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ECON 501 – Micro Theory I

RECITATION #2

MWG 3.D.1. Verify that the Walrasian demand function generated by the Cobb-Douglas utility function satisfies:

(i) Homogeneity of degree zero in (p,w):

x(αp,αw)=x(p,w) for any p,w and for any scalar α>0

(ii) Walras’ law:

p·x=w for all x that belong to x(p,w)

(iii) Convexity/Uniqueness: If the preference relation is convex, so that u(·) is quasiconcave, then x(p,w) is a convex set. Moreover, if the preference relation is strictly quasiconvex, so that u(·) is strictly quasiconcave, then x(p,w) consists of a single point.

Solution:

① Assume the Cobb-Douglas utility function to be 11 2U Ax x .1 Then, the utility maximization

problem (UMP) will be

1 2

11 2

,maxx x

Ax x , subject to 1 1 2 2w p x p x

Take FOCs, we can solve for the Walrasian demand function, and get

1

1

,w

x p wp

, and 22

(1 ),

wx p w

p

② Then we check condition (i). Increasing all prices and wealth by a common factor λ, we obtain

1 11 1

, , ,w w

x p w x p wp p

2 22 2

1 1, , .

w wx p w x p w

p p

1 All the results in this exercise still hold in the more general case of a Cobb-Douglas utility function

1 2U Ax x , see MWG 3.D.6 for more details.

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③ We can now check condition (ii),

211 1 2 2

1 2

1, , .

p wp wpx p x p w p x p w w

p p

Thus implying that Walras’ law is satisfied.

④ Let us now check condition (iii), i.e., convexity and uniqueness.

Convexity. If a preference relation is convex, the slope of the indifference curve becomes flatter as 1x

increases (rightward moves in the figure).

1x

2x

1x increasing

In order to check that this property holds for the Cobb-Douglas utility function, we need to test whether

the MRS (the slope of the indifference curve) is increasing in 1x , e.g., it goes from -5 for low values of

1x to -1/4 for higher values of 1x . Using the definition of the slope of indifference curves,

2 11,2

1

2

Udx x

slope MRSUdxx

we find that, in the case of the Cobb-Douglas utility function, this slope is is

3

1 11 1 2 2

1,21 2 1

2

(1 ) 1

Ux A x x x

MRSU A x x xx

We can now differentiate this expression with respect to 1x to check if the slope increases in 1x . In

particular,

22

1 1 1

( )0

1

xslope MRS

x x x

Therefore, the slope is indeed increasing in 1x , implying that the preference relation represented with the

Cobb-Douglas utility function is convex, i.e., indifference curves are bowed-in towards the origin. In fact,

preferences are not only convex, but strictly convex, since an increase in 1x yields a strict increase of the

slope of the indifference curve.

Uniqueness. From the expression of the Walrasian demands found in ①, we know that, for every combination of prices and wealth, , we obtain a unique quantity demanded for every good,

and . Hence, the Walrasian demand is unique. (Note that his should come as no surprise: when the

budget set is convex, i.e., the budget line does not exhibit different slopes, and preferences are strictly convex, indifference curves are tangent to the budget line at a single point, thus producing a unique optimal bundle as Walrasian demand.)

MWG 3.D.2. Verify that the indirect utility function

v(p,w)=[αlnα+(1-α)ln(1-α)]+lnw-αlnp1-(1-α)lnp2

satisfies the following properties:

(i) Homogeneous of degree zero in (p,w):

v(αp,αw)=v(p,w) for any p,w and for any scalar α>0

(ii) Strictly increasing in w and nonincreasing in pl for any l.

(iii) Quasiconvex: the set {(p,w) : v(p,w)≤v} is convex for any v.

(iv) Continuous in p and w.

Solution.

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① Tocheckcondition(i),weincreaseallpricesandwealthbyacommonfactorλ,whichyields

1 2

1 2

1 2

,

ln 1 ln 1 ln ln 1 ln

ln 1 ln 1 ln ln ln ln 1 ln 1 ln

ln 1 ln 1 ln ln 1 ln

, .

v p w

w p p

w p p

w p p

v p w

② Tocheckcondition(ii),

1 1

2 2

, 10,

,0,

, 10.

v p w

w wv p w

p p

v p w

p p

Showingthatv(p,w)isstrictlyincreasinginwealth,w,andnon‐increasinginprices.

③Wecanprovethattheset : ,Lp v p w v isconvex.

Recall that toproveQuasiconvexity,weneed to show that, for anyx,yXand [0,1] ,wehave

that

(1 ) max{ ( ), ( )}f x y f x f y

whiletoproveconvexity,weneedtoshowthatforanyx,yXand [0,1] ,wehavethat

(1 ) ( ) (1 ) ( )f x y f x f y

Define ln 1 ln 1 ln w ,andassume

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1 2

1 2

, ln 1 ln

, ln 1 ln

, (1 ) ,

v p w p p V

v p w p p V

p w p p w

Ifwecanprovethat ,v p w V ,thenthe set : ,Lp v p w v is convex.

1 2 1 1 2 2, ln 1 ln ln (1 ) 1 ln (1 )v p w p p p p p p

1 2

1 2

1 1 2 2

, ln 1 ln

(1 ) , (1 ) (1 ) ln 1 (1 ) ln (1 ) (1 )

, (1 ) , ln (1 ) ln 1 ln (1 ) ln

v p w p p V

v p w p p V

v p w v p w p p p p V

Therefore,

1 1 2 2

1 1 2 2

, ln (1 ) 1 ln (1 )

ln (1 ) ln 1 ln (1 ) ln

,

v p w p p p p

p p p p

V

v p w V

(According to the property of concavity of the ln(.), we have

1 2 1 2ln (1 ) ln (1 ) lnx x x x ,asdepictedinthefigurebelow)

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Therefore,the set : ,Lp v p w v is convex.

④Condition(iv)followsthefunctionalformof v .

MWG 3.D.6. Consider the three good setting in which the consumer has utility function

u(x)=(x1-b1)α(x2-b2)

β(x3-b3)γ

a) Why can you assume that α+β+γ=1 without loss of generality? Do so for the rest of the problem.

Solution. Define 1

1 1 2 2 3 3 ,u x u x x b x b x b

with

, , .

Then 1 and u

representsthesamepreferencesas u ,becausethefunction1

u u isamonotone

transformation.Thuswecanassumewithoutlossofgeneralitythat 1.

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b) Write down the first-order conditions for the UMP, and derive the consumer’s Walrasian demand and indirect utility functions. [This system of demands is known as the linear expenditure system, and it is due to Stone (1954)].

Solution.Useanothermonotonetransformationofthegivenutilityfunction,

1 1 2 2 3 3ln ln ln ln .u x u x x b x b x b

1 1 2 2 3 3 1 1 2 2 3 3

11 1 1

1 1 1 1

22 2 2

2 2 2 2

33 3 3 3 3 3 3

1 1 2 2 3 3

1 1 2 2

ln ln ln ( )

0

0

:

0

0

x b x b x b w p x p x p x

px x b

p x p bp

x x bFOCs p x p b

px x b p x p b

w p x p x p x

p x p x

3 3 1 1 2 2 3 3p x p b p b p b w

Let 1 1 2 2 3 3.p b p b p b p b

1 1 1 11 1 1

2 2 2 22 2 2

3 3 3 33 3 3

( )( )

( )( )

( )( )

1

w pb

w pbx b b w pb b

p p p

w pbx b b w pb b

p p p

w pbx b b w pb b

p p p

Getdemandfunction 1 2 31 2 3

, , , , ,x p w b b b w p bp p p

Pluginto u(x)=(x1-b1)α(x2-b2)

β(x3-b3)γ,thenweobtaintheindirectutilityfunction

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1 2 3

, .v p w w p bp p p

c) Verify that these demand functions satisfy the properties listed in Propositions 3.D.2 for the Walrasian demand function, and in Proposition 3.D.3 for the indirect utility function.

①Tocheckthehomogeneityofthedemandfunction,

1 2 31 2 3

1 2 31 2 3

, , , , ,

, , , , , .

x p w b b b w p bp p p

b b b w p b x p wp p p

TocheckWalraslaw,

1 1 2 2 3 3

31 21 1 2 2 3 3

1 2 3

, , , ,

.

p x p w p x p w p x p w p x p w

pp pp b p b p b w p b

p p p

p b w p b w

Theuniquenessisobvious.

②Tocheckthehomogeneityoftheindirectutilityfunction,

1 2 3

1

1 2 3

1 2 3

,

, .

v p w w p bp p p

w p bp p p

w p b v p wp p p

Tocheckthemonotonicity,

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1 2 3

1 1

2 2

3 3

,0,

,, 0,

,, 0,

,, 0.

v p w

w p p p

v p wv p w

p p

v p wv p w

p p

v p wv p w

p p

Thecontinuityfollowsdirectlyfromthegivenfunctionalform.

Inordertoprovethequasiconvexity,itissufficienttoprovethat,forany v and 0,w theset

3 : ,p v p w v isconvex.Considerthemonotonetransformation

1 2 3.ln , ln ln ln ln ln ln lnv p w w p b p p p

Sincethelogarithmicfunctionisconcave,theset

31 2 3: ln ln ln lnp w p b p p p v

is convex for every .v Since the other terms, ln ln ln , do not depend on ,p

thisimpliesthattheset 3 : ln ,p v p w v isconvex.Hencesois 3 : ,p v p w v .

(OrfollowthesamestepasinMWG 3.D.2 ③.)

MWG 3.D.7. There are two commodities. We are given two budget sets Bp0

,w0 and Bp

1,w

1 described, respectively, by

p0=(1,1) and w0=8

p1=(1,4) and w1=26

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The observed choice at (p0, w0) is x0=(4,4), and at (p1, w1) the consumer’s choice satisfies p·x1=w.

a) Determine the region of permissible choices for x1, if the choices x0 and x1 are consistent with maximization of preferences.

Before applying the definition of WARP to the following figure, we should clarify

WARP in this context: The Walrasian demand satisfies WARP if, for any two bundles x0

and x1, both of them being affordable under the new budget line Bp1,w1, the new bundle x1

is unaffordable under the old budget line, Bp0,w0. (Note that, relative to the definition of

WARP we discussed in class, here we are just switching the new and old budget line and,

as a consequence, the bundle that needs to be unaffordable.)

Since x0 is affordable under Bp1,w1, 1 0 1 1 0and ,p x w x x (indeed, bundle (4,4) lies

strictly below budget line Bp1,w1) the premise of WARP holds, and we can move to the

second step. For the second step of WARP to hold, we need that bundle x1 must be

unaffordable, i.e., 0 1 0.p x w Thus, 1x has to be on the bold thick line in the

following figure.

In the following four questions, we assume the given preference can be a differentiable

utility function .u

b) Determine the region of permissible choices for x1, if the choices x0 and x1 are consistent with maximization of preferences that are quasilinear with respect to the first good.

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If the preference is quasilinear with respect to the first good, then we can take a utility

function u so that 1

1u x

x

for every x (Exercise 3.C.5(b)). Hence the first-order

condition implies 2

2 1

t t

t t

u x p

x p

for each 0,1t . Since

0 12 20 11 1

p p

p p and u is concave,

0 12 2x x . Thus 1x has to be on the bold line in the following figure.

c) Determine the region of permissible choices for x1, if the choices x0 and x1 are consistent with maximization of preferences that are quasilinear with respect to the second good.

If thepreference isquasilinearwithrespect to thesecondgood, thenwecantakeautility

function u so that 2

1u x

x

for every x (Exercise 3.C.5(b)). Hence the first‐order

condition implies 1

1 2

t t

t t

u x p

x p

for each 1,0t . Since

0 11 10 12 2

p p

p p and u is concave,

wemusthave 0 11 1x x .Thus 1x hastobeontheboldlineinthefollowingfigure.

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d) Determine the region of permissible choices for x1, if the choices x0 and x1 are consistent with maximization of preferences for which both goods are normal.

Since 1 0 1p x w andtherelativepriceofgood1decreased, 1tx hastoincreaseifgood1is

normal. If good 2 is normal, then thewealth effect (positive) and the substitution effect

(negative)goinoppositedirectionwhichgivesusnoadditionalinformationabout 2x .Thus

1x hastobeontheboldlineinthefollowingfigure.

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e) Determine the region of permissible choices for x1, if the choices x0 and x1 are consistent with maximization of preferences when preferences are homothetic.

Ifthepreferenceishomothetic,themarginalratesofsubstitutionatallvectorsonarayare

the same, and they become less steep as the ray becomes flatter. By the first‐order

conditionsand0 1

11 10 12 2

,p p

xp p

has tobeon the right sideof the ray thatgoes through 0x .

Thus 1x hastobeontheboldlineinthefollowingfigure.