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FPM (Term I): Microeconomics Lecture #2

25th June 2013

Budgets, and Preferences

Lecture #2: 25th June 2013

Lecturer: Arnab Mukherji

Scribe: Dr. None

1 Preferences

So far we have been discussing how the set of feasible bundles, or the budget set, changes

with different price income situations as well as under rationing. Taxation is really like a

price change unless its a lump-sum tax, in which case it is like income shrinking. However,

it is time to return to our original and more complicated question of identifying the scale

that we are going to use to identify if we are doing the best possible. It turns out that no

formal scale is needed to be able to identify the best possible commodity bundle in the

budget, all we need to be able to say is if the consumer prefers a commodity bundle over

another, the dimension of preference itself is not important. We formalize this notion of

preferring a commodity bundle over another with the following notions of preference.

Indifference: for any two commodity bundles A = (xA1, xA

2) and B = (xB

1, xB

2), if

the consumer is equally satisfied with A as with B, then we say that the consumer

is indifferent between the two commodity bundles A and B and we denote this as

A B or (xA1 , xA2 ) (x

B1, xB

2 )

Strict Preference: for any two commodity bundles A and B, if the consumer prefers

bundle A to bundle B then we say that A is preferred to B and we denote this as

A B or (xA1, xA

2) (xB

1, xB

2).

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2 FPM (Term I): Lecture #2

Weak Preference: for any A and B, if the consumer is indifferent between A and

B or may prefer A to B then we say that the consumer weakly prefers A to B and

denote it as A B or (xA1 , xA2

) (xB1 , xB2

). Weak preference essentially rules out

the possibility for the consumer to prefer B over A.

Now we have a language to talk about preference over a pair of commodity bundles, or

more generally on a set of consumption bundles. We still havent made any assumptions

about consumers preferences on the commodity space. These are simply definitions

detailing when a consumer is indifferent between A and B or strongly, or weakly prefers

A over B. So far we have not made any assumption about the pattern of preferences

over the budget set, and one can have all kinds of weird inconsistent preferences. We

make some simple and arguable credible that are sufficient to drive or generate consistent

preference over budget sets. These are known as the axioms of consumer theory:

Complete: For any two commodity bundles A = (xA1 , xA2

) and B = (xB1, xB

2) it must

be the case that the consumer can identify if he likes A more than B (i.e. A B),

or B more than A (i.e B A), or is indifferent between A and B (i.e A B).

Reflexive: Any commodity is at least as good as itself: i.e. A A.

Transitive: For any three commodity bundles A,B, and C, ifA B and B C

then A B.

When all three properties hold, then the preferences are said to be ordered, since these

axioms are sufficient to allow us to rank all commodity bundles (not necessarily uniquely,

as there may be ties). Reflexivity is the least problematic in terms of credibility as all

it says is that a good is as good as itself. Completeness requires the consumer to not

only know all the possible combinations of goods, but it also requires the person to have

a preference over all these bundles, and most criticisms about its credibility are centered

on the information and processing of such information; but even this is usually not the

source of criticism for utility theory. Much of the criticism centers on the transitivity ax-

iom since it rules out all kinds of cycles in behavior, such as A being preferred to B being

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4 FPM (Term I): Lecture #2

We make a few assumptions to rule out these cases:

Monotonicity: The idea captures the notion that more is better: thus, if we are

looking at two commodity bundles A = (x1, x2) and B = (x1 + , x2), then it must

be the case that B A; ruling out A B. This idea is also another manifestation

of the idea that we are essentially concerned with the idea of decision making in a

world of scarcity and not one in which we are largely satiated.

The Strictly Preferred to set is a convex set. Does anyone remember the definition

of a convex set? The idea is that averages are preferred to extreme values always.

Discuss the difference between strict convexity and weak convexity.

An important implication of the Monotonicity assumption is that the indifference curve

must be downward sloping. How would we show this? Together the two assumption, in

addition to the axioms of preference, imply that the indifference curves are well-behaved

i.e. we see downward sloping indifference curves (also called indifference levels, surfaces,

etc.).Marginal Rate of Substitution The Marginal Rate of Substitution (MRS) for an indif-

ference curve is simply the slope of the tangent line at that point; as discussed, this is also

known as the derivative of the the indifference curve. MRS is short for the Marginal Rate

of Substitution ofx1 for x2 and should usually be written as MRSx1,x2 to be exactly right;

however, we will be sloppy and continue with MRS to minimize notational complication

and specifiy it should there be any confusion. Is the MRS a positive number or a negative

number?

Thus, wed define the MRS as:

MRS= limx10

x2x1

(1)

What does the MRS capture? The indifference curve is a set of points that the

consumer is indifferent between, thus, the MRS at a point captures the amount of x1

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FPM (Term I): Lecture #2 5

that the person is willing to give up for an additional amount of x2 so that his utility

remains unchanged. If one is willing to think of this as a rate of exchange, then the MRS

captures the rate of exchange of one good for another. Now, recall that the set of preferred

bundles for each commodity bundle is convex. If the set of preferred bundles is strictly

convex, then we have the case of diminishing Marginal Rate of Substitution ofx1 for x2

... intuitively it suggests that at higher and higher levels ofx1 the consumer is likely to

part with larger and larger amounts of x1 for the same marginal increment in x2. The

slope is declining (in absolute value) as we look at the slope for higher values of x1.