{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T00:15:33Z","timestamp":1760228133297,"version":"build-2065373602"},"reference-count":34,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2022,5,7]],"date-time":"2022-05-07T00:00:00Z","timestamp":1651881600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper incorporates more general cases with a new class of constantly adjusted concavity demand curves and includes three types of taxes. To verify the output symmetry under various forms of taxation, we simulate both linear and constant elasticity demand functions under the unit, demand ad valorem, and cost ad valorem taxes. If all the demand functions in the submarkets are linear, the total outputs are identical under both uniform pricing and third-degree price discrimination. Furthermore, if all the weak market demand curves are strictly \u201cRobinson-concave\u201d and all the strong market demand curves are strictly \u201cRobinson-convex\u201d or linear, then the total output under price discrimination exceeds that under uniform pricing, and vice versa. While different taxes lead to higher costs, the cost pass-through changes the prices of the products, and the change of total output still depends on the curvature of the demand curve. Therefore, the curvature of the demand curve remains the main determinant of changes in output. Our study provides a theoretical basis for market intervention in price discrimination.<\/jats:p>","DOI":"10.3390\/sym14050959","type":"journal-article","created":{"date-parts":[[2022,5,8]],"date-time":"2022-05-08T23:27:25Z","timestamp":1652052445000},"page":"959","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On Joan Robinson\u2019s Output Symmetry Theorems with Various Taxations under Third-Degree Price Discrimination: A Generalization"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8122-4342","authenticated-orcid":false,"given":"Shih-Shen","family":"Chen","sequence":"first","affiliation":[{"name":"School of Economics and Management, Fuzhou University of International Studies and Trade, Fuzhou 350202, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ken","family":"Hung","sequence":"additional","affiliation":[{"name":"A.R. Sanchez School of Business, Texas A & M International University, Laredo, TX 78041, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5767-4079","authenticated-orcid":false,"given":"Chien-Shu","family":"Tsai","sequence":"additional","affiliation":[{"name":"Department of International Business, National Kaohsiung University of Science and Technology, Kaohsiung 807618, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,5,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"100207","DOI":"10.1016\/j.iree.2020.100207","article-title":"Neutrality of Third-degree Price Discrimination","volume":"36","author":"Chao","year":"2021","journal-title":"Int. Rev. Econ. Educ."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"102713","DOI":"10.1016\/j.ijindorg.2021.102713","article-title":"Third-degree Price Discrimination in Oligopoly with Endogenous Input Costs","volume":"79","author":"Shaffer","year":"2021","journal-title":"Int. J. Ind. Organ."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1016\/j.geb.2021.11.008","article-title":"Third-degree Price Discrimination versus Uniform Pricing","volume":"131","author":"Bergemann","year":"2022","journal-title":"Games Econ. Behav."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Zhang, Z., Wang, Y., Meng, Q., and Han, Q. (2022). Impact of Third-Degree Price Discrimination on Welfare under the Asymmetric Price Game. Mathematics, 10.","DOI":"10.3390\/math10081215"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"149","DOI":"10.2307\/2233516","article-title":"A General Analysis of the Output Effect under Third-Degree Price Discrimination","volume":"98","author":"Shih","year":"1988","journal-title":"Econ. J."},{"key":"ref_6","unstructured":"Pigou, A.C. (1920). The Economics of Welfare, Macmillan. [1st ed.]."},{"key":"ref_7","unstructured":"Robinson, J. (1933). The Economics of Imperfect Competition, Macmillan."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"163","DOI":"10.2307\/1907267","article-title":"The Analysis of Output under Discrimination","volume":"18","author":"Edwards","year":"1950","journal-title":"Econometrica"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"84","DOI":"10.2307\/1056682","article-title":"Output under Discriminating Monopoly: A Revisit","volume":"37","author":"Silberberg","year":"1970","journal-title":"South. Econ. J."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"239","DOI":"10.2307\/1056735","article-title":"The Quantity of Output in Simple Monopoly and Discriminating Monopoly","volume":"41","author":"Finn","year":"1974","journal-title":"South. Econ. J."},{"key":"ref_11","first-page":"776","article-title":"Note on Output under Discrimination","volume":"24","year":"1977","journal-title":"Rev. Internazionale Sci. Econ. E Commer."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"490","DOI":"10.2307\/1882453","article-title":"The Theory of Limited and Unlimited Discrimination","volume":"54","author":"Leontief","year":"1940","journal-title":"Q. J. Econ."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"164","DOI":"10.2307\/1058608","article-title":"Output Changes under Third-Degree Price Discrimination: A Reexamination","volume":"48","author":"Smith","year":"1981","journal-title":"South. Econ. J."},{"key":"ref_14","first-page":"242","article-title":"Output and Welfare Implications of Monopolistic Third-Degree Price Discrimination","volume":"71","author":"Schmalensee","year":"1981","journal-title":"Am. Econ. Rev."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"96","DOI":"10.2307\/2230954","article-title":"Joan Robinson\u2019s Criterion for Deciding Whether Market Discrimination Reduces Output","volume":"86","author":"Greenhut","year":"1976","journal-title":"Econ. J."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1016\/0022-0531(78)90072-8","article-title":"Incremental Consumer\u2019s Surplus and Hedonic Price Adjustment","volume":"17","author":"Willig","year":"1978","journal-title":"J. Econ. Theory"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"892","DOI":"10.2307\/2232754","article-title":"Price Discrimination, \u2018Adjusted Concavity\u2019, and Output Changes under Conditions of Constant Elasticity","volume":"93","author":"Formby","year":"1983","journal-title":"Econ. J."},{"key":"ref_18","first-page":"705","article-title":"Monopoly Output under Alternative Spatial Pricing Techniques","volume":"62","author":"Greenhut","year":"1972","journal-title":"Am. Econ. Rev."},{"key":"ref_19","first-page":"133","article-title":"The Ad Valorem and Per Unit Tax of the Spatial Price Discrimination Models","volume":"7","author":"Yang","year":"2013","journal-title":"Int. J. Econ."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"54","DOI":"10.1016\/j.econlet.2009.09.024","article-title":"Third-degree Price Discrimination, Quality Choice, and Welfare","volume":"106","author":"Ikeda","year":"2010","journal-title":"Econ. Lett."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"511","DOI":"10.1111\/1467-6451.00085","article-title":"Third-Degree Price Discrimination with Interdependent Demands","volume":"46","author":"Layson","year":"1998","journal-title":"J. Ind. Econ."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1601","DOI":"10.1257\/aer.100.4.1601","article-title":"Monopoly Price Discrimination and Demand Curvature","volume":"100","author":"Aguirre","year":"2010","journal-title":"Am. Econ. Rev."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1231","DOI":"10.1111\/ecin.12075","article-title":"The Welfare Effects of Third-Degree Price Discrimination in A Differentiated Oligopoly","volume":"52","author":"Adachi","year":"2014","journal-title":"Econ. Inq."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"442","DOI":"10.1111\/1756-2171.12091","article-title":"Differential Pricing When Costs Differ: A Welfare Analysis","volume":"46","author":"Chen","year":"2015","journal-title":"RAND J. Econ."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1","DOI":"10.5018\/economics-ejournal.ja.2019-52","article-title":"Oligopoly Price Discrimination, Competitive Pressure and Total Output","volume":"13","author":"Aguirre","year":"2019","journal-title":"Economics"},{"key":"ref_26","unstructured":"Adachi, T. (2022, April 28). A Sufficient Statistics Approach for Welfare Analysis of Oligopolistic Third-Degree Price Discrimination. Available online: https:\/\/ssrn.com\/abstract=3006421."},{"key":"ref_27","first-page":"870","article-title":"Price Discrimination and Social Welfare","volume":"75","author":"Varian","year":"1985","journal-title":"Am. Econ. Rev."},{"key":"ref_28","first-page":"1259","article-title":"Third-Degree Price Discrimination and Output: Generalizing a Welfare Result","volume":"80","author":"Schwartz","year":"1990","journal-title":"Am. Econ. Rev."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"371","DOI":"10.2307\/1060277","article-title":"Taxation and the Output and Welfare of Third-Degree Price Discrimination","volume":"59","author":"Yang","year":"1993","journal-title":"South. Econ. J."},{"key":"ref_30","first-page":"1254","article-title":"Direction of Price Changes in Third-degree Price Discrimination","volume":"80","author":"Nahata","year":"1990","journal-title":"Am. Econ. Rev."},{"key":"ref_31","unstructured":"DeGraba, P. (1991). The Relationship between Optimal Third Degree Discriminatory Prices and the Optimal Uniform Price, Mimeograph, Cornell University."},{"key":"ref_32","first-page":"1011","article-title":"Bounding the Welfare Effects of Third-Degree Price Discrimination","volume":"83","author":"Malueg","year":"1993","journal-title":"Am. Econ. Rev."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1111\/j.1465-7295.1982.tb01158.x","article-title":"The Law of Demand, Positive Sloping Marginal Revenue, and Multiple Profit Equilibria","volume":"20","author":"Formby","year":"1982","journal-title":"Econ. Inq."},{"key":"ref_34","unstructured":"Liebman, J., Waren, A., Schrage, L.E., and Lasdon, L. (1986). Modeling and Optimization with GINO, Scientific Press."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/5\/959\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:07:40Z","timestamp":1760137660000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/5\/959"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,7]]},"references-count":34,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2022,5]]}},"alternative-id":["sym14050959"],"URL":"https:\/\/doi.org\/10.3390\/sym14050959","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,5,7]]}}}