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1. Problem Set IV: UMP, EMP, indirect utility, expenditure
Paolo Crosetto
[email protected]
February 22nd, 2010

2. Recap: indirect utility and marshallian demand
The indirect utility function is thevalue functionof the UMP:v(p,w) =maxu(x)s.t.pxwSince the end result of the UMP are the Walrasian demand functionsx(p,w),
the indirect utility function gives the optimal level of utility as a function of
optimal demanded bundles,
that is, ultimately, as a function of prices and wealth.
Summing up
In the UMP we assume a rational and locally non-satiated consumer with
convex preferences that maximises utility;
we hence nd the optimally demanded bundles at any(p,w);The level of utility associated with any optimally demanded bundle is the
indirect utility functionv(p,w).
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3. Recap: properties of the indirect utility function
The value function of a standard UMP, theindirect utility function v(p,w), is:
Homogeneous of degree zero inpandw(doubling prices and wealth doesn't
change anything);
Strictly increasing inwand nonincreasing inp
lfor anyl(all income is spent;
law of demand);
Quasiconvex inp: that is,f(p,w):v(p,w)¯vgis convex for any¯v(see
example inR
2
in lecture slides);
Continuous at allp0,w>0 (from continuity ofu(x)and ofx(p,w)).
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4. Cobb-Douglas Indirect Utility Function,a=0.5,w=1000
50
100
0
50
100
1
2
3
4
5
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5. Recap: expenditure function and hicksian demand
The expenditure function is thevalue functionof the EmP:e(p,u) =minpxs.t.u(x)uIn the EmP we nd the bundles that assure a xed level of utility while
minimizing expenditure
the expenditure function gives the minimum level of expenditure needed to
reach utilityuwhen prices arep.
Summing up
In the EmP we assume a rational and locally non-satiated consumer with
convex preferences that minimises expenditure to reach a given level of utility;
we denote the optimally demanded bundles at any(p,u)ash(p,u)[hicksian
demand];
The level of expenditure associated with any optimally demanded bundle is the
expenditure functione(p,u).
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6. Recap: properties of the expenditure function
Homogeneous of degree one inp(expenditure is a linear function of prices);
Strictly increasing inuand nondecreasing inp
lfor anyl(you spend more to
achieve higher utility, you cannot spend less when prices go up);
Concave inp(consumer adjusts to changes in prices doing at least not worse
than linear change);
Continuous inpandu(from continuity ofpxandh(p,u)).
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7. Cobb-Douglas Expenditure Function,a=0.5,u=1000
50
100
0
50
100
0
5000
10000
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8. Recap: basic duality relations
The bundle that maximises utility is the same that minimises expenditure
The indirect utility function gives the maximum utility obtainable with that
bundle
The wealth spent to obtain that utility is necessarily the minimum possible
And spending all that wealth generates the maximum level of utility.
Four important identities
1.v(p,e(p,u))u: the maximum level of utility attainable with minimal
expenditure isu;
2.e(p,v(p,w))w: the minimum expenditure necessary to reach optimal level
of utility isw;
3.xi(p,w)hi(p,v(p,w)): the demanded bundle that maximises utility is the
same as the demanded bundle that minimises expenditure at utilityv(p,w);
4.hi(p,u)xi(p,e(p,u)): the demanded bundle that minimises expenditure is
the same as the demanded bundle that maximises utility at wealthe(p,u).
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9. Recap: a new look at the Slutsky matrix
The hicksian demandh(p,u)is also called thecompensateddemand.
This reminds us of the Slutsky matrix, that gave us thecompensatedchanges
in demand for changes in prices.
¶h(p,u)
¶p
k
=
¶x(p,w)
¶p
k
+
¶x(p,w)
¶w
x
k(p,w)
In which the second term is exactly thelkentry of the Slutsky substitution
matrix we are by now familiar with.
This equation links the derivatives of the hicksian and walrasian demand
functions:
The two demands are the same when the wealth eect of a price change is
compensated away.
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10. Recap: Shephard's lemma
There are direct and straightforward relationships betweene(p,u)andh(p,u).
1.e(p,u)can be calculated by plugging the optimal demanded bundle under the
EmP,h(p,u), into the expression for calculating expenditurepx. Hence,
e(p,u) =ph(p,u).
2. h(p,u) =rpe(p,u).
This is mathematically theShephard's Lemma(though the Lemma was derived
from production theory, it is formally the same as the one exposed here).
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11. Recap: Roy's identity
The relationships betweenv(p,w)andx(p,w)are less straightforward, but of the
same kind:
1.v(p,w)can be calculated by plugging the optimal demanded bundle under the
UMP into the utility function, i.e.v=u(x(p,w)),
2.
an ordinal concept; in the case of expenditure we were dealing with a cardinal
concept, money.
In order to go from Walrasian demand to the Indirect Utility function we need
to sterilise wealth eects and take into account the ordinality of the concepts;
It can be proved that:
x
l(p,w) =
¶v(p,w)
¶p
l
¶v(p,w)
¶w
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12. Recap: nding one's way through all of this
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13. 1. Varian 7.4: UMP-EMP
Consider the indirect utility function given by
v(p1,p2,w) =
w
p1+p2
1.2.3.
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14. Solution I
Walrasian demand functions
Walrasian demand functions can be derived from the indirect utility function using
Roy's Identity:
x
l(p,w) =
¶v(p,w)
¶p
l

¶v(p,w)
¶w

1
In this case, plugging in the derivatives for the function,
x1(p,w) =

w
(p1+p2)
2
p1+p2
1

=
w
p1+p2
It can be veried that the same holds forx2(p,w). Hence the demand function is
given by
x1(p,w) =x2(p,w) =
w
p1+p2
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15. Solution II
Expenditure function
The expenditure function is the inverse of the indirect utility function with respect
to wealthw:
v(p,e(p,u)) =u
In this case, applying the above formula is enough to get the result:
e(p,u)
p1+p2
=u)e(p,u) = (p1+p2)u
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16. Solution III
Direct utility function
There is no easy automatic way to retrieve the utility function from indirect utility.
We need to 'invert' a maximum process, which is not trivial, or else to work on the
indirect utility and walrasian demand by 'inverting' the substitution.
In this case, we see a striking regularity: the indirect utility funtion is the same
as the demand functions.
It means that the optimal level of utility is reached when only one of the two
goods is consumed.
It is then the case of perfect complement goods, i.e. Leontie preferences.The resulting utility function is thenu(x) =minfx1,x2g
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17. 17 of 30

18. 2. MWG 3.D.6: Stone linear expenditure system
Consider the following utility function in a three-good setting:
u(x) = (x1b1)
a
(x2b2)
b
(x3b3)
g
Assume thata+b+g=1.
1.
demand and the indirect utility function.
2.2.1 x(p,w)is homogeneous of degree zero and satises Walras'
law;
2.2 v(p,w)is homogeneous of degree zero;2.3v(p,w)is strictly increasing inwand nonincreasing inplfor alll;2.4v(p,w)is continuous inpandw.
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19. UMP I
We will work better with a log transform of the utility function:
ˆu(x) =lnu(x) =aln(x1b1) +bln(x2b2) +gln(x3b3)
which will give us the following UMP:
maxˆu(x)s.t.pxw
Which, in turns, can be maximised using Lagrange method, to yield the following
FOCs:
a
x1b1
=lp1;
b
x2b2
=lp2;
g
x3b3
=lp3,px=w;l>0
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20. UMP II
Demand
The system can be solved to nd the walrasian demand function:
x(p,w) =
2
6
6
6
6
6
4
b1+
a(wpb)
p1
b2+
b(wpb)
p2
b3+
g(wpb)
p3
3
7
7
7
7
7
5
, in whichpb=
3
å
i=1
pibi
Indirect utility
Given this demand funtion, the indirect utility can be found by substitution:
v(p,w) =u(x(p,w)) =

a(wpb)
p1

a

b(wpb)
p2

b

g(wpb)
p3

g
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21. Properties ofx(p,w)
Homogeneity of degree zero
x(lp,lw) =
2
6
6
6
6
6
4
b1+
al(wpb)
lp1
b2+
bl(wpb)
lp2
b3+
gl(wpb)
lp3
3
7
7
7
7
7
5
=x(p,w)
Walras' law
px(p,w) =pb+ (wpb)

p1
a
p1
+p2
b
p2
+p3
g
p3

=
=pb+ (wpb)(a+b+g) =pb+wpb=w
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22. Properties ofv(p,w)I
Homogeneity of indirect utility
v(lp,lw) =

al(wpb)
lp1

a

bl(wpb)
lp2

b

gl(wpb)
lp3

g
which can easily be simplied to yieldv(p,w).
Derivatives
v(p,w)strictly increasing inw: rst simplify the indirect utility function to getv(lp,lw) = (wpb)

a
p1

a

b
p2

b

g
p3

g
and then simply dierentiate w.r.t.wto get
¶v(p,w)
¶w
=

a
p1

a

b
p2

b

g
p3

g
>0
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23. Properties ofv(p,w)II
Derivatives, continued
The derivatives w.r.t. prices imply long calculations, and yield:
¶v
¶p1
=v(p,w)


a
p1

¶v
¶p2
=v(p,w)


b
p2

¶v
¶p3
=v(p,w)


g
p3

That can be checked to be all<0, as required.
continuity
Continuity comes directly from the functional form: withp0, as assumed, there
are no asympthotes or kinks. Moreover, the utility function and the derived
walrasian demand being continuous, the indirect utility function has to be
continuous.
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24. 3. MWG 3.G.15: dual properties
Consider the utility function
u=2x
1
2
1
+4x
1
2
2
1. x1(p,w)andx2(p,w)
2. h(p,u)3. e(p,u)and verify thath(p,u) =rpe(p,u)4. v(p,w)and verfy Roy's identity.
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25. Walrasian demand
Solution strategy
To nd walrasian demand, just solve the UMP using Lagrange method.
The FOC system for this problem boils down to
1
2

x1
x2


1
2
=
p1
p2
;p1x1+p2x2=w
Yielding solution
x(p,w) =
2
6
4
p2w
4p
2
1
+p1p2
4p1w
p
2
2
+4p1p2
3
7
5
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26. Hicksian demand
Solution strategy
We need to nd Hicksian demand, knowingu(x)andx(p,w). This can be done in
two ways:
1. pof the Hicksian
demand knowingx(p,w). This is rather straightforward, but implies
integrating. The steps are:
Compute derivatives ofxl(p,w)w.r.t.plandw;
Apply the Slutsky equation to nd
¶h(p,w)
¶pl
;Integrate
R¶h(p,u)
¶pl
dplto gethl(p,w).2. h(p,w)x(p,e(p,u). This eliminates the need for
integration, but implies calculating the indirect utility function and from there
the expenditure function. The steps are:
Plug the demand functions intou(x)to getv(p,w);
Applyv(p,e(p,u)) =u, i.e. invertvw.r.t. wealthw;Applyh(p,w)x(p,e(p,u)), i.e. substitutewwithe(p,u)in the Walrasian
demand.We will follow road 2. This means answering further questions rst
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27. The road to Hicksian demand I
Plug the demand functions intou(x)to getv(p,w):
v(p,w) =u(x(p,w)) =2

p2w
4p
2
1
+p1p2
1
2
+4

4p1w
p
2
2
+4p1p2
1
2
Applyv(p,e(p,u)) =u, i.e. invertvw.r.t. wealthw;
2

p2e(p,u)
4p
2
1
+p1p2
1
2
+4

4p1e(p,u)
p
2
2
+4p1p2
1
2
=u
which, squaring both sides and then simplifying, gives two roots, one of which is
negative, the one remaining being:
e(p,u) =
1
4
u
2
p1p2
4p1+p2
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28. The road to Hicksian demand II
We are left with the last step, i.e. applyingh(p,w)x(p,e(p,u)):
i.e. we have to substitute thee(p,u)we found in the place ofw.
h1(p,u) =
1
4
p1p
2
2
u
2
(p2+4p1)(p1p2+4p
2
1
)
;h2(p,u) =
p
2
1
p2u
2
(p2+4p1)(4p1p2+p
2
2
)
That can be simplyed to yield
h1(p,u) =
1
4

p2u
4p1+p2

2
;h2(p,u) =

p1u
4p1+p2

2
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29. Expenditure function
Solution strategy
Again, we have two ways of nding the expenditure function:
1. v(p,w)fromx(p,w)andu(x), then invert it w.r.t.wto gete(p,u);2. e(p,u)directly fromh(p,u)plugging it in the objective functionpx.
As for us, we used road 1 and already worked oute(p,u)in the road towards
Hicksian demand, so no need to do it here.
You can easily check by yourself thath(p,u) =rpe(p,u)
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30. Roy's identity
Solution strategy
We can ndv(p,w)fromn eitherv(p,w) =u(x(p,w))or invertinge(p,u)w.r.t.
u; then, we just need to apply Roy's identity right hand side and check if the result
is the same as thex(p,w)we calculated beforehand. We have to check if this
holds:
x
l(p,w) =
¶v(p,w)
¶p
l

¶v(p,w)
¶w

1
, forl=1, 2
As for us, we already foundv(p,w). It's easy again to apply the formula and nd
that Roy's Identity holds
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