1
Ch14. Consumer’s Surplus
We have discussed how to derive a consumer’s demand function from her underlying
preferences or UF. We are now concerned with the reverse problem: how to estimate
unobservable preferences or UF from observed demand functions. In fact, we have
done this by recovering a CD UF from average expenditure shares, and the present
chapter will introduce more approaches to this problem.
14.1 Demand for a Discrete Good
Go back to the example in Ch 6, where UF is quasilinear:
u(X) = v (x
1
) + x
2
, good 1 is
a discrete good priced at p, good 2 is a $ amount spent on a composite good, and the
BL is
x
2
=
m

px
1
. Looking at figure 6.12, one sees that for
r
1
> r
2
> r
3
> …,
the
consumer is indifferent between 0 or 1 unit of good 1 at
p = r
1
, 1 or 2 units of good 1
at
p = r
2
, and so on. That is,
−
=
−
−
=
−
−
=
.........
)
3
,
3
(
)
2
,
2
(
)
2
,
2
(
)
,
1
(
)
,
1
(
)
,
0
(
3
3
2
2
1
r
m
u
r
m
u
r
m
u
r
m
u
r
m
u
m
u
⇒
()
−
+
=
−
+
−
+
=
−
+
−
+
=
+
..........
)
3
(
)
3
(
)
2
(
)
2
(
)
2
(
)
2
(
)
(
1
)
(
)
1
(
)
0
(
3
3
2
2
1
r
m
v
r
m
v
r
m
v
r
m
v
r
m
v
m
v
We have:
−
=
−
=
−
=
..........
)
2
(
)
3
(
)
1
(
)
2
(
)
0
(
)
1
(
3
2
1
v
v
r
v
v
r
v
v
r
…….
.
(Ex1)
Clearly, the reservation price measures the MU (i.e., the increment in the utility
needed to induce one more unit of good 1 for consumption).
We want to show the relation between prices and demand that if
n
units of good 1 are
demanded, then
1
+
≥
≥
n
n
r
p
r
.
Proof
: when she chooses to consume, say, 6 units of good 1 at
p
, then she prefers (
6,
m6p
) to (
x
1
, mpx
1
) for any other
x
1