Econ 350
Gorman-Lancaster Approach to
Estimating Demand for New Good
Gorman (1956, 1980) and Lancaster (1966, 1971) consider the problem of estimating the demand for a new good.
Goods
X
Characteristics
Z
Goods are packages of underlying characteristics. Characteristics are quantitative
(not qualitative).
1
Let bij be the quantity of the ith characteristic possessed by the j th good
Linearity:
(Zi from j)
Zi = bij Xj
Additivity: Zi = bij Xj + bik Xk
Zi =
Pn
j=1 bij Xj
(Zi from j and k)
i = 1, . . . , r
(Zi from all sources)
r characteristics, n goods.
Z = BX
B is r × n. Traditional model of consumer demand assumes B is a square matrix and
invertible,
X = B −1 Z.
2
General case has no such restriction. General problem:
Assume
U (Z)
U is quasiconcave in Z
PX ≤ M
Budget constraint in goods space
⎧
⎪
⎪
⎨ Z = BX; X ≥ 0, P > 0
max U(Z) such that
⎪
⎪
⎩ PX ≤ M
Direct substitution: max U (BX) such that P X ≤ M leads to corner solutions
and ßats.
Need a more systematic approach.
3
Budget set in goods space
{X | P X ≤ M, X ≥ 0}
convex set, P > 0.
Budget set in characteristics space
K = {Z | Z = BX, P X ≤ M, X ≥ 0}
feasible set.
Set convex. Both sets characterized by extreme points. For goods space these are
(i) the origin (0, . . . , 0)
(ii)
M
M
,...,
P1
Pn
4
Let X s be extreme points of goods space, s = 1, . . . , n. Mapped by Z s = BX s to
extreme points in characteristics space.
¶
M
bis
Zis =
P
µ s¶
M
Zs =
Bs
Ps
µ
for coordinate i
over all coordinates
Budget set in characteristics space:
1. A convex polytope (convex combinations of extreme points)
2. Has at most n + 1 extreme points
3. Every extreme point in characteristics space is an extreme point in goods space.
4. An extreme point in goods space is not necessarily an extreme point in characteristics space.
5
X3
M
P3
Z2
O
M
P1
M
P2
X2
Z1
X1
Figure 1: Case with r = 2, n = 3. Good 1 intensive in Z2 :
6
b11
b12
b13
<
<
.
b21
b22
b23
Two-stage maximization:
1. Compute minimal cost bundles of Z.
2. Maximize utility in Z space.
Implications: no longer have a representative agent, except under extreme conditions.
7
X3
M
P3
Z2
O
M
P1
M
P2
Z1
X1
Figure 2: Case with r = 2, n = 3, with
8
b11
b12
b13
<
<
.
b21
b22
b23
X2
Problem has two aspects:
1. DeÞne Goods Efficiency Set
2. Maximize over Goods Efficiency Set
Maximize U (Z) such that Z ∈ K.
(a) Every point on the efficiency frontier in C space is the image of a point on
budget in goods space.
(b) A point on efficiency frontier in goods space not necessarily on efficiency
frontier in characteristics space (see above).
9
Z2
Vertex Optima
Facet Optima
O
Z1
Figure 3: Indifference Curves.
10
Concept: Efficiency Frontier
1. As income expands, polytope expands proportionately.
2. As price of good 2 increases, it is less likely to be bought.
3. At a sufficiently high price, the good drops out.
Two effects:
1. Efficiency Substitution Effect
2. Personal Substitution Effect
Can get inferior goods even if all characteristics are normal.
∴ Normal in Characteristics. Not in goods.
11
Z2
Good 1
Good 2
Price at which
consumers are
indifferent
between 1 and 3
Drops out
of choice set
Good 3
O
Z1
Figure 4: Efficiency Frontier with
12
b11
b12
b13
<
<
.
b21
b22
b23
Return to the stated problem: how to estimate the demand for a new good? Recall
Knight’s Quotation. Need to make the future like the past.
Lancaster Approach:
1. Assume a common technology B across all consumers.
2. Need a model of preferences.
3. Why are people who are observationally identical buying different goods? (Model
of Preference Heterogeneity)
13
Consider the following example:
1. We observe X, Z
2. ∴ We can identify B
3. Estimate Preferences U (Z) allowing for heterogeneity in preferences (no more
representative consumer).
4. Forecast demand for new good as a vector of characteristics.
14
Is Good 3 purchased? NO
Is Good 30 purchased? YES
∴ We estimate the technology of purchase or not (purely a technical affair).
15
Z2
(3')
(1)
(3)
not purchased
(2)
O
Z1
Figure 5: Example with n = r = 2. Goods (1) and (2) are in the choice set.
16
Consider a case where only (1) and (2) are bought (and maybe only (1) or (2)).
⎛
⎞
⎜ b11 b12 ⎟
⎟
B=⎜
⎝
⎠
b21 b22
Assume B is nonsingular.
Rows: characteristics; Columns: goods
b11
b12
<
b21
b22
(as drawn)
Good 2 intensive in Z1 .
Good 1 intensive in Z2 .
Need to know distribution of preferences. Take the Two Good World, n = r = 2.
17
Example:
U = Z1α Z21−α ,
0≤α≤1
Distribution of Income M
FM (M)
Distribution of α
Fα (α)
Consumer buys both goods if on (1)—(2) face.
µ
α
1−α
¶
Z2
PZ
= 1
Z1
PZ2
18
Z2
(3')
(1)
2
O
(2)
Z1
Figure 6: Case where all three are bought.
19
Cobb-Douglas Þxes shares:
µ
α
1−α
¶
PZ Z1
= 1
PZ2 Z2
µ
α
(PZ2 Z2 )
1−α
so
PZ2 Z2 + PZ1 Z1 = M
PZ2 Z2 = (1 − α) M
PZ1 Z1 = αM
Z2 = (1 − α)
M
,
PZ2
20
Z1 = α
M
PZ1
¶
= PZ1 Z1
Buy good 1 only if we have
µ
In general,
α
1−α
¶µ
b21
b11
b22
Z2
b21
≤
≤
.
b12
Z1
b11
21
¶
≥
PZ1
.
PZ2
Therefore, we have three types of solutions: buy good 1 only, buy good 2 only or
buy both.
Map α −→
α
= V,
1−α
FV (V ) is cdf.
Assume it has density fV (V ) (absolutely continuous with respect to Lebesgue measure),
⎛
⎜ good 2 bought
Pr ⎜
⎝
exclusively
⎛
⎞
µ
µ
¶
¶
⎟
⎟ = Pr V ≤ PZ1 b12 ,
⎠
PZ b22
2
⎞
µ
µ
¶
¶
⎜ good 1 bought ⎟
PZ1 b11
⎜
⎟
Pr ⎝
⎠ = Pr V ≥ PZ b21 .
2
exclusively
22
Mixed Discrete-Continuous Model.
Z1 = α
M
PZ1
Z2 = (1 − α)
M
PZ2
Given the distribution of (α, M), can derive demands. Goal of Structural Estimation:
to identify Fα or FV .
For example, if V ∼ λe−λV = fV (V ),
Pr (good 2 bought) =
Z
PZ b
1 11
PZ b21
2
−λV
λe
0
23
¶¶
µ µ
PZ1 b11
.
dV = exp −λ
PZ2 b21
Can identify λ from
³
´
\
ln Pr (good 2 bought)
µ
¶µ ¶
−
= λ̂
PZ1
b11
PZ2
b21
where b denotes estimate.
24
Can recover Fα . How? Trivial:
⎛ ¯
¯ PZ1 b11
PZ1 b12
¯
⎜ ¯ PZ2 b21
PZ2 b22
¯
≤α≤
g⎜
⎝α ¯
b
P
PZ b12
¯ 1 + Z1 12
1+ 1
¯
P b
P b
Z2
22
Z2
22
¶
µ ¯
¯ PZ1 b11
b
α
P
Z
12
1
,
= g α ¯¯
≤
≤
PZ2 b21
1−α
PZ2 b22
i.e. if we know the distribution of a <
⎞
⎟
⎟
⎠
α
< b, we know the distribution of
1−α
a
b
≤α≤
.
a+1
b+1
25
In this technology,
Z = BX
if B −1 exists
B −1 Z = X
P B −1 Z = P X = M
¢
¡
P B −1 = PZ .
If we introduce a good with intensity
µ
¶
b13
, who buys it? And in what amounts?
b23
26
Suppose
b22
b23
b21
≤
≤
.
b12
b13
b11
Per unit cost of Z1 from good 3 is
P3
P3
. Per unit cost of Z2 from good 3 is
if
b13
b23
⎞
P3
⎜ b13 ⎟
−1
⎟
⎜
⎝ P3 ⎠ ≥ P B .
b23
⎛
Good drops out of the budget.
27
Z2
Breakeven Price
(3)
(1)
(2)
O
Z1
Figure 7: People buy good if
28
b22
b23
b21
≤
≤
.
b12
b13
b11