, the choices they face. In order to choose, you must necessarily have a set of preferences over the options you are presented with. Although utility theory was born in the 18^th century, this approach, which simply implied consumers being able to rank preferences (and not assign a numerical ‘utility factor’ to each choice) really took off at the beginning of the 20^th century because it offered an empirical, logical structure to microeconomics. Frisch was a pioneer in this in the early 1920s, but the definite birth of consumer behaviour as an economic science is probably attributable to Samuelson in the 1940s.

These preferences must fulfil a set of criteria in order to fulfil the necessary requirement of being classed as ‘rational’ (in a purely economic sense of the word).

• The first axiom is completeness. That is, whether one is indifferent to, or prefers, one set of options over another, they must always be able to make that choice. That is, a consumer can always rank a set of possibilities as either better, worse, equal or at least as good/bad as another.

 or  or both at the same time

• Transitivity. This simply means that consumers are able to order their preferences in a logical way- that is, if you prefer A to B and B to C, you must prefer A to C. It is useful to express this in binary logical form, with a set of symbols that represent:
  • A is preferred to B, …
  • …or B offers less utility than A,  
  • …or A and B are indifferent, 
  • …or A is at least as good as B, 
  • …and therefore, to be transitive, if  and , then .

• Continuity: in order for preference theory to be useful mathematically, we need to assume continuity. Continuity simply means that there are no ‘jumps’ in people’s preferences. In mathematical terms, if we prefer point A along a preference curve to point B, points very close to A will also be preferred to B. This allows indifference curves to be differentiated.

• Sometimes, and for purely formal purposes, a fourth axiom, reflexiveness, is mentioned:

Which simply implies that A is at least as good as itself.

This is the basic set of premises, or axioms that Economics requires of individuals in order to be rational economic agents. It also enable us to represent preferences graphically. However, a later set of axioms was studied and added by Samuelson as ‘preferable’, because their functions will also behave ‘preferably’, i.e., will be easier and more useful to work with:

• Because people are inherently insatiable, and we always want more of what we like, we derive the premise of monotonicity: weak monotonicity implies that if A contains more than B, A is at least as good as B. On the other hand, strong monotonicity implies that if A and B contain the same amount, but A contains more of at least one good, then A is strictly preferred to B.

• Convexity: Further developing our taste for variety, we inherently prefer baskets of goods, or choices, which contain a wider range. This can be represented with an indifference curve, which shows the rate at which we are willing to substitute units of A for B.   

As tends to happen with any axiom, there are plenty of examples of people violating them. A good example is when deciding what shade of blue to paint your living room. If presented with two shades, where one is very slightly lighter than the other, you will probably be indifferent to either if you cannot tell the difference. However, when presented with a thousand shades, each progressively lighter, and asked to choose between the original shade (midnight blue) and the last shade (azure), you will probably find you have a definite preference. This, of course, violates the premise of transitivity.
Video – Utility function:

Asymmetric information refers to transactions in which one of the parties has better information than the other one.  Adverse selection and moral hazard can result from the worst cases of asymmetric information in transactions between economic agents.

A key article on this subject is “The Market for Lemons: Quality Uncertainty and the Market Mechanism”, 1970, in which George Akerlof develops a classical example of second hand car market in which the seller is the only one that truly has complete information about the quality of the car, to simply explain the risk of asymmetric information.

Sellers of low-quality products will try to sell “lemons” –bad cars- as if they were high-quality cars. This will cause potential buyers to be suspicious of all these types of transactions and thus difficult the transactions of sellers that are trying to sell good quality second hand cars. Sellers with high-quality goods would then exit this market, leaving only an adverse selection of low-quality goods. Then, a distinction of quality will be made within the remaining low-quality cars, which will start the process again. In the end, the market of second hand cars will disappear.

This example can be generalized, and therefore applied to other economic matters, such as the labour market or the credit market.

The marginal rate of transformation (MRT) can be defined as how many units of good x have to stop being produced in order to produce an extra unit of good y, while keeping constant the use of production factors and the technology being used. It involves the relation between the production of different outputs, while maintaining constant the same level of production factors. It can be determined using the following formula:

The MRT is related to the production possibility frontier (PPF). The slope of the curves shows how a reallocation of the production can end with a different bundle of production, using the same quantity of inputs. Two of the most commonly used PPFs are depicted in the adjacent figure.

In the first graph, the MRT will change along the curve.

The second graph, which portrays the case of perfect substitutes output, that is the slope has an angle of 45º with each axis and therefore we have MRT = 1. When considering different substitutes goods, the slope will be different and the MRT can be defined as a fraction, such as  1/2 ,1/3, and so on. For perfect substitutes, the MRT will remain constant.

Not to be confused with: marginal rate of substitution and marginal rate technical substitution.

Video – Marginal rate of transformation:

The marginal rate of technical substitution (MRTS) can be defined as, keeping constant the total output, how much input 1 have to decrease if input 2 increases by one extra unit. In other words, it shows the relation between inputs, and the trade-offs amongst them, without changing the level of total output. When using common inputs such as capital (K) and labour (L), the MRTS can be obtained using the following formula:

The marginal rate of substitution (MRS) can be defined as how many units of good x have to be given up in order to gain an extra unit of good y, while keeping the same level of utility. Therefore, it involves the trade-offs of goods, in order to change the allocation of bundles of goods while maintaining the same level of satisfaction. It can be determined using the following formula:

Generally, if the price of something goes down, we buy more of it. This is down to two effects:

• Income effect: because it’s less expensive, we have more purchasing power because it is a smaller drain on our personal finances.
• Substitution effect: because it offers more utility per unit of money, other alternatives become less attractive.

What Eugen Slutsky managed to do was find an equation that decomposes this effect based on Hicksian and Marshallian demand curves.

Graphically:

Mathematically, it is based on the derivatives of Marshallian and Hickisan demands:

The left hand side of the equation is the total effect- that is, the derivative of x (quantity) respect p (price). It shows us how much the total quantity of x that we consume varies when we change price. The next part is the substitution effect- how much the variation is due to us finding similar options. It is obtained from the derivative of the Hicksian demand with regards price. The right hand side is the income effect, how much changes in our purchasing power affect the amount we consume of a certain good. It is the derivative of the Marshallian demand with regards wealth (multiplied by the quantity).

Whether the SE and the IE are positive or negative when prices rise depends on the type of good:

It is not always possible to tell what the total effect will be- if we are talking about inferior complementary goods, for example, the SE and the IE pull in opposite directions. The TE will depend on which effect is stronger.

Video – Income and substitution effects:

Video – Consumption duality:

Cost minimisation is a way of solving the optimisation problem regarding the utility function and the budget constraint, even though the most common way of doing this is by means of utility maximisation.

If we think about it, we don’t normally have a fixed budget for most purchases. We have a certain utility we expect to derive from them and we hope to spend as little as possible on them (but we don’t have a maximum budget).

In this case, it is utility that is fixed as a restriction, and cost that we can play around with. It’s like solving the consumer’s choice problem in a mirror image way to utility maximisation, and is associated with Hicksian demand curves.

The way we resolve this minimisation problem is very similar to utility maximisation, and is also done with a Lagrangian system, since there is a duality in consumption. In this case, we want to minimise our budget for a given utility:

Video – Cost minimisation:

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: