Forecasting 7.3 Indicators Michel Bierlaire Solution of the - PDF document

Forecasting – 7.3 Indicators Michel Bierlaire Solution of the practice quiz Consider the sample enumeration estimator of the market share of alter- native i in the population � S W ( i ) = 1 � ω n P n ( i | x n ; θ ) . (1) S n =1 For continuous variables, we assume that the relative change of the variable is the same for every individual in the population. For arc elasticities, we have ∆ x ink = ∆ x ipk = ∆ x ik , (2) x ink x ipk x ik and for point elasticities, we have an infinitesimal change, that is ∂x ink = ∂x ipk = ∂x ik , (3) x ink x ipk x ik where � S x ik = 1 x ink . (4) S n =1 1. The aggregate direct arc elasticity of the model with respect to the average value x ik is defined as = ∆ � W ( i ) x ik � W ( i ) E . (5) x ik � ∆ x ik W ( i ) Using (1), we can calculate ∆ � W ( i ) and we obtain � S = 1 ∆ P n ( i | x n , C n ) x ik � W ( i ) E w n . (6) x ik � S ∆ x ik W ( i ) n =1 1

We can now replace the term ∆ x ik x ik by ∆ x ink x ink (see (2)) � S = 1 ∆ P n ( i | x n , C n ) x ink � W ( i ) (7) E w n . x ik � S ∆ x ink W ( i ) n =1 By introducing P n ( i | x n , C n ) P n ( i | x n , C n ) , we obtain the definition of the disaggregate direct arc elasticity: � S = 1 ∆ P n ( i | x n , C n ) x ink P n ( i | x n , C n ) � W ( i ) E w n (8) x ik � S ∆ x ink P n ( i | x n , C n ) W ( i ) n =1 S � = 1 P n ( i | x n , C n ) w n E P n ( i ) (9) . x ink � S W ( i ) n =1 Finally, applying (1) again we obtain � S w n P n ( i | x n , C n ) � W ( i ) E P n ( i ) E = . (10) � S x ik x ink n =1 w n P n ( i | x n , C n ) n =1 This equation shows that the calculation of aggregate elasticities in- volves a weighted sum of disaggregate elasticities. However, the weight is not w n as for the market share, but a normalized version of w n P n ( i | x n , C n ) . 2. The derivation follows the same logic. The aggregate cross point elas- ticity of the model with respect to the average value x jk is defined as = ∂ � W ( i ) x jk � W ( i ) (11) E . x jk � ∂x jk W ( i ) Using (1), we can calculate ∂ � W ( i ) and we obtain � S = 1 ∂P n ( i | x n , C n ) x jk � W ( i ) E w n . (12) x jk � S ∂x jk W ( i ) n =1 We can now replace the term ∂x jk x jk by ∂x jnk x jnk (see (3)) � S = 1 ∂P n ( i | x n , C n ) x jnk � W ( i ) E w n . (13) x jk � S ∂x jnk W ( i ) n =1 2

By introducing P n ( i | x n , C n ) P n ( i | x n , C n ) , we obtain the definition of the disaggregate cross point elasticity: � S = 1 ∂P n ( i | x n , C n ) x jnk P n ( i | x n , C n ) � W ( i ) E w n (14) x jk � S ∂x jnk P n ( i | x n , C n ) W ( i ) n =1 � S = 1 P n ( i | x n , C n ) w n E P n ( i ) (15) x jnk � S W ( i ) n =1 Finally, applying (1) again we obtain � S w n P n ( i | x n , C n ) � W ( i ) E P n ( i ) = (16) E � S . x jk x jnk n =1 w n P n ( i | x n , C n ) n =1 3