We consider non-negative integer commodity vectors in the Arrow-Debray economic model. Our main result is a version of the Arrow-Debray equilibrium price theorem adapted to the case of integer commodity vectors. The proof is based on the geometric form of the Hahn-Banach theorem and significantly uses the specificity of the integer commodity space. Our proof works only for a single-point set of agents, and we do not know whether it can be modified for the general case using the same idea.

[1] Aliprantis C. D., Brown D. J., Burkinshaw O. Existence and Optimality of Competitive Equilibria, Springer-Verlag, Berlin. (1989). DOI 10.1007/978-3-662-21893-8
[2] Aliprantis C. D., Burkinshaw O. Positive Operators, Springer, Dordrecht, 2006.
[3] Arrow K. J., Debreu G. Existence of an equilibrium for a competetive economy, Econometrica 1954, 22, 265–290.
[4] Chen P.-A., Lu C.-J., Lu, Y.-S. An alternating algorithm for finding linear Arrow-Debreu market equilibria. Preprint. (2019). Available at arXiv:1902.01754. https://doi.org/10.48550/arXiv.1902.01754
[5] Fratini S. M. Interest, profit and saving in Arrow-Debreu equilibrium models. Review of Keynesian Economics, 8, (2020), no 1, 39–53.
[6] Кадець В. М. Курс функцiонального аналiзу та теорiї мiри. – Львiв: СПД Чижиков, 2011, – 600 с.
[7] Keller C., Tseng M. C. Arrow-Debreu meets Kyle: Price discovery for derivatives. Preprint. (2023). Available at arXiv:2302.13426. https://doi.org/10.48550/arXiv.2302.13426
[8] Popov M. M., Ukrainets O. Z. A maximal Riesz-Kantorovich theorem with applications to markets with an arbitrary commodity set. Mat. Studii. 62 (2024), no 2, 199–210. https://doi.org/10.30970/ms.62.2.199-210