Chapter 5:Applying Consumer Theory

• From chap 2&3, we learned that supply & demand curves yield a market equilibrium.

• From chap 4, we learned that a consumer maximizes his/her utility subject to constraints.

• This chapter does:– Derive demand curves from one’s u-max problem– How Δin income shifts demand (income elasticity)– Two effects of a price change on demand– Deriving labor supply curve using consumer theory– Inflation adjustment

• A consumer chooses an optimal bundle of goods subject to budget constraints.

• From the consumer’s optimum choice, we can derive the demand function:

x1= x1(p1, p2, Y)

• By varying own price (p1), holding both p2 and Y constant, we know how much x1 is demanded at any price.

→ Use this info to draw the demand curve.

5.1 Deriving Demand Curves

Figure 5.1 Deriving an Individual’s Demand Curve

Suppose that the price of beer changes while the price of wine remains constant.

Y = pbeerQbeer + pwineQwine

Original prices: pbeer=12, pwine=35Income: Y = 419The consumer can consume 12 (=419/35) units of wine or 35 (=419/12) units of beer if she consumes only one of the two.

Draw the budget line.

The price of beer changes: pbeer=6, pbeer=4 She can now consume 70 (=419/6) or 105(=419/4) units of beer.

Figure 5.1 Continued.

Change Pbeer holding Pwine and Y constant.

→ New budget constraint→ New optimal bundle of goods.

Tracing these optimal xbeer*, we can draw the demand curve for beer on Price-Quantity space.

5.2 How changes in Income shift demand curves

• How does demand curve change when income shifts, holding prices constant?

Figure 5.2 Effect of Budget Increase on an Individual’s Demand Curve

• Suppose that the income of the consumer increases.

• Income increases to $628 and $837 for same prices.

• She can now consume 18 (=628/35) units of wine or 52 (=628/12) units of beer if she consumes either one.

• Or she can now consume 24 (=837/35) units of wine or 70 (=837/12) units of beer if she consumes either one.

• The budget line expands outward, and she consumes more wine and beer because she can!

Figure 5.2 Continued.

Change Y holding Pbeer and Pwine constant.→ Budget line shifts outward→ New optimal bundle of goods

Demand curves shifts outward as Y increases if the good is normal.

Engel curve summarizes the relationship between income and quantity demanded, holding prices constant.

Income Elasticity of Demand

= How much quantity demanded changes when income increases.

% /

% /d d d d

d

in Q Q Q Q Y

in Y Y Y Y Q

Normal good η≥ 0 As Y rises, Qd also rises

Luxury η> 1 Qd increases by a greater proportion than Y

Necessity η< 1 Qd increases by a lesser proportion than Y

Inferior good η< 0 As Y rises, Qd decreases

Figure 5.3 Income-Consumption Curves and Income Elasticities

Figure 5.4 A Good that is both Inferior and Normal

5.3 Effects of a Price Change

• A decrease in p1 holding p2 & Y constant has two effects on individual’s demand:

Substitution effect: Change in Qd due to consumer’s behavior of substituting good 1 for good 2 (because x1 now relatively cheap), holding utility constant.

Income effect: Change in Qd due to effectively-increased income (lower p1 = higher buying power), holding prices constant.

Total effect = Substitution effect + Income effect

Total Effect

Suppose the consumer is maximizing utility at point A.

If the price of good x1 falls, the consumer will maximize utility at point B.This can be decomposed into two effects.

x1

x2

U1

A

U2

B

Total increase in x1

Substitution Effect

To isolate the substitution effect, we holdthe utility level constant but allow the relative price of good x1 to change

The substitution effect is the movementfrom point A to point C

The individual substitutes good x1 for good x2 because good x1 is now relatively cheaper

U1

x1

x2

A

Substitution effect

C

Income Effect

The income effect occurs because theindividual’s “real” income changes whenthe price of good x1 changes

The income effect is the movementfrom point C to point B

If x is a normal good,the individual will buy more because “real”income increased

What if x1 is an inferior good?

B

U1

U2

x1

x2

A C

Incomeeffect

Substitution effect

Total effect

Ordinary Goods and Giffen Goods

Ordinary Goods: As P decreases, Qd increases. ∂x1/∂p1 < 0 Giffen Goods: As P decreases, Qd decreases. ∂x1/∂p1 > 0

5.5 Deriving Labor Supply Curve

• We normally use consumer theory to derive demand behavior. But here, we derive labor supply curve using consumer theory.

• Individuals must decide how to allocate the fixed amount of time they have.

• The point here is “time is money.” When we do not work, we sacrifice or forgo wage income. That is, the opportunity cost of time is equal to the wage rate.

Utility function:

u= U(Y, N)

where N= Leisure time and Y is the consumption of other goods, which is equal to the labor income (wages).

Time constraint: H (labor time) + N = 24 hours

Max u = U(Y, N)

Subject to Y = w1 H = w1 (24 – N)

Model

The Budget Line

The time constraint: H + N =24

Leisure

Y = 24w

N (Leisure) H (Labor time)

Y = wH

The labor time determines how muchthe consumer can consumes the other goods.

Figure 5.8 Demand for leisure

Given 24hrs and wage w1

Original optimum at e1

To derive demand for leisure, increase wage to w2

New optimum at e2

A higher wage means a higher price of leisure

Demand curve for leisure on Price-Quantity space

Figure 5.9 Supply Curve of Labor

Substitution and Income Effects

• Both effects occur when w changes– Substitution effect: When w rises, the price for

leisure increases due to higher opportunity cost, and the individual will choose less leisure

– Income effect: Because leisure is a normal good, with increased income, she will choose more leisure

• The income and substitution effects move in opposite directions if leisure is a normal good.

Figure 5.10 Income and Substitution Effects of a Wage Change

The substitution effect is the movementfrom point A to point C

The individual chooses less leisure at B as a result of the increase in w

The income effect is the movementfrom point C to point B

Case 1: Substitution effect > Income effect

U1

U2

Leisure　（ N)

Consumption（ Y)

A

B

C

Substitution effectIncome effect

Total effect

Consumption（ Y)The substitution effect is the movementfrom point A to point C

The individual chooses more leisure at B as a result of the increase in w

The income effect is the movementfrom point C to point B

Case 2: Substitution effect < Income effect

U1

U2

Leisure（ N)

A

BC

Substitution effectIncome effect

Total effect

Application: Will you stop working if you win a lottery?

Figure 5.11 Labor Supply Curve that Slopes Upward and then Bends Backward

Tax revenue and Tax rates

Application: What is the optimal (i.e., maximizes the tax

revenue) marginal tax rate?

Sweden 58% (vs. actual 65%) Japan: 54 % (vs. 24 %)

Child-CareSubsidies:

The same resource for subsidy and the lump-sum payment. This means that the budgets lines go through e2.

5.4 Cost of Living Adjustments

• Nominal price: Actual price of a good• Real price: Price adjusted for inflation

• Consumer Price Index (Laspeyres index): Weighted average of the price increase for each good where weights are each good’s budget share in base year

Example

In the first case, both relative and real prices remain unchanged.

Real price = Nominal price / Price index, e.g., \240/2.00.

In the second case, it is not clear how we should compute the price index (P).

One reasonable way may be

where s: budget share

Year P1 P2

Price index

2000 \120 \500 100

2007 \240 \1,000 200

Year P1 P2

Price index

2000 \120 \500 100

2007 \108 \550 ??

1 1 2 21 2

1 2

p p p pP s s

p p

Price Index

Laspeyres index (Lp)

weight: base year quantity

= (Cost of buying the base-year’s bundles in the current year) / (Actual cost in the base year)

Paasche index (Pp)

weight: current year quantity

0 01 1 2 20 0 0 01 1 2 2

0 0 0 01 1 1 2 2 20 0 0 0

1 2

t t

p

t t

p x p xL

p x p x

p x p p x p

Y p Y p

1 1 2 20 01 1 2 2

1 1 1 2 2 20 01 2

t t t t

p t t

t t t t t t

t t

p x p xP

p x p x

p x p p x p

Y p Y p