Advanced Microeconomics 2013 (Slide 1)

7/29/2019 Advanced Microeconomics 2013 (Slide 1)

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Advanced Microeconomics 2013/Kultti

What is microeconomics about?

1 Theory about the individual responses to incentives.

2 Theory about competing preferences.
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What is the role of theory in microtheory?

A small set of assumptions/postulates about individualbehaviour.

In practice economists usually construct models which are

something less than theories.In this course the closest things to theory are consumerschoice and general equilibrium theory.

A good theory should yield some empirically testable

predictions/restrictions.
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Majority of models in microeconomics focuses on behaviourmediated by the price mechanism.

The basic tenet in microeconomics, like in all serious scientificenterprises, is that a mechanism or concept that cannot beformalised, i.e., presented mathematically, is nothing butblabbering.

In microeconomics we attempt to be precise about what wetalk about, and consequently the theory is mathematical.

Almost all microeconomically interesting phenomena are suchthat there are several forces/tendencies that go to differentdirections.

The main task is to evaluate the magnitude of the differentforces/tendencies.
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Revealed preference

CONSUMER/DECISION MAKER

The aim is to understand the behaviour of a consumer whohas some wealth and faces a set of prices for different goods.

The consumption set is given by a non-empty XRn+.

It is closed and convex, and contains the zero-vector (origin).

A consumption bundle is x = (x1,x2,...,xn1,xn) X.

The feasible set is given by B X.

It incorporates the constraints that the consumer faces.

Usually these are given by income y R+and a price vectorp= (p1,p2,...,pn1,pn) R

n+.
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Revealed preference

Revealed preference theory

The idea is that theory/model should be based on observables.

This idea is not generally sensible.

One can observe choices, prices and income.Basic idea: Given prices and income a consumers choicereveals that s/he regards the consumption bundle better thanany affordable alternative.

The consumer is assumed to be consistent in his/her choices,i.e., rational.
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Revealed preference

Definition

Weak axiom of revealed preference (WARP). Consider bundle x0

chosen at prices p0, and bundle x1 chosen at prices p1 where

x0

= x1

. If p0

x1

p0

x0

then p1

x1

< p1

x0

.

Assume that i) consumers choice function x(p,y) satisfiesWARP, and ii) that the consumer uses all his/her income(budged balancedness).
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Revealed preference

Proposition. Consumers choice function is homogeneous of degreezero in (p,y).

Proof.

Assume that the choices and corresponding prices and incomes are

given by x0, p0, y0, x1, p1 = tp0 and y1 = ty0 for t> 0. By budgetbalancedness we have

p1x1 = y1 = ty0 = tp0x0

Substitute tp0

for p1

to get tp0

x1

= tp0

x0

. Divide by t to getp0x1 = p0x0. Now if x0 = x1 then WARP indicates thatp1x0 > p1x1. Next substitute p1 for tp0 to get p1x1 = p1x0.
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Revealed preference

Assume that a consumers choice is x0 at

p0,y0

. Let prices

change to p and compensate this change to the consumer by

giving him/her income y = px0.

Now his/her choice is x

p,px0

.

l d f
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Revealed preference

Theorem

Consider a compensated price change. If the increase in incomeaffects choice (demand) positively, then an increase in the price ofgood i leads to decrease in the demand for good i.

R l d f
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Revealed preference

Proof.

Let p0

>> 0, y0

> 0 and x0

= x

p0

,y0

. Let p be any other price(vector) and let x = x

p,px0

. Since

px0 > px

WARP implies

that p0x> p0x0 if x= x0. Consequently,

p0x0 p0x (1)

Substituting px

p,px0

for y0 budged balance implies

px0 = px

p,px0

(2)

Subtract (1) from (2)pp0

x0

pp0

x

p,px0

(3)

R l d f
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Revealed preference

Proof.

(Continued) As (3) holds for any price p it holds in particular forp= p0 + tz where t> 0 and z Rn+. Now (3) becomes

tzx0 tzx

p0 + tz,

p0 + tz

x0

(4)

Dividing (4) by t yields

zx0 zx

p0 + tz,

p0 + tz

x0

(5)

Clearly there exists t(z) such that p0 + tz>> 0 for t (0, t(z)).

For t= 0 (5) holds as and equality. Thus, for fixed z we can definefunctionf(t) zx

p0 + tz,

p0 + tz

x0

(6)

that attains its maximum at t= 0.

R l d f
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Revealed preference

Proof.(Continued) Assuming differentiability of the choice function thismeans that f(0) 0. Taking the derivative of (6) and evaluating itat t= 0

f(0) = ni=1

nj=1

zixi

p0

,y0

pj+ xj

p0,y0

xi

p0

,y0

y

zj 0 (7)

This is the definition of a negative semidefinite matrix with ij-entryxi(p0,y0)

pj + xj

p

0

,y

0 xi(p0,y0)y . For j = i we know that

xi(p0,y0)pi

+ xi

p0,y0 xi(p0,y0)

y 0.

E l th
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Envelope theorem

ENVELOPE THEOREM

Consider a problem where function f is maximised given a setof constraints.

MaxxRn f(x; q) such that g1(x; q) = b1,...,gm(x; q) = bm()where q Rs is a vector of parameters and bi are constants.

The value function, v or v(q), associated with the problemgives the (maximum) value that function f achieves at thesolution to the problem x(q).

E l th
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Envelope theorem

Theorem

Let v(q) be the value function of problem (*). Let it bedifferentiable at point q and leti be the Lagrange multipliers

associated with the constraints at x(q

). Then

v(q)

qi=

f(x(q),q)

qi

m

j=1

jgj(x(q

),q)

qi

for all i = 1,..., s.

Envelope theorem
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Envelope theorem

Proof.v(q) = f(x(q),q) for all q which means that we can differentiatean identity. Then we get

v(q)

qi=

f(x(q),q)

qi+

n

k=1

f(x(q),q)

xk

xk(q)

qi

The first order conditions of problem (*) are

f(x(q),q)

xk=

m

j=1

jgj(x(q

),q)

xk

Envelope theorem
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Envelope theorem

Proof.

(Continued) Insert this into the above expression and change theorder of summation to get

v(q)

qi=

f(x(q),q)

qi+

m

j=1

j

n

k=1

gj(x(q),q)

xk

xk(q)

qi

It is known that gj(x(q),q) = bj for all q. Differentiating thisidentity one gets

n

k=1

gj(x(q),q)

xk

xk(q)

qi =

gj(x(q),q)

qi

Inserting this into the expression for v(q)

qithe result follows.
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Advanced Microeconomics 2013 (Slide 1)