Advanced Microeconomics - coin.wne.uw.edu.plcoin.wne.uw.edu.pl/jhagemejer/wp-content/uploads/2012_micro...Advanced Microeconomics ... Jan Hagemejer dvanced Microeconomics. Introduction

IntroductionThe consumer problem

Duality

Advanced Microeconomics

Consumer theory: optimization and duality

Jan Hagemejer

October 23, 2012

Jan Hagemejer Advanced Microeconomics


DualityIntroduction

Introduction

The plan:

1 The utility maximization

2 The expenditure minimization

3 Duality of the consumer problem

4 Some examples



Duality

The UMPWalrasian demandThe EMP

The consumer problem

In general, the consumer problem can be state as:

choose the best bundle that the consumer can a�ord

or: choose a bundle x � y for x and any y 's in the budget set B,given prices p and wealth w

or: if we have a utility function representing �, maximize utilitysubject to the budget constraint (given by p and w).

the correspondence between prices p, wealth w and the consumerchosen bundle is the demand correspondence.



Duality


The utility maximization problem (UMP)

We will asume that the consumer has a rational, continuous andlocally nonsatiated preference relation.

u(x) is a continuous utility function representing consumerpreferences

the consumption set is X = RL+

The utility maximization problem is de�ned as:

Maxx≥0 u(x)subject to p · x ≤ w

If u(x) is well behaved, then this problem has a solution x(p,w)which is the so-called Walrasian demand correspondence



Duality


The UMP (for interior solution)

The UMP is usually set up as a Kuhn-Tucker sort of problem.

Let us write down the Lagrange function:

L = u(x) + λ(w − p · x)

The �rst order conditions for an interior solution:

∂u(x)∂x1− λp1 ≤ 0

...∂u(x)∂xL− λpL ≤ 0

p · x − w ≤ 0

which gives (if interior solution)∂u(x)∂x

l∂u(x)∂x

k

= plpk

for all l , k

and hence: MRSlk = plpk



Duality


Marginal rate of substitution

Let's totally di�erentiate u = u(x) for a zero change in utility:

du = 0 =L∑

l=1

∂u(x)

∂xldxl

Lets assume that dxl 6= 0 and dxk 6= 0 and all other dxn = 0:

0 =∂u(x)

∂xldxl +

∂u(x)

∂xkdxk

Rearrange:

− dxl

dxk=

∂u(x)∂xl∂u(x)∂xk

= MRSlk

So at the optimal choice the ratio in which the consumer is willingto give away l for k is equal to the ratio of prices.



Duality


The UMP

The KT procedure says that either λ = 0 or the budget constraint isbinding (which is usually the case).

Also, we might have xl = 0 and it that case the relevant FOC is

satisi�ed with inequality, i.e. ∂u(x)∂xl

< λpL (corner solution)



Duality


The UMP without corner solutions

Example: Cobb-Douglas utility. The general case:

u(x) =L∏

i=1

xαi

i

with∑

i αi = 1.

The two good case:

u(x) = xα11xα22

with α1 + α2 = 1.

The UMP:max

x1,x2∈Xu(x) subject to p1x1 + p2x2 ≤ w



Duality


The Cobb Douglas continued

L = u(x) + λ0(w −L∑

l=1

plxl)

Why we do not need to care about corner solutions (xi = 0)? Why wouldthe only constraint be always binding (equality constraint)?

In the two good case:

L = xα11xα22

+ λ0(w −L∑

l=1

plxl)

Solution procedure.... (in class).

Solution of the problem (Walrasian demand function):

x1 = α1w

p1and x2 = α2

w

p2.

It also implies constant expenditure shares equal to αi .Jan Hagemejer Advanced Microeconomics


Duality


The UMP with corner solutions

If we suspect there may be corner solutions xl = 0 for some l , then weneed to set up the full Kuhn-Tucker problem with:

The budget constraint∑l

l=1plxl − w with the Lagrange multiplier

λ0

L inequality constraints xl ≥ 0 with the L Lagrange multipliers λl

L = u(x) + λ0(w −L∑

l=1

plxl) +L∑

l=1

λl(xl)

Then for the inequality constraints it is either (λl = 0 and xl > 0) or(λl > 0 and xl = 0). You have to check all the combinations!

Example: For the utility function u(x) =∑L

l=1alxl , l = 2, �nd the

demand function.



Duality



L = a1x1 + a2x2 + λ0(w −l∑

l=1

plxl) +l∑

l=1

λl(xl)

FOC's are:[x1] a1 − λ0p1 + λ1 = 0[x2] a2 − λ0p2 + λ2 = 0[λ0] p1x1 + p2x2 − w ≤ 0,[λ1] x1 ≥ 0, λ1 ≥ 0, λ1x1 = 0[λ2] x2 ≥ 0, λ2 ≥ 0, λ2x2 = 0

λ ≥ 0, λ0(w − p1x1 − p2x2) = 0,

Case 1: λ0 > 0, and all λl = 0, therefore all xl > 0 (interior solution)

from �rst 2 FOCs we have: a1p1

= λ0 and a2p2

= λ0 → a1/p1 = a2/p2 orp1p2

= a1a2

or better a1p1

= a2p2

(the expenditures on one unit of MU are

equal).

Only at that price ratio demand is a correspondence: p1x1 + p2x2 = w .All other cases are corner solutions.



Duality



Case 2:

λ0 > 0, and λ1 > 0, λ2 = 0 therefore x1 = 0 and x2 > 0 (corner solution)

The FOC's become:

p2x2 = w and therefore x2 = wp2.

a2 − λ0p2 = 0 and a1 − λ0p1 + λ1 = 0⇒ a1p1

= λ0 − λ1p1< λ0 = a2

p2, so

a1p1< a2

p2

Case 3:

λ0 > 0, and λ1 = 0, λ2 > 0 therefore x1 > 0 and x2 = 0 (corner solution)

The FOC's become:

p1x1 = w and therefore x1 = wp1.

a1 − λ0p1 = 0 and a2 − λ0p2 + λ2 = 0⇒ a2p2

= λ0 − λ2p2< λ0 = a1

p1, so

a2p2< a1

p1



Duality


The demand correspondence

In our problem the �nal demand is:

x(p) =

x1 = w

p1, x2 = 0. if a1

p1> a2

p2

x1, x2 : p1x1 + p2x2 = w if p1p2

= a1a2

x2 = wp2, x1 = 0 if a1

p1< a2

p2

As long as the bang for the buck is equal, we have the interior solution,otherwise only corner solutions.



Duality


Walrasian demand correspondence

The Walrasian demand correspondence x(p,w) assigns a set of chosenconsumption bundles for each price-wealth pair (p,w)

It can be multi-valued. If single valued we call it a demand function

Under the conditions of continuity and representation of u(x) theWalrasian demand correspondence possesses the following properties:

1 Homogeneity of degree zero in (p,w)2 Walras law: p · x = w (the budget constraint is binding)3 Convexity/uniqueness: if � is convex, so that u(·) is quasiconcave,

then x(p,w) is a convex set.

if � is strictly convex, so that u(·) is strictly quasiconcave, then

x(p,w) has just one element.



Duality


Walrasian demand correspondence



Duality


Properties of Walrasian demand

Wealth e�ects given the vector of prices p on the demand for good

l , partial derivative: ∂xl (p,w)∂w .

In matrix notation:

Dwx(p,w) =

∂x1(p,w)∂w...

∂xl (p,w)∂w...

∂xL(p,w)∂w

∂xl (p,w)∂w > 0, good is normal, if all > 0 then demand is normal

∂xl (p,w)∂w < 0, good is inferior.

Demand as a function of wealth x(p̄,w), Engel function

Wealth expansion path: Ep = {x(p̄,w) : w > 0}Income elasticity of demand: εw = ∂x(p,w)

∂ww

x(p,w) , necessity <1,

luxury >1



Duality


Wealth e�ects



Duality


Price e�ects

We can measure the e�ects of prices on the demand for goods.

The price e�ect is de�ned as:∂xl (p,w)∂pl

and usually < 0. If > 0 then

so-called Gi�en good.

In matrix notation

Dpx(p,w) =

∂x1(p,w)∂p1

∂x1(p,w)∂pL

.... . .

∂xL(p,w)∂p1

. . . ∂xL(p,w)∂pL

In that context we can de�ne the own- and cross-price elasticity of

demand ∂xl (p,w)∂pl

plxl (p,w) and

∂xl (p,w)∂pk

plxk (p,w) where k 6= l



Duality


Demand for good as a function of own price



Duality


O�er curve

OC - a locus of points demanded in over all possible values of one of theprices (in R2).



Duality


The Gi�en good



Duality


The indirect utility function

Once we have the optimal choice, x(p,w) we can plug it back intothe utility function.

u(x(p,w)) = v(p,w) is the indirect utility function

it says what the level of utility is, given prices and wealth and utilitymaximization



Duality


What for?

for example we can �nd thelevels of prices that generatesame utility given wealth

or the relationship wealthand utility at �xed prices



Duality


Expenditure minimization

We can go back and rede�ne our problem.

Instead of UMP, let us think of the consumer that has a desired levelof utility.

He wants to obtain this level of utility at the lowest possibleexpenditure.

the analogy of the production level and the cost minimization isobvious

The problem is set up as follows:

minx≥0 px subject to u(x) ≥ u

The solution is h(p, u), the demand for goods given prices andutility, the so called Hicksian demand (contrast it to x(p,w)).



Duality


The expenditure function

Once we have the solution to the problem, we can calculate the actualexpenditure:

e(p, u) =L∑

l=1

plh(p, u)

It is the �cost� of generating/obtaining a level of utility u given the set ofprices.

Why is it useful?

given the prices it determines a one-to-one relationship betweenmoney/expenditure and utility



Duality


Expenditure minimization



Duality


Hicksian demand and e�ects of a price change

∆wHicks is the the �Hicksian compensation�.



Duality

Duality of the consumer problem

The two problems can be related to each other:

1 x(p,w) = h(p, v(p,w))

2 x(p, e(p, u)) = h(p, u)

3 e(p, v(p,w)) = w

4 v(p, e(p, u)) = u



Duality

Some more nice properties

To recover Hicksian demand from expenditure function

If u(·) is a continuous utility function. For all p and u the Hicksiandemand h(p, u) is the derivative vector of the expenditure functionwith respect to prices.

h(p, u) = ∇pe(p, u)

To recover Walrasian demand from indirect utility function (Roy'sidentity - check for assumptions):

x(p,w) = − 1

∇wv(p,w)∇pv(p,w)



Duality

The Slutsky Equation

Suppose that u(·) is a continuous utility function representing a locallynonsatiated and strictly convex preference relation � de�ned on theconsumption set X = RL

+. Then for all (p,w), and u = v(p,w), we have

∂hl(p, u)

∂pk=∂xl(p,w)

∂pk+∂xl(p,w)

∂wxk(p,w)

and we can also rewrite it as:

∂hl(p, u)

∂pk︸︷︷︸Substitution e�ect

− ∂xl(p,w)

∂wxk(p,w)︸︷︷︸

Income e�ect

=∂xl(p,w)

∂pk



Duality

Recap


Documents

Advanced Microeconomics - coin.wne.uw.edu.plcoin.wne.uw.edu.pl/jhagemejer/wp-content/uploads/2012_micro...Advanced Microeconomics ... Jan Hagemejer dvanced Microeconomics. Introduction