Marshallian and Hicksian demands stem from two ways of looking at the same problem- how to obtain the utility we crave with the budget we have. Consumption duality expresses this problem as two sides of the same coin: keeping our budget fixed and maximising utility (primal demand, which leads us to Marshallian demand curves) or setting a target level of utility and minimising the cost associated with it (dual demand, which gives us Hicksian demand curves). We must also look at the Lagrangian functions where we obtain the first derivatives.

This leads us to the main difference between the two types of demand: Marshallian demand curves simply show the relationship between the price of a good and the quantity demanded of it. Hicksian demand curves show the relationship between the price of a good and the quantity demanded of it assuming that the prices of other goods and our level of utility remain constant.  This makes sense when we look at consumption duality: for dual (Hicksian) demand, we maintain a fixed level of utility, and so our level of wealth, or income, must remain constant. We simply cannot be as satisfied if we do not maintain equal purchasing power.

Marshallian and Hicksian demand curves meet where the quantity demanded is equal for both sides of the consumer choice problem (maximising utility or minimising cost). For prices above this equilibrium point, consumer wealth is higher with Hicksian demand curves than Marshallian demand curves, because to maintain utility constant, Hicksian demand curves assume real wealth remains unchanged. Marshallian demand assumes only nominal wealth remains equal. The opposite is true for prices below this point: Marshallian demand assumes that as nominal wealth remains the same but price levels drop (negative inflation), the consumer is better off. Hicksian demand assumes real wealth is constant, so the individual is worse off. This is why Marshallian demand curves are more ‘stable’: they reflect both rent effect and substitution effect. Hicksian demand curves only show substitution effects (utility is constant, therefore rent must remain constant), which means that demand varies with price only because other options become more attractive.

Formally,

For an individual problem, these are obtained from the first order conditions (maximising the first derivatives) of the Lagrangian for either a primal or dual demand problem.

Marshallian demand makes more sense when we look at goods or services that make up a large part of our expenses. Here, the income effect is very large. However, for smaller purchases, we are willing to spend more or less any amount as long as we derive the utility we expect to.

Video – Marshallian and Hicksian demand curves:

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:

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:

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:

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:

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:

Goods are something that provides its holder some kind of satisfaction, and therefore has a utility. There are different kinds of goods, and different classifications can be arranged and identified. We can differentiate between consumption goods (durable or perishable) and capital goods. Classification depending in responsiveness to incomes changes (normal goods and inferior goods) and to price changes (ordinary goods and Giffen goods) can also be made.

-Normal goods are those whose demand increases due to a rise in income levels, having therefore a positive correlation, which implies that the elasticity of this kind of goods is always higher than 0.  Normal goods are divided in two categories, Superior goods and Necessary goods.

·Superior goods, also known as luxury goods, are those goods that displace the demand of inferior goods after a rise in consumers’ income. They are a kind of normal goods as their demand increases when income does as well, however, the difference is that they occupy the share of inferior goods. The elasticity of this kind of goods is, for this, always higher than 1.

·Necessary goods are those whose demand increases when income does, however the increase in demand is less than proportional to the rise in income. Elasticity of this goods is always between 0 and 1.

-Inferior goods are those whose demand moves in opposite direction to the income variation of consumers. This occurs because consumers’ preferences change to other goods that are more highly regarded.

-Ordinary goods are those goods whose demands move in opposite direction to the price variation. This means an increase in the price of the good will mean a decrease of its demand and the other way around.

-Giffen goods are those goods whose demand moves in the same direction as the price variation, this meaning, raising the price of the good will increase its demand, and the other way around. The explanation to this kind of goods comes from the “Snob Effect”, the more expensive a good is, the less people can afford it, and therefore becoming more precious and highly valued.

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: