advanced microeconomics 2013 (slide 2)

7/29/2019 Advanced Microeconomics 2013 (Slide 2)

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Consumer theory

Preference based theory.

Let us next assume that consumers have unobservablepreferences over consumption bundles in X Rn+.

Formally preferences are a binary relation XX.

Most of the time

x1,x2 is denoted x1 x2.

Interpretation is that x1 is at least as good as x2.

We require some minimal features that preferences must

satisfy.
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Consumer theory

Completeness: For any x1,x2 X, either x1 x2 or x2 x1.

Transitivity: For any x1,x2,x3 X, if x1 x2 and x2 x3

then x1 x3.

A binary relation on X satisfying Axioms 1 and 2 is called a

preference relation.

Two other relations are derivable from i) x1 is strictlypreferred to x2, x1 x2, iff x1 x2 and

x2 x1

, ii) x1 is

indifferent to x2, x1 x2, iff x1 x2 and x2 x1.

It is straightforward that for any x1

,x2

X exactly one of thefollowing holds: x1 x2 or x2 x1or x1 x2.
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Consumer theory

In many text books a consumer with preference relation iscalled rational. This is incorrect since we need to make a

Behavioural assumption: Consumer chooses a maximalelement (according to ) in his/her feasible set B X.

Consider the set of l = {(ak)k=0 ,akR} of sequences of real

numbers. Binary relation l l is defined by(ak)

k=0 (bk)

k=0 iff there exists k such that for all k k it holds

that ak bk. Relation is transitive but not complete. Restricted

to the set c of converging sequences, is relation a preferencerelation?
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Consumer theory

Fix x0 X.

One can define so called upper and lower contour sets with

strict versions.

x0

=

x X : x x0

is called "at least as good as x0set.

Define the remaining four sets yourself.
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Consumer theory

To make preferences easier to deal with several additionalproperties are usually required of them.

Continuity: For all x Rn+ sets (x) and (x) are closed.

Strict monotonicity: For all x0,x1 Rn+ if x0 x1 then

x0 x1, and if x0 >> x1 then x0 x1.

Convexity: If x0 x1, then tx0 + (1 t)x1 x1 for t [0,1].If x0 = x1 and tx0 + (1 t)x1 x1 we have strict convexity.
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Consumer theory

Postulating these additional axioms the indifference curveshave the familiar form that feature diminishing marginal rateof substitution.

The really important thing to remember/understand is thatconsumers choice is intended to have a close relation to

his/her consumption. All of the above has been presentedwithout any reference to the time dimension. Consumption,however, is a flow, and one should think that consumptionbundle x = (x1,x2) actually means amount x1 of good 1 per

time unit (perhaps week), and amount x2

of good 2 per timeunit.

This way convexity may be seen as a natural assumption.
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Consumer theory

Usually theorising is done with utility functions instead ofpreference relations.

What has to be assummed about preferences in order to

guarantee a well behaved utility function?Function u: Rn+R is a utility function representingpreference relation iff for all x0,x1 Rn+ the following holds:

ux0 ux1 x0 x1
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Consumer theory

Theorem

If is complete, transitive, continuous and strictly monotonic thereexists a continuous u:Rn+R that represents.

h
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Consumer theory

Proof.

Just the idea: 1. Measure the distance along the 45-degree line toeach indifference curve. Associate to each bundle on an indifferencecurve the same number (as the bundle on the 45-degree line). 2.Show that this number exists and is unique. 3. Show that theconstructed utility function represents . 4. Show that u iscontinuous. Note that showing closedness of sets A and B in theproof (in the textbook) requires knowledge that scalar product is a

continuous mapping.

C h
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Consumer theory

Theorem

If u:Rn+R is a utility function representing preference relation

and f :R

R

is increasing then f (u) :R

n

+R

also represents.Let u:Rn+R represent. i) u is strictly increasing iff isstrictly monotonic. ii) u is quasiconcave iff is convex. iii) u isstrictly quasiconcave iff is strictly convex.

C h
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Consumer theory

When u is differentiable (as is usually assumed) uxi

is calledthe marginal utility of good i.

The ratio MRSijuxiu

xj

is called the marginal rate of

substitution of good i for good j. It is the slope of theindifference surface in the ij-co-ordinate system.

Notice that uxi

is a function and to get a numerical value itmust be evaluated at some point. The same holds for MRSij.

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Consumer theory

CONSUMERS PROBLEM

His/her aim is to choose a maximal element of his/her feasibleset.

Unless otherwise stated for the remainder of the presentationwe make the following

Assumption. Preference relation is complete, transitive,continuous, strictly monotonic and strictly convex on Rn+.

Generally the consumer is assumed to face a fixed set of pricesp= (p1,...,pn) and income y.

His/her problem is given by

maxxRn+

u(x) subject ton

i=1

pixiy (1)

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Consumer theory

The necessary conditions for interior solution x are given by

ux

i

(x)

uxj

(x) =pipj (2)

This is seen by forming the Lagrangean

= u(x) ni=1

pixiy (3)The first order conditions (FOCs) are given by

xi(x) =

u

xi(x)pi = 0

n

i=1

pixi y = 0 (4)

Notice that these equality conditions apply only to interiorsolutions.

More correctly one should use the- - -

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Consumer theory

Assume that u is continuous and quasiconcave, and that(p,y) >> 0.

If u is differentiable at x

and (x

,

) >> 0 satisfies (4) thenx solves the consumers problem.

More generally, if a maximisation problem is quasiconcave thenthe necessary conditions for maximum are also sufficient.

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Consumer theory

Example

Here the Assumption does not hold. Let u(x1,x2) = min{ax1,bx2}where a,b> 0. This utility function is not differentiable. Assume

that prices are p= (p1,p2) >> 0. The indifference curves have a90-degree kink on the line x2 =ab

x1, and the optimum is at a kink.Substituting into the budget constraint we get p1x1 + p2

ab

x1 = y.

Consequently, the consumers optimal choice is x1

= bybp1+ap2

and

x2

= aybp1+ap2

.

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Consumer theory

The important point to understand is that in the consumersproblem prices and income are fixed but arbitrary.

The solution can be thought of as a function of p and y andx(p,y) = (x1(p,y),...,xn(p,y)) is called the Marshaliandemand.

This only makes sense if the solution is unique but this is

guaranteed by Assumption.In the graph of the standard demand curve xi(pi,pi,y) onlypi is allowed to change.

Berges Maximum Theorem guarantees that the Marshalian

demand is a continuous function of (p,y).Roughly assuming that u is k times differentiable and someregularity conditions guarantees that the Marshalian demand isk1 times differentiable.

In the sequel differentiability is assumed when useful.

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Consumer theory

INDIRECT UTILITY AND EXPENDITURE FUNCTIONS

The objective is to study the maximised value of a consumersutility passing the optimal bundle that generates that value.

This is given by

v(p,y) = maxxRn+

u(x) subject to pxy

Given Assumption this is well-defined and

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

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Consumer theory

Theorem

If u is continuous and strictly increasing on Rn+ then v : Rn+1++ R

is/satisfiesi) Continuous on Rn++R+.

ii) Homogeneous of degree zero in (p,y).iii) Strictly increasing in y.iv) Decreasing in p.v) Quasiconvex in (p,y).

vi) xip0,y0 =v(p0,y0)/piv(p

0

,y0

)/y

for all i {1,2,...,n} if v is

differentiable at

p0,y0

andv

p0,y0/y= 0 (Roys identity).

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Consumer theory

Proof.

i) This follows from the Berges Maximum Theorem.ii) B = {x : pxy}= {x : tpx ty} for t> 0.iii) Let y > y. Now {x : pxy} {x : pxy}. Clearly there is

> 0 such that the -neighbourhood of x(p,y)is contained in{x : pxy}. Strict monotonicity of preferences implies the result.iv) Consider prices p0 p1, and notice that

p0p1

x

p0,y 0.

Thus, xp0,y is feasible at prices p

1, and consequently

v

p

1

,y v

p

0

,y

.

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Consumer theory

Proof.

(Continued) v) This is equivalent to the setL(p,y,k) = {(p,y) : v(p,y) k} being convex for all k> 0.Assume that

p0,y0

,

p1,y1 L(p,y,k), and consider

(pt,yt) = t

p0,y0

+ (1 t)

p1,y1

where t (0,1). If

(pt,yt) /

L(p,y,k) then u(x(pt,yt)) > k. Now ptx(pt,yt) = yt

and p0x(pt,yt) > y0 and p1x(pt,yt) > y1. The last twoinequalities are equivalent to tp0x(pt,yt) > ty0 and(1 t)p1x(pt,yt) > (1 t)y1. Summing these yieldsptx(pt,yt) > yt which is a contradiction.

vi) The Lagrangean of the consumers problem is given byL = u(x)(pxy). By the envelope theorem v

pi= L

pi=xi,

and vy

= Ly

= .

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Consumer theory

The important point is that the Theorem is a characterisation

of indirect utility function: Any function satisfying properties i)- vi) is an indirect utility function.

Since v is strictly increasing in y it can be inverted.

This provides another (dual) view of consumers problem given

by e(p,u)minxRn+px subject to u(x) u

0

.Expenditure function e :Rn+1++ R is a minimum valuefunction and has properties analogous to the indirect utilityfunction because expenditure and indirect utility function areeach others inverses.

Denote the expenditure minimising bundle for arbitrary utilitylevel u by xh(p,u) so that e(p,u) = pxh(p,u).

It is called the Hicksian demand.

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Consumer theory

Theorem

If u is continuous and strictly increasing on Rn+ then e: Rn+1++ Ris/satisfiesi) Zero when u has its lowest value onU = uR : u= u(x) ,x R

n+,

ii) Continuous onR

n

++U,iii) For p>> 0 strictly increasing and unbounded in u,iv) Increasing in p,v) Homogeneous of degree one in p,vi) Concave in p,

vii) Shephards lemma: e is differentiable in p at

p0

,u0

whenp0 >> 0, and xhi

p0,u0

=

e(p0,u0)pi

for i {1,2,...,n} if u isstrictly quasiconcave.

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Consumer theory

Proof.

i) Immediate.ii) This follows from Berges theorem of maximum.iii) Consider u0 < u1 and the corresponding bundles that minimiseexpenditures, namely x0 and x1. Assume that e

p,u1

e

p,u0

.

Now for t (0,1) sufficiently close to unity bundle tx1cost less than

e

p,u0

and yields utility higher than u0

which is a contradiction.Assume that e is bounded in u such that for all u e(p,u) e andassume that e is the least upper bound. Let {ui}

i=1 be a sequence

such that e(p,ui) e. Assume that {ui}i=1 is an increasing

sequence and consider the cost minimising bundles {xi}i=1 that

generate the utilities {ui}i=1. Because e converges to e also{xi}

i=1 converges to some x. By continuity of u we have u= u(x).

Bundle 2x yields more utility than x and costs 2e which is acontradiction.

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Co su e t eo y

Proof.

(Continued) iv) Follows from vii),v) Clearvi) Fix the utility level. Consider the convex combination of twoprices pt = tp0 + (1 t)p1 and the convex combination of the twocost minimising bundles xt = tx0 + (1

t)x1. Assume that the cost

minimising bundle at prices pt is x. Now p0x0 p0x andp1x1 p1x. Multiply the first inequality by t and the second by1 t and add to get tp0x0 + (1)tp1x1 ptx which is equivalentto tep

0,u+ (1 t)ep1,u e(p

t,u).

vii) The Lagrangean of the consumers problem is given byL = px

u0u(x)

. By the envelope theorem

epi

= Lpi

= xhi (p,u).

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y

The indirect utility function and expenditure function areclosely related.

Indeed, if u is continuous and strictly increasing then

Theorem

For all p>> 0,y 0,u U i) e(p,v(p,y)) = y and ii)v(p,e(p,u)) = u.

In other words, fixing p e and v are inverses.

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y

We know that when u(x1,x2) = min{ax1,bx2} thenx1

= bybp1+ap2

and x2

= aybp1+ap2

. The indirect utility function is

given by v(p,y) = abybp1+ap2

. The cost minimising bundle, i.e.,the Hicksian demand is such that x2 =

ab

x1. To achieve utility

level u the consumer needs bundleua ,

ub

, and the expenditurefunction is given by e(p,u) = p1

ua

+ p2ub

. Denoting e(p,u) = y

and inverting one gets u= abbp1+ap2

y. Denoting the utility levelby u= v(p,y) shows that even when the assumptions ofTheorem do not hold the expenditure and indirect utility

functions may be inverses.

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y

Under Assumption

Theorem

For p>> 0,y 0,u U, i {1,...,n} i) xi(p,y) = x

h

i (p,v(p,y)),and ii) xhi (p,u) = xi(p,e(p,u)).

Proof.

Evident, though hard to decipher from the text book.

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y

Properties of the consumer demand

Demand depends only on relative prices. This can be seen bydividing the price vector and income by one of the prices, saypi, in the consumers problem, and trying to figure outwhether the optimising bundle remains the same.

Real income refers to the maximum number of (real)commodities that can be bought.

Under Assumption xi(p,y) is homogeneous of degree zero inprices and income.

The Marshalian demand satisfies budget balancedness, i.e., theconsumer uses all his/her income. If not s/he could buy somemore which would increase utility.
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advanced microeconomics 2013 (slide 2)

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