Antonelli matrix

In economics, and especially in microeconomics, the Hessian matrix of the distance function is called the Antonelli matrix .

Definition and meaning

With n being the number of goods, the Antonelli matrix A is a matrix whose (i, j) -th entry is given by the expression ${\ displaystyle (n \ times n)}$

{\ displaystyle {\ frac {\ partial ^ {2} d ({\ overline {u}}, \ mathbf {q})} {\ partial q_ {i} \ partial q_ {j}}}}

given is. The matrix is represented as follows:

{\ displaystyle A ({\ overline {u}}, \ mathbf {q}) \ equiv \ left ({\ begin {array} {ccc} {\ frac {\ partial ^ {2} d ({\ overline {u }}, \ mathbf {q})} {\ partial q_ {1} ^ {2}}} & \ cdots & {\ frac {\ partial ^ {2} d ({\ overline {u}}, \ mathbf { q})} {\ partial q_ {1} \ partial q_ {n}}} \\\ vdots & \ ddots & \ vdots \\ {\ frac {\ partial ^ {2} d ({\ overline {u}} , \ mathbf {q})} {\ partial q_ {n} \ partial q_ {1}}} & \ cdots & {\ frac {\ partial ^ {2} d ({\ overline {u}}, \ mathbf { q})} {\ partial q_ {n} ^ {2}}} \ end {array}} \ right)}

.

In this case, referred to the so-called distance function for the specific level of utility and the n vector of the quantities of goods . The distance function indicates the number that has to be divided by so that the remaining combination of quantities of goods lies precisely on the indifference curve . ${\ displaystyle d ({\ overline {u}}, \ mathbf {q})}$ ${\ displaystyle {\ overline {u}}}$ ${\ displaystyle \ mathbf {q} = (q_ {1}, \ ldots, q_ {n})}$ ${\ displaystyle \ mathbf {q}}$ ${\ displaystyle {\ overline {u}}}$

The (i, j) th entry of the Antonelli matrix now indicates how much the price a household would be willing to pay for a multiple of i changes if its demand for good j is exogenously increased, but he does would like to remain on the same indifference curve.

Properties and relationship to related concepts

The Antonelli matrix is symmetrical under common assumptions about the distance function . It is also the pseudo-inverse of the Slutsky matrix .

literature

Michael Ahlheim: Measures of Economic Welfare. In: Salvador Barberà, Peter J. Hammond and Christian Seidl (eds.): Handbook of Utility Theory. Vol. 1. Kluwer Academic Publishers, Boston 1998, ISBN 0-7923-8174-2 , pp. 483-568.
Giovanni B. Antonelli: Sulla Teoria Mathematica della Economia Politica. Pisa 1886. [A translation into English is contained in John S. Chipman et al. Under the title On the Mathematical Theory of Political Economy . a. (Ed.): Preferences, Utility and Demand. Harcourt Brace Jovanovich, New York 1971, chapter 16.]
Angus Deaton: The Distance Function in Consumer Behavior with Applications to Index Numbers and Optimal Taxation. In: The Review of Economic Studies. 46, No. 3, 1979, pp. 391-405 ( JSTOR 2297009 ).
Nicholas Stern: A Note on Commodity Taxation: The Choice of Variable and the Slutsky, Hessian and Antonelli Matrices (SHAM). In: The Review of Economic Studies. 53, No. 2, 1986, pp. 293-299 ( JSTOR 2297653 ).

Individual evidence

↑ See Ahlheim 1998, p. 494; Deaton 1979, p. 394.
↑ Cf. Angus Deaton and John Muellbauer: Economics and consumer behavior. Cambridge University Press, Cambridge u. a. 1980, ISBN 0-521-22850-6 , p. 54 ff.
↑ Cf. Angus Deaton and John Muellbauer: Economics and consumer behavior. Cambridge University Press, Cambridge u. a. 1980, ISBN 0-521-22850-6 , p. 57.
↑ See Ahlheim 1998, p. 494; Deaton 1979, p. 394.
↑ See, in each case also on the derivation, Deaton 1979, p. 395 f. and Stern 1986, p. 295.

[1] See Ahlheim 1998, p. 494; Deaton 1979, p. 394.

[2] Cf. Angus Deaton and John Muellbauer: Economics and consumer behavior. Cambridge University Press, Cambridge u. a. 1980, ISBN 0-521-22850-6 , p. 54 ff.

[3] Cf. Angus Deaton and John Muellbauer: Economics and consumer behavior. Cambridge University Press, Cambridge u. a. 1980, ISBN 0-521-22850-6 , p. 57.

[4] See Ahlheim 1998, p. 494; Deaton 1979, p. 394.

[5] See, in each case also on the derivation, Deaton 1979, p. 395 f. and Stern 1986, p. 295.