Gorman-Lancaster Approach to Estimating Demand for New Good Gorman - - PowerPoint PPT Presentation

Econ 350

Gorman-Lancaster Approach to Estimating Demand for New Good

Gorman (1956, 1980) and Lancaster (1966, 1971) consider the problem of estimat- ing 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 jth good Linearity: Zi = bijXj (Zi from j) Additivity: Zi = bijXj + bikXk (Zi from j and k) Zi = Pn

j=1 bijXj

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−1Z. 2

General case has no such restriction. General problem: Assume U(Z) U is quasiconcave in Z PX ≤ M Budget constraint in goods space max U(Z) such that ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ Z = BX; X ≥ 0, P > 0 PX ≤ M Direct substitution: max U(BX) such that PX ≤ M leads to corner solutions and ßats. Need a more systematic approach. 3

Budget set in goods space {X | PX ≤ M, X ≥ 0} convex set, P > 0. Budget set in characteristics space K = {Z | Z = BX, PX ≤ 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 P1 , . . . , M Pn 4

Let Xs be extreme points of goods space, s = 1, . . . , n. Mapped by Zs = BXs to extreme points in characteristics space. Zs

i =

µM Ps ¶ bis for coordinate i Zs = µM Ps ¶ Bs

• ver 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 charac-

teristics space. 5

Z2 Z1 O X2 X3 X1 M P2

• M

P3

• M

P1

• Figure 1: Case with r = 2, n = 3. Good 1 intensive in Z2 : b11

b21 < b12 b22 < b13 b23 . 6

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 condi- tions. 7

Z2 Z1 O X2 X3 X1 M P2

• M

P3

• M

P1

• Figure 2: Case with r = 2, n = 3, with b11

b21 < b12 b22 < b13 b23 . 8

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 Z1 O

Vertex Optima Facet Optima

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 Z1 O

Good 1 Good 3 Good 2 Price at which consumers are indifferent between 1 and 3 Drops out

• f choice set

Figure 4: Efficiency Frontier with b11 b21 < b12 b22 < b13 b23 . 12

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
• f 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 Z1 O

(3') (1) (3) not purchased (2)

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)). B = ⎛ ⎜ ⎜ ⎝ b11 b12 b21 b22 ⎞ ⎟ ⎟ ⎠ Assume B is nonsingular. Rows: characteristics; Columns: goods b11 b21 < b12 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 = Zα

1 Z1−α 2

, 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 Z1 = PZ1 PZ2 18

Z2

2

Z1 O

(3') (1) (2)

Figure 6: Case where all three are bought. 19

Cobb-Douglas Þxes shares: µ α 1 − α ¶ = PZ1Z1 PZ2Z2 so (PZ2Z2) µ α 1 − α ¶ = PZ1Z1 PZ2Z2 + PZ1Z1 = M PZ2Z2 = (1 − α) M PZ1Z1 = αM Z2 = (1 − α) M PZ2 , Z1 = α M PZ1 20

Buy good 1 only if we have µ α 1 − α ¶ µb21 b11 ¶ ≥ PZ1 PZ2 . In general,b22 b12 ≤ Z2 Z1 ≤ b21 b11 . 21

Therefore, we have three types of solutions: buy good 1 only, buy good 2 only or buy both. Map α − → α 1 − α = V, FV (V ) is cdf. Assume it has density fV (V ) (absolutely continuous with respect to Lebesgue mea- sure), Pr ⎛ ⎜ ⎜ ⎝ good 2 bought exclusively ⎞ ⎟ ⎟ ⎠ = Pr µ V ≤ µPZ1 PZ2 ¶ b12 b22 ¶ , Pr ⎛ ⎜ ⎜ ⎝ good 1 bought exclusively ⎞ ⎟ ⎟ ⎠ = Pr µ V ≥ µPZ1 PZ2 ¶ b11 b21 ¶ . 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

PZ1 PZ2 b11 b21

λe−λV dV = exp µ −λ µPZ1 PZ2 b11 b21 ¶¶ . 23

Can identify λ from − ln ³ \ Pr (good 2 bought) ´ µPZ1 PZ2 ¶ µb11 b21 ¶ = ˆ λ where b denotes estimate. 24

Can recover Fα. How? Trivial: g ⎛ ⎜ ⎜ ⎝α ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ PZ1 PZ2 b11 b21 1 + PZ1 PZ2 b12 b22 ≤ α ≤ PZ1 PZ2 b12 b22 1 + PZ1 PZ2 b12 b22 ⎞ ⎟ ⎟ ⎠ = g µ α ¯ ¯ ¯ ¯ PZ1 PZ2 b11 b21 ≤ α 1 − α ≤ PZ1 PZ2 b12 b22 ¶ , i.e. if we know the distribution of a < α 1 − α < b, we know the distribution of a a + 1 ≤ α ≤ b b + 1. 25

In this technology, Z = BX if B−1 exists B−1Z = X PB−1Z = PX = M ¡ PB−1¢ = PZ. If we introduce a good with intensity µb13 b23 ¶ , who buys it? And in what amounts? 26

Suppose b22 b12 ≤ b23 b13 ≤ b21 b11 . Per unit cost of Z1 from good 3 is P3 b13 . Per unit cost of Z2 from good 3 is P3 b23 if ⎛ ⎜ ⎜ ⎝ P3 b13 P3 b23 ⎞ ⎟ ⎟ ⎠ ≥ PB−1. Good drops out of the budget. 27

Z2 Z1 O

(3) (1) (2) Breakeven Price

Figure 7: People buy good if b22 b12 ≤ b23 b13 ≤ b21 b11 . 28