C . Blackorby, D. Donaldson Mathematical Social Sciences 37 1999 139 –163
141
Thus, there are two circumstances in which Hicks’s result has normative significance. But these two possibilities are not encouraging. We live in many-consumer economies
and we cannot reasonably expect to find preferences that support an aggregate consumer, ´
nor can we bargain on the absence of income effects on all but the numeraire good for all consumers. In addition, these two possibilities force the evaluator to be indifferent to
income inequality, a standard but ethically unattractive feature of most cost-benefit analysis.
2. Partial equilibrium
Before presenting our main general equilibrium result, we discuss the problem in partial equilibrium. Following Sugden and Williams 1978, suppose that the govern-
ment operates a rail service between a city and one of its suburbs, as illustrated in Fig. 1.
b a
A project is planned which will reduce the price of rail trips from p to p b stands for
r r
‘before’, a stands for ‘after’. Cheaper transportation results in increased demand in the competitive suburban rental housing market, shifting the demand curve to the right—
b a
b
from D p , p to D p , p . This results in an increase in the price of housing from p
h r
h h
r h
h a
b
to p which, in turn, shifts the demand curve for rail trips to the left—from D p , p to
h r
r h
a
D p , p . The initial equilibrium is at C and K, and the equilibrium after the change is
r r
h
at E and I. Dupuit–Marshall consumers’ surpluses can be measured in many different ways. If
c
ˆ consumers’ surplus in the rail market is calculated first, it is s 5 ACFD. Once p has
r r
a
changed, the relevant housing demand curve is D p , p and consumers’ surplus in the
h r
h c
ˆ housing market is s 5 2 GILJ. Producers’ surplus the increase in landlords’ profit in
h p
the housing market is s 5 GIKJ, and consumers’ surplus plus producers’ surplus in
h
housing is
c p
c c
p
ˆ ˆ
ˆ s 1 s :5s 1 s 1 s 5 ACFD 2 GILJ 1 GIKJ 5 ACFD 2 ILK.
1
h r
h h
Alternatively, the consumers’ surplus in housing can be computed first, and, in this
c b
˜ case, it is s 5 2 GHKJ, the area to the left of the original demand curve D p , p .
h h
r h
a
The relevant demand curve for the rail-market surplus becomes D p , p , and
r r
h c
˜ consumers’ surplus is s 5 ABED. Using this method, consumers’ surplus plus
r
producers’ surplus in housing is
c p
c c
c
˜ ˜
˜ s 1 s :5s 1 s 1 s 5 ABED 2 GHKJ 1 GIKJ 5 ABED 1 HIK.
2
h r
h h
A third method for calculating the surpluses can be obtained by considering a price path which reduces the price of rail trips, keeping the housing market in equilibrium.
When this is done, price and quantity in the rail market move along the market demand
M
curve D p . The claim made by Hicks 1946a, 1946b and by Sugden and Williams
r r
1978 is that the area ACED to the left of the market demand curve is equal to the overall consumers’ surplus plus producers’ surplus in the housing market. This can be
seen intuitively by thinking of the change in p as a sequence of very small changes.
r
This makes the ‘corrections’ 2ILK and HIK in 1 and 2 very small, and the sum of consumers’ surplus and producers’ surplus approaches ACED, the shaded area in Fig. 1.
142 C
. Blackorby, D. Donaldson Mathematical Social Sciences 37 1999 139 –163
Fig. 1. Market demand curves and Dupuit–Marshall consumers’ surpluses.
In general, these three numbers are not the same; in order for them to coincide, the
1
consumers’ surpluses must be path independent, a requirement that is not often satisfied. If path independence is satisfied, the consumers’ surplus is well defined for all price
paths. To prove the result mathematically, we define the Dupuit–Marshall consumers’
b b
a a
surplus as a line integral along a continuous price path from p , p to p , p given by
r h
r h
1
See Silberberg 1972, Chipman and Moore 1976 and Section 4.
C . Blackorby, D. Donaldson Mathematical Social Sciences 37 1999 139 –163
143
b b
the functions P : [0,1]∞5 and P : [0,1]∞5
with p , p 5 P 0,P 0 and
r h
r h
r h
a a
p , p 5 P 1,P 1. Then
r h
r h
a a
p , p
r h
c
s 5 2
E
[D p , p dp 1 D p , p dp ]
r r
h r
h r
h h
b b
p , p
r h
1
5 2
E
[D P t,P tdP t 1 D P t,P tdP t]. 3
r r
h r
h r
h h
If the functions P and P are differentiable, then 3 can be written as
r h
1 c
9 9
s 5 2
E
[D P t,P tP t 1 D P t,P tP t]dt. 4
r r
h r
h r
h h
Producers’ surplus in the housing market is
a
p
h
1 p
s 5
E
S p dp 5
E
S P tdP t, 5
h h
h h
h h
h
b
p
h
where S is the housing supply function. It follows that the sum of overall consumers’
h
surplus and producers’ surplus in the housing market is given by
1 1
c p
s 1 s 5 2
E
D P t,P tdP t 1
E
[2D P t,P t 1 S P t]dP t. 6
h r
r h
r h
r h
h h
h M
M M
M M
Along the market equilibrium path, D P t,P t 5 S P t, where P t and P t
h r
h h
h r
h c
are equilibrium or market prices and, defining s as the overall consumers’ surplus along
m
the market-equilibrium path,
1 c
p M
M M
s 1 s 5 2
E
D P t,P tdP t
m h
r r
h r
a
p
r
M
5 : 2
E
D p dp 5 ACED. 7
r r
r
b
p
r
A cost-benefit test can be performed by adding the consumers’ surplus ACED to the change in government profit that results from the project.
2
A path-independent example is provided by the demand functions
2
These demand functions can be rationalized by the utility function Ur,h 5 min h2r,hj with income equal to
300.
144 C
. Blackorby, D. Donaldson Mathematical Social Sciences 37 1999 139 –163
300 600
]]] ]]]
D p , p 5 ,
D p , p 5 ,
8
r r
h h
r h
p 1 2p p 1 2p
r h
r h
and the supply function S p 5 20,
9
h h
b a
b
with p 5 10 and p 5 5. The decrease in p induces an increase in p from p 5 10 to
r r
r h
h a
p 5 12.5.
h
Then, computing consumers’ surplus in the rail market first,
5 12.5
12.5
300 600
c p
ˆ ]]
]]] s 1 s 5 2
E
dp 2
E
dp 1
E
20 dp
h r
h h
p 1 20 5 1 2p
r h
10 10
10 10
12.5 12.5
5 [300 ln p 1 20] 2 [300 ln5 1 2p ] 1 [20p ]
r 5
h 10
h 10
30 30
] ]
5 300 ln 2 300 ln
1 50 5 50. 10
25 25
When consumers’ surplus in housing is calculated first,
12.5 5
12.5
600 300
c p
˜ ]]]
]] s 1 s 5 2
E
dp 2
E
dp 1
E
20 dp
h h
r h
10 1 2p p 1 25
h r
10 10
10 12.5
10 12.5
5 2 [300 ln10 1 2p ] 1 [300 ln p 1 25] 1 [20p ]
h 10
r 5
h 10
35 35
] ]
5 2 300 ln 1 300 ln
1 50 5 50. 11
30 30
For any value of p , equilibrium in the housing market requires
r
600 ]]] 5 20,
12 p 1 2p
r h
so p
r
] p 5 15 2
, 13
h
2 and the market demand curve is given by
300
M
]]]] D p 5
5 10. 14
r r
p 1 30 2 p
r r
It follows that
5 c
p 10
s 1 s 5 2
E
10 dp 5 [10p ] 5 50,
15
m h
r r 5
10
which is the area to the left of the market demand curve.
C . Blackorby, D. Donaldson Mathematical Social Sciences 37 1999 139 –163
145
3
A path-dependent example is provided by the demand functions 5p
300 2 5p
h r
] ]]]
D p , p 5 1 5,
D p , p 5 2 5,
16
r r
h h
r h
p p
r h
and S p 5 20,
17
h b
a b
a
with p 5 10 and p 5 5. p increases from p 5 10 to p 5 11.
r r
h h
h
Calculating the rail-market surpluses first,
5 11
11 b
a
5p 300 2 5p
h r
c p
ˆ ]
]]] s 1 s 5 2
E
1 5 dp 2
E
2 5 dp 1
E
20 dp
S D
S D
h r
h h
p p
r h
10 10
10 5
11 11
50 275
] ]
5 2
E
1 5 dp 2
E
2 5 dp 1
E
20 dp
S D
S D
r h
h
p p
r h
10 10
10
5 59.66 2 21.21 1 20 5 58.45. 18
If the housing-market surplus is computed first,
11 5
11 b
a
300 2 5p 5p
r h
c p
˜ ]]]
] s 1 s 5 2
E
2 5 dp 2
E
1 5 dp 1
E
20 dp
S D
S D
h h
r h
p p
h r
10 10
10 11
5 11
250 55
] ]
5 2
E
2 5 dp 2
E
1 5 dp 1
E
20 dp
S D
S D
h r
h
p p
h r
10 10
10
5 2 18.83 1 63.12 1 20 5 64.29. 19
Equilibrium in the housing market requires, for any value of p ,
r
300 2 5p
r
]]] 2 5 5 20, 20
p
h
so p
r
] p 5 12 2
, 21
h
5 and the market demand curve is given by
60
M
] D p 5
1 4. 22
r r
p
r
The area to the left of the market demand curve is
3
These demand functions can be rationalized by the utility function Ur,h 5 5 lnr 2 5 1 h with income equal to 300.
146 C
. Blackorby, D. Donaldson Mathematical Social Sciences 37 1999 139 –163
5
60
c p
10
] s 1 s 5 2
E
1 4 dp 5 [60 ln p 1 4p ] 5 61.59. 23
S D
m h
r r
r 5
p
r 10
Note that all three answers are different because of path dependence. There is an additional important difficulty with the above result. It is that, in general,
p
the demand curves will shift as prices change because producers’ surplus s and the
h
change in net revenue from the project are changes in income. Consequently, the above argument requires there to be no income effects in both markets. If there are only two
markets, this cannot be true because budget constraints must be satisfied. Consequently, if all prices and incomes change, the consumers’ surplus calculated above cannot be
correct.
3. General equilibrium: the basic result
