The extensional realizability model of continuous functionals and three weakly non-constructive classical theorems

We investigate wether three statements in analysis, that can be proved classically, are realizable in the realizability model of extensional continuous functionals induced by Kleene's second model $K_2$. We prove that a formulation of the Riemann Permutation Theorem as well as the statement that all partially Cauchy sequences are Cauchy cannot be realized in this model, while the statement that the product of two anti-Specker spaces is anti-Specker can be realized

Taxation of Entrepreneurs Relative to Well Diversified Investors - A Swedish Perspective

The paper argues that cost of capital comparisons across closely held companies and entrepreneurial ventures on the one hand and widely held companies on the other, ought not to be based on an equal level assumption regarding the investors’ required rates of return, net of taxes. Theoretical considerations as well as empirical evidence show that the return requirements are much higher on investments by entrepreneurs in venture startups than on those by well diversified investors. Using earlier results based on the capital asset pricing model (CAPM), it is shown that the difference can be as high as a factor three. Given these circumstances, the paper concludes that by more or less neglecting this difference, the Swedish dual income tax system heavily discriminates against entrepreneurship and growth of small firms. This should be a lesson to other countries considering the mitigation of effects of capital mobility by a dual income tax, i.e. by using standardized measures of the amount of capital income to be taxed at lower rates than labour.Tax Neutrality; Entrepreneurship; Dual Income Taxation; Cost of Capital

(Mis-)Understanding Education Externalities

This article reviews the current state of research on education externalities. It finds that much of the confusion regarding their magnitude results from conceptual misunderstandings about their nature. The concepts of 'education', 'teaching', and 'knowledge' need to be distinguished for a better understanding. Whereas pure teaching yields externalities on the primary and secondary level, only the generation of knowledge may produce the spillovers which are typically linked to the tertiary level. The accumulation of education itself does not have such an effect. Education is argued to be a private good with well defined property rights. Individuals may exploit those and provide the production sector with the efficient amount of human capital. Following this rationale, it is demonstrated that empirical studies, contrasting estimates of private and social returns to education, are unsuitable to substantiate the existence of externalities. As a consequence, subsidies to tertiary programs are called into question.Public education finance; Education expenditures; Human capital externalities; Property rights; Endogenous economic growth; Private and social return to education

(Mis-)Understanding Education Externalities

This article critically evaluates the current state of research on education externalities. It occurs that much of the confusion regarding their magnitude results from conceptual misunderstandings pertaining to their nature. The essay distinguishes the concepts education, teaching, and knowledge. Whereas pure teaching yields externalities on the primary and secondary level, only the generation of knowledge may produce the spillovers which are typically linked to the tertiary level. The accumulation of education itself does not have such an effect. Education is argued to be a private good with well defined property rights. Individuals may exploit those and provide the production sector with the efficient amount of human capital. Following this rationale, it is demonstrated that empirical studies, contrasting estimations of private and social returns to education, are unsuitable to substantiate the existence of externalities. As a consequence, subsidies to tertiary programs are called into question.Public education finance; Education expenditures; Human capital externalities; Property rights; Endogenous economic growth; Private return; Social return; Public goods

On the mathematical and foundational significance of the uncountable

We study the logical and computational properties of basic theorems of uncountable mathematics, including the Cousin and Lindel\"of lemma published in 1895 and 1903. Historically, these lemmas were among the first formulations of open-cover compactness and the Lindel\"of property, respectively. These notions are of great conceptual importance: the former is commonly viewed as a way of treating uncountable sets like e.g. $[0,1]$ as 'almost finite', while the latter allows one to treat uncountable sets like e.g. $\mathbb{R}$ as 'almost countable'. This reduction of the uncountable to the finite/countable turns out to have a considerable logical and computational cost: we show that the aforementioned lemmas, and many related theorems, are extremely hard to prove, while the associated sub-covers are extremely hard to compute. Indeed, in terms of the standard scale (based on comprehension axioms), a proof of these lemmas requires at least the full extent of second-order arithmetic, a system originating from Hilbert-Bernays' Grundlagen der Mathematik. This observation has far-reaching implications for the Grundlagen's spiritual successor, the program of Reverse Mathematics, and the associated G\"odel hierachy. We also show that the Cousin lemma is essential for the development of the gauge integral, a generalisation of the Lebesgue and improper Riemann integrals that also uniquely provides a direct formalisation of Feynman's path integral.Comment: 35 pages with one figure. The content of this version extends the published version in that Sections 3.3.4 and 3.4 below are new. Small corrections/additions have also been made to reflect new development

Pincherle's theorem in Reverse Mathematics and computability theory

We study the logical and computational properties of basic theorems of uncountable mathematics, in particular Pincherle's theorem, published in 1882. This theorem states that a locally bounded function is bounded on certain domains, i.e. one of the first 'local-to-global' principles. It is well-known that such principles in analysis are intimately connected to (open-cover) compactness, but we nonetheless exhibit fundamental differences between compactness and Pincherle's theorem. For instance, the main question of Reverse Mathematics, namely which set existence axioms are necessary to prove Pincherle's theorem, does not have an unique or unambiguous answer, in contrast to compactness. We establish similar differences for the computational properties of compactness and Pincherle's theorem. We establish the same differences for other local-to-global principles, even going back to Weierstrass. We also greatly sharpen the known computational power of compactness, for the most shared with Pincherle's theorem however. Finally, countable choice plays an important role in the previous, we therefore study this axiom together with the intimately related Lindel\"of lemma.Comment: 43 pages, one appendix, to appear in Annals of Pure and Applied Logi

Banach Spaces as Data Types

We introduce the operators "modified limit" and "accumulation" on a Banach space, and we use this to define what we mean by being internally computable over the space. We prove that any externally computable function from a computable metric space to a computable Banach space is internally computable. We motivate the need for internal concepts of computability by observing that the complexity of the set of finite sets of closed balls with a nonempty intersection is not uniformly hyperarithmetical, and thus that approximating an externally computable function is highly complex.Comment: 20 page

Vertical Integration, Raising Rivals’ Costs and Upstream Collusion

This paper analyzes the impact vertical integration has on upstream collusion when the price of the input is linear. As a first step, the paper derives the collusive equilibrium that requires the lowest discount factor in the infinitely repeated game when one firm is vertically integrated. It turns out this is the joint-profit maximum of the colluding firms. The discount factor needed to sustain this equilibrium is then shown to be unambiguously lower than the one needed for collusion in the separated industry. While the previous literature has found it difficult to reconcile raising-rivals-costs strategies following a vertical merger with equilibrium behavior in the static game, they are subgame perfect in the repeated game studied here.collusion, foreclosure, raising rivals’ costs, vertical integration