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Chapter 4
Demand
I have enough money to last me the rest
of my life, unless I buy something.
Jackie Mason
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Chapter 4 Outline
4.1 Deriving Demand Curves4.2 Effects of an Increase in Income
4.3 Effects of a Price Increase
4.4 Cost-of-Living Adjustment
4.5 Revealed Preference
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4.1 Deriving Demand Curves
• If we hold people’s tastes, their incomes, and theprices of other goods constant, a change in the priceof a good will cause a movement along thedemand curve.
• We saw this in Chapter 2:
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4.1 Deriving Demand Curves
• In Chapter 3, we used calculus to maximize consumerutility subject to a budget constraint.
• This amounts to solving for the consumer’s system ofdemand functions for the goods.
• Example: q1 = pizza and q
2 = burritos
• Demand functions express these quantities in termsof the prices of both goods and income:
• Given a specific utility function, we can find closed-form solutions for the demand functions.
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4.1 Example: Deriving DemandCurves
• Constant Elasticity of Substitution (CES) utility function:
• Budget constraint:
• Y= p1q1 + p2q2
• In Chapter 3, we learned that the demand functions thatresult from this constrained optimization problem are:
• Quantity demanded of each good is a function of the pricesof both goods and income.
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4.1 Example: Deriving DemandCurves• Cobb-Douglas utility function:
• U(q1, q2) = q1a q2(1-a)
• Budget constraint:
• Y= p1q1 + p2q2
• In Chapter 3, we learned that the demand functionsthat result from this constrained optimization problemare:
• With Cobb-Douglas, quantity demanded of each goodis a function of only the good’s own-price and income.
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4.1 Deriving Demand Curves
• Panel a below shows the demand curve for q1,which we plot by holding Y fixed and varying p1.
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4.1 Deriving Demand CurvesGraphically
• Allowing the priceof the good on thex-axis to fall, thebudget constraint
rotates out andshows how theoptimal quantity ofthe x-axis goodpurchased
increases.• This traces out
points along thedemand curve.
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4.2 Effects of an Increase inIncome
• An increase in an individual’s income, holdingtastes and prices constant, causes a shift ofthe demand curve.
• An increase in income causes an increase indemand (e.g. a parallel shift away from theorigin) if the good is a normal good and adecrease in demand (e.g. parallel shift towardthe origin) if the good is inferior .
• A change in income prompts the consumer tochoose a new optimal bundle.
• The result of the change in income and thenew utility maximizing choice can be depictedthree different ways.
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4.2 Effects of an Increase inIncome
• The result of the change in income and thenew utility maximizing choice can be depictedthree different ways.
1.Income-consumption curve: using theconsumer utility maximization diagram, tracesout a line connecting optimal consumptionbundles.
2.Shifts in demand curve: using demanddiagram, show how quantity demanded increases
as the price of the good stays constant.3.Engle curve: with income on the vertical axis,
show the positive relationship between incomeand quantity demanded.
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4.2 Income-Consumption Curve andIncome Elasticities
• The shape of the income-consumption curvefor two goods tells us the sign of their incomeelasticities.
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4.2 Income-Consumption Curve andIncome Elasticities
• The shape of theincome-consumption andEngle curves can
change in waysthat indicate goodscan be both inferiorand normaldepending on an
individual’s incomelevel.
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4.3 Effects of a Price Increase
• Holding tastes, other prices, and incomeconstant, an increase in the price of a goodhas two effects on an individual’s demand:
1.Substitution effect : the change in quantitydemanded when the good’s price increases,holding other prices and consumer utilityconstant.
2.Income effect : the change in quantitydemanded when income changes, holding prices
constant.
• When the price of a good increases, the totalchange in quantity demanded is the sum ofthe substitution and income effects.
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4.3 Income and Substitution Effects
• The direction of the substitution effect isalways unambiguous.
• When price increases, individuals consume lessof it because they are substituting away from thenow more expensive good.
• The direction of the income effect dependsupon whether the good is normal or inferior;it depends upon the income elasticity.
•
When price increases and the good is normal,the income effect is negative.
• When price increases and the good is inferior,the income effect is positive.
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4.3 Income and Substitution Effectswith a Normal Good
• Beginning from budgetconstraint L1, anincrease in the price ofmusic tracks rotatesbudget constraint into
L2.
• The total effect ofthis price change, adecrease in quantityof 12 tracks per
quarter, can bedecomposed intoincome andsubstitution effects.
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4.3
CompensatedDemand Curve
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4.3 Compensated Demand Curve• Deriving the compensated, or Hicksian, demand curve
is straight-forward with the expenditure function:
• E is the smallest expenditure that allows theconsumer to achieve a given level of utility based ongiven market prices:
• Differentiating with respect to the price of the firstgood yields the compensated demand function for thefirst good:
• A $1 increase in p1 on each of the q1 units purchasedrequires the consumer increases spending by $q1 to keeputility constant.
• This result is called Shephard’s lemma.
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4.3 Slutsky Equation
• We graphically decomposed the total effect of a pricechange on quantity demanded into income andsubstitution effects.
• Deriving this same relationship mathematically utilizes
elasticities and is called the Slutsky equation.
• is elasticity of uncompensated demand and the totaleffect
• is elasticity of compensated demand and thesubstitution effect
• is the share of the budget spent on the good
• is the income elasticity
• is the income effect
*
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4.4 Cost-of-Living Adjustment
• Consumer Price Index (CPI): measure of the costof a standard bundle of goods (market basket) tocompare prices over time.
• Example: In 2010 dollars, what is the cost of a
McDonald’s hamburger in 1955?
• Knowledge of substitution and income effects allows
us to analyze how accurately the governmentmeasures inflation.
• Consumer theory can be used to show that the cost-of-living measure used by governments overstatesinflation.
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4.4 Cost-of-Living Adjustment(COLA)
• CPI in first year is the cost of buying the marketbasket of food (F) and clothing (C) that was actuallypurchased that year:
• CPI in the second year is the cost of buying the firstyear’s bundle in the second year:
• The rate of inflation determines how much additionalincome it took to buy the first year’s bundle in thesecond year:
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4.4 Cost-of-Living Adjustment(COLA)
• If a person’s incomeincreases automaticallywith the CPI, he canafford to buy the firstyear’s bundle in the
second year, butchooses not to.
• Better off in thesecond year becausethe CPI-based COLA
overcompensates inthe sense that utilityincreases.
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4.5 Revealed Preference
• Preferences predict consumer’s purchasing behavior
• Purchasing behavior infer consumer’s preferences