The Impact of Market Clearing Assumptions and Dynamics on Demand Elasticities
Proof of Equation (6)
We will prove the following result for the equilibrium estimator of the model defined by
equations (1) - (4).
p lim(α$ 1 − α 1 ) =
p lim{[( I − M 2∗e ) P]′ g}
γ . p lim{P$ ′M P$ }
(6)
1
where
M 2∗e = I − x 2∗e ( x 2∗e′ x 2∗e ) −1 x 2∗e′ , x 2∗e = M 1 x 2 e ,
M 1 = I − x1 ( x1′ x1 ) −1 x1′
We assume that equation (5) is the data generating process but we erroneously omit g$ from
(5) and therefore estimate the following via OLS:
Q = x1 β 1 + α 1 P$ + u3
( 7)
Using partitioned regression (Greene, 1993, p179), the OLS estimator of α 1 in (7) can be
written as:
α$ 1 = ( P$ ′M 1 P$ ) −1 P$ ′M 1 ( x1 β 1 + α 1 P$ + (1 / γ ) g$ + u1 )
(8)
where we have replaced Q by (5). Expanding (8) gives:
α$ 1 = ( P$ ′M 1 P$ ) −1 P$ ′M 1 x1 β 1 + ( P$ ′M 1 P$ ) −1 P$ ′M 1α 1 P$ + ( P$ ′M 1 P$ ) −1 P$ ′M 1 (1 / γ ) g$ + ( P$ ′M 1 P$ ) −1 P$ ′M 1u1
Since M 1 x1 = 0 the first term disappears, the second term simplifies and we get:
α$ 1 = α 1 + (1 / γ )( P$ ′M 1 P$ ) −1 P$ ′M 1 g$ + ( P$ ′M 1 P$ ) −1 P$ ′M 1 u1
Taking the plim of both sides we get:
p lim(α$ 1 ) = α 1 + (1 / γ )
p lim{ P$ ′M 1 g$} p lim{P$ ′M 1 u1 / n}
+
p lim{P$ ′M 1 P$ } p lim{ P$ ′M 1 P$ / n}
We assume p lim{ M 1 u1 / n} = 0 and thus the last term disappears and we can write:
p lim(α$ 1 − α 1 ) =
p lim{P$ ′M 1 g$ }
γ . p lim{P$ ′M P$ }
1
( 9)
Focus upon the numerator of (9):
P$ ′M 1 g$ = {( I − M ) P}′ M 1 ( I − M ) g = P ′{( I − M ) ′ M 1 ( I − M )}g
(10)
Consider the middle term in (10), since ( I − M ) is idempotent and expanding we get:
( I − M ) ′ M 1 ( I − M ) = M 1 − M 1 M − MM 1 + MM 1 M
Since MM 1 = M and M is idempotent we get:
(I − M ) ′ M1 (I − M ) = M1 − M1 M = M1 (I − M )
(11)
Substitute (11) into (10);
P$ ′M 1 g$ = P ′M 1 ( I − M ) g = P ′M 1 {x1 β$ g1 + x 2 e β$ g 2 }
(12)
where β$ gi are the regression coefficients gained by regressing g on x1 and x 2 e . Since
M 1 x1 = 0 , (12) becomes:
P$ ′M 1 g$ = P ′M 1 x 2 e β$ g 2
(13)
β$ g 2 can be determined using partitioned regression concepts:
β$ g 2 = ( x 2′ e M 1 x 2 e ) −1 x 2′e M 1 g
(14)
Substitute (14) into (13) and simplify recognising that M 1 is idempotent:
P$ ′M 1 g$ = P ′x 2∗e ( x 2∗e′ x 2∗e ) −1 x 2*e′ g = P ′( I − M 2*e ) g
Substitute (15) into (9) and we get our result:
p lim(α$ 1 − α 1 ) =
p lim{[( I − M 2∗e ) P]′ g}
γ . p lim{P$ ′M P$ }
1
(15)