Transcript Valuation 2: Environmental Demand Theory

Valuation 3: Welfare
Measures
• Willingness to pay and willingness to
act compensation
• Equivalent variation versus
compensating variation
• Price changes versus quantity changes
WTP and WTAC
• Consider price fall P*  P#
• Willingness to pay (WTP) to secure price
fall is known as equivalent variation
• Willingness to accept compensation
(WTAC) to forego price fall is known as
compensating variation
• WTP < WTAC, because of income effect
WTP and WTAC -2
• There are gains and loss, so four measures
–
–
–
–
WTP to secure a gain
WTAC to forego a gain
WTP to prevent a loss
WTAC to tolerate a loss
• People view gains and losses differently
• Confirmed by empirical studies, but not
uncontested
• Implies that surveys, policies need to be
carefully designed
Welfare Measures
• Compensating variation is the quantity of
income that compensates consumers for a
price change, that is, returns them to their
original welfare
• Equivalent variation is an income change
that yields the same utility change as the
price change
CV ( p0 , p1 )  e ( p1,U0 )  e ( p0 ,U0 )
EV ( p0 , p1 )  e ( p1,U1 )  e ( p0 ,U1 )
EV  CS  CV
Welfare Measures -2
• In case of quantity changes, compensating
and equivalent variation are defined as
CV (q0 , q1 )  e ( p , q1,U0 )  e ( p , q0 ,U0 )
EV (q0 , q1 )  e ( p , q1,U1 )  e ( p , q0 ,U1 )
• U0 results from (p,q0,y). If q0 increases to
q1, income has to be reduced by CV/p to
keep the same utility
?
EV  CV
Welfare Measures -3
?
EV  CV
• This need not hold for quantity changes.
• The reason is that the curvature of the
utility function can, in principle, increase or
decrease at higher levels of utility
• In practice, however, environmental goods
are relatively scarcer than market
commodities, so that one may expect the
compensating variation to be smaller than
the equivalent variation
Two Polar Cases
• An individual‘s utility u depends on the
consumption of x and a fixed quantity q
max u (x , q ) s.t.  pi xi  y
x
• This yields ordinary demand functions h
and an indirect utility function v
• An alternative way of defining
compensating and equivalent variation is
v ( p , q 1, y  C )  v ( p , q 0 , y );0  C  y
v ( p , q 1, y )  v ( p , q 0 , y  E );E  0
Perfect Substitution
• The direct utility
u (x , q )  u * (x1   (q ), x2,...xN )
• This yields indirect utility function
v ( p , q , y )  v * ( p1, p2,...pN , y  p1 (q ))
• Compensating, equivalent variation (p1=1)
v ( p , y   (q 1 )  C )  v ( p , y   (q 0 )); (q 1 )  C   (q 0 )
v ( p , y   (q 1 ))  v ( p , y   (q 0 )  E ); (q 1 )  E   (q 0 )
• Ergo, E=C
Zero Substitutability
• The direct utility
u (x , q )  u * * min(q , 1x1 ),..., min(q , N xN ) 
• There are areas where q is constraining;
the agent would be willing to pay a finite
amount to get more q and less x
• However, no amount of additional x would
compensate for a loss of q
• Here, equivalent variation is infinitely
larger than compensating variation
• This suggests that substitution is key
A More General Case
• An individual‘s utility u depends on the
consumption of x and a fixed quantity q
max u (x , q ) s.t.  pi xi   q  y
x ,q
• The dual yields the inverse compensated
demand curve
  ˆ( p , q , u )
• And an expenditure function
m ( p , q , u )  mˆ( p , ˆ( p , q , u ), u )  q ˆ( p , q , u )
A More General Case -2
m ( p , q , u )  mˆ( p , ˆ( p , q , u ), u )  q ˆ( p , q , u )
• Which implies
mq ( p , q , u )  ˆ
 ( p, q ,u )
• Compensating, equivalent variation
q1
0
ˆ
C  m ( p , q , u )  m ( p , q , u )    ( p , q , u )dq
0
0
1
0
q0
E  m ( p , q 0 , u 1 )  m ( p , q 1, u 1 ) 
q1
1
ˆ

(
p
,
q
,
u
)dq

q0
WTP v WTAC
• Compensating, equivalent variation
q1
0
ˆ(
C    p , q , u )dq
q0
E 
q1
1
ˆ(

p
,
q
,
u
)dq

q0
• Thus, E>C (E<C) for a normal (inferior) good
• Note, Hanemann switched signs, so CV>EV
A More General Case -3
• In the optimum
ˆ( p , q , u )  ˆ( p , q ,v ( p , q , y ))  ˆ( p , q , y )
• Define a „consumper surplus“
A
q1
 ˆ( p, q , y )dq
q0
• And an income elasticity
 ln ˆ( p , q , y )

 ln y
A More General Case -4
• Some trickery and approximation
2
2
L A
U A

 E C  
2y
2y
• However,
 ln ˆ( p , q , y ) (1   ) 



 ln y


• Where  is the income elasticity of
ordinary demand,  is the budget share of
q,  is the own-price elasticity of
compensated demand, and  is the
substitution elasticity of q
A More General Case -5
A
U A

 E C  
2y
2y
L
2
2
 ln ˆ( p , q , y ) (1   ) 



 ln y


• If =0 (no income effect), or if =,
E=A=C
• If, on the other hand, income elasticity is
high, or there few substitutes for q, then 
can vary substantially, even over small
ranges of q
Theory and Practice
• Horowitz and McConnell collect 208
observations of WTP and WTAC from 45
studies
• For all studies, the average ratio
WTAC/WTP is 7.2 (0.9)
• However, for public or non-market goods,
the ratio is 10.4 (2.5)
• For ordinary goods, it is 2.9 (0.3)
• For money, it is 2.1 (0.2)
• This is not inconsistent with theory
WTP v WTAC
1200
WTP
WTAC
1000
ratio
dollar
800
18
16
14
12
10
600
8
400
6
4
200
2
0
0
1
2
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7
8
9
10
11
12 13 14 15
Welfare Measures –4 (out)
• Compensating variation is defined as
CV (q0 , q1 )  e ( p , q1,U0 )  e ( p , q0 ,U0 )
• This is the same as
CV (q0 , q1 )  e ( p , q1,U0 )  e ( p , q1,U0 ) 
e ( p , q0 ,U0 )  e ( p , q0 ,U0 )   E
E  e ( p , q1,U0 )  e ( p , q0 ,U0 )
• If x is a weak compliment to q, and p is the
choke price, E = 0.