Abstract
We consider a generalized expenditure function and the corresponding Hicksian demand. First, we provide some economic interpretation of the problem at stake. Then, we obtain different properties of the solution: existence, Lipschitz behavior and differential properties. Finally, we provide a Slutsky-type property.
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Notes
In the paper, we use the following notations:
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\(x\ge y\) means: \(x_{h}\ge y_{h}\) for all \(h=1,\ldots ,\ell \).
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\(x\gg y\) means: \(x_{h}>y_{h}\) for all \(h=1,\ldots ,\ell \).
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\(e_{h}\) denotes the h-th vector of the canonical basis of \({\mathbb {R}}^{\ell }\).
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\(\mathbf {1}_{\ell }\) denotes the \(\ell \)-dimensional vector whose coordinates are all equal to one. Similarly, \(\mathbf {1}_{n}\) denotes the n-dimensional vector whose coordinates are all equal to one. When there is no confusion, we simply write \(\mathbf {1}\).
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Let \(v \in {\mathbb {R}}^{\ell }. \Vert v\Vert :=\sum _{h=1}^{\ell }|v_{h}|\) denotes the norm of the vector v.
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Take \(x:=\left( \dfrac{1}{4},\dfrac{1}{4}\right) \) and \(\left( x^{\nu }:=\left( 1+\dfrac{1}{\nu },\dfrac{1}{\nu }\right) \right) _{\nu \ge 1}\) as a counterexample.
For the sake of completeness, this is proved in “Appendix.”
A presentation of the expenditure minimization problem can be found in any intermediary or advanced microeconomics textbook. For the sake of completeness, we refer the reader to [9].
A good is considered public if its use by one agent does not prevent other agents from using it[...]. (Laffont[10])
Recall the definition in [9]: An externality is present whenever the well-being of a consumer or the production possibilities of a firm are directly affected by the actions of another agent in the economy. Many economic goods can be considered as positive externalities such as vaccination or network.
The first scalar product \(q\cdot x\) is the one of \({\mathbb {R}}^{\ell }\) where \(\ell =rn\), while the second scalar product \(p\cdot x_{h}\) is the one of \({\mathbb {R}}^{r}\).
See [11].
See [12].
This is possible when the neighborhood is small enough thanks to the openness of \({\mathcal {V}}\).
Like in standard microeconomics, for \(p \in {\mathbb {R}}^{\ell }_{++}\) and \(w>0\), the budget set B(p, w) is defined by: \(B(p,w):=\lbrace x \in {\mathbb {R}}^{\ell }_{++}: p\cdot x \le w \rbrace \).
See the appendix of [14].
We write: \(\lambda ^{\varepsilon _{q}}_{k}:=\lambda ^{\varepsilon _{q}}_{k}(p_q,v_q)\) for \(k \in \lbrace 1,\ldots , n\rbrace \) and \(q\in \mathbb {N}\) to simplify the notation.
See [11].
Thanks to Proposition 3.1, K(p, v) is nonempty.
Indeed, K(p, v) is a subset of M(p, v)
The vectors are, by convention, column vectors, and the transpose of a vector x is denoted by \(x^{T}\). We use the notation: \(\bar{\lambda }:=\lambda (\bar{p},\bar{v})\).
To simplify the notation, for all \(k=1,\ldots , r\), we write \(\alpha _{k}(p):=\lambda (p,\bar{v})\).
References
Barten, A., Böhm, V.: Consumer theory. In: Arrow, K.J., Intriligator, M. (eds.) Handbook of Mathematical Economics, vol. 2, pp. 381–429. North-Holland, Amsterdam (1982)
Diewert, E.: Duality approaches to microeconomic theory. In: Arrow, K.J., Intriligator, M. (eds.) Handbook of Mathematical Economics, vol. 2, pp. 535–599. North-Holland, Amsterdam (1982)
Penot, J.-P.: The bearing of duality in microeconomics. In: Kusuoka, S., Yamazaki, A. (eds.) Advances in Mathematical Economics, vol. 7, pp. 113–139. Springer, Tokyo (2005)
McKenzie, L.: Demand theory without a utility index. The Review of Economic Studies 24(3), 185–189 (1957)
Diamond, P.A., McFadden, D.: Some uses of the expenditure function in public finance. J. Public Econ. 3, 3–21 (1974)
Cornet, B., Vial, J.P.: Lipschitzian solutions of perturbed non linear problems. SIAM J. Control Optim. 24, 1123–1137 (1986)
Fiacco, A., McCormick, G.: Nonlinear Programming: Sequential Unconstrained Minimization Techniques. Classics in Applied Mathematics. SIAM, Philadelphia (1990)
Bonnisseau, J.-M.: Regular economies with non-ordered preferences. J. Math. Econ. 39, 153–174 (2003)
Mas-Colell, A., Whinston, M.D., Green, J.R.: Microeconomic Theory. Oxford University Press, New York (1995)
Laffont, J.J.: Fundamentals of Public Economics. The MIT Press, Cambridge (1988)
Eustaquio, R.G., Karas, E.W., Ribeiro, A.A.: Constraint qualifications for nonlinear programming. Technical report, Federal University of Parana (2010)
Arrow, K.J., Enthoven, A.C.: Quasi-concave programming. Econometrica 29(4), 779–800 (1961)
Berge, C.: Topological Spaces. Dover Publications Mineola, New York (1997)
Florenzano, M.: General Equilibrium Analysis: Existence and Optimality Properties of Equilibria. Kluwer Academic Publishers, Boston (2003)
Rader, T.: Nice demand functions. Econometrica 41, 913–935 (1973)
Biheng, N., Bonnisseau, J.-M.: Regular economies with ambiguity aversion. J. Math. Econ. 59, 24–36 (2015)
Acknowledgments
The author wishes to thank Jean-Marc Bonnisseau for his support and Alain Chateauneuf for his valuable comments. The author would also like to thank two anonymous referees and the editor for useful suggestions that improved the presentation of the paper. All remaining errors are his.
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Communicated by Boris S. Mordukovich.
Appendix
Appendix
Lemma 8.1
The set \({\mathcal {V}}\) is an open set.
Proof
Let \(v^{0}\in {\mathcal {V}}\). We want to construct a neighborhood of \(v^0\) contained in \({\mathcal {V}}\). There exists \(x^{0} \in {\mathbb {R}}^{\ell }_{++}\) such that: \(u_{k}(x^{0})\ge v_{k}^{0}\) for all \(k=1,\ldots , n\), and there exists \(y^{0}\in {\mathbb {R}}^{\ell }_{++}\) and \(k_{0}\in \lbrace 1,\ldots , n\rbrace \) such that: \(u_{k_{0}}(y^0)< v_{k_{0}}^{0}\). Proceed to define \(\underline{v}\) by \(\underline{v}:=v^{0}-\mathbf {1}_{n}\) and \(\bar{v}\) by \(\bar{v}_{k}:=u_{k}(x^{0}+\mathbf {1}_{\ell })\) for \(k=1,\ldots , n\). Finally, we define the sets \(A:=\lbrace v\in {\mathbb {R}}^{n}:u_{k_{0}}(y^0) <v_{k_{0}}\rbrace \) and \(\displaystyle B:=\prod _{k=1}^{n}\left]\underline{v}_{k},\bar{v}_{k}\right[\). By construction, the set \(A\cap B\) is a nonempty open neighborhood of \(v^0\) contained in \({\mathcal {V}}\). Since \(v^0\) was arbitrarily chosen, one concludes that the set \({\mathcal {V}}\) is an open set.\(\square \)
Lemma 8.2
The sets of solutions of Problem (3) and Problem (4) coincide.
Proof
We first prove that the set of solutions of Problem (3) is a subset of the one of Problem (4). Let y be a solution of Problem (3). Since \(z_1\) is feasible for Problem (3), one finds: \( p\cdot y \le p\cdot z_{1}\). So y is feasible for Problem (4) and y is clearly a solution to Problem (4). In fact, the set of feasible points of Problem (4) is obviously contained in the set of feasible points of Problem (3).
Let y be a solution of Problem (4), and y is feasible for Problem (3) by construction. Let z be feasible for Problem (3), either \(-p\cdot z <-p\cdot z_1\) or \(-p\cdot z \ge -p\cdot z_1\). In the first case, obviously, \(-p\cdot z\le - p\cdot y\). In the second case, z is feasible for Problem (4). Thus, \(-p\cdot z \le -p\cdot z_1\le - p\cdot y\). Since z was arbitrarily chosen, we conclude that y is a solution to Problem (3). \(\square \)
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Biheng, N. A Generalization of the Expenditure Function. J Optim Theory Appl 168, 661–676 (2016). https://doi.org/10.1007/s10957-015-0805-x
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DOI: https://doi.org/10.1007/s10957-015-0805-x