week 4 consumer theory - dr.serÇİn Şahİn - ana sayfa · · 2014-04-22week 4 consumer theory...

Week 4 Consumer Theory

Week 4

Consumer Theory

Serçin �ahin

Y�ld�z Technical University

16 October 2012


The Consumer's Problem

We view the consumer as having a consumption set, X = Rn+.

His preferences are described by the preference relation �de�ned on Rn

+.

the consumer's feasible alternatives consist a feasible set,

B ⊂ Rn+.

And consumer is motivated to choose the most preferred

feasible alternative:

x∗ ∈ B such that x∗ � x,∀x ∈ B .



Assumption: The consumer's preference relation � is

complete, transitive, continuous, strictly monotonic, and

strictly convex on Rn+.

Therefore, it can be represented by a real-valued utility

function, u, that is continuous, strictly increasing, and strictly

quasiconcave on Rn+.



An individual consumer is operating within a market economy.

A price pi > 0 prevails for each commodity i , i = 0, 1, ..., n.

The vector of market prices p� 0, as �xed from the

consumer's point of view.

The consumer is endowed with a �xed money income y ≥ 0.



Then the feasible set, B, which is called the Budget set is:

B ={x | x ∈ Rn

+,p.x ≤ y}



The consumer's utility-maximisation problem is written as

maxx∈Rn

+

u(x) s.t. p.x ≤ y

The solution vector depends on the parameters to the

consumer's problem. Because it will be unique for given values

of p and y , we can properly view the solution to the

consumer's maximisation problem as a function from the set of

prices and income to the set of quantities, X = Rn+, which is

known as ordinary, or Marshallian demand functions.

When income and all prices other than the good's own price

are held �xed, the graph of the relationship between quantitiy

demanded of xi and its own price pi is the standard demand

curve for good i



If we strengthen the requirements on u(x) to include

di�erentiability, we can use calculus methods.

maxx∈Rn

+

u(x) s.t. p.x ≤ y

If we form the Lagrangian:

L(x, λ) = u(x)− λ [p.x− y ]

Kuhn-Tucker conditions:



By strict monotonicity, these conditions reduce to

For any goods j and k , we can combine the conditions to

conclude that


Indirect Utility and Expenditure

The Indirect Utility Function

The relationship among prices, income and the maximised

value of utility can be summarised by a real-valued function,

v : Rn+ × R+ → R, which is called the indirect utility function.

v(p, y) = maxx∈Rn

+

u(x) s.t. p.x ≤ y

It is the maximum value corresponding to the consumer's

utility maximisation problem.

v(p, y) = u(x(p, y))




Properties of the Indirect Utility Function

If u(x) is continuous and strictly increasing on Rn+, then v(p, y) is



The Expenditure Function

To construct the expenditure function, we ask:

What is the minimum level of money expenditure the

consumer must make facing a given set of prices to achieve a

given level of utility?

e(p, u) ≡ minx∈Rn

+

p.x s.t. u(x) ≥ u

for all p� 0 and all attainable utility levels u.

If xh(p, u) solves this problem, the lowest expenditure

necessary to achieve utility u at prices p will be exactly equal

to the cost of the bundle xh(p, u), or

e(p, u) = p.xh(p, u)




The solution, xh(p, u), to the expenditure-minimisation

problem is precisely the consumer's vector of Hicksian

demands (or compensated demands ).




Properties of the Expenditure Function

If u(�) is continuous and strictly increasing, then e(p, u) is



Relations Between The Two

Let v(p, y) and e(p, u) be the indirect utility function andexpenditure function for some consumer whose utility functionis continuous and strictly increasing. Then for allp� 0, y ≥ 0, and u ∈ U :

1 e(p, v(p, y)) = y2 v(p, e(p, u)) = u

Mathematically, both the indirect utility function and theexpenditure function are simply the appropriately choseninverses of each other.

1 e(p, u) = v−1(p : u)2 v(p, y) = e−1(p : y)




Duality Between Marshallian and Hicksian Demand Functions

Under the assumption that consumer's preference relation � is

complete, transitive, continuous, strictly monotonic and strictly

convex on Rn+, we have the following relations between the

Hicksian and Marshallian demand functions for

p� 0, y ≥ 0, u ∈ U , and i = 1, ..., n:

xi (p, y) = xhi (p, v(p, y))

xhi (p, u) = xi (p, e(p, u))


Properties of Consumer Demand

Relative Prices and Real Income

'Money is a veil.'

The relative price of some good is, the number of units of

some other good that must be forgone to acquire 1 unit of the

good in question.

For any arbitrary good i and j , this is given by the price ratiopipj

because

The real income is the maximum number of units of some

commodity the consumer could acquire if he spent his entire

money income.




This is sometimes expressed by saying that the consumer's

demand behaviour displays an absence of money illusion.




Under the assumption that consumer's preference relation � is

complete, transitive, continuous, strictly monotonic and

strictly convex on Rn+, the consumer deand function

xi (p, y), i = 1, ..., n, is homogeneous of degree zero in all

prices and income, and it satis�es budget balancedness

p.x(p, y) = y ,∀(p, y)Homogeneity allows us to completely eliminate the yardstick of

money from any analysis of demand behaviour. This generally

done by arbitrarily designating one of the n goods to serve as

numeraire in place of money.



Income and Substitution E�ects




The Hicksian decomposition of the e�ect of a price changesuggests that the total e�ect (TE) of a price change can bedecomposed into two separate conceptual categories:

The substitution e�ect (SE) is that (hypothetical) change inconsumption that would occur if relative prices were to changeto their new levels but the maximum utility the consumer canachieve were kept the same as before the price change.The e�ect on quantity demanded of the change in purchasingpower resulting from the change in prices is called the incomee�ect (IE).




The Slutsky Equation: Let x(p, y) be the consumer's

Marshallian demand system. Let u∗ be the level of utility the

consumer achieves at prices p and income y . Then

Negative Own-Substitution Terms: Let xhi (p, u) be the

Hicksian demand for good i . Then




A good is called normal if consumption of it increases as

income increases, holding prices constant.

A good is called inferior if consumption of it declines as

income increases, holding prices constant.

The Law of Demand: A decrease in the own price of a

normal good will cause quantity demanded to increase.

If an own price decrease causes a decrease in quantity

demanded, the good must be inferior.




Symmetric Substitution Terms: Let xh(p, u) be the

consumer's system of Hicksian demands and suppose that e(·)is twice continuously di�erentiable. Then

Negative Semide�nite Substitution Matrix: Let xh(p, u)be the consumer's system of Hicksian demands, and let

called the substitution matrix, contain all the Hicksian

substitution terms. Then the matrix σ(p, u) is negativesemide�nite.




Symmetric and Negative Semide�nite Slutsky Matrix:

Let x(p, y) be the consumer's Marshallian demand system.

De�ne the ij-th Slutsky term as

and form the entire n × n Slutsky matrix of price and income

responses as follows:

Then s(p, y) is symmetric and negative semide�nite.



Some Elasticity Relations

The income elasticity of demand for good i measures the

percentage change in the quantity of i demanded per 1 per

cent change in income, and denoted by the symbol ηi .

The price elasticity of demand for good i measures the

percentage change in the quantity of i demanded per 1 per

cent change in the price pj , and denoted by the symbol εij .

If j = i , εii is called the own-price elasticity of demand forgood i .If j 6= i , εij is called the cross-price elasticity of demand forgood i with respect to pj .




The income share or proportion of the consumer's income,

spent on purchases of good i is denoted by the symbol si .

Aggregation in Consumer Demand: Let x(p, y) be theconsumer's Marshallian demand system. Then the followingrelations must hold among income shares (si ), price (εi ) andincome elasticities (ηi )of demand:

1 Engel aggregation:n∑

i=1

siηi = 1

2 Cournot aggregation:n∑

i=1

siεij = −sj , j = 1, ..., n

week 4 consumer theory - dr.serÇİn Şahİn - ana sayfa · · 2014-04-22week 4 consumer theory...

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