Economics 102 Lecture 5 Choice Rev

7/28/2019 Economics 102 Lecture 5 Choice Rev

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Lecture 5: Theory of the Consumer:

Choice

Optimal choice

Consumer demand

Examples of demand

Utility functions from demand

functions Implications of the MRS condition

An application: Taxes

Formal derivation of demand curves


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Object: find the bundle in the budget set that is onthe highest indifference curve

The bundle on the highest indifference curve thatjust touches the budget line is labeled .This is the optimal choice for the consumer.

It is the best bundle that she can afford, althoughthere are other bundles that she prefers.

See the following diagrams

),( *2*

1 xx

x1

x2


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x1

x2Utility

Utility x2

x1


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x1

x2

Utility

Utility

x1

x2


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Utility

x1

x2

Utility

x1

x2


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Utility

x1

x2

Utility

x1

x2

Affordable, but notthe most preferred

affordable bundle.


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x1

x2

Utility

Affordable, but not

the most preferred

affordable bundle.

The most preferred

of the affordable

bundles.

x1

x2

Utility


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Utility

x1

x2

Utility

x1

x2


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Utilityx1

x2

x1

x2


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1

x1

x2

Affordable

bundles

x1

x2

Affordable

bundles


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x1

x2

Affordable

bundles

More preferred

bundles

Affordable

bundles

x1

x2

More preferred

bundles


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1

x1

x2

x1*

x2*

x1

x2

x1*

x2*

(x1*,x2*) is the most

preferred affordable

bundle.


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1

At this choice, the indifference curve is tangent tothe budget line.

Tangency condition doesnt hold for all cases, but itdoes hold for the more interesting cases.

What is always true is that at the optimal point, theindifference curve cant cross the budget line

Exception: boundary optimum


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In general, tangency condition is only anecessary condition for optimality. It is not a sufficient condition as indicated in the

case of bent indifference curves or curves withconvex and nonconvex parts. Fig 5.4

Tangency condition of the budget line andindifference curve is sufficient in the case ofconvex preferences.

In general, there may be more than one bundlethat satisfies the tangency condition. However,for strictly convex preferences (i.e., no flat spotson IC), there is only one optimal choice on eachbudget line.

7


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When x1* > 0 and x2* > 0 the optimal choice

bundle is INTERIOR.

If buying (x1*,x2*) costs P m then the budget

is exhausted.

x1

x2

x1*

x2*

(x1*,x2*) is interior .

(a) (x1*,x2*) exhausts the

budget; p1x1* + p2x2* = m.

(b) The slope of the indiff.

curve at (x1*,x2*) equals

the slope of the budget

constraint.


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1

(x1*,x2*) satisfies two conditions:

(a) the budget is exhausted;

p1x1* + p2x2* = m

(b) the slope of the budget constraint, -

p1/p2, and the slope of the indifference

curve, the MRS, containing (x1*,x2*) are

equal at (x1*,x2*).

Economic interpretation of the tangency condition:

MRS - as a rate of exchange where the consumer is justwilling to stay put.

Market is offering a rate of exchange equal to the ratio ofprices.

At equilibrium (optimal choice), the consumer must be ata rate where MRS is equal to the rate of exchange.

If MRS is not equal to the price ratio, there could be somescope for exchange of one good for another that isaffordable to the consumer.

Thus, whenever MRS is not equal to the price ratio, thenthe consumer is not at the optimal choice.


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1

Demanded bundle or ordinary demand Optimalchoices of goods 1 and 2 at some set of prices andincome. It is the most preferred affordable bundle at thegiven prices and budget.

Demand function Function that relates the optimalchoice, the quantity demanded to the different values ofprices and incomes As prices and incomes change, the optimal bundle demanded

would also change.

Written as functions of both prices and income:

Different preferences lead to different demand functions

),,,(

),,,(

212

211

mppx

mppx

When x1* > 0 and x2* > 0 and (x1*,x2*)exhausts the budget, and indifference curveshave no kinks, the ordinary demands areobtained by solving:

(a) p1x1* + p2x2* = y

(b) the slopes of the budget constraint, -p1/p2, and of the indifference curvecontaining (x1*,x2*) are equal at (x1*,x2*).


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Suppose that the consumer has Cobb-

Douglas preferences.

ThenMU

U

xax xa b1

11

12

MUU

xbx xa b2

21 2

1

baxxxU(x 2121 ),

So the MRS is

At (x1*,x2*), MRS = -p1/p2 so

./

/

1

2

1

21

2

1

1

2

1

1

2

bx

ax

xbx

xax

xU

xU

dx

dxMRS

ba

ba

axbx

p

px

bp

apx2

1

1

22

1

21

*

*

* *. (A)


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(x1*,x2*) also exhausts the budget so

p x p x m1 1 2 2* *

. (B)

So now we know that

*

1

2

1*

2x

ap

bpx

(A)

p x p x m1 1 2 2* * .(B)


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2

So now we know that

xbp

apx2

1

21

* * (A)

p x p x m1 1 2 2* *

. (B)Substitute

So now we know that

xbp

apx2

1

21

* * (A)

p x p x m1 1 2 2

* *

. (B)

p x pbp

apx m1 1 2

1

21

* *.

Substitute

and get

This simplifies to .


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2

xbm

a b p2

2

*

( ).

Substituting for x1* in

p x p x m1 1 2 2* *

then gives

xam

a b p1

1

*

( ).

So we have discovered that the most preferred affordable

bundle for a consumer with Cobb-Douglas preferences

U x x x xa b( , )1 2 1 2

is

( , )( )

,( )

.* * ( )x x am

a b p

bm

a b p1 2

1 2


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2

x1

x2

xam

a b p1

1

*

( )

x

bm

a b p

2

2

*

( )

baxxxxU 2121 ),(

Property of the Cobb-Douglas - consumer always spends afixed fraction of his income on each good.

The size of the fraction depends on the exponents of thefunction.

fraction spent on good 1:

substituting the demand function for good 1:

the same is true for good 2

This is why it is often convenient to represent theexponents of the C-D function as equal to one.

mxp /11

ba

a

p

m

ba

a

m

p

m

xp

1

111


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If either x1* = 0 or x2* = 0 then the ordinary

demand (x1*,x2*) is at a corner solution to

the problem of maximizing utility subject to

a budget constraint.

For instance, the demand for perfect

substitutes

Perfect substitutes

A consumer will purchase the cheaper one. If

they have the same price, then the consumer

doesnt care which one he purchases.

Demand function for good 1:

.when,0

;when,m/pand0betweenno.any

;when,/

21

211

211

1

pp

pp

pppm

x


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2

x1

x2

MRS = -1

x1

x2

MRS = -1

Slope = -p1/p2 with p1 > p2.


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2

x1

x2

MRS = -1


x1

x2

2

*

2p

yx

0*1 x

MRS = -1



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2

x1

x2

xy

p1

1

*

x2 0*

MRS = -1

Slope = -p1/p2 with p1 < p2.

So when U(x1,x2) = x1 + x2, the most

preferred affordable bundle is (x1*,x2*)

where

0,),(

1

*

2

*

1p

yxx

and

2

*

2

*

1 ,0),(p

yxx

if p1 < p2

if p1 > p2.


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2

x1

x2

MRS = -1

Slope = -p1/p2 with p1 = p2.

y

p1

y

p2

x1

x2

All the bundles in the

constraint are equally the

most preferred affordable

when p1 = p2.

y

p2

y

p1


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Perfect Complements: Optimal choice would always lieon the diagonal, no matter what the prices are.

No matter what the prices, consumer is purchasing thesame amount of good 1 for each good 2 (or at the fixed rateat which they complement one another).

Adding the budget constraint to this condition and solvingalgebraically just gives us the optimal choice bundle.

it is as if the consumer were just spending all of her money

on a single good that has a combined price.

)/( 2121

2211

ppmxxx

mxpxp

x1

x2U(x1,x2) = min{ax1,x2}

x2

= ax1


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2

x1

x2

MRS = 0

U(x1,x2) = min{ax1,x2}

x2 = ax1

x1

x2

MRS = -

MRS = 0


x2

= ax1


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3

x1

x2

MRS = -

MRS = 0

MRS is undefined


x2 = ax1

x1


x2

= ax1


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3

x1


x2 = ax1

Which is the most

preferred affordable bundle?

x1


x2

= ax1

The most preferred

affordable bundle


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3

x1


x2 = ax1

x1*

x2*

x1


x2

= ax1

x1*

x2*

(a) p1x1* + p2x2* = m


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3

x1


x2 = ax1

x1*

x2*

(a) p1x1* + p2x2* = m

(b) x2* = ax1*

(a) p1x1* + p2x2* = m; (b) x2* = ax1*.

Substitution from (b) for x2* in (a) gives p1x1* + p2ax1* = m

which gives

A bundle of 1 commodity 1 unit and a commodity 2 units costs p1 + ap2;

m/(p1 + ap2) such bundles are affordable.

.app

amx;app

mx21

*221

*1


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3

x1


x2 = ax1

xm

p ap1

1 2

*

x

am

p ap

2

1 2

*

Discrete goods

good 1 - is a discrete good

good 2 is money spent on everything else

To look at optimal choice, two ways:

compare the bundles (1, m-p1), (2, m-2p1), (3,m-3p1) and get the one which yields the highestutility

Indifference curve analysis: Typically, as theprice decreases further, the consumer willchoose to consume more units of good 1.


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Concave Preferences

Optimal choice would be a boundary choice.

Non-convex preferences imply that if you dont

like to consume two things together (the

opposite of convex preferences), then youll

spend all of your money on one or the other


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3

x1

x2

x1

x2


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3

x1

x2

The most preferred

affordable bundle

Notice that the tangency solutionis not the most preferred affordable

bundle.

Neutrals and Bads

Neutral good - Consumer spends all her money

on the good she likes and doesnt purchase any

of the neutral good.

Bad - Consumer spends all her money on thegood.


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Using observed demand behavior, can we determine the kind ofpreferences that generated that demand?

Example: A Cobb-Douglas utility function may be inferred ifdemand behavior shows a certain constancy in the shares ofexpenditures for each good

We can use this fitted utility function to analyze the effects ofpolicy changes.

In general: Given observations of choice behavior, we try to determine

what is maximized. After determining this, we predict choice behavior in new

situations and or evaluate proposed changes in the economicenvironment. Analysis can be extended by estimating preferences for certain

groups of individuals and analyzing differential impact of policychanges on them.

If everyone is facing the roughly the same

prices for two goods and everyone is

optimizing and at an interior solution, then

everyone must have the same marginal rates

of substitution for the two goods.

The quantities consumed would differ

At quantities consumed however, the MRS

would be the same.


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Price ratios can be used to value possible

changes in consumption bundles.

Since prices measure the rate at which

people are just willing to substitute one

good for another, they can be used to value

policy proposals that involve making

changes in consumption.

If the government wants to raise a certain amount ofrevenue, is it better to impose a quantity tax or anincome tax?

Budget constraint with the quantity tax:

Optimal choice should satisfy the budgetconstraint:

The revenue raised by the tax is:

mxpxtp 2211 )(

mxpxtp *

2

*

1 21)(

*

1

*txR


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Budget constraint with the income tax:

substituting for R*:

This budget line has the same slope but will pass through

the point .

It is therefore an affordable choice for the consumer

*

2211 Rmxpxp

*2211 1

txmxpxp

),( *2*1 xx

However is not an optimal choicewith the income tax.

With the quantity tax, the MRS is

Income tax allows us to trade at

Income tax budget line cuts the indifferencecurve at the optimal choices with the quantitytaxes, implying that there will be a point in thebudget line that will be preferred to it.

),( *2*

1 xx

21 /)( ptp

21/ pp


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Therefore , an income tax is definitely better

than the quantity tax since you can raise the

same amount of revenue while leaving the

consumers better off.


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Limitations:

Assume only one consumer.

Assume that consumers income doesnt

change after the imposition of the tax, i.e.,

his income generating behavior doesnt

change.

Ignored the supply response.

Ways to derive the demand

curves:

By using MRS condition and the

budget constraint By unconstrained maximization

By constrained maximization


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MRS condition and the budget constraint

Represent the consumers preferences by autility function :

Optimal choice must satisfy the condition

MRS can be expressed as the negative of theratio of derivatives of the utility function.Substituting and canceling the minus signs :

),( 21 xxu

2

121 ),(

p

pxxMRS

2

1

221

121

/),(

/),(

p

p

xxxu

xxxu

Optimal choices should also satisfy the budget constraint:

These give two equations in two unknowns, which can be solved.

One way is to express the budget constraint in terms of one ofthe goods and substitute the definition into the MRS condition.

substituting into equation we get:

This has just one unknown variable x1, and can be solved for as afunction of the prices and income. The budget constraint thenyields the solution for x2 as function of prices and income

mxpxp 2211

1

2

1

2

2 xp

p

p

mx

2

1

212121

112121

/))/(/,(

/))/(/,(

p

p

xxpppmxu

xxpppmxu


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Unconstrained maximizationi)

Unconstrained equivalent: define one variable in terms of the

other

ii)

Substitute in the utility function to get the unconstrained max

problem

iii)

mxpxp

xxu

2211

21

such that

),(max

1

2

1

2

12 )( xp

p

p

mxx

))/(/,(max 12121 xpppmxu

Differentiate with respect to x1, set the result equal to zero to

derive the first order condition of the form:

iv)

Differentiating equation in (ii) yields

v)

Substituting into (iv) gives us

Utilizing the condition that the optimal choice must satisfy thebudget constraint gives us two equations in two unknowns

0))(,())(,(

1

2

2

121

1

121

dx

dx

x

xxxu

x

xxxu

2

1

1

2

p

p

dx

dx

2

1

2

**

1

**

/),(

/),(

21

21

p

p

xxxu

xxxu


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Constrained maximization

Construct the Lagrangian function:

Differentiate to get the first order conditions:

Solve the three equations in three unknowns by matrix algebra

).(),( 221121 mxpxpxxuL

.0

0),(

0),(

2211

2

2

*

2

*

1

2

1

1

*

2

*

1

1

mxpxpL

px

xxu

x

L

px

xxu

x

L

Or reduce the equations by noting that:

This equation simply says that the MRS must equal the priceratio.

The budget constraint gives us the other equation so we are backto two equations in two unknowns

2

1

2

*2

*1

1

*2

*1

),(

),(

p

p

x

xxu

x

xxu

Economics 102 Lecture 5 Choice Rev

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