Competitive equilibrium between carbon loss and sequestration driven by erosion: Stratified responses of microbial metabolism and mineral protection

In today’s fast-paced digital world, consumers demand instant access to online content and are intolerant of delays, making website speed a key competitive advantage in attracting web traffic. Google’s Speed Update and Core Web Vitals have further emphasized the significance of website speed in web traffic competition. This study examines how firms strategically compete for web traffic by managing website speed, focusing specifically on two distinct strategies: response-based and capacity-based. Under response-based competition, firms first set their desired website speed (or equivalently, website response time), subsequently determining the necessary website capacity. In contrast, in capacity-based competition, firms initially select the website capacity level, which in turn determines the website response time. We analyze a duopoly scenario in which two firms compete for web traffic. Although website speed and capacity are functionally related, surprisingly, firms sometimes compete more aggressively under response-based competition. Interestingly, the aggression of response-based competition can sometimes increase firms’ profits. We also show that when firms freely choose the decision process, firms sometimes engage in a mode of competition in equilibrium, which yields a lower profit for the capacity provider (e.g., computing capacity provider) than the alternative mode. We further show how the cloud provider can increase profit by strategically inducing firms to engage in a preferred mode of competition. This is achieved by lowering the unit price of renting capacity related to that mode of competition. This strategic price reduction can lead to faster websites for consumers, an increase in the provider’s revenue, and consequently an increase in the cloud provider’s profit under a cost-efficiency condition. The profit of firms can sometimes increase too, implying a win-win-win for all the parties, namely, firms, consumers, and the provider.

Approximating Competitive Equilibrium by Nash Welfare

The vacuum impedance Z₀ = √(μ₀/ε₀) ≈ 376.73 Ω and the speed of light c = 1/√(μ₀ε₀) are two complementary invariants derived from the same pair of electromagnetic constants {μ₀, ε₀}: one encodes dynamics, the other kinematics. This paper shows that an identical structural duality arises within the Ze framework (Tkemaladze, 2026a), where a binary event stream is partitioned into N_T T-events and N_S S-events. We define Ze permittivity ε_Ze = N_T/T = 1 − v (temporal accumulation, analogous to ε₀) and Ze permeability μ_Ze = N_S/T = v (spatial flow, analogous to μ₀). Two invariants follow: Ze impedance Z_Ze = √(μ_Ze/ε_Ze) = √(v/(1−v)) and Ze speed c_Ze = τ/T = √(1−v²). We show that Z_Ze is universal: it is independent of stream type (i.i.d. Bernoulli, Markov, deterministic) when v is held fixed. The Ze generation map v₁ → v₂ = 2(1−v₁)/(2−v₁)² (Tkemaladze, 2026b) translates into an impedance map Z₁ → Z₂, with a unique stable fixed point Z* = 0.9161 (corresponding to v* = 0.4563). The matching condition Z_Ze = 1 (v = 0.5) coincides exactly with the Nash equilibrium of Ze competition (Tkemaladze, 2026c): the maximum-entropy state. The paper proposes five falsifiable predictions connecting Ze impedance to measurable properties of causal event streams.

The author of this article uses demand-and-supply diagrams to study a production economy with a numéraire, providing elementary proofs to fundamental economic results. He demonstrates that the competitive outcome lies within the core of the economy’s market game and illustrates core convergence to the competitive equilibrium when the market thickens. He further explains the impossibility of efficient trade under private preferences and incentive constraints. His graphical approach makes economic theory more accessible and unified for learning and teaching.

We consider a class of N-player games and mean-field games of singular controls with ergodic performance criterion, providing a benchmark case for irreversible investment games featuring mean-field interaction and strategic complementarities. The state of each player follows a geometric Brownian motion controlled additively through a nondecreasing process, whereas agents seek to maximize a long-term average reward functional with a power-type instantaneous profit under strategic complementarity. We explore three different notions of optimality, which in the mean-field limit, correspond to the mean-field control solution, mean-field coarse correlated equilibria, and mean-field Nash equilibria. We explicitly compute equilibria in the three cases and compare them numerically in terms of yielded payoffs and existence conditions. Finally, we show that the mean-field control and mean-field equilibria can approximate the cooperative and competitive equilibria, respectively, in the corresponding N-player game when N is sufficiently large. Our analysis of the mean-field control problem features a novel Lagrange multiplier approach, which proves crucial in establishing the approximation result, whereas the treatment of mean-field coarse correlated equilibria necessitates a new, specifically tailored definition for the stationary setting. Funding: The authors acknowledge financial support from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) [Project-ID 317210226– SFB 1283]. F. Cannerozzi acknowledges financial support from the European Union—NextGenerationEU [Grant NRRP-CUP G53D23006840001].

Abstract With the rise of emerging risks, model uncertainty poses a fundamental challenge in the insurance industry, making robust pricing a first‐order question. This paper investigates how insurers' robustness preferences shape competitive equilibrium in a dynamic insurance market. Insurers optimize their underwriting and liquidity management strategies to maximize shareholder value, leading to equilibrium outcomes that can be analytically derived and numerically solved. Compared to a benchmark without model uncertainty, robust insurance pricing results in significantly higher premiums and equity valuations. Notably, our model yields three novel insights: (1) The minimum, maximum, and admissible range of aggregate capacity all expand, indicating that insurers' liquidity management becomes more conservative. (2) The expected length of the underwriting cycle increases substantially, far exceeding the range commonly reported in earlier empirical studies. (3) While the capacity process remains ergodic in the long run, the stationary density becomes more concentrated in low‐capacity states, implying that liquidity‐constrained insurers require longer to recover. Together, these findings provide a potential explanation for recent skepticism regarding the empirical evidence of underwriting cycles, suggesting that such cycles may indeed exist but are considerably longer than previously assumed.

The TWC Economy as a Dissipative Structure: Landauer, Prigogine, and the Thermodynamics of Endogenous Growth

This paper provides a synthetic survey of applications of differential, algebraic, and general topology in mathematical economics, with particular emphasis on empirically verifiable results. We argue that topological machinery is intrinsically necessary on three levels. First, Smale’s proof of Walrasian equilibrium existence without convexity assumptions, and the local uniqueness result for generic economies, require Sard’s theorem and the preimage theorem; the classical Arrow–Debreu existence proof uses fixed-point methods (Kakutani) but not the full differential apparatus. Second, identifiability of structural models — including optimal-transport labour market models, partially identified models, and nonparametric IV regression — is equivalent to specific topological properties of the parameter space (connectedness, contractibility, vanishing of higher homotopy groups). Third, applying persistent homology to financial time series yields leading crisis indicators: the persistence landscape norm of H₁ barcodes rises 3–8 weeks before market crashes. Together, these results trace a path from Brouwer’s theorem and Chichilnisky–Heal homotopy groups through the outer Hausdorff metric in location theory to the topology of interbank liability networks in systemic risk analysis.

The article explores the transformation of the classical model of market equilibrium under the influence of structural shifts in the modern economy. It argues that the classical model of competitive equilibrium, based on the assumptions of rationality and flexibility of prices, loses its explanatory power in the context of technological heterogeneity, structural shifts, and changes in the nature of economic behavior. Based on current data from international research and market analytics, the article examines three key aspects: the labor market (shift in the supply curve), the impact of AI on industry equilibria, and the phenomenon of “K-shaped” divergence. The analysis showed that the current equilibrium is characterized by a bifurcation, where macroeconomic stability coexists with microeconomic fragmentation.

ABSTRACT Valence state engineering has emerged as a powerful strategy to optimize catalytic performance by modulating the electronic structure of metal active sites. However, the valence state regulation in high‐entropy compounds (HECs) remains elusive due to their complex multi‐element components and electronic interactions. Here, the valence states of different metals in two‐dimensional (2D) high entropy oxide (HEO) (FeNiMoRuV)O 2− x are precisely modulated through controlled pyrolysis of corresponding 2D high entropy hydroxide (HEHO) (FeNiMoRuV)(OH) 2 under varying temperatures. Temperature‐controlled pyrolysis selectively reduces the oxidation state of Ru, while simultaneously increasing the valence state of other constituent metals (Fe, Ni, Mo, and V), suggesting a competitive redox equilibrium. Notably, these low‐valence Ru sites with oxygen vacancy in 2D HEO significantly reduce Ru–O bond energy and promote the generation of O–*O intermediates, thereby enabling oxygen evolution with a lattice oxygen mediated‐oxygen vacancy site mechanism. 2D HEO with low‐valence Ru exhibits superior electrolytic water performance (HER/OER) compared to HEHO and other HEO with high‐valence Ru, achieving a current density of 1000 mA cm −2 at 1.923 V, which exceeds the commercial Pt/C||RuO 2 system. Therefore, this study reveals the valence state regulatory mechanism of HECs and provides a solid hammer for the catalytic mechanism of valence state engineering.

A dual-site extension is proposed to the previously derived ideal adsorbed solution theory analytical solution for two coadsorbing single-site BET isotherms. The proposed dual-site BET (DSBET) isotherm model is described in detail, along with multiple possible simplifications for real systems, including modeling humid carbon capture (CO2/H2O mixtures) using amine-functionalized chemisorbents and activated alumina. These simplifications allow nonideal binary equilibrium predictions of single-site Langmuir or dual-site Langmuir isotherm models (type-1 isotherms) coadsorbing with type-2 or type-3 isotherms as either a single-site BET or a dual-site BET isotherm model. The GAB (Guggenheim-Anderson-de Boer) isotherm could also be utilized in the proposed model via a change of variables. Many of the same advantages seen by the extended dual-site Langmuir isotherm are found for the dual-site BET isotherm. These include at least two distinct multicomponent equilibrium predictions per binary gas pairing and the ability to predict nonideal coadsorption. The choice of nonlinearity parameters (and their corresponding adsorption-site) is discussed along with their implications in coadsorption equilibrium prediction. A generalized single-site and dual-site BET isotherm model was also derived to extend multicomponent loading predictions to any number of components in a coadsorbing mixture. The DSBET isotherm model is able to predict nonideal adsorption equilibrium for both competitive and cooperative systems. This is demonstrated using hypothetical binary mixtures of DSBET isotherms and their IAST solutions, as well as loading deviation plots quantifying their difference from the unary DSBET prediction. Multiple DSBET isotherm model examples are shown for both hypothetical and experimental systems. Experimental examples include CO2/H2O mixtures on an amine-functionalized polymer (Lewatit VP OC 1065) and F-200 activated alumina.

We investigate the optimal time-consistent use of foreign exchange interventions (FXI) in a small open economy model driven by endowment and portfolio flow shocks, with endogenous FX market depth and a lower bound constraint on FX reserves. In a competitive equilibrium, large capital flows increase conditional exchange rate volatility and make FX markets more shallow. Unlike in the unconstrained case, the central bank's optimal interventions are not solely targeted at offsetting inefficient fluctuations in the UIP premium but also incorporate a forward-looking element due to the risk of depleting reserves. We show that this environment engenders optimal time-consistent FXI policy that is state-dependent. FX sales are more effective than FX purchases, and the policy may respond less or more than one-for-one to capital outflows, depending on their size and the economy's net foreign asset position. Adopting the policy delivers sizable welfare gains, significantly exceeding those from a simple rule directed at stabilizing current capital flows, but only if the initial level of FX reserves is sufficiently far from its effective lower bound.

• This paper examines constrained optimal carbon pricing in a general equilibrium model with incomplete asset markets. A carbon policy consists of state-dependent carbon taxes and a sharing rule for tax revenue recycling. • The social cost of carbon (SCC) is defined as the present value of the future marginal costs of additional CO2 emissions, discounted at (personalized) prices. • For the case of complete markets, we state simple, sufficient conditions that ensure that setting carbon taxes equal to the SCC results in a Pareto-efficient competitive equilibrium. When markets are incomplete, constrained Pareto-efficient carbon taxes generically differ from the SCC. • To examine the potential quantitative importance of these differences, we consideran Aiyagari-style model with a climate change externality. We prove that the deviations of constrained optimal carbon taxes from the SCC can be arbitrarily large. This paper examines constrained optimal carbon pricing in a general equilibrium model with incomplete asset markets. A carbon policy consists of state-dependent carbon taxes and a sharing rule for tax revenue recycling. The social cost of carbon (SCC) is defined as the present value of the future marginal costs of additional CO2 emissions, discounted at (personalized) prices. For the case of complete markets, we state simple, sufficient conditions that ensure that setting carbon taxes equal to the SCC results in a Pareto-efficient competitive equilibrium. When markets are incomplete, constrained Pareto-efficient carbon taxes generically differ from the SCC. To examine the potential quantitative importance of these differences, we consider an Aiyagari [1994]-style model with a climate change externality. We prove that (i) the SCC cannot be estimated from aggregate damage functions and market prices alone, and (ii) the deviations of constrained optimal carbon taxes from the SCC can be arbitrarily large.

Iron overload diseases (IOD) are harmful conditions that may lead to a significant decrease in quality of life. The only three IOD approved chelators have significant adverse effects that hinder therapeutic adherence. The search for new chelators may benefit from drug repositioning (DR), a strategy that aims to identify new applications for approved drugs. Antiresorptives (AR) are drugs that inhibit bone resorption. Here, the iron binding and antioxidant effects of four bisphosphonates (etidronate, alendronate, tiludronate, and zoledronate) and strontium ranelate AR were studied in buffer and cell models, in order to verify their potential as alternative treatments of IOD in the absence of bone disease. Competition equilibrium tests between the bisphosphonates and ferric calcein or ferric transferrin showed a moderate ability to scavenge iron. Bisphosphonates showed antioxidant activity against iron-induced reactive species generation in the presence of ascorbate. Etidronate and tiludronate helped to prevent cell death by iron-dependent oxidative stress. Although measurable, the effect of physiological levels of calcium did not prevent the desired chelating and antioxidant effects of the bisphosphonates. Our results show that etidronate and tiludronate have valuable physicochemical properties that could be employed in a DR strategy for the treatment of IOD.

By allowing individuals to use goods they do not produce, economic exchange is recognized as driving the wide diversity of economic activities seen in human societies. Since productivity also depends on innate abilities, we ask whether economic exchange could have influenced human evolution and promoted adaptive genetic diversity. We model a system where individuals produce and exchange goods under a Walrasian equilibrium, with abilities determined by an evolving quantitative genetic trait. We then analyse how exchange shapes the evolutionary pressures on this trait. Our analysis demonstrates that exchange consistently promotes negative frequency-dependent selection, which favours the maintenance of genetic diversity. Exchange also generates stable long-term adaptive polymorphism when the production of goods requires different abilities. Importantly, we establish that the mode of exchange matters: markets, where individuals can switch trading partners, promote genetic diversity under broader conditions than when exchange occurs in isolated pairs. Finally, we show that genetic diversity and economic specialization can facilitate the emergence of the other under a wider range of conditions. Our findings suggest that economic exchanges play a crucial role in fostering biological diversity and offer insights into how a culturally determined mode of organization may have shaped human evolution.

The paper deals with the practical aspects of the use of crown jewel obligations and the crown jewel defense in ongoing mergers and acquisitions, with the aim of maintaining equilibrium and a healthy competitive environment, protecting consumers, and preventing monopolistic practices or other types of anti-competitive behavior. It explains that a crown jewel encompasses the most strategically important and valuable assets of an economic entity—assets that play a crucial role in shaping market positioning and significantly impact the profitability of an incumbent firm’s business. While the study devotes significant attention to European Commission and the USA mergers and acquisitions regulations and the role of crown jewel obligation in this transections, the practical steps of the targeted companies to curb and avoid hostile takeover is described. Article provides the European Commission decisions and U.S. Department of Justice rulings concerning major corporate mergers, particularly those involving crown jewel obligations and hostile takeover litigation. Key words: Crown Jewel Obligation, Crown Jewel Defense, Merger, Acquisition, Divestiture, Hostile Takeover, Up-Front Buyer, Strategic Assets, Stakeholders, Stocks, Board of Directors.

In A Mathematical Formulation of the Ricardian System, Pasinetti (1960) lays out the foundations of what has been dubbed the canonical classical model. He proves the model to be logically consistent and determinate in all of its macro-economic features, and derives the equilibrium values of all key variables independently of demand conditions. The model thus provides macroeconomic foundations to the classical theory of distribution. This paper examines the decentralized, competitive mechanism underlying the macroeconomic outcomes. First, we model a classical economy with capitalists, workers, and landlords and define the notion of a Classical Competitive Equilibrium (CCE). A unique CCE exists in a large class of convex classical economies, and the resulting income distribution coincides with that of Pasinetti’s canonical classical model. Second, we use an agent-based model in order to examine more explicitly the decentralized competitive mechanisms at play in the classical economy. We show that a simple competitive interaction between boundedly rational agents with localized knowledge generates classical gravitational dynamics with the key distributive variables oscillating around their equilibrium values.

Abstract We introduce markup equilibrium—an extension of Walrasian equilibrium in which consumers pay a fixed percentage markup over producer prices. In quasilinear markets, markup equilibria exist despite nonconvexities. They are resource-feasible and envy-free, incur no budget deficit, and require little more communication and computation than ordinary Walrasian equilibrium. The associated markup mechanism is asymptotically incentive-compatible. We also introduce a Bound-Form First Welfare Theorem, which states that for any feasible allocation, the welfare loss compared to the first-best is bounded, using any price vector, by the sum of the resulting (i) budget surplus and (ii) rationing losses suffered by the participants. Using producer prices, this bound implies that any markup equilibrium with a small markup and few unallocated goods is nearly efficient.

Abstract This study analyzes exchange economies in which all players have the same utility function and all goods except money are indivisible. In this context, a necessary and sufficient condition for the existence of a competitive equilibrium is obtained based on the fact that certain associated cooperative games have a non-empty core. Based on this result, we find two conditions in the cardinal case, each of which is equivalent to the existence of an equilibrium in an economy. By introducing the $$\alpha $$ -concavity property, we obtain another sufficient (although not necessary) condition for the existence of an equilibrium. The study also proposes an algorithm to determine whether a cardinal economy has an equilibrium and, if so, to obtain the equilibrium price.