4 - Demand

Question 1

If people’s tastes, their income, and the price of other goods are held constant, what does a movement along the demand curve correspond to?

Answer 1

A change in price.

Question 2

In chapter 3, we showed how to maximize utility subject to a budget constraint. How can we trace out the demand curves?

Answer 2

The demand functions would be in the form:
q1 = D1(p1,p2,Y)
q2 = D2(p1,p2,Y)
We can trace out the demand function for one good by varying its price while holding other prices and income constant.

Question 3

For Perfect Complements give the:

• Utility function
• Solution type
• Demand function for q1
• Demand function for q2

Answer 3

min(q1,q2)
Interior
q1 = Y/(p1+p2)
q2 = Y/(p1+p2)

Question 4

For CES, ρ ≠ 0, ρ<1, σ = 1/(ρ-1) give the:

• Utility function
• Solution type
• Demand function for q1
• Demand function for q2

Answer 4

(q1^ρ + q2^ρ)^(1/ρ)
Interior
q1 = (Yp1^σ) / (p1^(σ+1) + p2^(σ+1))
q2 = (Yp2^σ) / (p1^(σ+1) + p2^(σ+1))

Question 5

For the Cobb-Douglas give the:

• Utility function
• Solution type
• Demand function for q1
• Demand function for q2

Answer 5

q1^(α)q2^(1-α)
Interior
q1 = αY/p1
q2 = (1-α)Y/p2

Question 6

For Perfect Substitutes where p1=p2=p give the:

• Utility function
• Solution type
• Demand function for q1
• Demand function for q2

Answer 6

q1 + q2
Interior
q1+q2 = Y/p

Question 7

For Perfect Substitutes where p1

Answer 7

q1 + q2
Corner
q1 = Y/p1
q2 = 0

Question 8

For Perfect Substitutes where p1>p2 give the:

• Utility function
• Solution type
• Demand function for q1
• Demand function for q2

Answer 8

q1 + q2
Corner
q1 = 0
q2 = Y/p2

Question 9

For Quasillinear where Y > a^(2)p2/(4p1) give the:

• Utility function
• Solution type
• Demand function for q1
• Demand function for q2

Answer 9

aq1^(0.5) + q2
Interior
q1 = (ap2/2pq)^2
q2 = Y/p2 - (a^(2)p2/4p1)

Question 10

For Quasillinear where Y < a^(2)p2/(4p1) give the:

• Utility function
• Solution type
• Demand function for q1
• Demand function for q2

Answer 10

aq1^(0.5) + q2
Corner
q1 = Y/p1
q2 = 0

Question 11

How can we derive the demand curve graphically?

Answer 11

If we increase the price of a product while holding other prices, the consumer’s tastes, and income constant we cause the consumer’s budget constraint to rotate, prompting the consumer to chose a new optimal bundle. This change in quantity demanded is the information we need to draw the demand curve.

Question 12

How can we graphically visualize deriving the demand curve with two graphs one on top of the other, the top one a IC and BC graph between two goods, the bottom one a demand curve for the good on the x-axis of the previous graph.

Answer 12

• Panel a): q1 on x-axis, q2 on y-axis (held constant). The various budget constraints (which correspond to various p1’s) rotate inward as the price goes up, reaching lower indifference curves at each new, lower optimal bundle.
• Panel b): q1 on x-axis, p1 on y-axis. The downward sloping demand curve traces the q1 obtained from panel a) on the x-axis, with the corresponding p1.

Question 13

In our graphical explanation of how we derive the demand curve, what does panel a) also show?

Answer 13

The price-consumption curve

Question 14

What is the price-consumption curve?

Answer 14

The line through the optimal bundles that the consumer would consume at each price of q1, when p1 and Y are held constant.

Question 15

What does the upward sloping nature of the price-consumption curve tell us?

Answer 15

Because the price-consumption curve tell is upward sloping, we know that the their consumption of both q1 and q2 will increase as the p1 falls.

Question 16

Given our explanation for how to graph a demand curve, how can we use the same information in the price-consumption curve to draw a consumer’s demand curve, for q1?

Answer 16

Corresponding to each possible p1 on the vertical axis of panel b), we record on the horizontal axis the q1 demanded by the consumer from the price-consumption curve.

Question 17

How does our explanation for how to graph a demand curve relate to the inverse relationship between price and utility?

Answer 17

We can use the relationship between the points in panel a and b in the explanation for how to graph a demand curve to show that consumer’s utility is higher at lower prices.

Question 18

What is the effect of an increase in income, holding tastes and prices constant?

Answer 18

An increase in an individual’s income,holding tastes and prices constant, causes a shift of the demand curve. An increase in income causes a parallel shift of the budget constraint away from the origin, prompting a consumer to choose a new optimal bundle with more of some or all of the goods.

Question 19

What are the 3 graphs we use to analyze a change in income? (The 3 graphs all have the quantity of the good on the x-axis.)

Answer 19

The 3 graphs all have the quantity of the good on the x-axis.

• The y-axis of the first graph is q2 and this graph has indifference curves and budget constraints (representing different income levels).
• The y-axis of the second graph is p1 and this graph is the demand curve.
• The y-axis of the third graph is Y and this graph represents the Engel curve

Question 20

There are the 3 graphs we use to analyze a change in income? The y-axis of the first graph is q2 and this graph has indifference curves and budget constraints (representing different income levels). What do the various equilibriums represent or what are they called?

Answer 20

The income-consumption curve (or income-expansion path) show how consumption of q1 and q2 rise as income rises. As income goes up, consumption of both goods increases.

Question 21

How can we show the relationship between the quantity demanded and income directly rather than by shifting demand curves to illustrate to illustrate the effect?

Answer 21

We can plot the Engel curve, which show the relationship between q1 and Y, holding prices constant, with q1 on the x-axis and Y on the y-axis.

Question 22

How are income elasticities useful in analyzing how increases in income affect demand?

Answer 22

Income elasticities tell us how much the quantity demanded of a product changes as income increases. We can use income elasticities to summarize the shape of the Engel curve or the shape of thei ncome-consumption curve.

Question 23

When is a good said to be an inferior good?

Answer 23

A good is called an inferior good is less of it is demanded as income rises: ξ<0.

Question 24

When is a good said to be a normal good?

Answer 24

A good is called a normal good if more of it is demanded as income rises: ξ≥0.

Question 25

When is a good said to be a luxury good?

Answer 25

A good is called a luxury good if the quantity demanded of a normal good rises more than in proportion to a person’s income: ξ>1.

Question 26

When is a good said to be a necessity?

Answer 26

A good is called a necessity if the quantity demanded of a normal good rises less than or in proportion to a person’s income: 0≤ξ≤1.

Question 27

What does the shape of the income-consumption curve tell us?

Answer 27

The sign of their income elasticities: whether the income elasticities for those goods are positive or negative.

Question 28

The horizontal and vertical dotted lines through the original equilibrium quantities divide the new, outward-shifted budget line into three sections. Explain how the section where the new optimal bundle is located determines the consumer’s income elasticities of q1 and q2. Suppose that the consumer’s indifference curve is tangent to the new budget line at a point in the upper-left section (to the left of the vertical dotted line).

Answer 28

The consumer’s income-consumption curve, which goes from the original equilibrium q1 and q2 to the new equilibrium (upper-left), he buys more q2 and less q1 as his income rises, so q2 is a normal good (ξ≥0) for the consumer and q1 is an inferior good (ξ<0).

Question 29

The horizontal and vertical dotted lines through the original equilibrium quantities divide the new, outward-shifted budget line into three sections. Explain how the section where the new optimal bundle is located determines the consumer’s income elasticities of q1 and q2. Suppose that the consumer’s indifference curve is tangent to the new budget line at a point in middle section (to the right of the vertical dotted line and above the horizontal lines).

Answer 29

The consumer’s income-consumption curve, which goes from the original equilibrium q1 and q2 to the new equilibrium (middle), is upward sloping, so he buys more of both q2 and q1 as his income rises, so q1 and q2 are normal goods (ξ≥0).

Question 30

The horizontal and vertical dotted lines through the original equilibrium quantities divide the new, outward-shifted budget line into three sections. Explain how the section where the new optimal bundle is located determines the consumer’s income elasticities of q1 and q2. Suppose that the consumer’s indifference curve is tangent to the new budget line at a point in the lower-right section (to the right of the vertical dotted line and below the horizontal dotted line).

Answer 30

The consumer’s income-consumption curve, which goes from the original equilibrium q1 and q2 to the new equilibrium (middle), is downward-sloping, so he buys more q1 and less q2 as his income rises, so q1 is a normal good (ξ≥0) and q2 is an inferior good (ξ<0).

Question 31

How does a consumer’s consumption ultimately react to an increase in income?

Answer 31

When a consumer’s income goes up, their budget constraint shifts outward. Depending on their tastes (the shape of the indifference curves), they may buy more q1 and less q2, less q1 and more q2, or more q1 and more q2. Therefore, either both goods are normal, or one good is normal and the other is inferior.

Question 32

Why is it impossible for all goods to be inferior?

Answer 32

If both goods were inferior, the consumer would buy less of both goods as their income rises - which makes no sense. Were they to buy less of both, they would be buying a bundle that lies inside their original budget constraint. Even at their original, relatively low income, they could have purchased that bundle but chose not to.

Question 33

We just argued using graphical and verbal reasoning that a consumer cannot view all goods as inferior. How can we derive the stronger result: The weighted sum of a consumer’s income elasticities equals one?

Answer 33

We start with:
p1q1 + p2q2 + … + pnqn = Y
By differentiating this equation with respect to income, we obtain:
p1dq1/dY + p2dq2/dY + … + pnqqn/dY = 1
Multiplying and dividing each term by qiY:
(p1q1/Y)(dq1/dY)(Y/q1) + (p1q1/Y)(dq1/dY)(Y/q1 + … + (pnqn/Y)(dqn/dY)(Y/qn) = 1
If we define the budget share of Good i as θi = piqi/Y and note that the income elasticities are ξi = (dqi/dY)(Y/qi), we can rewrite this expression to show that the weighted sum of the income elasticities equals one:
θ1ξ1 + θ2ξ2 + … + θnξn = 1

Question 34

How can we use the weighted income elasticities formula θ1ξ1 + θ2ξ2 + … + θnξn = 1 to make predictions about income elasticities?

Answer 34

If we know the budget share of a good and a little bit about the income elasticities of some goods, we can calculate bounds on other, unknown income elasticities, which can be very useful to governments and firms.

Question 35

Holding tastes, other prices, and income constant, an increase in a price of a good has which two effects on an individual’s demand?

Answer 35

• Substitution effect

- Income effect

Question 36

What is the substitution effect?

Answer 36

The change in the quantity of a good that a consumer demands when the good’s price rises, holding other prices and the consumer’s utility constant. If the consumer’s utility is held constant as the price of the good increases, the consumer substitutes other goods that are now relatively cheaper for this now more expensive good.

Question 37

What is the income effect?

Answer 37

The change in the quantity of a good that a consumer demands because of a change in income, holding prices constant. An increase in price reduces a consumer’s buying power, effectively reducing the consumer’s income or opportunity set and causing the consumer to buy less of at least some goods.

Question 38

What is a doubling of the price of all goods the consumer buys equivalent to?

Answer 38

A drop in the consumer’s income to half its original level. Even a rise in the price of only one good reduces a consumer’s ability to buy the same amount of all goods previously purchased.

Question 39

How do changes in product prices affect consumption?

Answer 39

When the price of a product rises, the total change in the quantity purchased is the sum of the substitution effect and the income effect

Question 40

Describe how we should think about the substitution effect?

Answer 40

The substitution effect is the change in the quantity demanded from a compensated change in p1. which occurs when we increase the consumer’s income by enough to offset the rise in price so that their utility stays constant.

Question 41

How can we determine the substitution effect from an increase in price?

Answer 41

To determine the substitution effect, we draw an imaginary budget constraint that is parallel to L2 (which is an inwardly rotated version of L1) and tangent to their original indifference curve I1. This imaginary budget constraint L* has the same slope as L2 because both curves are based on the new, higher p1.

Question 42

How can we visualize the substitution effect graphically?

Answer 42

If p1 rises relative to p2 and we hold the consumer’s utility constant by raising their income to compensate them, their optimal bundle shifts from e1 to e*. The corresponding change in q1 is the substitution effect.

Question 43

How does an increase in price also induce an income effect?

Answer 43

The consumer also faces an income effect because the increase in p1 shrinks their opportunity set. The income effect is the change in the quantity of a good a consumer demands because of a change in income, holding prices constant.

Question 44

How can we visualize the income effect graphically?

Answer 44

The parallel shift of the budget constraint from L* to L2 captures this effective decrease in income. The change in q1 due to the movement from e* to e2 is the income effect.

Question 45

What is the total effect of a change in price?

Answer 45

Total effect = substitution effect + income effect

Question 46

Why is the substitution effect unambiguous?

Answer 46

Because indifference curves are convex to the origin: Less of a good is consumed when its price rises given that a consumer is compensated so that they remain on the original indifference curve. The substitution effect causes a movement along an indifference curve.

Question 47

How should we think about the income effect?

Answer 47

The income effect causes a shift to another indifference curve due to a change in the consumer’s opportunity set. The direction of the income effect depends on the income elasticity. If a good is a normal good, their income effect is negative as their income drops.

Question 48

What is the total effect of a price increase for an inferior good?

Answer 48

If a good is inferior, the income effect and the substitution effect move in opposite directions. For most inferior goods, the income effect is smaller than the substitution effect. As a result, the total effect moves in the same direction as the substitution effect, but the total effect is smaller.

Question 49

What is the total effect of a price decrease for a Giffen good?

Answer 49

For a Giffen good, a decrease in its price causes the quantity demanded to fall because the income effect more than offsets the substitution effect.

Question 50

What does the demand curve for a Giffen good look like?

Answer 50

It’s a demand curve with an upward slope.

Question 51

What is a Marshallian demand curve?

Answer 51

The regular or Uncompensated demand curve.

Question 52

Describe the Marshallian demand curve.

Answer 52

Along the demand curve, we hold other prices,, income, and the consumer’s tastes constant, while allowing their utility to vary. We can observe this type of demand curve by seeing how the purchases of a product changes as its prices increases.

Question 53

Unless otherwise stated, which demand curve are we referring to?

Answer 53

The Marshallian or Uncompensated demand curve.

Question 54

What does the Compensated demand curve show?

Answer 54

How the quantity demanded changes as the prices rises, holding utility constant, so that the change in the quantity demanded reflects only the pure substitution effect from a price change.

Question 55

Why is it called the compensated demand curve?

Answer 55

Because we would have to compensate an individual - give the individual extra income - as the price rises so as to hold the individual’s utility constant.

Question 56

What else is the compensated demand curve also called?

Answer 56

The Hicksian demand curve.

Question 57

What is the compensated demand function for the first good?

Answer 57

q1 = H(p1,p2,U̅) where we hold utility constant.

Question 58

Why can’t we observe the compensated demand curve directly?

Answer 58

Because we do not observe utility levels.

Question 59

What is a consequence of the fact that the compensated demand reflects only substitution effects?

Answer 59

Because the compensated demand reflects only substitution effects, the Law of Demand must hold: A price increase causes the compensated demand for a good to fall.

Question 60

Compare the Marshallian and the Hicksian demand curves graphically.

Answer 60

Their compensated and uncompensated demand curves must cross at the original price where the original budget line, L, is tangent to I along which utility is U̅. At that price, and only at that price, both demand curves are derived using the same budget line. The compensated demand curve is steeper than the uncompensated curve around this common point, because it reflects only the substitution effect. The uncompensated demand curve is flatter because the (normal good) income effect reinforces the substitution effect.

Question 61

How can we use the expenditure function E = E(p1,p2,U̅), where E is the smallest expenditure that allows the consumer to achieve utility level U̅, given market prices?

Answer 61

If we differentiate the expenditure function with respect to the price of the first good, we obtain the compensated demand function for that good:
∂E/∂p1 = H(p1,p2,U̅) = q1

Question 62

What is an informal explanation for ∂E/∂p1 = H(p1,p2,U̅) = q1?

Answer 62

That if p1 increases by $1 on each of the q1 units that the consumer buys, then the minimum amount the consumer must spend to keep their utility constant must increase by $q1.

Question 63

What is a second, more intuitive explanation for ∂E/∂p1 = H(p1,p2,U̅) = q1?

Answer 63

This expression can also be interpreted as the pure substitution effect on the quantity demanded because we are holding the consumer’s utility constant as we change the price.

Question 64

What does the usual price elasticity of demand, ε, capture?

Answer 64

The usual price elasticity of demand, ε, captures the total effect of a price change - that is, the change along an uncompensated demand curve.

Question 65

The usual price elasticity of demand, ε, captures the total effect of a price change - that is, the change along an uncompensated demand curve. We can break this price elasticity of demand into two terms involving elasticities that capture the substitution and income effects. How do we measure the substitution effect?

Answer 65

We measure the substitution effect using the pure substitution elasticity of demand, ε, which is the percentage that the quantity demanded falls for a given percentage increase in price if we compensate the consumer to keep the consumer’s utility constant. That is, ε is the elasticity of the compensated demand curve.

Question 66

The usual price elasticity of demand, ε, captures the total effect of a price change - that is, the change along an uncompensated demand curve. We can break this price elasticity of demand into two terms involving elasticities that capture the substitution and income effects. How do we measure the income effect?

Answer 66

The income effect is the income elasticity, ξ, times the share of the budget spent on that good, θ.

Question 67

What is this relationship among the price elasticity of demand, ε, and the income elasticity of demand, ξ, called?

Answer 67

The Slutsky equation

Question 68

What is the Slutsky equation?

Answer 68

Total effect = substitution effect + income effect

ε = ε* + (-θξ)

Question 69

What does the Slutsky equation ε = ε* + (-θξ) imply?

Answer 69

If a consumer spends little on a good, a change in its price does not affect the person’s total budget significantly. Thus, the total effect, ε, hardly differs from the substitution effect, ε*, because the price change has little effect on the consumer’s income.

Question 70

What is required for a Giffen good to have an upward-sloping demand curve?

Answer 70

For a Giffen good to have an upward-sloping demand curve, ε must be positive. The substitution elasticity, ε*, is always negative:Consumers buy less of a good when its price increases, holding utility constant. Thus, for a good to be a Giffen good and have an upward-sloping demand curve, the income effect, -θξ, must be positive and large relative to the substitution effect.

Question 71

For a Giffen good to have an upward-sloping demand curve, ε must be positive. The substitution elasticity, ε*, is always negative, thus, for a good to be a Giffen good and have an upward-sloping demand curve, the income effect, -θξ, must be positive and large relative to the substitution effect. When is this likely to happen?

Answer 71

The income effect is more likely to be a large positive number if the good is very inferior (that is, ξ is a large negative number, which is not common) and the budget share, θ, is large (closer to one than zero).

Question 72

What is one reason we don’t see upward-sloping demand curves?

Answer 72

The goods on which consumers spend a large share of their budget, such as housing, are usually normal goods rather than inferior goods.

Question 73

By knowing both the substitution and income effects, we can answer questions that we could not answer if we knew only the total effect of a price change. What is one particularly important use of consumer theory?

Answer 73

To analyze how accurately the government measures inflation. Many long-term contracts and government programs include cost-of-living adjustments (COLA’s), which raise prices or income in proportion to an index of inflation.

Question 74

How can we can use an example with only two goods, clothing and food, to show how the CPI is calculated.

Answer 74

In the first year, consumers buy C1 units of clothing and F1 units of food at prices pc1 and pf1. We use this bundle of goods, C1 and F1, as our base bundle for comparison. In the second year, consumers buy C2 and F2 units at prices pc2 and pf2. The government knows from its surveys of prices that the price of clothing in the second year is pc2/pc1 times as large as the price in the previous year.

Question 75

One way we can average the price increases of each good in calculating the CPI is to weight them equally. But do we really want to do that? What’s an alternative way of weighting them?

Answer 75

To assign a larger weight to the price changes of goods with relatively large budget shares. In constructing its averages, the CPI weights using budget shares.

Question 76

The CPI for the first year is the amount of income it took to buy the market basket that was actually purchased that year:
Y1 = pc1C1 + pf1F1.
What is the cost of buying the first year’s bundle in the second year?

Answer 76

Y2 = pc2C1 + pf2F1.

That is, in the second year, we use the prices for the second year but the quantities from the first year.

Question 77

How do we calculate the rate of inflation here?

Answer 77

We determine how much more income it took to buy the first year’s bundle in the second year, which is the ratio:
Y2/Y1 = (pc2C1 + pf2F1) / (pc1C1 + pf1F1)

Question 78

What does the ratio Y2/Y1 = (pc2C1 + pf2F1) / (pc1C1 + pf1F1) reflect?

Answer 78

How much prices rise on average.

Question 79

The ratio Y2/Y1 = (pc2C1 + pf2F1) / (pc1C1 + pf1F1) reflects how much prices rise on average. By multiplying and dividing the first term in the numerator by pc1 and multiplying and dividing the second term by pf1, what do we find that this index is equivalent to?

Answer 79

Y1/Y2 = [(pc2/pc1)pc1C1 + (pf2/pf1)pf1F1)] / Y1
= (pc2/pc1)θc + (pf2/pf1)θf
where θc = pc1C1/Y1 and θf = pf1F1/Y1 are the budget shares of clothing and food in the first, or base, year.

Question 80

What is the CPI a weighted average of?

Answer 80

The CPI is a weighted average of the price increase for each good, pc2/pc1 and pf2/pf1, where the weights are each good’s budget share in the base year, θc and θf.

Question 81

Klaas signed a long-term contract when he was hired. According t the COLA clause in his contract, his employer increases his salary each year by the same percentage that the CPI increases. Klaas spends all his money on clothing and food. His budget constraint in the first year is Y1 = pc1C + pf1F, which we rewrite as:
C = Y1/pc1 - (pf1/pc1)F
Describe this curve in the first year.

Answer 81

The intercept of the budget constraint, L1, on the vertical (clothing) axis is Y1/pc1, and the slope of the constraint is -(pf1/pc1). The tangency of his indifference curve I1 and the budget constraint L1 determine his optimal consumption bundle in the first year.

Question 82

Klaas signed a long-term contract when he was hired. According t the COLA clause in his contract, his employer increases his salary each year by the same percentage that the CPI increases. Klaas spends all his money on clothing and food. His budget constraint in the first year is Y1 = pc1C + pf1F, which we rewrite as:
C = Y1/pc1 - (pf1/pc1)F
Describe this curve in the second year.

Answer 82

In the second year, his salary rises with the CPI to Y2, so his budget constraint in that year, L2, is
C = Y2/pcp2 - (pf2/pc2)F
The new constraint, L2, has a flatter slope than L1, and goes through the original optimal bundle.

Question 83

Klaas signed a long-term contract when he was hired. According t the COLA clause in his contract, his employer increases his salary each year by the same percentage that the CPI increases. Klaas spends all his money on clothing and food. His budget constraint in the first year is Y1 = pc1C + pf1F, which we rewrite as:
C = Y1/pc1 - (pf1/pc1)F
In the second year, his salary rises with the CPI to Y2, so his budget constraint in that year, L2, is
C = Y2/pcp2 - (pf2/pc2)F
Why does the new constraint, L2, have a flatter slope than L1?

Answer 83

The new constraint, L2, has a flatter slope, -(pf2/pc2), than L1 because the price of clothing rose more than the price of food.

Question 84

Klaas signed a long-term contract when he was hired. According t the COLA clause in his contract, his employer increases his salary each year by the same percentage that the CPI increases. Klaas spends all his money on clothing and food. His budget constraint in the first year is Y1 = pc1C + pf1F, which we rewrite as:
C = Y1/pc1 - (pf1/pc1)F
In the second year, his salary rises with the CPI to Y2, so his budget constraint in that year, L2, is
C = Y2/pcp2 - (pf2/pc2)F
Why does the new constraint, L2, go through the original optimal bundle?

Answer 84

The new constraint goes through the original optimal bundle because by increasing his salary according to the CPI, the firm ensures that Klaas can buy the same bundle of goods in the second year that he bought in the first year.

Question 85

Given our CPI adjusted worker contract example, Klass can buy the same bundle after his raise, but does he? Why not?

Answer 85

No. His optimal bundle in the second year is e2, where the indifference curve I2 is tangent to his new budget constraint, L2. The movement from e1 to e2 is the total effect from the changes in the real prices of clothing and food. This adjustment to his income does not keep him on his original indifference curve I1.

Question 86

Klaas is better off in the second year than in the first. How so?

Answer 86

The CPI adjustment overcompensates him for the change in inflation in the sense that his utility increases. Klass is better off because the prices of clothing and food did not increase by exactly the same amount.

Question 87

How does the CPI adjustment raise Klaas’ utility?

Answer 87

After a CPI adjustment, Klaas’ budget constraint in the second year, L2, would be exactly the same as in the first year, L1, so he would choose exactly the same bundle, e1, in the second year as he chose in the first year. Because the price of food rose by less than the price of clothing, L2 is not the same as L1. Food became cheaper relative to clothing. Therefore, by consuming more food and less clothing, Klass has a higher utility in the second year.
Had clothing become relatively less expensive, Klass would have raised his utility in the second year by consuming relatively more clothing. Thus, it doesn’t matter which good becomes relatively less expensive over time for Klaas to benefit from the CPI compensation; it’s necessary only for one of the goods to become a relative bargain.

Question 88

Do CPI adjustment’s match inflation?

Answer 88

A CPI adjustment overcompensates for inflation.

Question 89

What is a true cost-of-living index?

Answer 89

An inflation index that holds utility constant over time.

Question 90

How can we identify how big of an increase in Klaas’ salary would leave him exactly as well off in the second year as he was in the first?

Answer 90

We can answer this question by applying the same technique we used to identify the substitution and income effects. We draw an imaginary budget line, L, that is tangent to I1 so that Klaas’ utility remains constant but has the same slope as L2. The income Y, corresponding to that imaginary budget constraint is the amount that leaves Klaas’ utility constant.

Question 91

How can we construct Klaas’ true cost-of-living index?

Answer 91

We can use the income that just compensates Klaas for the price changes, Y*, to construct a true cost-of-living index.

Question 92

We have demonstrated that the CPI has an upward bias in the sense that an individual’s utility rises if we increase the person’s income by the same percentage by which the CPI rises. If we make the CPI adjustment, what are we implicitly assuming (incorrectly)?

Answer 92

That consumers do not substitute toward relatively inexpensive goods when prices change, but they keep buying the same bundle of goods over time.

Question 93

What do we call this overcompensation from CPI adjustments?

Answer 93

A substitution bias.

Question 94

The CPI calculates the increase in prices as Y2/Y1. We can rewrite this expression as:
Y2/Y1 = (Y/Y1)(Y2/Y)
Describe this expression.

Answer 94

The first term to the right of the equal sign, Y/Y1, is the increase in the true cost-of-living. The second term, Y2/Y, reflects the substitution bias in the CPI. It is greater than 1 because Y2>Y*, so the CPI overestimates the increase in the cost-of-living.

Question 95

What is the exception to the general rule of CPI adjustments overcompensating due to substitution bias?

Answer 95

There is no substitution bias if all prices increase at the same rate so that relative prices remain constant.

Question 96

Given that CPI adjustments generally overcompensate due to the substitution bias, when is this bias greatest?

Answer 96

The faster some prices rise relative to others, the more pronounced the upward bias caused by the substitution that occurs toward less expensive goods.

Question 97

We’ve seen that we can predict a consumer’s purchasing behavior if we know that person’s preferences. We can also do the opposite. What do we mean by this?

Answer 97

We can infer a consumer’s preferences from observing the consumer’s buying behavior. If we observe a consumer’s choice at many different prices and income levels, we can derive the consumer’s indifference curves using the theory of revealed preference.

Question 98

What is the utility of revealed preference?

Answer 98

We can use this theory to demonstrate the substitution effect. Economists can use this approach to estimate demand curves merely by observing the choices consumers make over time.

Question 99

What is the first basic assumption of the theory of revealed preference?

Answer 99

That a consumer choose bundles to maximize utility subject to a budget constraint: The consumer chooses the best bundle that the consumer can afford.

Question 100

What is the second basic assumption of the theory of revealed preference?

Answer 100

That the consumer’s indifference curve is convex to the origin so that the consumer picks a unique bundle on any budget constraint.

Question 101

If such a consumer chooses a more expensive bundle of goods, a over a less expensive bundle,b, then we say that the consumer prefers bundle a to b.

Answer 101

We say that bundle a is revealed to be preferred to bundle c if the consumer chooses a over c directly, or if we learn indirectly that the consumer prefers a to b and b to c.

Question 102

We know that a consumer prefers bundle a to any other bundle in the shaded area, labeled “Worse Bundles”, by a sequence of direct or indirect comparisons. Due to the more-is-better property, the consumer prefers bundles in the are above and to the right of a. What does this imply?

Answer 102

Thus, the indifference curve through a must lie within the white area between the worse and better bundles.

Question 103

If we learn that a consumer chooses bundle a when faced with L1, bundle d when faced with L3, and bundle e when faced with L4, we can expand their better bundle area. How does this expanded better bundle area help us with revealed preference?

Answer 103

We know that their indifference curve through a must lie within the white area between the better and worse bundle areas. Thus, if we observe a large number of choices, we can determine the shape of their indifference curves, which summarize their preferences.

Question 104

One of the clearest and most important results from consumer theory is that the substitution effect is negative: The Law of Demand holds for compensated demand curves. This result stems from utility maximization, given that indifference curves are convex to the origin. The theory of revealed preference provides an alternative justification without appealing to unobservable indifference curves or utility functions. Suppose that a consumer is indifferent between bundle a, which consists of Ma music tracks, and Ca candy bars,, and bundle b with Mb music tracks and Cb candy bars. That is, the bundles are on the same indifference curve. The price of candy bars, C, remains fixed at pc, but the price of songs changes. We observe that when the price for M is pma, the consumer chooses bundle a - that is, a is revealed to be preferred to b. Similarly, when the price is pmb, he chooses b over a. Continue the reasoning.

Answer 104

Because the consumer is indifferent between the two bundles, the cost of the chosen bundle must be less than or equal to that of the other bundle. Thus, if he chooses a when the price is pma, then 
pmaMa + pcCa ≤ pmaMb + pcCb or
pma(Ma-Mb) + pc(Ca-Cb) ≤ 0
And if he chooses b when the price is pmb, then 
pmbMb + pcCb ≤ pmbMa + pcCa
or 
pmb(Mb-Ma) + pc(Cb-Ca) ≤ 0 
Adding the two equations we learn that
(pma-pmb)(Ma-Mb) ≤ 0

Question 105

In reference to revealed preference, what does the equation (pma-pmb)(Ma-Mb) ≤ 0 show and what does it allow us to do?

Answer 105

That the product of the difference in prices times the difference in quantities of music purchased is nonpositive. That result can be true only if the price and quantity move in opposite directions: When the price rises, the quantity falls. Thus, we are able to derive the substitution effect result without using utility functions or making any assumptions about the curvature of indifference curves.