By Louis Philips, Lester D. Taylor (auth.), Louis Phlips, Lester D. Taylor (eds.)
During this testomony to the celebrated profession of H.S. Houthakker a few Professor Houthakker's neighbors, former colleagues and previous scholars provide essays which construct upon and expand his many contributions to economics in aggregation, intake, progress and exchange. among the individual participants are Paul Samuelson, Werner Hildenbrand, John Muellbauer and Lester Telser. The e-book additionally comprises 4 formerly unpublished papers and notes via its extraordinary dedicatee.
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N}. Consider the quotient Q(a) One then obtains N n Mih ,,,i (h) h=l =-_ _ N _ sign (a) n Mi i (h) h=l h,,, sign (a) = __ 48 Thus Q(a) is independent of the permutation a. Consequently, we showed that det M(i 1, ... ,iN ) = Q • det M(i 1, ... ,iN ) . Note that the matrix K1 depends only on f3, and hence on the parameter a. 1178. For S = 3 we obtain The matrix K1 is positive definite if f32 > 9/8 and det K1 > O. 2159. For S = 4 we obtain 2 M= 3 4 3 4f32 5f34 5 6f36 4 5f34 6f38 7 f312 5 6f36 7 f312 8f318 By using the first row and calculating the 4 subdeterminants, one obtains that det K1 where C(u) = u 4(u - I )4C(U) , = 384u6 + 312u 5 - 615u4 - 996u 3 - 374u 2 + 700u + 625, u = f32.
Again we can approach this question empirically or theoretically. Using the UK Family Expenditure Survey for the years 1969 - 83, we estimated the moments of the expenditure distribution (for different household groups) and checked the estimated matrix of moments for positive definiteness. l For values of S less than 4, the estimated matrices turned out to 1 These estimations and calculations have been made by H. P. Schmitz, SFB 303, Bonn. We would like to thank him for his help. 46 be positive definite.
This remark is similar to a result in Freixas and Mas-Colell (1987) since, when S = 2, the image of any function in 2(b) is contained in a two-dimensional subspace of the commodity space. In order to understand which property of the income distribution drives the matrix of moments to become positive definite, we now compute this matrix for a lognormal distribution: These calculations should be taken only as illustrations. In fact, our assumption is that the cross-section demand functions on the relevant domain are in the linear span of the base functions.
Aggregation, Consumption and Trade: Essays in Honor of H.S. Houthakker by Louis Philips, Lester D. Taylor (auth.), Louis Phlips, Lester D. Taylor (eds.)
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