Consumer Theory

Introduction

Budget Constraint and Preferences

Representation of preferences: utility function Consumer Choice: Individual and aggregate demand.

Demand properties.

Market demand and Equilibrium

Model: Simplified representation of reality.

Exist different types of goods in the economy Model needs to account for exogenous variables (taken as

given) and endogenous variables (determined by the forces described by the model).

Economic Principles

The Optimization principle: People try to choose the best patterns of consumption they can afford.

The Equilibrium principle: Prices adjust until the amount that people demand of something is equal to the amount that is supplied.

• Demand curve: Curve that relates demanded quantity to the price.

Reservation price: a person’s maximum willingness to pay for something.

(highest price that a given person will accept and still purchase the good)

• Supply curve: Curve that relates the quantity supplied to the price

Supply curve varies in the long run and short run. Determined by firm’s production costs.

Consumers choose THE BEST bundle of

goods they CAN AFFORD

‘ the one they like more’

‘budget constraint’

“can afford” Household’s consumption possibilities are constrained by its

budget and the prices of the goods and services it buys.

A budget line describes the limits to a household’s consumption choices.

1. Relative price is the price of one good divided by the price of another good.

2. A household’s real income is the household’s income expressed as the quantity of goods that the household can afford to buy.

Budget constraint• Goods the consumer wants to consume: many… but we

simplify to 2 goods

CONSUMPTION BUNDLE: (x1, x2)

• We can observe price of the goods, p1 and p2, and money the consumer has, m.

BUDGET CONSTRAINT:

p1x1 + p2x2 ≤ m

Money spend in good 1

Money spend in good 2

• BUDGET SET:

all consumption bundles that satisfy the budget constraint. • BUDGET LINE:

set of all bundles that cost exactly m

p1x1 + p2x2 = m

We want to know:

a. slope of the budget lineb. how budget line changes when prices changec. how budget line changes when income changesd. What is the numerarie good

1. Taxes and subsidies:

Quantity tax: consumer pays certain amount to the government for each unit purchased. Price becomes (p+t)

Value tax (ad valorem tax): proportional tax on the price of a good. Price becomes p(1+τ)

Subsidy: Opposite of a tax

Lump-sum tax: independent of price and quantity, (m-T)

2. Rationing: Level of consumption of some good fixed to be no larger than some amount.

Policy decisions that affect the budget constraint

Consumers choose THE BEST bundle of

goods they CAN AFFORD

‘ the one they like more’: Preferences A household’s preferences determine the benefits or satisfaction a person receives consuming a good or service.

‘budget constraint’

Preferences• Consumers choose CONSUMPTION BUNDLES

• Notation: Consumer is INDIFFERENT between X and Y : (x1, x2)~ (y1, y2) Consumer STRICTLY PREFERS X to Y: (x1, x2) › (y1, y2) If consumer prefers or is indifferent between two bundles we

say he WEAKLY PREFERS X to Y: (x1, x2) ≥ (y1, y2)

(x1, x2)≥ (y1, y2) and (x1, x2)≤ (y1, y2) imply (x1, x2)~ (y1, y2)

(x1, x2)≥ (y1, y2) and no (x1, x2)~ (y1, y2) imply (x1, x2)› (y1, y2)

Assumptions about preferences

1. Complete: any two bundles can be compared.

2. Reflexive: any bundle is at least as good as itself

3. Transitive: if (x1, x2)≥ (y1, y2) and

(y1, y2)≥ (z1, z2) then we assume that

(x1, x2)≥ (z1, z2)

Indifference curves

• Weakly preferred set: all consumption bundles that are weakly preferred to (x1, x2).

• Boundaries of the weakly preferred set: INDIFFERENCE CURVES.

All bundles of goods that leave the consumer indifferent to the given bundle.

Indifference curves representing distinct levels of preference cannot cross ( that would contradict transitivity).

Well-behaved preferences

1. More is better: Monotonicity assumption.

(we assume we have goods, no bads). Indifference curves have negative slope.

2. Averages are preferred to extremes: Assume that for any t є (0,1), if (x1, x2)~ (y1, y2), then (t x1 + (1-t) y1, t x2 + (1-t) y2)≥ (x1, x2)

(set of bundles weakly preferred to X is a convex set)

Examples of indifference curves

a. Perfect substitutes: consumer is willing to substitute one good for the other at a constant rate.

Ex. 1 l. And 5 l. bottles of water.

b. Perfect complements: goods that are always consumed together in fixed proportions.

Ex. Right and left hand globes.

c. Bads: goods that the consumer doesn’t like.

d. Neutrals: goods that the consumer doesn’t care about.

e. Satiation: exists an overall best bundle and the ‘closer’ the consumer is to that best bundle, the better off he is in terms of his own preferences.

Ex. Chocolate and ice-cream

f. Discrete goods: instead of curves we have a ‘set of points’.

Degree of substitutability

The shape of the indifference curves reveals the degree of substitutability between two goods.

Marginal rate of substitution

The marginal rate of substitution (MRS) measures the rate at which a person is willing to give up good y, (the good measured on the y-axis) to get an additional unit of good x (the good measured on the x-axis) and at the same time remain indifferent (remain on the same indifference curve).

The magnitude of the slope of the indifference curve measures the marginal rate of substitution.

Marginal rate of substitution (MRS)

Negative number: give up from one of the goods and get more from the other to stay on the same indifference curve.

A diminishing marginal rate of substitution is the key assumption of consumer theory.

A diminishing marginal rate of substitution is a general tendency for a person to be willing to give up less of good y to get one more unit of good x, and at the same time remain indifferent, as the quantity of good x increases.

Indifference curves exhibit DIMINISHING MRS: “the more you have of one good, the more willing you are to give some of it in exchange for the other good”.

Utility• “ numeric measure of a person’s happiness”: Total utility is the

total benefit a person gets from the consumption of goods. Generally, more consumption gives more utility.

• Utility function assigns a number to every possible consumption bundle such that more-preferred bundles get assigned larger numbers than less preferred bundles.

u (x1, x2) ≥ u (y1, y2) if and only if (x1, x2) ≥ (y1, y2)

Ordinal utility: only ranking matters. A monotonic transformation of a utility function is a utility function that represents the same preferences as the original function.

Cardinal utility: attach a significance to the magnitude of utility.

• Marginal utility: change in utility when consumer gets additional unit of good 1 (holding constant amount of good 2 he has).

As the quantity consumed of a good increases, the marginal utility from consuming it decreases. We call this decrease in marginal utility as the quantity of the good consumed increases the principle of diminishing marginal utility.

• Marginal rate of substitution: Slope of the indifference curve. How consumer is willing to substitute a small amount of good 2 for good 1.

MRS = ratio of marginal utilities

Ratio of marginal utilities is independent of the particular transformation of the utility function you choose to use.

The consumer’s best affordable point:

– Is on the budget line.

– Is on the highest attainable indifference curve.

– Has a marginal rate of substitution between the two goods equal to the relative price of the two goods.

CONSUMER CHOICE

• Well behaved preferences: consumer will choose bundles ON the budget line.

• Choice: highest possible indifference curve. Set of preferred bundles does not intersect with the set of bundles he can afford.

• Optimality: Indifference curve tangent to the budget line.

Possible problems:

- kinky tastes, boundary optimum, non-convex preferences

• Slope of the budget line (p1/p2): how many units of good 2 has the consumer to give up to get an extra unit of good 1.

• Marginal rate of substitution: Rate at which the consumer is willing to substitute one good for the other.

If indifference curves are strictly convex, then there will be only one optimal choice on each budget line.

MRS= -p1/p2

Demand function: function that relates the optimal choice (the quantities demanded) to the different values of prices and income.

Taxes

• Quantity tax: price of the good is increased by the amount of the tax,

p1 becomes (p1+t)

• Income tax: new income=m-T

• Question: for which type of preferences will the consumer be just as happy facing an income or a quantity tax?

Demand

• Demand function: give the optimal amounts of each of the goods as a function of the prices and income faced by the consumer.

x1= x1( p1,p2,m) x2= x2( p1, p2,m)

• Want to know: how demand changes with prices and income.

Changes in income• Normal good: demand increases with income• Inferior good: demand decreases with income

(whether a good is inferior or not depends on the income level we are examining)

Income expansion path: in the (x1,x2) axis, how optimal choice changes with income.

Engel curve: in the (x1,m) axis, how demand changes with income.

• Luxury good: if demand goes up by a greater proportion than income.• Necessary good: if demand goes up by a lesser proportion than

income.

Changes in prices

• Giffen good: decrease in price leads to a decrease in the demand for the good.

• Ordinary good: demand decreases with increase in price.

Price offer curve: in the (x1,x2) axis, demand of the good as function of the price of the good.

Demand curve: in the (x1,p1) axis, demand as function of price.

• If demand of good 1 goes up when p2 increases, we say that good 1 is a substitute for good 2.

• If demand of good 1 decreases when p2 increases, we say that good 1 is a complement for good 2

REVEALED PREFERENCES

• Objective: discover people’s preferences from observing their behavior.

• “choice reveals preference”: Revealed preferences (what we choose is revealed preferred to anything else that is available)

Directly revealed preferred: Let (x1,x2) be the chosen bundle when prices are (p1,p2), and let (y1,y2) be a bundle such that p1x1+p2x2≥ p1y1+p2y2. Then, if the consumer is choosing the most preferred bundle he can afford, we must have (x1,x2)>(y1,y2)

Indirectly revealed preferred: when we apply transitivity to revealed preferences.

Weak axiom of revealed preferences

If (x1,x2) is directly revealed preferred to (y1,y2), and the two bundles are not the same, then it can not be that (y1,y2) is directly revealed preferred to (x1,x2).

i.e. if (x1,x2) is chosen at prices (p1,p2) and (y1,y2) is

purchased at prices (q1,q2), then if p1x1+p2x2≥ p1y1+p2y2 it must NOT be the case that q1y1+q2y2≥ q1x1+q2x2.

Strong axiom of revealed preferences

If (x1,x2) is revealed preferred to (y1,y2) and (y1,y2) is different from (x1,x2), then (y1,y2) cannot be directly or indirectly revealed preferred to (x1,x2).

If observed choices satisfy SARP, we can always find nice, well-behaved preferences that could have generated the observed choices.

Index numbersExamine cost of a bundle in 2 different periods: b (base period) and t• Iq= (w1x1t+w2x2t)/ (w1x1b+w2x2b) If weights are base period prices: Laspeyres Index If weights are t period prices: Paasche Index

• Price indexes: Iq= (w1p1t+w2p2t)/ (w1p1b+w2p2b) Period t quantities as weights: Paasche Period b quantities as weights: Laspeyres

• Consumer price index (CPI): comparison of the cost of a standard bundle of goods over time. Each month, government reports how much it costs to buy the bundle of goods that an average consumer purchased in a base year.

SLUTSKY EQUATION

Consumer’s choice response to a change in price.

The law of demand states that, other things remaining the same, the higher the price of a good, the smaller is the quantity demanded.

Change in price of a good:

(1)Changes rate at which you can exchange one good for another_ SUBSTITUTION EFFECT

(2) Changes total purchasing power of your income_ INCOME EFFECT

• Steps:

(a) Change relative prices and adjust income to keep purchasing power constant ( pivot budget line around original consumed bundle)

(m’-m)= x1(p,m) (p’-p) Substitution effect= x1(p’,m’)-x1(p,m)

by revealed preferences, Δx1s≥0 whenever price decreases.

(b) Adjust purchasing power with new prices (shift pivoted line out to the new demanded bundle with original income)

Income effect= x1(p’,m)-x1(p’,m’)

• Total change in demand: Slutsky indentity

Δx1=Δx1s + Δx1n=

= (x1(p’,m’)-x1(p,m))+(x1(p’,m)-x1(p’,m’))

Law of demand

If the demand for a good increases when income increases, then the demand for that good must decrease when its price increases

(normal good, income and substitution effect reinforce each other)

Hicks substitution effect

• Roll around chosen point indifference curve instead of keeping consumption bundle as feasible (M’ is enough to get a bundle indifferent to the original one)

Buying and selling

Consumer starts with an endowment of goods (w1,w2)

Gross demand: amount of the good the consumer actually ends up consuming

Net demand: difference between what the consumer ends up consuming and the initial endowments of goods.

Budget constraint:

p1x1+p2x2= p1w1 + p2w2

p1 (x1-w1) + p2 (x2-w2) = 0

• If (xi-wi)>0 Net buyer or net demander • If (xi-wi)>0 Net seller or net supplier

Change in endowment similar to a change in income Price changes: budget line pivots around the endowment

• Net demand:

d1 (p1,p2) = x1(p1,p2) – w1 if positive = 0 otherwise

• Net supply:

s1 (p1,p2) = w1 – x1(p1,p2) if positive = 0 otherwise

Slutsky equation revised

• Two income effects: - ordinary income effect - endowment income effect

• Total change in demand = = substitution effect+ + ordinary income effect + + endowment income effect

• Endowment income effect= = change in demand when income changes x change in income when

price changes.

TotalTotal

benefitbenefitAmountpaid

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Consumer surplus is the value of a good minus the price paid for it, summed over the quantity bought. It is measured by the area under the demand curve and above the price paid, up to the quantity bought.

• Quasilinear utility, discrete goods: U(x,y)= v(x)+y If r units of good x are consumed, then r n<p<r n+1

Gross consumer surplus= v(n)= sum (rn) When x=n, utility is given by: U(n, m-pn)= v(n)+m-pn

Net consumer surplus = v(n)-pn

• Alternative approach: CS= (r1-p) + (r2-p)+…+(rn-p)= v(n)-np

• Let R be the amount of money the consumer needs to receive to exchange the n units of good x consumed for money.

v(0)+m+R= v(n)+m-pn R=v(n)-pn

Compensating and equivalent variations

Alternative to estimate utility changes.

(1) Compensating variation: How much money we would have to give the consumer after the price change to make him just as well off as he was before the price change.

(2) Equivalent variation: How much money would have to be taken away from the consumer before the price change to leave him as well off as he would be after the price change. Maximum the consumer is willing to pay to avoid the price change.

REVIEW(1) Budget constraint

p1x1 + p2x2 = m Slope= -p1/p2

(2) Preferences

• Assumed to be complete, reflexive and transitive• Well-behaved: monotonic (more is better) and convex

( averages preferred to extremes)• MRS= rate at which consumer is willing to substitute one good

for the other (slope of the IC, diminishing MRS)

(3) Utility

• Ordinal representation of preferences• MRS= -MU1/MU2

(4) Choice

• Optimality: Indifference curve tangent to the budget line (MRS=-p1/p2)

• Demand function: function that relates the optimal choice to the different values of prices and income

(5) Demand

• Changes in income: normal and inferior goods.• Changes in prices: ordinary and giffen goods.• Substitutes and complements

(6) Revealed preferences

• Discover people’s preferences from their behavior (what we choose is revealed preferred to anything that was feasible)

• WARP, SARP

(7) Slutsky equation

Change in price of a good:• Change in the rate at which you can exchange one good for

another: substitution effect.• Change total purchasing power of your income: income effect

(8) Endowments: add an additional income effect since value of the endowment is affected by the change in price.

Market demand

• xi= xi(p1,p2,m) consumer i demand of good x

• Market demand: X=X(p1,p2,m1,…,mn) depends on prices and distribution of incomes.

• Representative consumer market demand: X=X(p1,p2,M) where M is the sum of incomes of all individual consumers.

• P(X)= inverse demand function. Measures MRS or marginal willingness to pay of every consumer purchasing the good.

Price elasticity of demand

The price elasticity of demand is a units-free measure of the responsiveness of the quantity demanded of a good to a change in its price when all other influences on buyers’ plans remain the same.

• Calculating Elasticity

The price elasticity of demand is calculated by using the formula:

Percentage change in quantity demanded --------------------------------------------------------------

Percentage change in price

• If the quantity demanded doesn’t change when the price changes, the price elasticity of demand is zero and the good as a perfectly inelastic demand.

• If the percentage change in the quantity demanded equals the percentage change in price, the price elasticity of demand equals 1 and the good has unit elastic demand.

• Between the two previous cases, the percentage change in the quantity demanded is smaller than the percentage change in price so that the price elasticity of demand is less than 1 and the good has inelastic demand.

• If the percentage change in the quantity demanded is infinitely large when the price barely changes, the price elasticity of demand is infinite and the good has perfectly elastic demand.

Elasticity

• Measure of how responsive is demand to changes in prices and income.

• Price elasticity of demand: percent change in quantity divided by the percent change in price.

εp= (Δq/Δp) * (p/q)

• Elasticity and revenue: Revenue= price * quantity ΔR=pΔq + qΔp

When is ΔR/Δp>0? When ε(p)>-1

The change in total revenue due to a change in price depends on the elasticity of demand:

• If demand is elastic, a 1 percent price cut increases the quantity sold by more than 1 percent, and total revenue increases.

• If demand is inelastic, a 1 percent price cut decreases the quantity sold by more than 1 percent, and total revenues decreases.

• If demand is unitary elastic, a 1 percent price cut increases the quantity sold by 1 percent, and total revenue remains unchanged.

The total revenue test is a method of estimating the price elasticity of demand by observing the change in total revenue that results from a price change (when all other influences on the quantity sold remain the same).

• If a price cut increases total revenue, demand is elastic.

• If a price cut decreases total revenue, demand is inelastic.

• If a price cut leaves total revenue unchanged, demand is unit elastic.

• Constant elasticity of demand: pq= R , q= R/p • Elasticity and marginal revenue:

ΔR=pΔq + qΔp MR=ΔR/ Δq =p+ q*(Δp/Δq)=p(1+ (1/ε))

If objective is to maximize profits, producer wants to be on the elastic part of the demand.

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Unit elastic

Inelastic demand

Inelasticdemand;price cutdecreasesrevenue

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Income elasticity Income elasticity of demand= percent change in quantity over

percent change in price.

Normal good: Income elasticity >0 Inferior good: Income elasticity <0 Luxury good: Income elasticity >1

Weighted average of income elasticities:

(Δm/m) = (p1x1/m) Δx1/x1 + (p2x2/m) Δx2/x2

1= expenditure share on each good * income elasticity of each good.

Question: In a two good model, if one good is an inferior good, the other must be a luxury good. True or false?

Markets

– A market is any arrangement that enables buyers and sellers to get information and do business with each other.

– A competitive market is a market that has many buyers and many sellers so no single buyer or seller can influence the price.

Equilibrium

• Supply curve: how much individuals (producers) are willing to supply of a good at each possible market price.

We can derive the market supply.

• Demand curve: how much individuals are willing to demand of a good at each possible market price. ( how much they are willing to pay to get the good).

We can derive the market demand.

Competitive market: each supplier and demander takes prices as given and chooses his best response to the market prices.

• Equilibrium price: price where the supply of goods equals the demand.

S(p)= market supply D(p)= market demand Equilibrium price solves the equation S(p)=D(p)

Special cases: (i) Fixed supply (ii) Horizontal supply (firm wants to supply any amount of the

good at a constant price) (iii) Inelastic demand (iv) Horizontal demand

Price Determination

• At equilibrium price, quantity demanded equals quantity supplied

– below equilibrium price, shortage P– above equilibrium price, surplus P

– at equilibrium price, no tendency for change

– equilibrium quantity — quantity bought and sold at equilibrium P

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Taxes and equilibrium

• When prices are introduced in equilibrium, there are two prices to take into account: the price the supplier receives and the price the consumer is paying. These prices differ in the amount of the tax.

introduce tax in supply or demand: as far is prices finally paid by supplier and consumer, what matters is the amount of the tax, not who is responsible for paying it.

• Special cases:

(i) Perfectly elastic supply: all tax is passed along to the consumers

(ii) Perfectly inelastic supply: none of the tax is passed along.(iii) If supply is nearly horizontal, much of the tax can be passed. If

the supply is nearly vertical, very little of the tax can be passed along.

Deadweight loss due to taxes

Taxes change equilibrium quantities and produce a revenue for the government.

Question: what is the deadweight loss of a tax when supply is vertical?