price theory

dodo1995

Lecture5-TheoryofDemand.pdf

Home >Business & Finance homework help >Economics homework help >price theory

Price Theory

Lecture 5: Theory of Demand

Topics for today’s lecture . . .

1. Income & substitution effects

2. Normal & inferior goods

3. Deriving demand

4. Measuring welfare

Income & substitution effects

Optimal choice with a changing price

Emily has an income of $500. Her preference over baskets containing groceries (good x) and

the composite good (good y ) are represented by the utility function U(x , y) = xy + 100x .

The associated marginal utilities are MUx = y + 100 and MUy = x .

How does Emily’s behaviour change when the price of groceries falls from an initial price of

Px1 = $10 per item, to a final price of Px2 = $5 per item?

What factors drive this change in behaviour?

A fall in the price of groceries

200

30 60

500

BL1 BL2

100

U1 U2

0 x

If the price of groceries is Px1 = $10 per

item, Emily’s optimal consumption basket

contains:

• xA = 30 items of groceries, and,

• yA = 200 units of the composite good.

If Px2 = $5 per item, Emily’s optimum is:

• xB = 60 items of groceries, and,

• yB = 200 units of the composite good.

(You should check these.)

Definition: Law of demand

All else being equal, the quantity demanded of a good falls as the price of the good rises.

The law of demand implies that demand curves are downward sloping. The term ‘law’ is used

because this relationship is pervasive throughout the economy.

Two consequences of a change in price

500

BL1 BL2

100

0 x

The fall in the price of groceries changes

the tradeoff along the budget line.

• Along BL1, each unit of groceries has an opportunity cost of 10 units of the

composite good.

• Along BL2, the opportunity cost falls to 5 units of the composite good.

The shift in the budget line also expands

the budget set, increasing the available

consumption possibilities.

Definition: Substitution effect

The change in the amount of a good that would be consumed as the price of that good

changes, holding constant all other prices and the level of utility.

If the price of groceries falls, Emily can achieve the same level of utility by substituting

groceries for other goods (i.e., by buying more groceries and less of other goods).

The decomposition consumption basket

200

30 60

BLD

500

BL1 BL2

U1 U2

0 x

We are looking for a basket that delivers

Emily the same utility as basket A, where

the slope of her indifference curve is:

−Px2 = −$5.

The decomposition budget line is tangent

to U1, and parallel to BL2.

The decomposition consumption basket

D is located at the point of tangency.

Exercise: Decomposition consumption basket

Recall that Emily’s utility function is U(x , y) = xy + 100x , with the associated marginal

utilities MUx = y + 100 and MUy = x .

1. Calculate the utility that Emily receives at the initial optimum (xA = 30 and yA = 200).

Use your answer to write an equation for the indifference curve that contains basket A.

2. Derive an expression for Emily’s marginal rate of substitution. Use your answer to write

the tangency condition at the final price (Px2 = 5).

3. Solve the equations for the indifference curve and tangency condition simultaneously to

find the decomposition consumption basket for the price change.

Exercise solutions

1. The utility Emily derives from basket A is,

U(30, 200) = xAyA + 100xA = 30× 200 + 100× 30 = 9000.

Therefore, every basket on the same indifference curve as basket A must satisfy the

equation,

xy + 100x = 9000.

2. Emily’s marginal rate of substitution is,

MUx MUy

= y + 100

x .

The tangency condition at the final prices can therefore be stated as,

MUx MUy

= Px2 Py

⇒ y + 100

x = 5.

Exercise solutions

3. To find the decomposition consumption basket we need to solve the tangency condition

and the indifference curve equation simultaneously. Rearranging the tangency condition

to solve for y , y + 100

x = 5 ⇒ y = 5x − 100.

Substituting for y into the indifference curve equation,

x(5x − 100) + 100x = 9000 ⇒ 5x2 = 9000 ⇒ xD = √

1800 ≈ 42.43.

Substituting for x into the tangency condition yields,

yD = 5 √

1800− 100 ≈ 112.13.

The substitution effect

112

200

30 60

BLD

500

BL1 BL2

U1 U2

0 x

The substitution effect is the change in

the consumption of groceries between

baskets A and D:

SE = xD − xA = 42.43− 30 = 12.43.

If preferences display a diminishing

marginal rate of substitution then:

• A price fall leads to a positive substitution effect.

• A price rise leads to a negative substitution effect.

Definition: Income effect

The change in the amount of a good that a consumer would buy as purchasing power

changes, holding all prices constant.

The income effect accounts for that part of the change in the quantity of groceries

purchased, that isn’t accounted for by the substitution effect.

The income effect

112

200

30 60

BLD

500

BL1 BL2

U1 U2

0 x

The budget lines BL2 and BLD have the

same slope.

When Px falls, the expenditure associated

with the final budget line BL2, is higher

than that associated with BLD .

The income effect is the change in the

consumption of groceries between baskets

D and B:

IE = xB − xD = 60− 42.43 = 17.57.

Quiz 1

BL2 BL1

0 x

For Oscar, right shoes (good x), and left

shoes (good y ), are perfect complements.

The price of right shoes increases, shifting

the budget line from BL1 to BL2.

The income and substitution effects

associated with this price change are:

(a) SE = 0 and IE = 3.

(b) SE = 3 and IE = 0.

(d) SE = −3 and IE = 0.

Normal & inferior goods

Normal goods

42 60

BLD

500

BL1 BL2

U1 U2

0 x

A good is normal if, all else being equal,

the quantity demanded rises as income

rises.

For Emily, groceries are a normal good as

an outward parallel shift in her budget line

leads to increased consumption of

groceries.

If a good is normal, the income and

substitution effects have the same sign.

Goods that are neither normal nor inferior

xB = xD

BLDBL1

BL2

U1 U2

0 x

A good is neither normal nor inferior if, all

else being equal, the quantity demanded

does not change as income rises.

In this example, the final consumption

basket, and the decomposition

consumption basket, contain the same

quantity of groceries.

If a good is neither normal nor inferior,

the income effect is zero.

Inferior goods

xB xD

BLDBL1

BL2

U1 U2

0 x

A good is inferior if, all else being equal,

the quantity demanded falls as income

rises.

In this example, the final consumption

basket contains less of the composite

good than the decomposition

consumption basket.

If a good is inferior, the income and

substitution effects have opposite signs.

Giffen goods

xAxB xD

BLD

BL1

BL2

0 x

A Giffen good is a good for which, all else

being equal, the quantity demanded rises

as the price of the good rises.

For a Giffen good, the income effect has

the opposite sign to the substitution

effect, and is larger in magnitude.

A Giffen good violates the Law of

Demand.

Quiz 2

Initially, the price of coffee is $3.50 per cup. At this price Horace purchases 8 cups per week.

When the price of coffee rises to $4 per cup, Horace reduces his consumption to 6 cups per

week. If the decomposition consumption basket for this price change contains 5 cups of

coffee then coffee is,

(a) a normal good.

(b) neither a normal nor an inferior good.

(d) a Giffen good.

Quiz 3

Initially, the price of coffee is $3.50 per cup. At this price Horace purchases 8 cups per week.

When the price of coffee falls to $2.50 per cup, Horace reduces his consumption to 5 cups

per week. If the decomposition consumption basket for this price change contains 10 cups of

coffee then coffee is,

(a) a normal good.

(b) neither a normal nor an inferior good.

(d) a Giffen good.

Deriving demand

Deriving demand from preferences

Tarquin’s preferences over baskets containing cappuccinos (good x), and the composite good

(good y ), are represented by the utility function U(x , y) = 40 √

x + y .

His corresponding marginal utilities are,

MUx = 20√

x and MUy = 1.

Tarquin’s income is $500.

How can we use this information to derive Tarquin’s demand for cappuccinos?

Optimal choice & price changes

500

BL1

125

BL2

250

BL3

500

U1 U2

0 x

Tarquin’s optimal consumption basket

varies with the price of cappuccinos:

• If Px = $4 then x = 25 cappuccinos and y = 400 units.

• If Px = $2 then x = 100 cappuccinos and y = 300 units.

• If Px = $1 then x = 400 cappuccinos and y = 100 units.

(You should check these.)

Plotting the demand curve

40010025

500

BL1 BL2 BL3

0 x

The x coordinate of Tarquin’s optimum is

the number of cappuccinos he demands

at the corresponding price.

The path described by these points is

Tarquin’s demand curve.

Notice that Tarquin’s income, and the

price of the composite good, do not vary

along the demand curve.

Exercise: Deriving demand functions

Recall that Tarquin’s preferences over baskets containing cappuccinos (good x), and the

composite good (good y ), are represented by the utility function U(x , y) = 40 √

x + y . The

corresponding marginal utilities are,

MUx = 20√

x and MUy = 1.

1. Using Px to represent the price of a cappuccino, and I to represent Tarquin’s income,

write the equation of Tarquin’s budget line. (Note that Py = 1 for the composite good.)

2. Derive tarquin’s marginal rate of substitution and use your answer to state the tangency

condition.

3. Solve the budget line equation and tangency condition simultaneously to find Tarquin’s

demand for cappuccinos and the composite good. Are cappuccinos a normal good?

Exercise solutions

1. Given that we only know one of the prices, the equation of Tarquin’s budget line is,

Pxx + y = I .

No further substitutions are possible.

2. Tarquin’s marginal rate of substitution is,

MUx MUy

= 20√

x ×

1 =

20√ x .

Given that Py = 1, the tangency condition can be stated as,

MUx MUy

= Px Py

⇒ 20√

x = Px .

Exercise solutions

3. Our goal is to find Tarquin’s optimal quantities of x and y as functions of Px and I only.

Rearranging Tarquin’s tangency condition to solve for x ,

20√ x = Px ⇒

√ x =

Px ⇒ x =

400

P2x .

For Tarquin, cappuccinos are neither normal nor inferior as demand for cappuccinos does

not vary with income. Substituting for x into the equation of Tarquin’s budget line,

( 400

P2x

) + y = I ⇒ y = I −

400

Px .

Changes in income

2 3

100

BL1

175

BL2

250

BL3

0 x

This figure shows Jared’s optimal basket

of movies (good x), and the composite

good (good y ), at three levels of income.

• I1 = $100.

• I2 = $175.

• I3 = $250.

Income and demand

2 3 4

100

175

250 y

A B

DI=100

DI=250

DI=175

0 x

Along each of the three budget lines, the

price of movies is Px = 25.

We can plot the x-coordinates of each of

Jared’s optimums against this price.

Each of these points lies on a different

demand curve, corresponding to the

different income levels.

We cannot learn anything further without

observing how Jared responds to a price

change.

Measuring welfare

The problem with measuring consumer utility

There are no ‘natural’ units for measuring an individual’s utility:

• Suppose that a individual’s preferences are represented by the utility function U(x , y) = f (x , y).

• For any constant α, and positive constant β, the utility function U ′(x , y) = α+ βf (x , y) also represents the individual’s preferences, because it ranks baskets in the same way.

Without units for measuring utility, it is not generally possible to make meaningful

interpersonal comparisons (and therefore not generally possible to aggregate utility).

This problem is a consequence of Arrow’s impossibility theorem.

Monetary measures of welfare

An alternative to measuring utility is to construct a monetary measure of the benefit an

individual derives from her/his consumption.

• A monetary measure has unambiguous units (eg. ‘dollars’).

• Monetary measures allow interpersonal comparisons.

• We can sum monetary measures across the members of society, to construct a measure of social welfare.

If we measure welfare in monetary terms, then welfare can be ‘transferred’ from one

individual to another (eg. via taxation and transfers).

Definition: Consumer surplus

A consumer’s ‘willingness to pay’ for a good or service, less the amount she/he actually pays.

Consumer surplus can be illustrated as the area between the demand curve, and the price.

Emily’s demand for groceries

0 x

Returning to Emily’s optimal choice

problem from earlier in the lecture:

Emily’s demand for groceries is described

by the function x = 300/Px .

• When Px = $10, Emily demands 30 units of groceries (basket A).

• When Px = $5, Emily demands 60 units of groceries (basket B).

Consumer surplus

0 x

The fall in the price of groceries,

increases Emily’s consumer surplus by an

amount equal to the shaded area.

• Emily receives increased surplus from each unit she was already consuming.

• And adds surplus from each additional unit she demands at the lower price.

The shaded area is approximately $208.

(This can be found by integrating the

demand function between $5 and $10.)

The problems with consumer surplus as a measure of welfare

Utilising a monetary measure does not allow us to avoid the implications of Arrow’s

impossibility theorem:

• Each dollar of benefit is not equal: Individuals typically experience diminishing marginal utility in consumption.

• Interpersonal comparisons may be problematic: Implicitly, monetary measures of welfare treat a dollar of benefit to a low-income consumer as equivalent to a dollar of benefit to

high-income consumer.

Quiz 4

0 x

This figure illustrates Henry’s demand for

wine. If the price of wine is Px = $28 per

bottle, Henry’s consumer surplus is,

(a) $63.

(b) $84.

(d) $175.

Quiz 5

2.5

0 x

This figure illustrates Henry’s demand for

wine. If the price of wine increases from

Px = $35 to Px = $42 per bottle, the

change in Henry’s consumer surplus is,

(a) −$1.75

(b) −$15.75

(d) −$28.00

Questions?

Key concepts from today’s lecture

You can use these concepts (as search terms) to conduct further research into the topics

covered in today’s lecture:

• Law of demand

• Substitution effect

• Income effect

• Decomposition consumption basket

• Normal goods

• Inferior goods

• Giffen goods

• Labour & leisure

• Labour supply curve

• Demand curve

• Social welfare

• Consumer surplus