Lecture Slides 1BB 2012

8/2/2019 Lecture Slides 1BB 2012

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Lecture Slide 1

Econ 310

Princeton University

February 2012


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about the class

I 10 problem sets: do them and understand them.

I work them out by yourself; go over the answers to make sure

you understand them.

I read material before class, even if it is difficult to understand

I check blackboard regularly

I you are graded on the best 8 problem sets.


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what is (micro)economics

build models consisting oftrade-off talking rational economic agents(totreps)1

I totrep has preferences: for every pair of alternatives

(consumption goods, jobs,...), the preference tells us what totrep

would choose.I totreps preferences are rational that means there are no

inconsistencies.

I totrep always chooses his most preferred alternative among the

available options.

use these models to analyze economic institutions and policies.

1This term is taken from David Kreps, Notes on the theory of choice."


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Plan

I analyze totreps behavior.

I optimal consumption choices of a consumer.I optimal price and output decisions of a monopolist.I ...

I analyze the interaction of totreps (equilibrium analysis.)

I what happens when totreps interact in a competitive

market?I what happens when one totrep sells a used car to another?I

what happens when oligopolistic firms run by totrepscompete?

I ...


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Purpose:

I Develop an understanding of how economic institutions

(regulations, taxes, etc.) affect economic outcomes.

I When an economist says: this is what happens when we

institute a minimum wage, she really means: this is what

happens to the choices of a population of totreps if we institute a

minimum wage.

I Obvious question: are the choices of totreps a good

approximation of what really happens?

I Why this approach?


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Part 1: preference and utility


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Outline:

I define preferences

I introduce utility functions

I what does it mean for a utility function to represent a

preference?

I book: this part is a bit different from the book; read chapter 3.

Key Takeaway: if totreps preferences have no inconsistencies then we candescribe them with a utility function.


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preferences

I there is a (finite) set of objects X = {x,y, z,...}

I for example: X could be the occupations, or consumption plans.

I totrep makes pairwise comparisons, expressing a preference for

some objects over others.

The symbol denotes a binary relation on the set of choices X:

interpret x y as x is weakly preferred to y.

totrep is indifferent between x and y ifx y and y x: we write

x y.

totrep strictly prefers x over y ifx y but we dont have y x: we

write x y.


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rationality

Assumption 1: The preference is complete: for all x,y in the set X we

have x y or y x (possibly both).

Assumption 2: The preference is transitive: x y and y z implies

x z.


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utility

A utility function is a convenient way to describe totreps preferences.

A utility function u associates to each object in X a real number u(x).

The utility function u represents the preference if

x y implies u(x) u(y) and u(x) u(y) implies x y

TheoremThe preference can be represented by a utility function if and only if it

satisfies Assumptions 1 and 2.


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example

X = {MD,JD,MBA, PhD}

I MD JD;MD MBA;MD PhD

I JD MBA;JD PhD

I MBA JD;MBA PhD

complete? transitive?

MD JD MBA PhD


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example continued...

We have a complete and transitive preference

MD JD MBA PhD

Two (of many) possible utility functions:

I Utility function #1:

u1(MD) = 1, u1(JD) = u1(MBA) = 0, u1(PhD) = -.01

I Utility function #2:

u2(MD) = 1, u2(JD) = u2(MBA) = .99, u2(PhD) = -1000


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utility

TheoremThe preference can be represented by a utility function if and only if itsatisfies Assumptions 1 and 2.

We will prove this theorem. It has two parts:

I Only if part: If the preference can be represented by a utility

function, then the preference must be complete and transitive.

We will do this first (very easy).

I If part: If the preference is complete and transitive then we can

find a utility function that represents it. This is the more difficultpart. We will construct a particular utility function for the

preference that does the job.


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Only if part

I

We have a utility function u that assigns each element ofX a realnumber.

I The utility function represents a preference . That is: x y if

and only ifu(x) u(y).

First, we prove that the preference must be complete:

I take x,y then we have u(x) u(y) or u(y) u(x).

I In the first case, we have x y since u represents the preference.

I In the second case we have y x since u represents thepreference.

I So, must be complete. Easy!


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Only if part continued

Next, we prove that the preference must be transitive:

I take x,y, z with x y and y z. We must prove that x z.

I Since u represents the preference we know that u(x) u(y) and

u(y) u(z)

I But this implies that u(x) u(z).

I Since u represents the preference this, in turn, implies x z.

I Therefore, is transitive. Done!


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an intermediate step

if the preference is transitive then x z and y x imply y z.

I first, we show that y z: since x z we know that x z and

therefore transitivity implies y z.

I next, we show that we cannot have z y: ifz y then y x

and transitivity would imply z x. But x z implies that we

dont have z x and therefore z y is impossible.

I so: y z and we dont have z y. This means y z. Done!

f


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If partNow, we prove that a complete and transitive preference can be

represented by a utility function.

Define Z(x) to be all those elements ofX that are strictly worse than x

Z(x) = {z 2 X|x z}

Here is what we know about Z(x):

(i) Ifx y then Z(x) = Z(y). Why? ifx z then, since y x itfollows that y x. The intermediate step now implies that

y z. Conversely, ify z it follows that x z

(ii) Ifx y then Z(y) is a subset ofZ(x): every element ofZ(y) is in

Z(x). Why? ify z and x y then it follows from theintermediate step that x z.

(iii) Ifx y then Z(y) is a strict subset ofZ(x): at least one elementofZ(x) is not in Z(y). Why? y is in Z(x) because x y. But y isnot in Z(y).

if


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...if part...

Here is our utility function:

u(x) = number of elements in Z(x)

utility ofx: is the number of choices that are worse than x.

We must prove that for this choice ofu:

(i) x y implies u(x) u(y)

(ii) u(x) u(y) implies x y.

if


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...if part...

(i) x y implies u(x) u(y)

I either x y or x y.

I

ifx

y then (as we have shown) Z(x) = Z(y). But that meansu(x) = u(y). Good!

I ifx y then (as we have shown) Z(y) is a strict subset ofZ(x).But that means u(x) > u(y). Good!

if t


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...if part....

(ii) u(x) u(y) implies x y.

I we are trying to prove that u(x) u(y) implies x y.

I which is the same as: ifx is not preferred to y then u(x) is notgreater or equal to u(y).

I which is the same as: ifx is not preferred to y then u(y) > u(x)

I by completeness: ifx is not preferred to y then y x

I So, to complete the proof we must show that y x implies

u(y) > u(x).

I But we have already done that on the previous page! Done!


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Part 2: consumers

tli


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outline

I introduce the framework of consumer choice

I describe preferences by indifference curves

I compute marginal rates of substitution

I book: we follow the book. read chapters 3 and 4 in Varian.

consumption vectors


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consumption vectors

I in consumer theory, the alternatives are consumption vectors.

I 2 goods; a vector (x1, x2) 2 R2 specifies the quantities of good 1

and good 2.

I we call totrep a consumer if his preferences are over consumption

vectors

I set of alternatives: all non-negative vectors in R2.

utility functions


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utility functions

Maintained assumption: the consumer has complete and transitive

preferences on the non-negative vectors in R2 that can be represented

by a utility function.

The utility function u represents the consumers preference if

(x1, x2) (x01, x

02) implies u(x1, x2) u(x

01, x

02)

and, conversely,

u(x1, x2) u(x01, x

02) implies (x1, x2) (x

01, x

02)

transformations of utility functions


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transformations of utility functions

Fact: ifu represents a preference then so does any strictly increasing

function ofu

Why? Iff is a strictly increasing function then

u(x1, x2) u(x01, x

02) if and only iff(u(x1, x2)) f(u(x

01, x

02))

for example:

I u and 2 u represent the same preference

I u and u3 represent the same preference

I Ifu is always positive then u and ln u represent the same

preference

indifference curves: an example


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indifference curves: an example

utility function is

u(x1, x2) = x1 x2

Indifference curve: all the combinations ofx1 and x2 that give the

same utility:

u(x1, x2) = x1 x2 = utility level k

Therefore

x2 = g(x1) =k

x1

indifference curves are a family of hyperbolas


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indifference curves


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indifference curves

I for each consumption vector (x1, x2) the indifference curvethrough (x1, x2) divides R

2+ into those consumptions that are

better than (x1, x2) and those that are worse than (x1, x2).

averages are better


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averages are better

I the indifference curves in our

example have a particular shape: the

average of two points on the

indifference curve is better than the

original points.

I We say that the consumerspreferences are convex: the set of

vectors better than (x1, x2) is aconvex set.

I Can you think of situations where

convexity makes sense?

I Can you think of situations where it

does not make sense?

marginal rate of substitution


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marginal rate of substitution

I the marginal rate of substitution is the slope of the indifference

curve

I the slope of a curve is the derivative of the function thatdescribes the indifference curve.

marginal rate of substitution in the example


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marginal rate of substitution in the example

Recall

I u(x1, x2) = x1 x2.

I the function g : R+ ! R+ where g(x1) = k/x1 describes theindifference curve

the derivative ofg is

g 0(x1) = -k(x1)2

since k is the utility level, we have

k = x1 x2

substitute for k:

-k

(x1)2= -

x1 x2(x1)2

= -x2x1

= MRS

MRS: an alternative derivation


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MRS: an alternative derivation

I for u(x1, x2) = x1

x2 we found that

MRS = -x2x1

I Notice x2 =u(x1,x2)

x1 and x1 =u(x1,x2)

x2

Therefore:

MRS = -x2

x1= -

u(x1,x2)x1

u(

x1,x2)

x2

Is this true for all utility functions?

...yes... as long as the MRS is well defined


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y g

Indifference curve is a function g that solves the equation

u(x1,g(x1)) = k

The indifference curve g gives us for each x1 the corresponding

x2 = g(x1) so that utility is k.

Therefore: if we change x1, the utility must stay unchanged:

u(x1,g(x1))

x1+

u(x1,g(x1))

x2 g 0(x1) = 0

and solving for g 0 we get

g 0(x1) = -

u(x1,g(x1))x1

u(x1,g(x1))x2

= -

u(x1,x2)x1

u(x1,x2)x2

= MRS

other examples of utility functions...


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p yThe good x2 is money that is spent on many other goods. Utility is

linear in money:

u(x1, x2) = x2 + v(x1)

and v is a concave function.

MRS = -

u(x1,x2)x1

u(x1,x2)x2

= -v 0(x1)

MRS is independent ofx2.

Each indifference

curve is a verticallyshifted version of a

single indifference

curve

..examples contd...


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p

The goods are red and green pencils. The consumer cares about the

total number of pencils

u(x1, x2) = x1 + x2

MRS = -1

the indifference

curves are straight

lines with slope-

1

left shoe, right shoe


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g

the goods are perfect complements; the consumer needs one of each

u(x1

, x2

) = min{x1

, x2}

In this case, we cant use our formula to compute the MRS

I the indifference

curves are

L-

shaped.

I ifx1 x2 the

additional right

shoe delivers

no utility.I ifx2 x1 the

additional left

shoe delivers

no utility.

useless stuff


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the consumer has use for only one of the goods

u(x1, x2) = x1

Again, we cant use the formula to compute the MRS because the

marginal utility of good 2 is zero.

I indifference

curves are

vertical lines

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