Consumer Theory: ObjectivesConsumer Theory: Objectives

Derive and understand:

1.How Rational People make Choices

2. How this should Guide our own Decisions

Consumer Behavior

• We are now studying the foundations of demand theory.- Why do demand curves slope downward?- Why do they shift with changes in prices for substitutes and complements?- Why do they shift with changes in income?- What normative significance can we give to demand based on underlying consumer preferences?

Preferences

• Preferences are complete if for any two consumption points x and x', either x ≤x' (x is at least as good as x') or x' ≥ x (x' is at least as good as x), or both.

• Preferences are reflexive if for all x, x ≥ x (x is at least as good as itself).

Preferences

• Preferences are transitive if x ≥ x' and x' ≥ x'' implies that x ≥ x''.

• Preferences are strongly monotonic if for any two commodity points x = (x1, x2) and x' = (x'1, x'2) if x1 ≤ x'1, x2 ≤ x'2, and x ≠ x', then x' is preferred to x.

• Preferences are continuous if the set of all choices that are at least as good as a choice x' and the set of all choices that are no better than x' are both closed sets.

From Preferences to Utility Function

Representation Theorem:

If a consumer has a preference relation that is complete, reflexive, transitive, strongly monotonic, and continuous, then these preferences can be represented by a continuous utility function u(x) such that u(x) > u(x') if and only if x > x'.

• Consider the following (ordinal) utility function for Food (F) and Clothing ( C ) for Emily:

U(F,C) = FC• Each indifference curve gives the combinations of F and C that yield the same

level of satisfaction to Emily (e.g. 25, 50, 100).

• Think of slicing the utility function at different levels and projecting into F,C space.

• Her marginal rate of substitution of Food for Clothing is given by the slope at any point on an indifference curve

MRSFC = - dC/dF|U = a constant[MRSFC is the amount of Clothing Emily is willing to give up for one addition unit

of food, holding utility constant]

• Marginal rate of substitution of F for C is equal to the marginalutility of F divided by the marginal utility of C at a point on anindifference curve, i.e holding utility constant (dU = 0)

dU(F,C) = UFdF + UCdC = 0Where UF = Marginal utility of F

UC = Marginal utility of C

UF/Uc = -dC/dF|U = constant = MRSFC

• For U = FC: UF = C, UC = F -> MRSFC = C/F• Set U = 25: C = 5, F = 5; MRSFC = 5/5 = 1C = 2.5, F= 10; MRSFC = 2.5/10 = 0.25

Declining MRS of F for C as F increases holding U constantDeclining marginal utility of Food as F increases U constant

Budget ConstraintBudget constraints limit an individual’s ability to consume in light of the prices they must pay for various goods and services. This is where scarcity comes in and what helps to make economics the dismal science• The Budget Line– The budget line indicates all combinations of two commodities for which total money spent equals a given fixed level of income (ignore borrowing and saving for now)

Consumers choose a combination of goods thatwill maximize the satisfaction they can achieve, given the limited budget available to them.• The maximizing market basket must satisfy twocoditions:1) It must be located on the budget line.2) Must give the consumer the most preferred combination of goods and services.

Max U(q1, q2, q3,…qn)S.T. Σpiqi = I

• Let: U = FC

• I = PFF + PCC

• Write C in terms of F and substitute• C = I/PC – PFF/PC

• MaxF U = F(I/PC – PFF/PC) = FI/PC – PFF^2/Pc

• dU/dF = I/Pc – 2PFF/Pc = 0• F = I/2PF

• C = I/2PC -> MRS = C/F = PF/PC

• A Corner Solution– If the MRS is, in fact, significantly greater than the

price ratio, then a small decrease in the price of frozen yogurt will not alter the consumer’s market basket.

– At point B, the MRS of ice cream for frozen yogurt is greater than the slope of the budget line.

– This suggests that if the consumer could give up more frozen yogurt for ice cream he would do so.

– However, there is no more frozen yogurt to give up since all of her income is going to ice cream

How Rational Consumers Choose: A Graphical How Rational Consumers Choose: A Graphical ApproachApproach

Good 1

Good 2

Direction of increasing Utility

Indifference Curve: tells us between which bundles the consumer is indifferent

Slope: Marginal Rate of Substitution of Good 1 for Good 2 (M.R.S.)

Budget Line: p1x1 + p2x2 = I

Slope: Ratio of Prices (p1/p2)

0

2 ingredients:

-Preferences: for each consumer, tells us tastes of

consumers

-Budget constraints: tells us prices of goods and a consumer’s income

Budget Set: tells us what the consumer can afford, given his income I and prices p1, p2

Optimal bundle

MRS = p1/p2

Graphical Example Graphical Example Price of Beer: pB = 6, Price of Tacos: pT = 1, Income: I = 18

18

Tacos

Beer31

12

Indifference curve corresponding to a utility of 5 (say)

-1/pT

1/pB -ab

a/b = M.R.S.

pB/pT = price ratio

If you spend $1 more on beer rather than on tacos:

-you lose -1/pT units of Tacos

-you gain 1/pB units of beer

Numerical ExampleNumerical Example

x = Units of Tacos, y = Units of Beer

M.R.S.(x,y) = .5x/y : MRS of Beer for Tacos

Price of Beer: 6, Price of Tacos: 1, Income: 18

Need to Solve: .5x/y = 6 and x + 6y = 18

Which gives: x =12, y =1

Given formula, say

ApplicationsApplications1. You have just bought a mansion in Middletown.

Suddenly, after payment, the price of all houses in your new neighborhood increase. Are you better off after the change? Would you be better off if the price had decreased instead (ignore transaction costs, fees…)?

2. The government decided that potato consumption should be subsidized: for each pound bought, you get $1 from the government. The government then decides, to balance its budget, to levy the lowest tax -per capita- that achieves this goal. Are you overall better off – or worse off?

B

X = (x,y)-bundle chosen before price increase

B = initial budget line; I = initial indifference curve

X = (x,y)-bundle chosen after price increase

B’= final budget line; I = final indifference curve

X

I I’

B’

y= All others goods ($)

x= Housing (sq.)

(Housing) price increase(Housing) price increase

X’

: All indifference curves to follow are for sake of

illustration!

(they are not derived)

B

X= (x,y)-bundle chosen before price decrease

B= initial budget line; I= initial indifference curve

X= (x,y)-bundle chosen after price decrease

B’= final budget line; I= final indifference curve

X

I I’

X’

B’

y= All others goods ($)

x= Housing (sq.)

(Housing) price decrease(Housing) price decrease

B’

B

B’’I

I’

I’’

X’

X’’

X

20 30 40 75

60

90

120

150

70

50

y= all other goods ($)

x= potatoes (pds)

B : Income: I = $150; (potato) price: $3X : chosen bundle (initial situation)B’ : Income: I = $150; price: $2 = 3 - 1($1 subsidy)X’ : chosen bundle (when subsidy is applied, but not the tax)B’’: Income: I = 150 - Tax; price: $2(Total Subsidy = Tax)X’’: chosen bundle (eventual situation)

Initial budget line

Final budget line

Income and Substitution Effects• Substitution Effect

• When the price of a good decreases relative to others goods, holding purchasing power constant, consumers substitute toward the good that is relatively cheaper.

• Note that relative prices change, so intuitively, the slope of the budget line will change.

• Note that purchasing power is held constant.• Thus, we hold the consumer to the same level of utility (i.e.,

consumer can only purchase the same level of utility as before).

Suppose the price of CD’s falls, rotating the budget

line out.

Rotating the Budget line Movie

CD

• Remember the slope of this budget line is -PC/PM.• When PC falls, the slope becomes less negative.• First rotate budget line out, (i.e., from white line to blue line).• Then parallel shift it back to the green line.

Substitution Effect Movie

CD

• Next draw in your original indifference curve.• The original point is C1.• When the price of CD’s falls, the substitution effect causes consumption to increase to C2.• The substitution effect looks the same for all three cases.

C1 C2

Income Effect• The direction of the income effect depends on

whether the good is a normal good (i.e., increased purchasing power increases consumption) or an inferior good (i.e., increased purchasing power decreases consumption).

• The income effects help to distinguish our three cases.

Case I: Income Effect: Normal Good

Movie

CD

U1

U2

• Increased purchasing power increases consumption from C2 to C3.

C2 C3

Income Effect:

Movie

CD

U1

U2

• Increased purchasing power increases consumption from C2 to C3.• Put C3 to the right of C2.

C2 C3C1

Total Movement is C1 to C3

Movie

CDC1 C3

P1

C1

P2

D

PCD

CDC3

C2

u1

u2

Note: Utility rises as price falls

Case II: Inferior Good.Substitution Effect is Stronger

Movie

CD

U1

• Derive the Substitution Effect just as before:

•i.e., from C1 to C2

C2C1

Income Effect:

Movie

CD

U1

U2

• Increased purchasing power decreases consumption from C2 to C3.• Put C3 to the left of C2 but to the right of C1.

C2C3C1

Be careful that your indiff. curves don’t cross!

Derivation of Demand:Total effect is C1 to C3

Movie

CDC1 C3

P1

C1

P2

D

PCD

CDC3C2

u1

u2

Note: Again, utility rises as price falls

Demand stilldownward sloping.

Case III: Super-Inferior (Giffen) GoodIncome Effect is Stronger

Movie

CD

U1

• Derive the Substitution Effect just as before:

•i.e., from C1 to C2

C2C1

Income Effect:

Movie

CD

U1

U2

• Increased purchasing power decreases consumption from C2 to C3.•Put C3 to the left of C2 and to the left of C1.

C2C3 C1

Be careful that your indiff. curves don’t cross!

Derivation of Demand:Total effect is C1 to C3

Movie

CDC1C3

P1

C1

P2

D

PCD

CDC3

C2

u1

u2

Note: Again, utility rises as price falls.

Here, we getan upwardsloping Demand

• Many public programs to help the poor “target” subsidies at particular goods and services (food stamps, housing subsidies)

• These programs increase the individual’s real income, but the money must be spent on the targeted goods and services

• If the gov’t transferred an equivalent amount of income without restrictions, the individuals would be better off

• However, these programs are paternalistic and reflect concerns that the funds will be wasted on “beer”

• In the economic model of the utility-maximizing consumer, the consumer’s utility function associates a numerical value to each conceivable choice. Given prices and monetary resources, the consumer chooses the best (utility-maximizing) bundle from among all those she can afford.– This optimal choice occurs when, given his choice, what an additional unit of a

commodity is worth to him equals the price of the commodity

• The model of the utility-maximizing consumer is rationalized by economists as an as if model. No one believes that consumers actually maximize a utility function. But if the consumer’s choice behavior conforms to relatively simple rules, the consumer acts as if she maximizes utility. And important consequences follow.– Unhappily, systematic violations of these simple rules can be observed in real life.

Consumer marketers and advertising executives are well compensated for their skills in manipulating how consumers frame their choice

– Economists continue to use the model of the utility-maximizing consumer, in the belief that the violations are usually insignificant or in the hope that the conclusions drawn from models so constructed are not grossly affected by violations

Points to remember:Points to remember: