### G-83-09

# The Fundamental Lemma of Demand Theory (A Synthesis from the Intrinsic Viewpoint)

## Edouard Wagneur and C Bronsard

A regular consumption unit is defined by (Ω, ⩽, W_{S}), where Ω is an open connected subset of ℝ^{n} and ⩽ is given by a utility function u: Ω → ℝ of differentiability class C^{2} which satisfies: ∀ × εΩ:

A1: Du(x) >> 0 (1 - jet monottonicity)

A2: Du(x) ⋅ h = 0 ⤇ D^{2}u(x) (h,h) < 0 (2 - jet regular concavity).

and W_{s} is parameter depending (the unit of account) vector field on Ω with W^{i}_{s}(x) ≥ 0, i = 1 ... n. To every point x ε Ω we can associate a price vector P together with an income m = xp and the (n+1) - uple (p,m) is uniquely defined up to the value s of the unit of account. More precisely, to each consumption bundle x and value s ε R^{+} of the unit of account, there is a unique price-income vector (p(x,s), m(x,s)). This defines a map ϕ or inverse demand system [Antonelli (1886), Hicks-Allen (1934), Hotelling (1935), Wold (1944), Samuelson (1950), Hicks (1956), Salvas-Bronsard et al. (1977), Anderson (1980), Huang (1983)]. this map can be locally characterized by a matrix H which defines an Antonelli (local) structure.

Conversely, to each point (p,m) ε ϕ ( Ω × ℝ^{+}) there corresponds a unique (n+1) - uple (x,s). This defines a map σ (= ϕ^{-1}) or demand system [to previous authors we add Slutsky (1915) and for a recent survey, Barten-Böhm (1982)]. this map can be locally characterized by a matrix K which defines a Slutsky (local) structure.

Such characterizations are intrinsic to the consumption unit. However, when it faces an institutional framework where either quantities (MDP procedures) or prices (competitive equilibrium) are given, these intrinsic characterizations are consistent with this institutional framework [Bronsard-Leblanc (1980)].

Nevertheless, these two previous frameworks can be seen as two extreme (or polar) cases, the general situation being of mixed type [Samuelson (1965), Bronsard-Salvas-Bronsard (1980), Chavas (1983)] or of truncated mixed type (i.e. of quantity rationing type [Tobin-Hauthakker (1950-51), Pollak (1969), Drèze (1977), Neary-Roberts (1980), Bronsard-Salvas-Bronsard (1980), Chavas (1983)].

using a natural generalization of the Hotelling-Wold identity, we present a unified treatment of the literature on inverse demand systems and their duality with demand systems. Moreover, the general setting mentioned can be formulated within a unique model which also includes polar cases and can be summarized in the following fundamental lemma of demand theory:

(FL) H^{✝}_{I} = K_{I} = K_{11} - K_{12} K^{-1}_{22} K_{21}

where H_{I} is the upper left (k,k) submatrix of H, H^{✝}_{I} its relfexive g-inverse, K_{I} the (k,k) Slutsky matrix of mixed type (truncated or not) and is either the (n,n) standard intrinsic Slutsky matrix or an (l,l) intrinsic Slutsky matrix of mixed type (truncated or not).

An interesting application of FL can be given in the case of a monopoly equilibrium under some price rigidity: the same equilibrium can be implemented either as a keynesian equilibrium or as a price discriminatory one.

Published **April 1983**
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27 pages