# Microeconomics - consumer preferences

QUESTION 1.(a)

A utility function is a function U (x) that assigns a number to every consumption bundle x € X.

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Utility function U () represents preference relation º if for any x and y, U (x) ≥ U (y) if and only

if x º y - **Microeconomics - consumer preferences** introduction. That is, function U assigns a number to x that is at least as large as the number it assigns to y if and only if x is at least as good as y. The nice thing about utility functions is that if we know the utility function that represents a consumer’s preferences, we can analyze these preferences by deriving properties of the utility function.

Consider a typical indifference curve map, and assume that preferences are rational. The line drawn in figure is the line x2 = x1, but any straight line would do as well. Notice that we could identify the indifference curve Ix by the distance along the line x2 = x1 we have to travel before intersecting Ix. Since indifference curves are downward sloping, each Ix will only intersect this line once, so each indifference curve will have a unique number associated with it. Further, since preferences are convex, if x >y, Ix will lay above and to the right of Iy (i.e. inside

X2

X1

a

Iy), and so Ix will have a higher number associated with it than Iy.

We will call the number associated with Ix the utility of x. Formally, we can define a function

u (x1, x2) such that u (x1, x2) is the number associated with the indifference curve on which (x1, x2) lies. It turns out that in order to ensure that there is a utility function corresponding to a particular preference relation, we need to assume that preferences are rational and continuous. The assumption that preferences are rational agrees with how we think consumers should behave, so it is no problem.

The assumption that preferences are continuous is what we like to call a technical assumption, by which we mean that is that it is needed for the arguments to be mathematically rigorous (read: true), but it imposes no real restrictions on consumer behavior. Indeed, the problems associated with preferences that are not continuous arise only if we assume that all commodities are infinitely divisible (or come in infinite quantities). Since neither of these is true of real commodities, we do not really harm our model by assuming continuous preferences. Utility is an ordinal concept.

Utility is an ordinal (i.e. ordering or ranking) concept.

For example, if U(x) = 6 and U(y) = 2 then bundle x is strictly preferred to bundle y. However, x is not necessarily “three times better” than y.

QUESTION 1: (b)

Price of the product: The law of demand states that other demand determinants remaining constant, when the price of a given product falls demand rises and vice-versa. It means, if the price of a given product falls, the market demand for the given product will rise and if the price rises, the market demand for the product will fall. However, the product under consideration should be a normal product and not an inferior good or prestige goods.

Income: The absolute size of income determines the quantum of goods and services produced in an economy. Thus greater the size of income of the people, greater will be the demand for goods and services and vice-versa. In addition the rate of growth of incomes will determine the incremental growth in demand for goods and services. Thus higher the rate of growth of income, higher will be the quantity demanded by the people.

Tax: The tax structure and the tax exempt level of income determines the disposable income of the people. It is the disposable income of people that determines the market demand. Higher the tax exempt

level of income and less steeply graded the tax structure, greater will be the disposable income the people and greater will be the market demand.

Climate: Market demand for certain products changes along with the change in the climate. While the demand for soft drinks rises dramatically in the hot summers, the demand for wine increases during winters. Seasonal products like woolen wear and rain wears will have demand only during their respective seasons and almost nil demand when the season is over. Climate change does influence optimal choice of consumers.

QUESTION 1.(c)

There is no perfect competition in reality and constant advertising plays an important role in changing or altering the purchase behavior of consumers. One of the most important objective of advertising is to inform the people of the availability of goods and services so that among the alternatives available the consumer can make a rational choice. Advertising is the characteristic feature of imperfect competition. A large number of goods that we consume in the modern times are either produced by monopolistic or oligopolistic firms. Constant advertising by these firms creates brand consciousness. It creates demand where it doesn’t exist. It creates new desires where the consumers may not need it. Apart from groceries that do not require much advertising as it is a basic need even a topline product will have little or no demand, if the product is not advertised. Today advertisements are more attractive and appeals emotionally to the consumers which affects the rational thinking of consumers to some extent and creates desires.

(“MICROECONOMICS- CONSUMER PREFERENCES”) (“Page # 3”)

QUESTION 2.(a)

Ordinal scales require respondents to express their feelings of relative magnitude about the given topic. Ordinal scales activate both the assignment and order scaling properties and allow researchers to create a hierarchical pattern among the possible raw data responses (or scale points) that determine “greater than/less than” relationships. Data structures that can be derived from ordinal scale measurements are in the forms of medians and ranges as well as modes and frequency distributions. An example of a set of ordinal scale descriptors would be “complete knowledge,” “good knowledge,” “basic knowledge,” “little knowledge,” and “no knowledge.” While the ordinal scale measurement is an excellent design for capturing the relative magnitudes in respondents’ raw responses, it cannot capture absolute magnitudes.

An interval scale activates not only the assignment and order scaling properties but also the distance property. This scale measurement allows the researcher to build into the scale elements that demonstrate the existence of absolute differences between each scale point. Normally, the raw scale descriptors will represent a distinct set of numerical ranges as the possible responses to a given question (e.g., “less than a mile,” “1 to 5 miles,” “6 to 10 miles,” “11 to 20 miles,” “over 20 miles”). With interval scaling designs, the distance between each scale point or response does not have to be equal. Disproportional scale descriptors (e.g., different-sized numerical ranges) can be used. With interval raw data, researchers can develop a number of more meaningful data structures that are based on means and standard deviations, or create data structures based on mode, median, frequency distribution, and range. Ratio scales are the only scale measurements that simultaneously activate all four scaling properties (i.e., assignment, order, distance, and origin). Considered the most sophisticated scale design, they allow researchers to identify absolute differences between each scale point and to make absolute comparisons between the respondents’ raw responses. (2001, Ref 1 )

Cardinal scales represent more aspects of the domain than the ordering. On a ratio scale, the ratio v(a)/v(b) between the numbers assigned to the objects a and b represents some real relation between a and b. In one class of cases (e.g., length) this relation may be thought of as involving the aggregation of a number v(a) of unit objects to form a complex object which is equal to a, while another number v(b) of units are required to form an object equal to b. In other contexts the represented operation is intuitively more analogous to arithmetical division; it may for example be the immediately judged proportion or share of b, which is felt to be taken up by a (“How intense is the pain a, expressed as a share of the intensity of b?”).

The only transformations of a scale which preserve ratios are multiplications with a constant. Ratio scaling requires rather strict formal properties of the domain. (Ref. 2)

QUESTION 2.(b)

Let the two goods be x1 and x2. U(x1,x2) is the utility function. Then,

Perfect Substitutes Utility Function is

U(x1, x2) = ax1+bx2

Perfect Complements Utility Function is

(“MICROECONOMICS- CONSUMER PREFERENCES”) (“Page # 4”)

U(x1, x2) = min(x1,x2)

QUESTION 2.(c)

Let L be the Lagrangian optimization, X and Y be the two goods whose Marshallian demand functions are to be found out.

Combining & yields,

Substituting into and solving for yields:

Cobb-Douglas Utility Function

U(X, Y) = X Y (a>0,b>0)

a

b

U(X,Y) =[1/( Py Px + Px)] [1/ Px Py + Py] (a>0,b>0)

One theory counter to Marshall is that price is already known in a commodity before it reaches the market, negating his idea that some abstract market is conveying price information. The only thing the market communicates is whether or not an object is exchangeable or not (in which case it would change from an object to a commodity). This would mean that the producer creates the goods without already having customers — blindly producing, hoping that someone will buy them (“buy” meaning exchange money for the commodities). Modern producers often have market studies prepared well in advance of production decisions; however, misallocation of factors of production can still occur.

(“MICROECONOMICS- CONSUMER PREFERENCES”) (“Page # 5”)

Keynesian economics also runs counter to the theory of supply and demand. In Keynesian theory, prices can become “sticky” or resistant to change, especially in the case of price decreases. This leads to a market failure.

Gregory Mankiw’s work on the irrationality of actors in the markets also undermines Marshall’s simplistic view of the forces involved in supply and demand.

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QUESTION 2.(d)

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Definition of Revealed Preference: Given some vectors of prices and chosen bundles (p ; x ) for t = 1,..

t

t

t

t

:… T, we say x is directly revealed preferred to a bundle x (written x RDx) if p x >= p x. We say x is

t

revealed preferred to x (written x Rx) if there is some sequence r, s, t u, v such that p x >= p x , p x >=

u

p x; …. ; p x >= p x. In this case, we say the relation R is the transitive closure of the relation RD.

t

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s

s

( Ref. 3, p:2)

Definition of Weak Axiom of Revealed Preference: If x RDx then it is not the case that x RDx.

Algebraically, p x >= p x -> p x < p x. (Ref. 4, p:2)

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s

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Definition of Strong Axiom of Revealed Preference (SARP): If x Rx then it is not the case that x Rx.

Algebraically, SARP says x Rx implies p x < p x. (Ref. 5, p:3)

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Definition Generalized Axiom of Revealed Preference (GARP): The data (p ; x )satisfy the Generalized

Axiom of Revealed Preference (GARP) if x Rx implies p x <= p x. (Ref. 6, p:7)

Definition 6 (Homothetic Axiom of Revealed Preference) A set of data (p ; x ) for t = 1,….,, T satisfy the Homothetic Axiom of Revealed Preference (HARP) if for every sequence r; s; t; : : : ; u; v

p x /p x p x /p x ……… p x /p x >= 1 (Ref. 7,p:12)

(“MICROECONOMICS- CONSUMER PREFERENCES”) (“Page # 6”)

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Suppose we are given a definite set of observed budgets and choices (p ,x ) for t = 1,…, T that are consistent with GARP and a new price p and expenditure y. The possible bundles x that could be demanded at (p ; y ) by the consumers would be:

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Clearly all that is necessary is to describe the set of x for which the (expanded) data set (p , x ) for t = 0,…., T satisfy GARP. Varian [1982a] calls this the set of supporting bundles. In an analogous way, one can choose a new bundle x and ask for the set of prices at which this bundle could be demanded. This is the set of supporting prices. Formally,

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S(x ) = {p: ( p , x ) satisfy GARP for t = 0,…, T}

Of course, one could also ask about demanded bundles or prices that are consistent with utility functions with various restrictions imposed such as homotheticity, separability, specific forms for Engel curves and so on. (Ref 8, p:12)

QUESTION 3.

In microeconomics, the utility maximization problem is the problem consumers face: “i.e the way they should spend their money in order to maximize their utility?

Suppose their consumption set

has L commodities. If the prices of the L commodities are

and the consumer’s wealth is w, then the set of all affordable packages, the budget set, is

.

The consumer would like to buy the best package of commodities it can afford. If

is the consumer’s utility function, then the consumer’s optimal choices x(p, w) are

.

Finding x(p, w) is the utility maximization problem.

(“MICROECONOMICS- CONSUMER PREFERENCES”) (“Page # 7”)

The solution x(p, w) need not be unique. If u is continuous and no commodities are free of charge, then x(p, w) is nonempty. Proof: B(p, w) is a compact space. So if u is continuous, then the Weierstrass theorem implies that u(B(p, w)) is a compact subset of . By the Heine-Borel theorem, every compact set contains its maximum, so we can conclude that u(B(p, w)) has a maximum and hence there must be a package in B(p, w) that maps to this maximum.

If a consumer always picks an optimal package as defined above, then x(p, w) is called the Marshallian demand correspondence. If there is always a unique maximizer, then it is called the Marshallian demand function. The relationship between the utility function and Marshallian demand in the Utility Maximization Problem mirrors the relationship between the expenditure function and Hicksian demand in the Expenditure Minimization Problem.

In microeconomics, the expenditure minimization problem is the dual problem to the utility maximization problem: “how much money do I need to be happy?”. This question comes in two parts. Given a consumer’s utility function, prices, and a utility target,

how much money would the consumer need? This is answered by the expenditure function.

what could the consumer buy to meet this utility target while minimizing expenditure? This is answered by the Hicksian demand correspondence.

Expenditure function

Formally, the expenditure function is defined as follows. Suppose the consumer has a utility function u defined on L commodities. Then the consumer’s expenditure function gives the amount of money required to buy a package of commodities at given prices p that give utility greater than u * ,

where

is the set of all packages that give utility at least as good as u * .

Hicksian demand correspondence

Secondly, the Hicksian demand correspondence h(p,u * ) is defined as the cheapest package that gives the desired utility. It can be defined in terms of the expenditure function with the Marshallian demand correspondence

h(p,u * ) = x(p,e(p,u * )).

(“MICROECONOMICS- CONSUMER PREFERENCES”) (“Page # 8”)

If the Marshallian demand correspondence x(p,w) is a function (i.e. always gives a unique answer), then h(p,u * ) is also called the Hicksian demand function. (Ref. 9)

Start with the UMP.

max u (x)

s.t : p · x ≤ w.

The solution to this problem is x (p,w), the Walrasian demand functions. Substituting x (p,w)

into u (x) gives the indirect utility function v (p,w) ≡ u (x (p,w)). By differentiating v (p,w) with

respect to pi and w, we get Roy’s identity, xi (p,w) ≡ -vpi/vw

Now the EMP.

min p · x

s.t. : u (x) ≥ u.

The solution to this problemis the Hicksian demand function h (p, u), and the expenditure function

is defined as e (p, u) ≡ p · h (p, u). Differentiating the expenditure function with respect to pj gets

you back to the Hicksian demand, hj (p, u) ≡ əe(p,u)/əpj

.The connections between the two problems are provided by the duality results. Since the same

bundle that solves the UMP when prices are p and wealth is w solves the EMP when prices are p

and the target utility level is v (p,w), we have that

x (p,w) ≡ h (p, v (p,w))

h (p, u) ≡ x (p, e (p, u))

Applying these identities to the expenditure and indirect utility functions yields more identities:

v (p, e (p, u)) ≡ u

e (p, v (p,w)) ≡ w. (Ref. 10, p:64)

A Note on Duality

Consider a price vector p and wealth w. The bundle that solves the UMP, x. = x (p,w) is found at

the point of tangency between the budget line and the consumer’s utility isoquant. The consumer’s

utility at this point is given by u. = u (x.). Thus x. is the point of tangency between the line

p · x = w and the curve u (x) = x..

Now, consider the EMP when the target utility level is given by u.. The bundle that solves

the EMP is the bundle that achieves utility u. at minimum cost. This is located by finding the

point of tangency between the curve u (x) = u. and a budget line (which is what (3.3) says). But,

we already know from the UMP that the curve u (x) = u. is tangent to the budget line p · x = w

at x. (and is tangent to no other budget line). Hence x. must solve the EMP problem when the

target utility level is u.! Further, since x. lies on the budget line, p · x. = w. So the minimum

cost of achieving utility u. is w. Thus the UMP and the EMP pick out the same point.

Let me restate what I’ve just argued. If x. solves the UMP when prices are p and wealth is w,

then x. solves the EMP when prices are p and the target utility level is u (x.). Further, maximal

(“MICROECONOMICS- CONSUMER PREFERENCES”) (“Page # 9”)

utility in the UMP is u (x.) and minimum expenditure in the EMP is w. This result is called the

“duality” of the EMP and the UMP. (Ref. 11, p:53)

Consider a consumer who has wealth w and faces initial prices p0. Utility at this point is given

v ¡p0,w¢.

If prices change to p1, the consumer’s utility at the new prices is given by:

v ¡p1,w¢.

Thus the consumer’s utility increases, stays constant, or decreases depending on whether:

v ¡p1,w¢. v ¡p0,w¢ is positive, equal to zero, or negative.

While looking at the change in utility can tell you whether the consumer is better of or not, it

cannot tell you how much better off the consumer is made. This is because utility is an ordinal

concept. The units that utility is measured in are arbitrary. Thus it is meaningless to compare,

for example, v ¡p1,w¢. v ¡p0,w¢ and v (p2,w) . v (p3,w). And, if v () and y () are the indirect

utility functions of two people, it is also meaningless to compare the change in v to the change in

y.

Bibliography:

1. .Joseph F. Nair, Jr, Robert P. Bush, David J. Ortinau (2001), http://www.mhhe.com/business/marketing/hair/student/summary/ch12slo.mhtml, Marketing research, Mc Graw Hill Company.

2.http://www.phil.gu.se/qol/quantolife, Quantifying Quality of Life

3.http://www.sims.berkeley.edu/~hal/Papers/2005/revpref.pdf, Revealed Preference, Hal R. Varian

4.ibid

5.ibid

6.ibid

7.ibid

8.ibid

9.http://www.answers.com/topic/expenditure-minimization-problem, Expenditure Minimization Problem, http://www.answers.com/ , Search- Utility maximization problem.

10..http://ksghome.harvard.edu/~nmiller/notes/notes3.pdf, The Traditional Approach to Consumer Theory

11….ibid

.