Roy's identity

Roy's identity (pronunciation like French roi, [ʁwa-] ) is an important phrase in microeconomics . He notes an important relationship between the Marshallian demand and the indirect utility function . It was named after the French economist René Roy.

Presentation and meaning

Despite the in many ways analogy between the concept of the indirect utility function and that of the expenditure function, at first glance there is no direct analogy to Shephard's lemma , according to which the derivation of the expenditure function according to the price corresponds to the corresponding Hicksian demand function. However, a slight modification still provides a certain degree of comparability. The relationship is known as Roy's identity .

Roy's identity: Be steady and strictly monotonously increasing. Be further differentiable in one place and . Then applies to all ( ): ${\displaystyle u(\cdot )}$ ${\displaystyle v(\mathbf {p} ,y)}$ ${\displaystyle (\mathbf {p} _{0},y_{0})}$ ${\displaystyle \partial v({\mathbf {p} }_{0},y_{0})/\partial y\neq 0}$ ${\displaystyle i}$ ${\displaystyle 1\leq i\leq n}$

${\displaystyle x_{i}(\mathbf {p} _{0},y_{0})=-{\frac {\partial v(\mathbf {p} _{0},y)/\partial p_{i}}{\partial v(\mathbf {p} _{0},y)/\partial y}}}$

With regard to the two requirements of the theorem on the utility function, it would even suffice to assume that it is continuous and that the underlying preference-indifference relation fulfills the property of local unsaturation and is convex. Note: It refers to an order of preference as local non-saturated if for any and for each environment to one exists, applies to: (see also order of preference ). However, the stronger assumption of strict monotony, which is more commonly used in practice (together with that of the differentiability of the indirect utility function) also enables a simpler proof of the relationship. ${\displaystyle u(\cdot )}$ ${\displaystyle R}$ ${\displaystyle x_{a}\in X}$ ${\displaystyle \epsilon }$ ${\displaystyle U_{\epsilon }}$ ${\displaystyle x_{a}}$ ${\displaystyle z\in U_{\epsilon }}$ ${\displaystyle zPx_{a}}$

proof

The Lagrangian of the utility maximization problem (see the article Marshall's demand function ) reads . The envelope theorem implies to apply that ${\displaystyle {\mathcal {L}}(\mathbf {x} ,\lambda )=u(\mathbf {x} )+\lambda (y-\mathbf {p\cdot x} )}$

${\displaystyle \mathrm {(1)} \quad \forall i:{\frac {\partial v(\mathbf {p} ,y)}{\partial p_{i}}}={\frac {\partial {\mathcal {L}}\left(\mathbf {x} ^{*},\lambda ^{*}\right)}{\partial p_{i}}}=-\lambda ^{*}x_{i}^{*}}$

such as

${\displaystyle \mathrm {(2)} \quad \forall i:{\frac {\partial v(\mathbf {p} ,y)}{\partial y}}={\frac {\partial {\mathcal {L}}\left(\mathbf {x} ^{*},\lambda ^{*}\right)}{\partial y}}=\lambda ^{*}}$

(2) inserted in (1) then immediately yields Roy's identity.

See also

• Slutsky decomposition

literature

• Geoffrey A. Jehle and Philip J. Reny: Advanced Microeconomic Theory. 3rd ed. Financial Times / Prentice Hall, Harlow 2011, ISBN 978-0-273-73191-7 .
• Andreu Mas-Colell, Michael Whinston, and Jerry Green: Microeconomic Theory. Oxford University Press, Oxford 1995, ISBN 0-19-507340-1 .
• René Roy: La Distribution du Revenu Entre Les Divers Biens. In: Econometrica. 15, 1974, pp. 205-225.
• Hal Varian : Microeconomic Analysis. WW Norton, New York and London 1992, ISBN 0-393-95735-7 .

Web links

• Microeconomics script (PDF; 1.3 MB) (alternative proof of Roy's identity using Shephard's Lemma in Section 4, p. 73.)

Individual evidence

1. ↑ See Peter Hammond and Knut Sydsaeter: Mathematics for economists: basic knowledge with practical relevance. Pearson Germany, 2008, ISBN 3-8273-7357-3 , p. 601.
2. ↑ See Jehle / Reny 2011, p. 29; with slightly weaker assumptions Mas-Colell / Whinston / Green 1995, pp. 73 f.
3. ↑ See Mas-Colell / Whinston / Green 1995, p. 59.