lista de teoremas

8/6/2019 lista de teoremas

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Lista de Teoremas j JEHLE & RENY j Chapter 1

Existence of a real value function representing the preference re-lation

Theorem 1 If the binary relation % is complete, transitive, continuos andstricly monotonic, there exists a continuous real-valued function, u : Rn+ ! R;which represents .

Invariance of the utility function to positiv monotonic transforma-

tions

Theorem 2 Let% be a preference reation onRn+ and supose u(x) is a utilityfunction that represents it. Thenv(x) also represents% if and only if v(x) =

f(u(x)) for every x, where f :R

!R

is strictly increasing on the set of valuestaken on by u.

Properties of preferences and utility functions

Theorem 3 Let% be represented by u : Rn+ ! R. Then:1. u(x) is strictly increasing if and only if% is strictly monotonic.2. u(x) is quasiconcave if and only if is% convex.3. u(x) is strictly quasiconcave if and only if is% strictly convex.

Suciency of consumers FOC

Theorem 4 Suppose that u(x) is continuous and quasiconcave onRn+ and that

(p; y) >> 0. If u is dierentiable atx, and (x; ) >> 0 solves:

@L

@x1=

@u(x)

@x1 p1 = 0

...@L

@xn=

@u(x)

@xn pn = 0

y p x = 0

Thenx solves the consumers maximization problem at pricesp and incomey.

Dierenciable demand

Theorem 5 Letx >> 0 solve the consumers maximization problem al pricesp0 >> 0 and income y0 > 0. If

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u is twice continuously dierentiable onRn++;

@u(x

)=@xi > 0 for some i = 1;:::;n and

the bordered Hessian of u has a nonzero determinant at x

Thenx(p; y) is dierentiable at (p0; y0)

Properties of the indirect utility function

Theorem 6 If u(x) is continuous and strictly increasing onRn+ then v(p; y):

v(p; y) = maxx2Rn

+

u(x) s:t: p x y

is

1. Continuous onRn++ Rn+2. Homogeneous of degree zero in (p; y)3. Strictly increasing in y4. Decreasing inpMoreover it satisces5. Quasiconvex in (p; y)6. Roys Identity: Ifv(p; y) is dierenciable at(p0; y0) and@v(p; y)=@y 6= 0,then

xi(p0; y0) =

@v(p0; y0)=@pi@v(p0; y0)=@y

i = 1;:::;n

Properties of the expenditure function

Theorem 7 If u() is continuous and strictly increasing, thene(p; u):

e(p; u) = minx2Rn

+

p x s:t: u(x) u

1. Zero when u takes on the lowest level of utility in U:2. Continuous on its domainRn++ U:3. For allp >> 0, strictly increasing and unbounded above in u.4. Increasing inp5. Homogeneous of degree 1 inp.6. Concave inp.If in adittion, is u() strictly quasiconcave, we have:

7. Sheppards lemma: e(p; y) is dierentiable inp at (p0; u0) withp0 >> 0and

@e(p0; u0)

@pi= xhi (p

0; u0) i = 1;:::;n

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Relations beteen indirect utility function and expenditure function

Theorem 8 Letv(p; y) ande(p; u) be the indirect utility function and expendi-ture function for some consumer whose utility function is continuous and strictlyincreasing. Then for allp >> 0; y 0; and u 2 U :

1. e(p; v(p; y)) = y2. v(p;e(p; u)) = u

Duality between marshallian and hicksian demand functions

Theorem 9 Under Theorem 3 we have the following relations between the Hick-sian and Marshallian demand functions for p >> 0; y 0; and u 2 U andi = 1;:::;n:

1. xi(p; y) = xhi (p; v(p; y))

The Marshallian demand at pricesp and income y is equal to the Hicksiandemand at pricesp and the utility level that is the maximum that can be achievedat pricesp and income y.

2. xhi (p; u) = xi(p;e(p; u))The Hicksian demand at any pricesp and utility level u is the same as the

Marshallian demand at those prices and an income level equal to the minimunexpenditure necessary at those prices to achieve that utility level.

Homogeneity and budget balancedness

Theorem 10 Under Theorem 3 the consume demand funciton xi(p; y) i =1;:::;n, is homogeneous of degree zero in all prices and income, and it satiscesbudget balancedness, p x(p; y) = y for all (p; y):

The slutsky equation

Theorem 11 Letx(p; y) be the consumers Marshallian demand system. Letu be the level of utility the consumer achieves at pricesp and income y: Then:

@xi(p; y)

@pi | { z } TE

=xhi (p; u

)

@pj | { z } SE

xj(p; y)@xi(p; y)

@yi = 1;:::;n

| {z } IE

Negative own-substitution terms

Theorem 12 Letxhi (p; u) be the Hicksian demand for good i.Then

xhi (p; u)

@pi 0 i = 1;:::;n

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The law of demand

Theorem 13 A decrease in the own price of a normal good will cause quantitydemanded to increase. If an own price decrease causes a decrease in the quantitydemanded, the good must be inferior.

Symmetric substitution terms

Theorem 14 Letxh(p; u) be the consumers system of Hicksian demands andsupose thate() is twice continously dierentiable. Then:

xhi (p; u)

@pj=

xhj (p; u)

@pii; j = 1;:::;n

Negative semidenitive substitution matrix

Theorem 15 Letxh(p; u) be the consumers system of Hicksian demands, andlet

(p; u)

0BB@

xh1 (p;u)@p1

: : : xh

1 (p;u)@pn

.... . .

...xhn(p;u)@p1

xhn(p;u)@pn

1CCA

called the substitution matrix contain all the Hicksian substitutions terms.Then the matrix(p; u) is negative semidefnite

Symmetric and Negative semidenitive slutsky matrix

Theorem 16 Letx(p; y) be the consumers Marshallian demand system. De-ne the ijth Slutsky term as

xi(p; y)

@pj+ xj(p; y)

xi(p; y)

@y,

and form the entire n n slutsky matrix of price and income responses asfollows:

s(p; u) 0BB@

x1(p;u)@p1

+ x1(p; y)x1(p;y)

@y: : : x1(p;u)

@pn+ xn(p; y)

x1(p;y)@y

... . . . ...xn(p;u)@p1

+ x1(p; y)xn(p;y)

@y xn(p;u)

@pn+ xn(p; y)

xn(p;y)@y

1CCA

Then s(p; u) is symmetric and negative semidenite.

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Aggregation in consumer demand

Theorem 17 Letx(p; y) be the consumers Marshallian demand system. Leti; ij andsi for i,j =1,...,n be as dened as before. Then the following relationsmust hold among income shares, price and income elasticities of demand:

1.Engel aggregation:Xn

i=1sii = 1:

2. Cournot aggregation:Xn

i=1siij = si; j = 1;:::;n


Constructing a utility function from an expenditure function

Theorem 18 Let E : Rn++ R+ ! R+ satisfy properties 1 through 7 of anexpenditure function given in theorem 7. Let A(u) be as in . Then the function

u : Rn+ ! R+ given by

u(x) max fu 0 j x 2 A(u)g

is increasing, unbounded above,and quasiconcave.

The expenditure function derived utility u, is E

Theorem 19 LetE(p; u), dened onRn++ R+, satisfy properties 1 to 7 of anexpenditure function and let u(x) be derived from E as in theorem 18. Then foral nonnegative prices and utility.

E(p; u) = minx

p x s:t: u(x) u

That is, E(p; u) is the expenditure function generated by derived utility u(x).

Duality between direct and indirect utility

Theorem 20 Suppose thatu(x) is quasiconcave and dierentiable onRn++ withstrictly positive partial derivatives. Then for all x 2 Rn++; v(p;p x), the in-direct utility function generated by u(x), achieves a minimum in p on Rn++and

u(x) = minp2Rn

++

v(p;p x) s:t: u(x) u

Duality and system of inverse demands

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Theorem 21 Let u(x) be the consumers direct utlity function. Then the in-verse demand function for good i is associated with income y = 1 is given by:

pi(x) =@u(x)=@xiPn

j=1 xj(@u(x)=@xj)

Budget balancedness and symmetry imply homogeneity

Theorem 22 Ifx(p; y) satises budget balancedness and its Slutsky matrix issymmetric, then it is homogenious of degree zero in p and y:

Integrability theorem

Theorem 23 A continously dierentiable function x: Rn+1++ Rn+ is the demand

function generated by some increasing, quasiconcave utility function if (and onlyif, when the utility is continuous, strictly increasing and strictly quasiconcave)it satises budget balancedness, symmetry, and negative semideniteness.


Separable production functions

The eslasticity of substitution

(Global) returns to scale

(Local) returns to scale

The cost function

The short run, or restricted cost function

The prot function

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