54 02 Consumption

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Review of MarketsEquilibrium

Question: What determines the price andquantity of a good? Answer: In a competitivemarket, the equilibrium price and quantity aredetermined by the market demand curve and

market supply curve. At the equilibrium, thequantity demanded equals the quantitysupplied. P* is the equilibrium price and Q* isthe equilibrium quantity.

Market DemandFor the next four classes, we shallfocus on demand. What informationdoes a market demand curve provideto us? The market demand curveanswers a long series of hypothetical

questions:If the price were _____, howmany units wouldhouseholds purchase?

Where does the market demand curve come from?Now, let us get to the roots of the market demand curve; that is, where does themarket demand curve come from? The answer: the market demand curve is thehorizontal sum of each individual households demand curve:

Where does an individual households demand curve come from?Now, let us get to the roots of an individual households demand curve. Wheredoes an individual households demand curve come from? In other words, howdoes an individual household decide upon how much of a good to purchase?How much beer will a household purchase? How much pizza will it purchase?How many DVDs? What process does an individual household use to makethese decisions? Economics call the process utility maximization. To explainthis, we shall begin with a hypothetical problem that you might face and analyzehow you might go about making your decision.

P

Q

S

D

Market for Beer

P*

Q*

P

Q

D

If P = .50

If P = 1.00

If P = 1.50

If P = 2.00

Market Demand Curve for Beer

P

Q

D

If P = .50

If P = 1.00

If P = 1.50

If P = 2.00

Market Demand Curve for BeerHousehold A Household B

QQ

PP


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Introduction To Utility MaximizationSuppose that when you go to the Post Office today you find a letter from your mom.Inside you discover a $10 bill. In the letter, mom stipulates that you must spend theentire $10 this evening on pizza and beer. It appears that she believes what you havebeen telling her; that is, she believes that you are devoting all your time to academicpursuits.

Now, you are faced with a question:How much of the $10 should you spend on pizza and how much on beer?

GoalPresumably, your goal is to make yourself as happy as possible. Economistsrefer to happiness as well being or utility. There is one catch, however. You onlyhave $10 to spend. We have a constrained maximization problem:

Maximize Utility (Well Being or Happiness)Subject To Budget Constraint

What other information is relevant to you decision? Clearly, the prices of beerand pizza are important. Suppose that

Price of Beer = PBeer = $2 per canPrice of Pizza = PPizza = $1 per slice

Budget ConstraintFirst, we shall focus on your budget constraint. The budget constraint illustratesall of the combinations of beer and pizza that you can afford to purchase; that is,the budget constraint illustrates all of your affordable combinations of beer andpizza.

Affordable CombinationsBeer Pizza

Spend all $10 on pizza 0 cans 10 slicesSpend $2 on beer and $8 on pizza 1 can 8 slices

Spend $4 on beer and $6 on pizza 2 cans 6 slicesSpend $6 on beer and $4 on pizza 3 cans 4 slicesSpend $8 on beer and $2 on pizza 4 cans 2 slicesSpend all $10 on beer 5 cans 0 slices

To illustrate your budgetconstraint, we place pizza onthe horizontal axis and beer onthe vertical axis. Note that youcannot afford to purchase acombination that lies aboveyour budget constraint. Forexample, you cannot purchase

6 slices of pizza and 3 cans ofbeer because they would cost atotal of $6 + $6 = $12 and youonly have $10.

2

4

6

8

10

1 2 3 4 5

Pizza

Beer

Spend all $10 on beer

Spend $8 on beer and $2 on pizza




Spend all $10 on pizza


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Indifference CurveNext, we shall focus on your utility. Utility is the economists term for happinessor well being. An indifference curve illustrates all the combinations of beer andpizza that keep you just as well off; that is, an indifference curve illustrates all thecombinations of beer and pizza that keep you at a constant value of utility (orhappiness or well being).

For example, consider one particularcombination of beer and pizza: 6 slices ofpizza and 2 cans of beer. We can draw anindifference curve through thiscombination. What generalizations can wemake about the shape of your indifferencecurve? Your indifference curve must bedownward sloping. Why? If you were toconsume less pizza, you would have toreceive more beer to remain just as well off.Alternatively, if you were to consume lessbeer, you would have to receive more pizza

to remain just as well off. The principle isstraightforward. If you were to consume less of one good, you must be givenmore of the other good to remain just as well off.

Key Point:Your indifference curve is downward sloping.

Indifference Curves and Contour LinesRecall that an indifference curve illustrates all of the combinations of beer andpizza that keep you just as well off; that is, an indifference curve illustrates all ofthe combinations of beer and pizza that keep you at a constant value of utility (orhappiness or well being).

What is an indifference curve? What is a contour line?An indifference curve illustrates A contour line illustratesall the combinations of all the combinations of

beer and pizza that keep your x1and x2that keep the function

utility at the same constant value. at the same constant value

NB: An indifference curve is just acontour line of the utility function.

Note that you have one indifferencecurve for each value of utility. Asyour utility increases, you move tohigher indifference curves reflecting

the fact that you are consuming morebeer and pizza. The higher theindifference curve, the greater yourutility, your happiness, or your wellbeing.

Key Point:When you move to a higher indifference curve, your utility increases.

2

4

6

8

10

1 2 3 4 5

Pizza

Beer

2

4

6

8

10

1 2 3 4 5

Pizza

Beer

Utility = U1

Utility = U2

U2> U

1


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Goal RevisitedRecall that your goal is to make yourself as happy as possible; that is, your goal isto maximize your utility. But note that you have a budget constraint, you onlyhave $10 to spend. You a facing a constrained maximization problem:

Maximize UtilitySubject To Budget Constraint

How can we describe the solution to yourproblem? Superimpose your budgetconstraint line and indifference curves ontothe same graph. You want to find thehighest indifference curve that still touchesyour budget constraint line. This illustratesthe combination of beer and pizza thatmaximizes your utility when you only have$10 to spend. To achieve more utility youwould have to move to a higherindifference curve; but then you would notbe on your budget constraint meaning that

you would be spending more than the $10 you have.

Indifference Curves and TastesWe can make two observations that are common to all households:

An indifference curve for every individual is downward sloping. As an individual moves to a higher indifference curve, his/her level of

utility rises.While the indifference curves of all individuals share these two characteristics,the exact shape of an indifference curve varies from person to person reflectingthe fact that each of us have different tastes. The diagrams below illustrate anindifference curve of a hypothetical beer lover and a hypothetical pizza lover. If

you were to take a can of beer from a beer lover, you must give him/her manymore slices of pizza to keep him/her just as well off. Similarly, if you were totake a slice of pizza from a pizza lover, you must give him/her many more cansof beer to keep him/her just as well off. The indifference curves both of the beerand pizza lovers are downward sloping, but they are shaped very differently.

2

4

6

8

10

1 2 3 4 5

Pizza

Beer

2

4

6

8

10

1 2 3 4 5

Pizza

Beer1 2 3 4 5

Beer

2

4

6

8

10

Pizza

Beer Lover

1 less can of beer

4 more slices of pizza

Pizza Lover

1 less slice of pizza

3 more cans of beer


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6

Generalization: The Households Constrained Utility Maximization ProblemNext, we shall generalize the concepts that we just introduced.

NotationWe denote our two goods as Good X and Good Y; income is denoted by I.

Two goods: Good X and Good Y

X = units of good X Y = units of good YPX= price of good X PY= price of good Y

IncomeI = income

The ProblemThe household seeks to maximize its utility subject to its budget constraint:

Maximize Utility = U(X, Y)Subject To PXX + PYY = I

Budget ConstraintWe represent the budget constraint with the following equation:

PXX + PYY = I

what you what you yourspend on spend on incomegood X good Y

Y-interceptThe Y-intercept of the budgetconstraint represents thesituation in which you buynone of good X; that is, X = 0.In this case, you are

spending all of your incomeon good Y; consequently,

you can purchaseI

PYunits of

good Y. To show this morerigorously, recall yourbudget constraint:

PXX + PYY = I

Since X = 0,PYY = I

and

Y =

I

PY

Spend all income on good Y

Spend all income on good X

Y

X

I/PY

I/PX

PXX + PYY = I


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X-interceptThe X-intercept of the budget constraint represents the situation in whichyou buy none of good Y; that is, Y = 0. In this case, you are spending all of

your income on good X; consequently, you can purchaseI

PXunits of good X.

To show this more rigorously, recall your budget constraint:

PXX + PYY = ISince Y = 0,

PXX = I

and

X =I

PX

SlopeOnce we calculate the intercepts, it is easy to calculate the slope. First, notethat the slope is negative. Now compare the rise and run between the twointercepts:

Slope =RiseRun =

I

PYI

PX

= PXPY

Spend all income on good Y

Spend all income on good X

Y

X

I/PY

I/PX

PX

X + PY

Y = I

slope = PX/PY

Utility Function: Utility = U(X, Y)In general, you wish to maximize your utility; that is, you wish to maximize yourhappiness or well being. Your utility depends on how many units of the goodsyou are consuming; that is,

Utility = U(X, Y) X = units of good XY = units of good Y

There is one problem, however. You only have a limited amount of income tospend. In other words, you have a budget constraint. We can describe yourproblem in the following way:


We have already illustrated your budget constraint. Now we shall turn to yourutility function.


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Note that different individuals will have very different utility functions. Anindividuals utility function is a very personal thing; it depends on his/her tastes.We know that the tastes of different people vary greatly. Some people love beer,others hate it; some people love lobster, others despise it; etc. So when we talkabout a utility function we are always talking about one individual.

We shall discuss three ways that we use to describe utility functions: Marginal Utility Indifference Curves Marginal Rate of Substitution

Marginal Utility: Vary the consumption of one good while holding the consumptionof all other goods constant

Verbal Definition:The marginal utility of a good tells us by (approximately)how much utility changes when the consumption of the good changes by a smallamount while the consumption of each other good remains the same.

Marginal Utility of Good X: MUX

Verbal Definition: MUXtells us by (approximately) how much utilitychanges when the consumption of good X changes by a small amountwhile the consumption of each other good remains the same.

For example, suppose that MUX= 3:

X = +1 U +3 while consumption each other good constantX = +2 U +6 while consumption each other good constantX = 1 U 3 while consumption each other good constant

We can now convert our verbal definition of marginal utility into a rigorousmathematical one by noting that in general

U MUXX while consumption all other goods constant

Dividing through by X:UX MUX while consumption all other goods constant

Now, take the limit as X approaches 0:

limX0

UX = MUX

What type of derivative is this? It is a partial derivative because the consumptionof each other good remains constant:

UX= MUX

We now have a mathematically rigorous definition of the marginal utility ofgood X: the marginal utility of good X equals the partial derivative of the utility

function with respect to good X:

MUX=UX

Using the same logic we can show that the marginal utility of good Y equals thepartial derivative of the utility function with respect to good Y:

MUY=UY


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Diminishing Marginal UtilityRecall that

U MUXXThe marginal utility of good X indicates byhow much your utility changes when youconsume one unit of good X. Diminishing

marginal utility suggests that as you consumemore and more of good X, the marginal utilityof good X diminishes; that is as you consumemore and more of good X, the increase inutility provided by one additional unit of goodX diminishes.

This property appears to make sense. Suppose that good X is steak and youbegin to eat a steak dinner tonight. When you first sit down for dinner, you arefamished and that first bite of steak tastes great. The second bite tastes prettygood also. But as you eat more and more steak you begin to get filled up. Asthis occurs, each additional bite of steak certainly tastes good, but not as good asthe first few bites. The first bite tastes great providing you with much additionalutility; the last bite provides not nearly as much additional utility. Thisphenomenon is one illustration of diminishing marginal utility.

Indifference Curve: Vary the consumption of both goods so as to keep utility constantAn indifference curve illustrates all thecombinations of good X and good Y thatkeep the household just as well off; that is,an indifference curve illustrates all thecombinations of good X and good Y thatkeep the household at the same constantlevel of utility. A household is indifferentbetween any two points that lie on the same

indifference curve.

An indifference curve is downward sloping.It is easy to understand why. If you takesome of good Y away from an individual,then you must give him/her more of good X to keep him/her just as well off. Inother words, if you take some of one good away from someone, you must givethat person more of the other good to keep him/her just as well off.

MUX

X

Y

X

Utility = U(X,Y) = constant

less Y

more X


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Amherst CollegeDepartment of EconomicsEconomics 54Fall 2005

Thursday, September 15: The Utility Maximizing Household

Review of Key Mathematical PointsPartial derivative: y = f(x1, x2)

The partial derivative tells us by (approximately) how much y changes when onex changes by a small amount while the other xs remain the same.

Total differential approximation: y = f(x1, x2)

y yx1

x1 +yx2

x2

change in y change in ycaused by the caused by thechange in x1 change in x2

Contour line: y = f(x1, x2)

A contour line illustrates all of thecombinations of x1and x2that keep y at a given

constant value.

Slope of Contour Lines Tangent: y = f(x1, x2)

The slope of a contour lines tangent equals thenegative of the ratio of partial derivatives:

Slope of contour lines tangent =

yx1

yx2

Constrained Utility Maximizing ProblemMaximize Utility = U(X, Y)Subject To PXX + PYY = I

Budget Constraint: PXX + PYY = I

X-intercept: Y=0PXX = IX =I

PX

If the household were to spend its entire income ongood X and consequently nothing on good Y.

Y-intercept: X=0PYY = IY = IPY

If the household were to spend its entire income ongood Y and consequently nothing on good X.

Slope = PXPY

contour lines tangent

x1

x2contour line: y = f(x1,x2) = constant

Y

X

I/PY

I/PX

PXX + PYY = I

slope = - PX/PY

Budget Constraint


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Describing the Utility FunctionMarginal Utility: Vary one good while holding each of the other goods constant

Verbal definition:By (approximately) how much the household's utility changeswhen the consumption of one good changes by a small amount while theconsumption of all other goods remain the same.

MUX=

U

X MU

Y=

U

Y

Indifference Curve: Vary both goods so as to keep utility constantVerbal definition:An indifference curve illustrates all thecombinations of good X and good Y that keep the householdjust as well off; that is, all the combinations of good X andgood Y that provide the household with a constant level ofutility.

An indifference curve is a contour line of a households utilityfunction. A household is indifferent between any two pointsthat lie on the same indifference curve.

Marginal Rate of Substitution (MRS)Verbal Definition:The marginal rate of substitution, MRS, indicates how ahousehold can substitute good Y for good X so as to keep utility constant.

For example, suppose that MRS = 4:

X = 1 Y +4 to keep utility constantX = 2 Y +8 to keep utility constantX = +1 Y 4 to keep utility constant

Note that in general:

Y MRS X to keep utility constantKeep this result in mind.

How is the marginal rate of substitution related to marginal utilities?

Claim: MRS =MUXMUY

To justify this claim, review the total differential approximation:y = f(x1, x2)

y yx1

x1 +yx2

x2

change in y change in ycaused by the caused by thechange in x1 change in x2

Now let us apply this to the utility function: yU, x1X, and x2Y:

y = f(x1, x2) y yx1

x1+yx2

x2

U = U(X,Y) Utility UXX +

UYY

Y

X



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Since the marginal rate of substitution keeps utility constant, U = 0; therefore,

0 UXX +

UYY to keep utility constant

Moving the second term to the left hand side:

UYY

UXX to keep utility constant

Dividing both sides by UY:

Y

UX

UY

X to keep utility constant

Next note that MUX=UXand MUY=

UY; therefore,

Y

UX

U

Y

X = MUX

MUYX to keep utility constant

Now, recall that based on the verbal definition of the marginal rate ofsubstitution we showed that

Y MRS X to keep utility constant

Clearly,

MRS =

UX

UY

=MUXMUY

Geometric Interpretation of the Marginal Rate of SubstitutionClaim: slope of indifference curves tangent = MRS

This follows from the fact that an indifference curve is acontour line of the utility function.

To see why, recall that when y = f(x1, x2), a contour line

illustrates all the combinations of x1and x2that keep y

at a given constant value. The slope of a contour linestangent equals the negative of the ratio of partialderivatives:

Slope of contour lines tangent =

y

x1yx2

contour lines tangent

x1

x2contour line: y = f(x1,x2) = constant


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Now, we shall apply this to utility functionsand indifference curves. Recall that Utility =U(X, Y). An indifference curve illustrates allthe combinations of good X and good Y thatkeep utility at a given constant value. Anindifference curve is a contour line of the utility

function. Therefore, the slope of anindifference curves tangent equals thenegative of the ratio of partial derivatives:

Slope of indifferent curves tangent =

UX

UY

= MUXMUY

Now, recall how the marginal rate of substitution is related to the partialderivatives of the utility function, and accordingly marginal utilities:

MRS =

UX

UY

=

MUX

MUY

Therefore,

Slope of indifferent curves tangent = MRS

Diminishing Marginal Rate of SubstitutionIt is easy to argue that the households indifference curve must be downwardsloping: if you take some of one good away from the household, you must give itmore of the other good to keep it just as well off.

In general, there are twoways to draw a downward

sloping curve, however. Itmay be bowed in toward theorigin or bowed out awayfrom the origin. Typically,indifference curves arebowed in toward the origin.To explain why, let us returnto our beer and pizzaexample. Consider two possible shapes for your indifference curve: bowed intoward the origin and bowed away from the origin. Consider two combinationsof beer and pizza that lie on each indifference curve. Combination A containslittle beer; combination B much beer.

IntuitionWhen should it be easy to substitute for beer and when should it bedifficult? It should be easy to substitute for beer if you have many cansto begin with; it should be difficult to substitute if you have only just afew cans initially:

Combination A Combination Blittle beer much beer

Difficult to substitute for beer Easy to substitute for beer

Y

X


slope = - MRS

Pizza Pizza

Beer Beer

B: much beer

A: little beerA: little beer

B: much be


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Constrained Utility Maximizing Problem: Finding the SolutionMaximize Utility = U(X, Y)Subject To PXX + PYY = I

To find the solution to the households constrained utilitymaximizing problem, we superimpose both the budget

constraint and the indifference curves on a single graph: First, we draw the budget constraint line. We know

that the solution lies on the budget constraint linebecause the household must balance its budget;this gives us one equation:

PXX + PYY = I

Note that the slope of the budget constraint line

equals the negative of the price ratio, PXPY

.

Second, we find the highest indifference curve that still touches the budgetconstraint line. Either an interior solution or a corner solution can result. Ineither case, we have a second equation:

Interior solution Corner solutionThe indifference curve is tangent The solution is one of the two

to the budget constraint line; intercepts: either the Y-intercepttherefore, the slope of the or the X-intercept. Therefore,

indifference curve equals the either X = 0 or Y = 0. We haveslope of the budget line: a second equation. Therefore,

either Y =I

PY or X =

IPX

. In the

diagram above, X = 0 and Y =I

PY.

slope of the slope of theindifference = budget

curves tangent constraint line

MRS = PXPY

MRS =PXPY

We have a second equation.

With two equations, we can solve for the two unknowns.

Solution

X

Y

Budget constraint

Slope = -PX/PY

I/PX

I/PY

Indifference Curve

MRS = PX/PY

Y

X

I/PY

I/PX

PXX + PYY = I

slope = - PX/PY

Budget Constraint

Y

X

I/PY

I/PX

PXX + PYY = I

slope = - PX/PY


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First, calculate the marginal rate of substitution. Recall that the marginal rate ofsubstitution equals the ratio of marginal utilities; also, recall that the marginal utilitiesequal the partial derivatives of the utility function:

MRS =MUXMUY

=

UX

U

Y

=X1Y

XY1 =

YY+1

XX+1 =

Y+1

X+1=

Y1

X1 =

YX

Substitute for the MRS in equation (2), MRS =PXPY

:

YX=

PXPY

,

Now, solve for Y:

Y =PXPY

X.

Question:Is this the demand function for good Y?Answer:No!Y is expressed in terms of X. Remember that the demandfunction for good Y cannot be expressed in terms of good X, but rather

only in terms of prices and income.

Substitute for Y in equation (1), PXX + PYY = I:

PXX + PYPXPY

X = I.

Simplify the left-hand sides second term:

PXX +PX X = I.

Multiply both sides by :PXX + PXX = I,

Factor out PXX from the left-hand side of the equation:

(+)PXX = I

Solve for X:

X =

+I

PX

This is the demand function for good X because X is expressed only interms of prices and income.

What about the demand function for good Y? From above recall that Y =PXPY

X;

substitute

+I

PX in for X:

Y =PXPY

+I

PX,

Simplify the expression:

Y =

+I

PY.

So, we have calculated the demand functions for a Cobb-Douglas utility function:

X =

+I

PX and Y =

+

IPY

Note that our assumption of an interior solution is justified in this case becauseindifference curves are bowed in toward the origin and both X and Y must be positive.


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Tuesday, September 20: Properties of Demand Functions

Review of Key PointsConstrained Utility Maximizing Problem: Finding the Solution


To illustrate the solution to the households constrained utility maximizingproblem, we superimpose both the budget constraint and the indifference curveson a single graph:

First, we draw the budget constraint line. To meet the budget constraint,the solution must lie on the budget constraint line. Note that the slope ofthe budget constraint

line equals thenegative of the price

ratio, PXPY

.

Second, we find thehighest indifferencecurve that still touchesthe budget constraintline.

Demand FunctionsReview the households constrained utility maximizing problem:


An individual households demand function expresses the X and Y that solvesthe households constrained utility maximizing problem in terms of the prices ofthe goods and the households income:

X(PX, PY; I) = households demand function for good X

Y(PX, PY; I) = households demand function for good Y

PreviewWe shall explore three properties of demand functions:

change all prices and income by the same proportion; change income while keeping prices constant; change one price while keeping income and all other prices constant.

Solution

X

Y

Budget constraint

Slope = -PX/PY

I/PX

I/PY

Y(PX, PY; I)

X(PX, PY; I)

Indifference curve

MRS = PX/PY

Interior Solution


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Change All Prices and Income by the Same ProportionWhat happens if all prices and income double?

max Utility = U(X, Y) PX2PX max Utility = U(X, Y)

s.t. PXX + PYY = I PY2PY s.t. 2PXX + 2PYY = 2I

I 2I

Focus on the budget constraint and calculate the intercepts:Before: PXX + PYY = I

X-intercept: Y = 0I

PX

Y-intercept: X = 0I

PY

After: 2PXX + 2PYY = 2I

X-intercept: Y = 02I

2PX =

IPX

Y-intercept: X = 02I

2PY =

I

PY

Since the intercepts have not changed, the budget constraint line has notchanged.

The budget constraint linehas not been affected; hence,the solution to thehouseholds utilitymaximizing problem will notbe affected. The combinationof good X and good Y thatsolves the problem initiallywill also solve the problemafter all prices and incomedouble. In other words, thequantity of each gooddemanded is unchanged when all prices and income double. How can weexpress this mathematically?

X(2PX, 2PY; 2I) = X(PX, PY; I) Y(2PX, 2PY; 2I) = Y(PX, PY; I)

Of course, there is nothing sacred about doubling all prices and income. The quantitydemanded is unchanged when all prices and income triple, or increased by a factor offive, or increase by 1 percent, etc. Technically, this property of demand functions iscalled homogeneity of degree 0.

Solution

X

Y

Budget constraint

Slope = -PX/PY

I/PX

I/PY

Y(PX, PY; I)

X(PX, PY; I)

Indifference curve

MRS = PX/PY

Interior Solution


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Change Income While All Prices Remain the SameWhat happens if income increases while all prices remain the same?

max Utility = U(X, Y) max Utility = U(X, Y)

s.t. PXX + PYY = I I I + I s.t. PXX + PYY = I + I

I > 0

When income increases and all prices remain the same, there is an increase in thehouseholds real purchasing power.

Focus on the budget constraint and calculate the intercepts and slope:Before: PXX + PYY = I

X-intercept: Y = 0I

PX

Y-intercept: X = 0I

PY

slope =riserun =

IPY

IPX

= PXP

Y

After: PXX + PYY = I + I

X-intercept: Y = 0I + I

PX

Y-intercept: X = 0I + I

PY

slope =riserun =

I + IPY

I + IP

X

= PXPY

Both the X- and Y-interceptshave increased, but the slopehas remained the same.Therefore, the budgetconstraint line has shifted outin a parallel fashion. Thehouseholds real purchasingpower has increased; theoutward parallel shift of thebudget constraint line meansthat the household can afford

to purchase more of good Xand also more of good Y. Achange in real purchasing power is referred to as an income effect:

Parallel Shift in the Real Purchasing IncomeBudget Constraint Line Power Changes Effect

X

Y

Budget constraintSlope = -PX/PY

I/PX

I/PY

(I+I)/PX

(I+I)/PY


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Normal and Inferior GoodsTypically, when a households purchasing power increases, it responds byconsuming more of most goods. When purchasing power increases, ahouseholds consumption of a normal good increases. While most goods arenormal goods, there are some exceptions. For example, when a householdsincome increases, its consumption of Old Milwaukee Beer will typically decrease.

A good whose consumption decreases with an increase in purchasing power iscalled an inferior good:

Increase in real purchasing power

Outward parallel fashion shift of the budget constraint line

Normal Good Inferior Good

Increase in purchasing power Increase in purchasing powerincreases consumption of X decreases consumption of X

XI > 0

XI < 0

We can use partial derivatives to describe normal and inferior goods. Good X is

a normal good if the partial derivative of its demand function with respect to I ispositive; X is inferior if the partial derivative is negative. Note that the partialderivative is appropriate here because only income is changing: all prices areremaining constant.

Engel CurveAn Engel curve plots the quantity of a gooddemanded in terms of income. If a good isnormal, the Engel curve has positive slope; ifinferior, it has a negative slope. The figure to theright illustrates an Engel curve for a normalgood.

X

Y


I/PX

I/PY

(I+I)/PX

(I+I)/PY

X

Y


I/PX

I/PY

(I+I)/PX

(I+I)/PY

X

I

Engel Curve


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Change the Price of One Good While Income and All Other Prices Remain the SameWhat happens if the price of Good X increases while income and all other prices remainthe same?

max Utility = U(X, Y) max Utility = U(X, Y)

s.t. PXX + PYY = I PXPX+ PX s.t. (PX+ PX)X + PYY = I

PX> 0

Focus on the budget constraint and calculate the intercepts and slope:Before: PXX + PYY = I

X-intercept: Y = 0I

PX Y-intercept: X = 0

IPY

slope =riserun =

IPY

IPX

=

PXPY

After: (PX+ PX)X + PYY = I

X-intercept: Y = 0 IPX+ PX

Y-intercept: X = 0 IPY

slope =riserun =

IPY

I

PX+ PX

= PX+ PX

PY

The Y-intercept has remained the same because income and the price of good Yhas remained the same; remember that the Y-intercept represents the case inwhich the householdspends all its income ongood Y. Since income andthe price of good Y havenot changed, the amountof good Y the householdcan afford to purchase isunchanged when onlygood Y is purchased.

The X-intercept hasdecreased because whileincome has remained thesame, the price of good Xhas risen; the X-intercept

represents the case in which the household spends all its income on good X.Since income has remained the same and the price of good X has risen, thehousehold cannot afford to purchase as much of good X when only good X ispurchased.

The budget constraint line has become steeper. It has rotated about the Y-intercept.

A

C

X

Y

I/PXI/(PX+PX)

I/PY

slope = -P

slope = - (PX+PX)/PY

New budget line

PXPX+PX

Initial budge


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The effect of an increase in the price of good X while income and all other pricesremain the same on the households utility maximizing combination of good Xand good Y is illustrated in the above diagram by the movement from point A topoint C. The household is made worse off when the price of good X increaseswhile income and all other prices remain the same. The household must move toa lower indifference curve.

Intuitive Explanation of the Substitution and Income Effects

The price of Good X, PX, has increased

Substitution Effect Income Effect

Since the price of Good Y, PY, Since income, I,

has remained the same has remained the same

Good X has become more Decrease in

expensive relative to Good Y real purchasing power

Household should substitute Good Y for the more expensive Good X X is normal X is inferior

Less X Less X More X

Price Compensated Budget Line: A Way to Separate the Substitution and IncomeEffects

The movement from A to C reflects both thesubstitution effect and the income effect. Wewould like to find a point B that would allowus to separate movement from A to C into itstwo parts: the movement from A to B would

capture the substitution effect and the movement from B to C the income effect.How do we find point B?

Allow the price of good X to increase. The budget line becomessteeper rotating about the Y-intercept; we move from the red budgetline to the blue budget line. The households utility maximizingcombination for good X and good Y moves from point A to point C.

Now, fictitiously provide the household with additional income tocompensate it for the loss in purchasing power caused by theincrease in the price of good X. In other words, we start with theblue budget line and then as we give the household additionalincome, an outward shift of the budget line results.

How much additional income should the household receive? Justenough to enable it to remain just as well off as it was initially; justenough income to keep the household on its initial indifferencecurve. The green budget line illustrates this.

A C

B

Substitution and

Income Effects

Substitution

Effect

Income

Effect


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7

The greenbudget line iscalled thepricecompensatedbudget line

because itcompensatesthe householdfor the loss inpurchasingpower causedby the increasein the price ofgood X. Themovement from A to B captures the substitution effect; the movement from B toC the income effect.

Now, let us tie the graph and our intuition together by focusing on Good X.

Recall that our intuition suggests the following:

The price of Good X, PX, has increased






Household should substitute Good Y for the more expensive Good X X is normal X is inferior


In ourexample,bothgood Xand goodY arenormalgoods:

A

B

C

X

Y

I/PXI/(PX+PX)

I/PY

slope = -PX/PY


Price compensated

budget line

New budget line

AB: substitution effect

BC: income effect

slope = -(PX+PX)/PY

PXPX+PX

Initial budget line

A

B

C

X

Y

I/PX

I/PY

slope = -PX/PY


BC: income effect

PXPX+PX

Initial budget line

SubstitutionIncomeEffectEffect


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Individual Households Demand Curve: Where Does It Come From?An individual households demandcurve for good X answers the followingseries of hypothetical questions:

If the price of good X were ____,how many units would the

household purchase given thatall else remains the same?

In this context, what does the phrase allelse remains the same mean? It meansthat income and the price of good Y remain the same.

We are now in aposition to understandwhere the individualhousehold demandcurve comes from. Allwe must do is to draw

the graph for thedemand curveimmediately below thehouseholdsindifference curvebudget line graph.Initially, the price ofgood X is PX. The

vertical red lineindicates the quantityof good X thehousehold demands,

the quantity of good Xin the householdsutility maximizingcombination of good Xand Y. Therefore, wehave found one pointon the individualhouseholds demandcurve, the pointcorresponding to onespecific price of good X, PX. This is indicated by the red point on the bottom diagram. To

find another point, we increase the price from PXto PX+ PX. In the top diagram, the

budget line rotates about the Y-intercept becoming steeper reflecting the higher price forgood X. The vertical blue line indicates the quantity of good X demanded at this higherprice. This provides us with a second point on the households demand curve, the point

corresponding to a price of PX+ PX, as indicated by the blue point on the bottomdiagram. An individual households demand curve is a direct result of that householdsdesire to maximize its utility subject to its budget constraint.

X

D: X(PX, PY, I)

NB: PYand I remains constant

Price of Good X

A

C

X

Y

I/PXI/(PX+PX)

I/PY

slope = -PX/PY


New budget line

Initial budget line

X

Price of Good X

PX+PX

PX

D


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Shape of an Individual Households Demand Curve: Must a Demand Curve Be DownwardSloping?

Good Xs demand curve answers thefollowing series of hypotheticalquestions:

If the price of good X were

____, how many units wouldthe household purchase giventhat all else remains the same?

In this context, what does the phraseall else remains the same mean? Itmeans that income and the price of good Y remain the same.

The demand curve is downward sloping whenever an increase in the price of good Xdecreases the quantity of good X demanded:

PX X

What happens when the price of good X increases while income and the price of othergoods remains constant?






Household should substitute Good Y

for the more expensive Good X X is normal X is inferior


When the price of good X increases, the substitution effect always reduces the quantity ofgood X. If good X is normal, the income effect also leads to a decrease in the quantity ofgood X. Therefore, for a normal good the demand curve will always be downwardsloping because the substitution and income effects reinforce each other.

What if good X were an inferior good? When the price of good X increases, thesubstitution effect reduces the quantity of quantity of good X, but the income effectincreases the quantity. For an inferior good, the substitution and income effects opposeeach other. If the substitution effect dominates, the demand curve will be downwardsloping. If the income effect dominates, the demand curve will be upward sloping,however. We call a good whose demand curve is upward sloping a Giffin good. Notethat a Giffin good must be inferior, but an inferior good need not be a Giffin good.

X

D: X(PX, PY, I)

NB: PYand I remains constant

Price of Good X


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Let us summarize. When the price of good X increases

If Good X is normal If Good X is inferiorSubstitution Effect Income Effect Substitution Effect Income Effect

less X less X less X more X

Demand curve is Demand curve isdownward sloping downward sloping unless

always the income effect dominatesthe substitution effect


28/37


Thursday, September 22: Substitution and Income Effects Continued

Review of the Properties of Demand FunctionsIncrease All Prices and Income by the SameProportion

When all prices and income are increasedby the same proportion, the budgetconstraint line is unaffected.

Consequently, the utility maximizingsolution and the quantity of each gooddemanded is unchanged.

Increase in Income While All Prices Remain the Same

Increase in real purchasing power

Outward parallel shift of the budget constraint

Normal Good Inferior Good

Increase in purchasing power Increase in purchasing powerincreases consumption of X decreases consumption of X

Increase in the Price of Good X While Income and All Other Prices Remain the Same







for the more expensive Good X X is normal X is inferior


Individual households demand curves will be downward sloping: if the good is a normal good; even if the good is an inferior good, as long as the substitution effect

dominates the income effect, as long as a good is not too inferior.

Y

Budget cSlope = -

I/PX

I/PY

(I+I)/PX

(I+I)/PY

Solution

X

Y

Budget constrai

Slope = -PX/PY

I/PX

I/PY

Y(PX, PY; I)

X(PX, PY; I)

Indifference curve

MRS = PX/PY

Interior Solution


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2

Illustrating the Income and Substitution Effects: Price Compensated Budget Line

Allow the price of good X to increase. The budgetline becomes steeper rotating about the Y-intercept.

Now, fictitiously provide the household withadditional income to compensate it for the loss inpurchasing power caused by the increase in the

price of good X. When the household receivesmore income, an outward parallel shift of the budget line results.

How much additional income should the household receive? Just enoughto enable it to remain just as well off as it was initially; just enough incometo keep the household on its initial indifference curve.

A

B

C

X

Y

I/PXI/(PX+PX)

I/PY

slope = -PX/PY


Price compensated

budget line

New budget line


BC: income effect

slope = -(PX+PX)/PY

PXPX+PX

Initial budget line

Describing Indifference CurvesIndifference curves are

always downward sloping; typically bowed in toward the origin.

Elasticity of Substitution

The elasticity of substitution, denoted by ,indicates how bowed in the indifferencecurve is.1The extreme cases are a fixedproportion and linear. The elasticity ofsubstitution for a

fixed proportion indifference curveis 0;

linear is infinite.

1The rigorous definition of the elasticity of substitution is:

Elasticity of Substitution = =

d(YX)dMRS

(YX)MRS

For now, however, it suffices to define it informally as representing how bowed in indifferencecurves are. We shall return to the rigorous definition later in the course.

A C

B

Substitution and

Income Effects

Substitution

Effect

Incom

Effect

Infinity

= 0

> 0

Y

X


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3

Size of the Substitution and Income EffectsSubstitution Effect when the Price of Good X Increases

The size of the substitution effect depends on how bowed in the indifferencecurves are. As we shall see later in the course, the elasticity of substitutionindicates how bowed in the indifference curve is. When the elasticity ofsubstitution is low, the indifference curve is very bowed in, and the substitution

effect is small. When the elasticity of substitution is high, the indifference curveis not very bowed in, and the substitution effect is large:

Elastiticity of Substitution Low Elasticity of Substitution High

Indifference Curve Very Bowed In Indifference Curve Not Very Bowed In

Y Y

X X

Substitution Effect Substitution Effect

LargeSmall

Income Effect when the Price of Good X IncreasesThe size of the income effect depends on how much of the good X you areconsuming. If you are only consuming a small amount of good X, the increase inthe price of good X will only have a small impact on your purchasing power. Onthe other hand, if you are consuming a large amount of good X, the increase inthe price of good X will have a large impact on your purchasing power.

Consuming a small amount of Good X Consuming a large amount of Good X

Increase in the price of Increase in the price ofGood X has a small impact Good X has a large impact

on purchasing power on purchasing power

Income effect is small Income effect is large

For example, when the price of caviar increases, I suspect that it has no impact onyour purchasing power. Why, because you are not purchasing any caviar. Anincrease in the price of caviar has no effect on your purchasing power becauseyou are purchasing none. On the other hand, an increase in the price of beerwould probably have a large impact on the average college student because the

average student purchases much beer.


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Gross Substitutes and ComplementsGross substitutes

Two goods are gross substitutes if households can substitute one for the other.For example, beer and wine are gross substitutes. When the price of beerincreases, households consume less beer and substitute wine; that is,

PBeerUp Wine Up

Gross complementsTwo goods are gross complements if households consume those goods together.For example, burgers are fries are gross complements. When the price of burgersincreases, households place fewer orders for burgers and fries and consequentlyconsume fewer fries; that is,

PBurgersUp Fries Down

Substitutions and Income Effects Applied to Substitutes and ComplementsGross substitutes and complements tell us what happens to the consumption ofone good when the price of another good changes. Accordingly, we shall ask thefollowing question: what happens to the consumption of good Y when the price

of good X increases?Substitution Effect Income Effect






for the more expensive Good X Y is normal Y is inferior

More Y

2 Less Y More Y

When the price of good X increases and we are concerned with the consumptionof good Y, we see that the substitution and income effects oppose each otherwhenever good Y is normal.

If the substitution effect dominates, we have gross substitutes. If the incomeeffect dominates, we have gross complements (assuming normal goods):

Substitution effect dominates Income effect dominates

PXUp Y Up PXUp Y Down

Gross substitutes Gross complements

2If we are considering more than two goods, the substitution effect need not increase theconsumption of one good when the price of another good increases. With two goods, however,the substitution effect must increase the consumption of one good when the price of anotherincreases. Look at the diagram to convince yourself of this.


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A ParadoxQuestion: can there be two goods that look like gross substitutes when the priceof one good increases, but complements when the price of the other goodincreases? The answer to this question is yes. To understand how this can occur,consider a household with school age children. Focus attention on thehouseholds purchases of educational software (Educ) and computer games

(Games) for the children. The parents are concerned with their childrenseducation; consequently, they are proponents of educational software, but frownupon computer games. Initially, the household purchases much educationalsoftware and few computer games; that is, initially,

Few Games purchased Much Educ purchased

Now, consider the following two observations:

Observation 1:When the price of computer games rises, concerned parentswould reduce their consumption of computer games and purchase moreeducational software:

PGamesUp Educ Up.

Educational software and computer games look like substitutes.

Observation 2:When the price of educational software rises, concernedparents may reduce their consumption of computer games so that theycould still afford to purchase what is vital to their childrens future,educational software:

PEducUp Games Down.

Educational software and computer games look like complements.

To resolve this paradox consider income and substitution effects. Recall that thesize of the income effect depends on how much of the good the householdpurchases. The household purchases few computer games and mucheducational software:

PGamesUp PEUp

Few Games purchased Much Educ purchased

Income effect small Income effect large

Substitution effect dominates Income effect dominates

PGamesUp Educ Up PEducUp Games Down

Gross substitutes Gross complements

By considering income and substitution effects we have resolved the paradox.


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6

Market Demand CurvesRecall that the market demand curve answersa long series of hypothetical questions:

If the price were _____, how manyunits would consumers purchase?

Also, recall that the market demand curve isthe horizontal sum of each individualhouseholds demand curve. Now we knowwhere each individual households demandcurve comes from; therefore, we know wherethe market demand curve comes from. Sinceeach individual households demand curve isdownward sloping (unless the good is aGiffin good, that is, unless the good is inferiorand the income effect dominates the substitution effect), the market demand curve isdownward sloping:

Elasticities: Ways to Describe Market Demand Functions

There are three ways that are commonly used to describe demand functions: (Own) price elasticity of demand Cross price elasticity of demand Income elasticity of demand

When considering elasticity, it is important to keep in mind that we are alwaysmeasuring percent changes not absolute changes.

P

Q

D

If P = .50

If P = 1.00

If P = 1.50

If P = 2.00

Market Demand Curve for Beer

P

Q

D

If P = .50

If P = 1.00

If P = 1.50

If P = 2.00

Market Demand Curve for BeerHousehold A Household B

QQ

PP


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(Own) Price Elasticity of Demand (EX,PX)

Verbal Definition: EX,PXindicates how sensitive the quantity demanded of a good is to

that goods own price.

EX,PX = the percent change in the quantity of good X demanded resulting from

a 1 percent change in the price of good X

=percent change in the quantity of good X demanded

percent change in the price of good X

How can we express the own price elasticity of demand more rigorously? To understandhow, first review what we mean by percent changes:

If X increases from 200 to 220, there is a 10 percent increase. How did wecalculate that? We took the change in X, 20, divided by the initial value, 200, andthen multiplied the quotient by 100:

X: 200 220

Percent change in X =20200 100 = .1100 = 10 percent

We can now generalize this:

Percent change in X =XX 100

Similarly, we can calculate the percent change in the price of X:

Percent change in PX=PXPX

100

Now, recall that

EX,PX =percent change in the quantity of good X demanded

percent change in the price of good X

Substituting in the expressions for the percent changes:

EX,PX =

X

X

100

PXPX

100

Simplifying,

EX,PX =

XX

PXPX

=X

PXPXX

Taking limits as PXapproaches 0:

EX,PX =X

PXPXX

Note that EX,PX< 0, sinceX

PX


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Terminology: Elastic, Unit Elastic, and Inelastic DemandThe terms elastic, unit elastic, and inelastic demand all refer to the own priceelasticity of demand. Verbally, demand is elastic when the quantity demanded isvery sensitive to the goods own price; inelastic when the quantity demanded isnot very sensitive to the goods own price. Unit elastic is the dividing point.What makes the definitions of these terms a little confusing is that the own price

elasticity of demand is a negative number. One way to make these definitions ofthe terms a little less confusing is to define them in terms of absolute values:

Elastic Unit Elastic Inelastic

|EX,PX| > 1 |EX,PX| = 1 |EX,PX| < 1

Cross Price Elasticity of Demand (EX,PY)

Verbal Definition: EX,PYindicates how sensitive the quantity demanded of a good is to the

price of another good.

EX,PY = the percent change in the quantity of good X demanded resulting from

a 1 percent change in price of good Y


percent change in the price of good Y

How can we express the cross price elasticity of demand more rigorously? Tounderstand how, first review what we mean by percent changes:


EX,PY =

XX 100

PYPY

100

Simplifying,

EX,PY =XX

PYPY

=X

PYPYX

Taking limits as PYapproaches 0:

EX,PY =X

PYPYX

Income Elasticity of Demand (EX,I)

Verbal Definition: EX,Iindicates how sensitive the quantity demanded of a good is toincome.

EX,I = the percent change in the quantity of good X demanded resulting from

a 1 percent change in income


percent change in income


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9

How can we express the cross price elasticity of demand more rigorously? Tounderstand how, first review what we mean by percent changes:


EX,I =

XX 100

I

I 100

Simplifying,

EX,I =

XX

II

=XI

IX

Taking limits as I approaches 0:

EX,I =XI

IX

Relationships Among Demand ElasticitiesAll price and income elasticities for a single good sum to zero

This is a very fancy way to say something that we observed earlier. Tounderstand why, first use the verbal notion of elasticities to calculate the percentchange in the quantity of good X demanded when all prices and income increaseby one percent:

PXchanges by 1% PYchanges by 1% I changes by 1%

Percent changein quantity of = EX,PX + EX,PY + EX,IX demanded

Recall that if all prices andincome increase by the sameproportion, the householdsbudget constraint line isunaffected; hence, if all pricesand income change by thesame proportion, thehouseholds utility maximizingcombination of good X andgood Y is unaffected. Morespecifically, if all prices andincome increase by one

percent, the quantity of good X demanded is unaffected:

Percent changein quantity of = EX,PX + EX,PY + EX,I = 0X demanded

Solution

X

Y

Budget constraint

Slope = -PX/PY

I/PX

I/PY

Y(PX, PY; I)

X(PX, PY; I)

Indifference curve

MRS = PX/PY

Interior Solution


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Weighted average of the income elasticities of all goods sums to 1.This too is a fancy way of saying something that is quite simple: if you give ahousehold more income, the household will spend it. To understand why, let

I = additional income given to a householdX = additional units of good X purchasedY = additional units of good Y purchased

Claim: PXX + PYY = IAmount spent on Amount spent on Additional

X more units of X Y more units of Y income

PXX + PYY = I

This equation simply says that if you give a household more income, itwill spend it.

Now, consider this equation:

PXX + PYY = I

Divide both sides by I:PXX

I +PYY

I = 1

Multiply and divide the first term by XI; multiply and divide the second term byYI:

PXX

I XIXI +

PYY

I YIYI = 1

Rearrange terms:PXX

I XI

IX +

PYY

I YI

IY = 1

Take the limit as I approaches 0:PXX

I XI

IX +

PYY

I YI

IY = 1

Now, substitute EX,IforXI

IXand EY,Ifor

YI

IY:

PXX

I EX,I +PYY

I EY,I = 1

54 02 Consumption

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