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ECON 4910 Lecture notes 4
Consumer theory
The consumer maximization problem
Prices and income are pi and m.
Max U ( x) s.t.  pi xi  m
(1)
iN
The solution gives the Marshallian demand functions:
xi  xi ( p, m), i  N
(2)
Substitution in the utility function gives the indirect utility function:
U ( x)  U ( x1 ( p, m),.., xN ( p, m))  V ( p, m)
(3)
The minimum expenditure problem
Min
 p x s.t. U ( x)  U
iN
o
i i
(4)
The Lagrangian for the problem:
L    p j x j   ( U ( x)  U o )
jN
(5)
The first order conditions:
L
  pi   U i( x)  0
xi
(6)
Solving for endogenous variables as functions of the exogenous variables gives the
Hicksian compensated demand functions:
xi  hi ( p,U o ), i  N
(7)
Substitution in the budget condition gives the Expenditure functions:
 p x   p h ( p, U
iN
i i
iN
i i
o
)  E ( p, U o )
The Envelope Theorem gives us the following relationship (remember Shephard’s
Lemma):
(8)
2
 (  p j x j )
jN
pi

L
  xi 
pi
(9)
E ( p, U )
E ( p, U )
  xi 
 hi ( p, U o )
pi
pi
o
o
Welfare measures of price changes
Marshallian consumer surplus, MCS
We look at one price, pi, only, and a change from pio to pi1 .
MCS 
pi  p1i

xi ( pi , pi , m)dpi
(10)
pi  pi0
( pi  price vector without pi ) .
Compensating variation, CV
CV  E ( p , pi ,U )  E ( p , pi ,U )  
0
i
o
1
i
o
pi  p1i

hi ( pi , pi ,U o )dpi
pi  pi0
(11)
(CV  ()0 for pi0  () pi1 )
The relationship (9) between the Expenditure function and the Hicks demand function is
used.
From Economists’ Mathematical Manual, Chapter 9, Integration we find:
Definite integrals

b
a
f ( x)dx  ab F ( x)  F (b)  F (a) if F ( x)  f ( x) for all x in a, b
The reference is the utility level before a change. CV measures the difference in income
for the two price level situations given that we remain on the same utility level. Therefore
CV is the minimum (maximum) amount we will accept (pay) for the change to occur.
Using the definition of indirect utility function (3) CV can be expressed implicitly as
follows:
V ( pi0 , pi , m)o  V ( pi1 , pi , m  CV )o
(12)
3
The change CV in income would compensate for the price change so utility remains at
the initial level.
.
Equivalent variation, EV
EV  E ( p , pi ,U )  E ( p , pi ,U )  
0
i
1
1
i
1
pi  p1i

hi ( pi , pi ,U 1 )dpi
pi  pi0
(13)
( EV  ()0 for pi0  () pi1 )
Notice that as long as nominal income m is constant we have that
E ( pi0 , pi ,U o )  E ( pi1 , pi ,U 1 )  m ). We can therefore write (13) as:
EV  E ( pi0 , pi ,U 1 )  E ( pio , pi ,U o )
(14)
EV measures the difference in income that supports the two utility levels keeping the
initial prices. The equivalent variation is therefore the maximum (minimum) that we are
willing to pay (accept) not to change the price from pio to pi1 .
Using the indirect utility function (3) we have:
V ( pi0 , p i , m  EV )1  V ( pi1 , p i , m)1
(15)
EV is the change in income that is equivalent to the welfare effect of the price change.
Willingness To Pay (WTP) and Willingness To Accept (WTA)
WTP = CV > 0, EV > 0 (change gives higher utility)
WTA = CV < 0, EV < 0 (change gives lower utility)
Size relationships between the measures
CV< MCS <EV
NB! For welfare decrease all measures becomes negative
Welfare measures of environmetal changes
Compensating surplus
4
V ( p, q o , m)o  V ( p, q1 , m  CS )o ,
(16)
CS  E ( p, q o ,U o )  E ( p, q1 ,U o )
CS is the compensation for the income effect of a environmental change, keeping the
original utility level.
Equivalent surplus
V ( p, q o , m  ES )1  V ( p, q1 , m)1 ,
ES  E ( p, q o ,U 1 )  E ( p, q1 ,U 1 )
ES is the change in income equivalent to the change in environment quality, keeping the
utility level after the change.
Problem 1, 2 Kolstad Chapter 15
s
Ux
sx
U2
ES
y = ps s = 4
2
CS
A
A(5)
A(10)
U2: s = 2 for A = 10 → CV = 4 – 2 = 2
Ux: sx = 2*4 for A =10/2 = 5 → EV = 8 – 4 = 4
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Problem 7, Kolstad Chapter 15
CV < EV, independently of sign of change:
p
xi=hi(pio,pi-1,Uo)
xi=hi(pi1,pi-1,U1)
po
p1
Uo
U1
x
x
o
x
1
Figure 1a. Price decrease
CV,EV > 0; CV< EV
p
xi=hi(pi1,pi-1,U1)
xi=hi(pio,pi-1,Uo)
p1
po
U1
Uo
x
1
x
x
o
Figure 1b. Price increase
CV, EV < 0; CV < EV