Chapter 1 Consumer Theory Part II - Lakehead Universityflash.lakeheadu.ca/~kyu/E5113/NCT1B.pdf · 2019-01-10 · Outline 1 Introduction to Duality Theory Indirect Utility and Expenditure

Chapter 1 Consumer Theory Part IIEconomics 5113 Microeconomic Theory

Kam Yu

Winter 2019

Outline

1 Introduction to Duality TheoryIndirect Utility and Expenditure FunctionsOrdinary and Compensated Demand Functions

2 Properties of Consumer DemandRelative Prices and Real IncomeIncome and Substitution EffectsImplications

3 ElasticitiesDefinitionsAverage Elasticities

Kam Yu (Lakehead) Chapter 1 Consumer Theory Part II Winter 2019 2 / 25

Introduction to Duality Theory Indirect Utility and Expenditure Functions

Relations between V and E

Let V (p, y) and E (p, u) be the indirect utility function and theexpenditure function derived from a continuous and strictly increasingutility function. Then for all p� 0, y ≥ 0, and u ≥ 0,

1 E (p,V (p, y)) = y ,

2 V (p,E (p, u)) = u.

Notes:

1 Reading assignment: See page 42 in JR for a proof.

2 For a given price vector p, the indirect utility function v : R+ → R+

maps income y into utility u,

v(y) = u.

Since v is strictly increasing in y , the inverse function exists and isthe expenditure function. That is v−1(u) = e(u) = y .


Introduction to Duality Theory Indirect Utility and Expenditure Functions

Example (Also see Example 1.4 in JR)

Consider the Cobb-Douglas utility function U(x1, x2) = xα1 x1−α2 , and

0 ≤ α ≤ 1.

The ordinary demand functions are (see exercise 20, chapter 7 in Yu)

x∗1 = d1(p, y) =αy

p1, x∗2 = d2(p, y) =

(1− α)y

p2.

The indirect utility function is

V (p, y) =

(α

p1

)α(1− αp2

)1−αy . (1)

To find the expenditure function, let u = V (p, y). Solving for y usingequation (1) gives

E (p, u) = y =(p1

α

)α( p2

1− α

)1−αu. (2)


Introduction to Duality Theory Ordinary and Compensated Demand Functions

Relations between d and h

Let d(p, y) and h(p, u) be the ordinary and compensated demandfunctions derived from a continuous and strictly increasing utility function.Then for all p� 0, y ≥ 0, and u ≥ 0,

h(p,V (p, y)) = d(p, y), (3)

d(p,E (p, u)) = h(p, u). (4)

Notes:

1 Reading assignment: See page 45–46 in JR for a proof.

2 For a given price vector p, the compensated demand function forgood i is a composite function, that is, hi = di ◦ e, or

hi (u) = di (e(u)) = di (y).

3 Similarly,di (y) = hi (v(y)) = hi (u).


Introduction to Duality Theory Ordinary and Compensated Demand Functions

Example Continued

Apply Shephard’s lemma to the expenditure function in equation (2),we get the compensated demand functions

h1(p, u) =

(α

1− αp2

p1

)1−αu, h2(p, u) =

(1− αα

p1

p2

)α

u.

Using equations (1) and (3), the ordinary demand functions are

d1(p, y) =

(α

1− αp2

p1

)1−α( αp1

)α(1− αp2

)1−αy =

αy

p1,

d2(p, y) =

(1− αα

p1

p2

)α( αp1

)α(1− αp2

)1−αy =

(1− α)y

p2

as before.

Exercise: Use the ordinary demand functions and the expenditure functionin the example to demonstrate equation (4).


Properties of Consumer Demand Relative Prices and Real Income

Homogeneity Revisited

Recall that the budget constraint is unaffected by multiplying apositive scalar to prices and income. That is for any α > 0,αpTx ≤ αy is the same as pTx ≤ y .

As a result the ordinary demand function d and the indirectly utilityfunction V are homogeneous of degree zero in (p, y).

Let α = 1/pi , where pi is the price of any good i (called thenumeraire, such as gold or silver). Then the price vector becomes(

1

pi

)p =

(p1

pi, . . . ,

pi−1

pi, 1, . . . ,

pnpi

).

The ratio pj/pi is called the price of good j relative to good i , and isexpressed in quantity of good i needed to buy one unit of good j .

Similarly, y/pi is the real income in terms of good i .


Properties of Consumer Demand Income and Substitution Effects

Decomposition of the Effect of a Price Change

Suppose that in period 0 a consumer with income y buy two goodswith prices p0. Let x0 be optimal bundle.

In period 1 the price of good 1 goes down from p01 to p1

1 . Theoptimal bundle now is x1.

The total effect of the price change on consumption is x1 − x0.

We can decompose the total effect into two components, onerepresents a relative price change and the other real income change.

The first component, called substitution effect, is obtained bykeeping the utility, or standard of living, constant at the period 0level, u0, with the price change.

Effectively it is the difference between the optimal bundle given bythe compensated demand function, xs = h(p1, u0), and the originaloptimal bundle x0.

The residual change, x1 − xs , called income effect, reflects the realincome change due to the price change.



52 CHAPTER 1

x1

(a)

(b)

x2

x1

p1

x20

x21

x2s

x10

p10

x11

p11

x1s

x10 x1

1x1s

!p10/p2

0

!p11/p2

0

!p11/p2

0

y/p20

u0

u1

SE

SE

TE

TE

IE

IE

SE IE

TE

"x1

"p1

x1h(p1, p2

0, u0)

x1(p1, p20, y)

Figure 1.20. The Hicksian decomposition of a price change.

substitution effects on good 1 and good 2, and we regard them as due ‘purely’ to thechange in relative prices with no change whatsoever in the well-being of the consumer.Look now at what is left of the total effect to explain. After hypothetical changes fromx01 and x

02 to x

s1 and x

s2, the changes from xs1 and x

s2 to x

11 and x

12 remain to be explained.

Notice, however, that these are precisely the consumption changes that would occur if, atthe new prices and the original level of utility u0, the consumer were given an increase inreal income shifting his budget constraint from the hypothetical dashed one out to the final,post price-change line tangent to u1. It is in this sense that the Hicksian income effect cap-tures the change in consumption due ‘purely’ to the income-like change that accompaniesa price change.



The Slutsky Equation

Effects of a price change pj on xi is nicely summarized by the equation

∂di (p, y)

∂pj=∂hi (p, u)

∂pj− dj(p, y)

∂di (p, y)

∂y.

The term on the left is the total effect. The first term on the right isthe substitution effect, and the second income effect.

Intuition for the income effect:

Consider a small change in price of good j equal to ∆pj .This effectively change the real income by ∆y = −(∆pj)xj .Express this in ∆pj = −∆y/xj .Then the effect on good i is

∆xi∆pj

= −xj∆xi∆y

.



Proof of the Slutsky Equation

Recall the duality relation hi (p, u) = di (p,E (p, u)).

Differentiate both sides of the equation with respect to pj :

∂hi (p, u)

∂pj=∂di (p, y)

∂pj+∂di (p, y)

∂y

∂E (p, u)

∂pj. (5)

By Shephard’s lemma, ∂E (p, u)/∂pj = hj(p, u).

At (p, y), the ordinary demand and compensated demand are thesame, that is,

hj(p, u) = dj(p,E (p, u)) = dj(p, y).

Substitute this into the last term in equation (5) and rearrange, weget the Slutsky equation.


Properties of Consumer Demand Implications

More Properties of Consumer Demand

Consider the case that i = j in the Slutsky equation:

∂di (p, y)

∂pi=∂hi (p, u)

∂pi− di (p, y)

∂di (p, y)

∂y.

Because the expenditure function E is concave in p, the compensateddemand function of good i is decreasing in its own price, that is,∂hi (p, u)/∂pi ≤ 0.

The income effect in the Slutsky equation can be positive or negative.

We define good i to be1 a normal good if ∂di (p, y)/∂y > 0,2 an inferior good ∂di (p, y)/∂y < 0.

Law of Demand: If good i is normal, then the ordinary demandfunction is decreasing in its own price, that is, ∂di (p, y)/∂pi < 0.

Question: What is the contrapositive of the law of demand?


Properties of Consumer Demand Implications

Symmetric Substitution

The right-hand side of equation (5) is often called the i-jth Slutskyterm,

sij(p, y) =∂di (p, y)

∂pj+ xj

∂di (p, y)

∂y.

The term on the left-hand side of equation (5), by Shephard’s lemma,is

∂hi (p, u)

∂pj=∂2E (p, u)

∂pj∂pi.

It follows that the n × n Slutsky matrix formed by the Slutsky termsis equal to the Hessian of the expenditure function with respect to theprice vector,

S(p, y) = ∇2pE (p, u).

Therefore S is symmetric and negative semi-definite.

S is also observable.


Elasticities Definitions

Income Elasticity of Demand

Prices, quantities, and income have units attached to them, like $/kg(food), $/Tax Return (H&R Block), etc. Sometimes we want theconsumers’ response in price and income changes expressed in a“dimensionless” number.

Define income elasticity of demand for good i as

ηi =% change in quantity demanded for good i

% change in income

=∆di (p, y)/di (p, y)

∆y/y

=∂di (p, y)

∂y

y

di (p, y).



Properties of Income Elasticity

Income elasticity can be positive or negative:

inferior goods: η < 0

necessities: 0 ≤ η ≤ 1

luxuries: η > 1



Luxury, Normal, and Inferior Goods

Observations during the 2008 recession:



Price Elasticity of Demand

Price elasticity:

εij =% change in quantity demanded for good i

% change in price of good j

=∂di (p, y)

∂pj

pjdi (p, y)

.

If i = j , then εi is called the own-price elasticity of demand for good i .

If i 6= j , then εij cross-price elasticity of demand for good i withrespect to good j .

Technically, εi is always negative because of the law of demand.Often we only care about the absolute value, |εi |, that is, the valuewithout the negative sign.



Own-Price Elasticity

Definitions:

inelastic: 0 ≤ |εi | < 1unitary elastic: |εi | = 1elastic: |εi | > 1

Some examples:

Good or Service Elasticity |εi |Cigarettes 0.42Salmon 2.47Gasoline 0.50Chicken 1.67Peak hour bus services 0.23



Substitutes and Complements

Two goods are substitutes if the price of one good goes up result in anincrease in demand for the other good. Examples:

Zippers and buttons

Butter and margarine

Natural gas and fuel oil

Pork and chicken

Two goods are complements if the price of one good goes up result in andecrease in demand for the other good. Examples:

Movie tickets and restaurant meals

Lettuce and salad dressing

Blue-ray players and HDTV



Examples of Cross-Price Elasticity


Elasticities Average Elasticities

Engel Aggregation

Define the expenditure share of good i to be

si =pixi∑nj=1 pjxj

=pixiy.

It is obvious that si ≥ 0 and∑n

i=1 si = 1.

Engel aggregation:n∑

i=1

siηi = 1.

The intuition behind this result is that the weighted average changein consumption with respect to income change is equal to the changein income itself.



Proof of Engel Aggregation

Recall that straight monotonicity of the utility function implies that thebudget constraint is binding. This translates into the so-called budgetbalancedness condition: For all p� 0 and y ≥ 0,

n∑i=1

pidi (p, y) = y . (6)

Differentiate both sides with respect to y , we get

n∑i=1

pi∂di (p, y)

∂y= 1.

Multiply and divide each term on the left by ydi (p, y) gives the result.



Cournot Aggregation

The Cournot aggregation states that for j = 1, . . . , n,

n∑i=1

siεij = −sj .

It means that the weighted average of the effect of change in pj on allgoods depends on good j ’s income share.



Proof of Cournot Aggregation : Differentiate both sides of the budgetbalancedness in (6) with respect to pj , we have

xj + pj∂dj(p, y)

∂pj+

n∑i 6=j

pi∂di (p, y)

∂pj= 0.

The last equation implies that

n∑i=1

pi∂di (p, y)

∂pj= −xj .

Multiple each side by pj/y , we obtain

n∑i=1

piy

∂di (p, y)

∂pjpj = −sj .

Multiplying and dividing each term on the left by xi gives the result.



From Statistics Canada’s Survey of Household Spending

Canadian households spent an average of $62,183 on goods andservices in 2016, up 2.8% from 2015.

Spending on shelter accounted for 29.0% of this total, followed bytransportation (19.2%) and food (14.1%).

Provincially, the highest average spending on goods and services wasreported by households in Alberta ($74,044), followed by Ontario($66,220) and Saskatchewan ($65,411). Households in NewBrunswick ($50,175) reported the lowest average spending.

On average, couples with children spent $88,273 on goods andservices, compared with $34,674 for one-person households.


Documents

Chapter 1 Consumer Theory Part II - Lakehead Universityflash.lakeheadu.ca/~kyu/E5113/NCT1B.pdf · 2019-01-10 · Outline 1 Introduction to Duality Theory Indirect Utility and Expenditure