The preferences of consumers

CONSUMER PREFERENCES

We can define a basket or a bundle as a group of goods or services available for consumers to buy. A basket could simply be made up of 3 oranges and 2 apples and another one may be composed by 5 pencils and 2 pens. Baskets with two goods can be represented on a two dimensional graph with the help of coordinates using each axis to represent the quantity of each good. In order to understand which bundle is preferred to a specific consumer we need to examine consumer preferences. In general, in order to deal with consumers’ preferences we consider 3 assumptions that make it easier for us to reason in a logical way:

Preferences are complete: the consumer can perfectly tell which basket he or she prefers. If he prefers basket A to basket B we write A>B, if instead B is preferred to A we write B<A while if the consumer is indifferent whether he consumes A or B we write A=B.

• Preferences are transitive: meaning that if you have three bundles: A, B and C, if A>B, B>C then A>C

• More is better: meaning that a consumer will always prefer having a basket full of goods than one with fewer goods. This means that if you have basket A which contains 3 oranges and 5 apples and basket B that contains 7 oranges and 5 apples the consumer will surely like more basket B because the quantity of apples is the same while the quantity of oranges is higher in bundle B.

These assumptions makes possible to us to represent preferences thanks to utility functions which measure the utility (satisfaction) the consumer receives when consuming a specific bundle.

CONSIDERING ONLY ONE GOOD

Suppose that a person’s utility function is U(x) = √x (square root of x) where x is the quantity consumed of a good. This function respects all the assumptions described before: preferences are complete because there is a level of utility for any x, the more is better is respected because the curve always increases as x increases and therefore an higher x (quantity) means higher utility and because of that preferences are also transitive. We define the marginal utility of good x MUx as the rate at which utility changes as the quantity consumed changes. It is defined as:

Graphically the marginal utility of a good is represented by the slope of the line that is tangent to the utility function. The utility a consumer gets from consuming more depends on how much of the good has already been consumed. The marginal utility is mathematically the derivative of the utility function with respect to x. Therefore for the utility function U(x) = √x the marginal utility has the form: MUx = 1/(2√x). Notice that the marginal utility curve declines as x increases but remains positive. This shows the principle of diminishing marginal returns. This is because the more of a good is consumed the less satisfaction is got from an additional unit of good consumed. To understand why, suppose you are hungry, if I offer you a slice of pizza the satisfaction you get from it is really high but, if you have eaten some slices of pizza and you are not hungry anymore, if I offer you an additional slice of pizza, you would still be happy to receive that slice but not as much as before. This is exactly the practical meaning of the assumption that more is better: the more pizza I give to you the happier you will be even if the utility per slice decreases. Intuitively, this hypothesis in reality is not always true. If you continue to eat slices of pizza, at some point you will surely start to feel bad and therefore, for each slice of pizza I offer you, you will be less happy meaning that total utility is decreasing and the marginal utility of slices of pizza has become negative.

CONSIDERING MULTIPLE GOODS

Let’s now consider the case when the consumer can consume two goods and not only one. We will use as examples slices of pizza and glasses of coke. On the horizontal axis we will measure the number of slices of pizza consumed that we are going to indicate with x1 while on the vertical axis we will measure the number of glasses of coke consume that we are going to indicate with x2. If utility is measured by the function: U=√x1x2. For example if a consumer is consuming 4 slices of pizza and 9 glasses of coke his utility will be √(4*0) =√36 = 6. Because now we are considering on the same graph the quantity of slices of pizza (x1), the quantity of glasses of coke (x2) and the utility the consumer gets we need to use a three-dimensional graph measuring on the z-axis the utility of the consumer. The highest is the point the more the consumer will like the bundle associated with it.

When considering more than one good the marginal utility of a good is the rate at which the total utility of the consumer changes when changing the quantity of that specific good while keeping the quantities of all other goods fixed. With only two goods, when computing the marginal utility of a good you have to keep the quantity of the other one as fixed. In our case the marginal utility of slices of pizza measures how the satisfaction of the consumer changes when the quantity of slices of pizza he consumes increases. It is indicated with MUx and it is computed as:

In our case MU1 is equal to: MU1 =√x2/(2√x1). If x1=4 and x2=9 MU1=√9/(2√4)= 3/4

The marginal utility of glasses of coke measures how the satisfaction of the consumer changes when the quantity of glasses of coke changes when the quantity of glasses of coke he consumes increases. It is indicated with MU2 and it is computed as:

In our case MU2 is equal to: MU2 =√x1/(2√x2). If x1=4 and x2=9 MU2=√4/(2√9)= 2/6 = 1/3.

THE CONCEPT OF AN INDIFFERENCE CURVE

Since working with three dimensional graphs is not convenient, we need to reduce these graphs to two dimensional ones. We can do so by representing the projection of the three dimensional graph in two dimensions. By doing so we get a series of different curves. The farther the curve is from the origin the highest is the utility it represents, meaning that the height on the three dimensional graph is higher than the height corresponding to the curves closer to the origin. Each curve represents a certain level of utility for the consumer. This means that, along that curve, the utility the consumer gets by consuming any bundle on it is exactly the same. The consumer is therefore indifferent when deciding which bundle to consume if both lie on the same curve. Because of that these curves are called indifference curves. Indifference curves have the following properties:

• They are all sloping downward if the consumer likes both goods. This because if you increase the quantity of one of the good, in order to keep the consumer indifferent, you have to take away some quantity of the other good from him. Otherwise, because of the more is better assumption, the consumer would be better off and therefore will not remain on the same indifference curve.

• The curves can not intersect with each other because each one represents one specific level of utility and, if they could intersect, the bundle where they intersect would lie on two indifference curves meaning that the consumer assigns to the same group of goods two different values.

• Every bundle must lie only on one indifference curve as a consequence of the previous property.

THE MARGINAL RATE OF SUBSTITUTION

We denote as marginal rate of substitution the consumer’s willingness to substitute one good for another while remaining indifferent so to remain on the same indifference curve. If both goods have positive marginal utilities the exchange the consumer is willing to make between the two goods is represented by the slope of the indifference curve. That is because the slope tells us how much you have to change x2 (measured on y axis) when changing x1 (measured on the x axis) in order to remain on the indifference curve. The marginal rate of substitution is calculated mathematically as the ratio of the marginal utilities of the goods in the bundle so:

The law of diminishing marginal rate of substitution tells us that, for most bundles, the MRS diminishes as the consumption of the good on the horizontal axis increases (so when the consumption of the other good diminishes). To understand why suppose that you consider once again slices of pizza and coke: if you have a bundle with a lot of pizza but no coke you will be willing to sacrifice some pizza for water but if you’ve already drunk enough you will be less willing to give up your precious slices of pizza.

PREFERENCES WITH PERFECT SUBSTITUTES

Sometimes consumers value goods as perfect substitutes meaning that they are indifferent whether they consume one good or the other (butter and margarine are a perfect example for that). Because the consumer is willing to substitute the two goods with the same proportions (ex. 80gr of margarine for 100gr of butter) the marginal rate of substitution stays the same everywhere on the indifference curve. This implies that the indifference curve itself is linear. A general function to describe indifference curves for perfect substitutes has the form: U = ax1 + bx2 where a and b are positive constants. MRS is given by:

so in our case MRS=a/b

PERFECT COMPLEMETNS

Sometimes consumers can not substitute one good for the other or the use of one good without the other is pointless. Right and left shoes offer a perfect example of that: you do not get any utility if you do not have one right shoe for any left shoe. In this case we are therefore considering goods that are perfect complements. These pairs of goods are consumed together in fixed proportions (ex. Two lenses for each pair of glasses). Because by adding 1 good without increasing also the other won’t raise utility, the indifference curves are L shaped and the general equation describing them is min {ax1 ; bx2} where a and b are positive constants. Obviously, because the consumer does not substitute one good for another there is no marginal rate of substitution.

COBB-DOUGLAS

When a consumer is willing to have both goods but not in fixed proportions we have Cobb-Douglas utility functions. A general expression of a Cobb-Douglas utility function is: U=Ax1α x2β. Marginal utilities in this case are computed as follow:

MU1= α*A*[(x1)^(α-1)]*[x2^(β)]

MU2= β*A*[(x1)^(α)]*[(x2)^(β-1)]

Both MU1 and MU2 are positive when A, α and β are positive and therefore the initial assumption of the more is better is satisfied. Because the consumer likes both goods so that both the marginal utilities are positive the indifference curve will be sloping downward. Cobb-Douglas utility functions have diminishing marginal rate of substitution.

QUASILINEAR UTILITY FUNCTION

The preferences of a consumer who buys the same quantity of a good regardless of his income are represented by a quasilinear function. It is a function that is made up of two components (the quantities of the goods) and in at least one of these components is linear. These kind of functions are peculiar because as we move to the north of the graph the marginal rate of substitution of each indifference curve is exactly the same meaning that the curves are parallel to each other. The general equation for this kind of function is U=v(x1) + bx2 where b is simply a positive constant and v(x1) is itself a function dependent on x1 which is increasing in x1 (functions as x12 or √x are perfect examples of that.