Preferences and Utility

Transcription

1 Introduction to Microeconomics Preferences and Utility Introduction An indifference curve is a graph that shows different bundles of goods which a consumer is indifferent. This means that every point along the line represents a combination of goods that provide the same level of satisfaction or utility. Another way to think about is that utility represent preferences and is not the result of preferences. Indifference curves are an important concept in microeconomics because they allow us to characterize demand patterns over commodity bundles. If there are two products and we assume they are normal goods then if we increase the consumption of one product we must give up some of the other product to maintain the same level of utility. Likewise, if we decrease our consumption of one product then we must consume more of the other product to ensure our utility is unchanged. We usually assume there are two goods because this makes it much easier to conceptualise and graph. In addition we usually assume goods are normal. Originally utility theory was conceptualized by Cournot and Gossen. Utility theory was formalized and championed independently by three proponents, Stanley Jevons in Britain, Carl Menger in Austria and Leon Walrus in Switzerland. This was the beginning of the neoclassical or marginal revolution. Later on Edgeworth and Pareto further developed this theory and formalized it mathematically. Now indifference curves and utility theory is ubiquitous in microeconomic theory. In this PDF we examine the axioms of rational choice followed by the assumptions underlying indifference curves and some specific and common utility functions. The income and substitution effects are examined separately in the Income and Substitution Effect pdf. Axioms of Rational Choice The following properties of continuity, completeness and transitivity define the preference relationship and the basic set of postulates of rational behaviour. Continuity If A is preferred to B then points sufficiently close to A are also preferred to B. This is important for a mathematical development of the theory of choice as it rules out discontinuous preference relationships. Completeness If A and B are any two options for an individual then they can always specify one of the possibilities below 1. A > B (A preferred to B) 2. B > A (B preferred to A) 3. A = B (A equal to B) 1

2 Transitivity If A is preferred to B and B is preferred to C then A must be preferred to C. Properties of Utility Functions If A > B and B > C then it must be that A > C Non-uniqueness of utility: The number of utils or units of utility given to a specific utility curve is irrelevant as we are only interested in the ranking between utility functions. Units of utility do not have any meaningful application except that of ranking alternative options. For instance U(A) = 100 > U(B) = 50 is no different to U(A) = 100,000 > U(B) =5, as long as U(A) > U(B) for all A and B. However when U(A) = 100 > U(B) = 50 it does not mean that the benefit derived from A is twice as large as that of B. Incomparable across Individuals: Utility is not comparable across individuals because we only have ordinal utilities instead of cardinal utilities. Essentially how do we know that someone feels the same about something and places the same level of satisfaction on an activity as we ourselves do. There is much debate on this area of economics as it borders the metaphysical and is hard to confirm empirically. Ordinal Utility Functions: Ordinal utility functions are unique only up to an order preserving monotonic transformation. This means that we can change the function and numbers, but as long as the ranking is preserved they will still produce the same rankings. These functions are simply a ranking of different levels of utility. We would like to compare utilities but the reason we cannot compare them is because the ordinal utilities do not tell us anything about the strength of preferences. If we could use cardinal utility functions then we could measure the relative strength of preferences but we cannot use cardinal utility functions. Cardinal Utility Functions: Allow us to compare utility across individuals by utilising a universal index of common numbers, or a common unit of measurable utility, which would enable us to compare individuals in the same units. Due to the introspective nature of satisfaction we cannot easily create then verify an index is shared between two people, thus we are forced into using ordinal utilities. Indifference Curves: Indifference curves represent utility functions and have the following properties. Strict Convexity Convexity is not sufficient and it is important we have strict convexity as this ensures that the consumer prefers a balance consumption bundle. Strict convexity is equivalent to the assumption of diminishing marginal returns. Non Satiation Non-satiation is the assumption that is required for maximisation of utility. This assumption can be stated as more is always preferred to less. There are never unfulfilled consumption opportunities available through unused income. It means that people will consume all their wealth over the given periods in order to maximise utility. Continuity Rules out knife edge parts of the indifference curve ensuring a smooth curve. Essentially is rules out any gaps in the indifference sets. 2

3 Completeness Indifference curves provide a complete ranking of all levels of utility. Transitivity Ensures that utility curves are ranked according to preferences and utility level. Utility Curve Assumptions (Extended) Convexity: The diagram illustrates that a linear combination of the two points is always preferred to the utility obtainable at either point A or B. Additionally any linear combination of A and B will be tangent to a higher utility curve. Essentially this means that there is a preference for diversity. Convexity of the Utility Curve A C B Non-Satiation: Non-satiation is the assumption that is required for maximisation of utility. This assumption can be stated as more is always preferred to less. There are never unfulfilled consumption opportunities available through unused income. It means that people will consume all their wealth over the given periods in order to maximise utility. A C U B D In the above consumption space it is evident that in the more is preferred to less. For instance by starting in area B and moving into area D the individual is better off because they can get more for a given level of. Similarly the individual is also better off at A than B because they can get more for the same level of. The individual will be indifferent between the spaces A and D in the diagram because the number of goods changes for both and. By locating in D the individual has given up some in exchange for more therefore they are not better off and on the same utility curve. Area C is the optimal area where the consumer has 3

4 more of both and. The individual will always want to locate in C as it will put them on a higher utility curve. We can summarize these relationships as follows: C > (A = D) > B There is no combination of good that could on the indifference curve in quadrants B or C, because it would have more (C) or less (B) of both products which means the utility is higher or lower. Therefore the graph must slope downward from quadrant A to D. Utility Functions for Specific Preferences Cobb-Douglas An example of a commonly use Cobb Douglas Production Function Utility = U(x, y) = x α y 1 α In this utility function the relative sizes of α and 1-α represent the relative importance of x and y respectively. Also because utility functions are unique only up to a monotonic transformation it is often convenient to have α+(1-α)=1 Perfect Substitutes U3 The indifference curves for perfect substitutes will have constant slopes as they can be trades at a constant rate because they are perfectly interchangeable. An example of perfect substitutes would be energon batteries and superlife batteries which both have the same life spans and all that is really different is the brand name. A person would be willing to give up the same amount of good y for one more unit of good x no matter how much x and y are being consumed. An example of a utility function that is linear because of perfect substitutes is as follows. 4

5 Utility = U(x, y) = αx + βy The marginal rate of substitution for perfect substitutes will become, MRS = α β Which is a constant MRS because both α and β are both numbers. As x increases the MRS will not increase or diminish but stay constant because the formula for the MRS does not include x and therefore has no impact from a change in x. Perfect Compliments Goods that are perfect compliments may be coffee and milk, bread and butter or guns and bullets. These goods are all consume in proportion and this is why we get these L shaped indifference curves. For instance it may be a ratio of 4 ounces of coffee for 1 ounce of milk. Utility = U(x, y) = min (αx, βy) If we let ounces of coffee be denoted as x and ounces of milk y then we would have the following. The utility is given by the smaller of the two terms in the parentheses. Utility = U(x, y) = min (4x, y) At the vertices of the indifference curves the relationship between the goods must be the same as the ratio. In our example the ratio at the corner of intersection will be; Constant Elasticity of Substitution α β = 4 1 = 4 U 5

6 The CES functions are more general than the three utility functions already mentioned above. In fact all three utility functions above are special cases of the CES function. Where δ 1, δ 0 Utility = U(x, y) = xδ δ + yδ δ Utility = U(x, y) = lnx + lny If δ is equal to 1 then the first equation will result in perfect substitutes. If δ is equal to 0 it will correspond to the Cobb Douglas function in the second equation. The shape of the common CES function shown in the diagram above is for the case of δ = 1. Utility = x 1 y 1 = 1 x 1 y In this particular situation σ = 1 (1 δ) = 1 2 and the graph shows sharply curved indifference curves which are between the fixed proportions complements case where σ = 0 and the case of perfect substitutes where σ =. The negative signs show that the marginal utilities of x and y are positive and diminishing. Homothetic Preferences Homothetic: The MRS only depends on the ration of the amounts of the two goods and not on the quantities of the two goods. As an example we examine the MRS of a Cobb Douglas function. Utility = U(x, y) = x α y 1 α αxα 1 y 1 α MRS = U(x, y) = (1 α)x α y α = α β y x This function is homothetic because the MRS only depends on the ratio of y x. This means that a homothetic function maintains the same slope and if multiplied by a constant it will move closer to or further from the origin. In the above function the constant α β is a number which has no effect on the slope and depending on the value will determine how far away from the origin the indifference curve is. When we scale the function up by different numbers we just get higher utility curves which are copies of the original indifference curve. Therefore with homothetic preferences we can understand the individual s behaviour by examining one or two indifference curves because we know preferences will not change much at very different levels. 6