Advanced Microeconomics 2013 (Slide 1)
Transcript of 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.
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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)
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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.
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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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