Equilibrium Price Research Articles

Review of wind futures by Gaussian and Lévy jump risks

Wind power futures, once a hedging tool in energy-related derivatives markets were discontinued due to low liquidity. However, recent increases in electricity price spreads have introduced new financing challenges for renewable energy projects in Europe, leading to heightened price risks and a renewed demand for such instruments. Also, the regulation on renewable electricity for renewable fuels of non-biological origin (RFNBO) — compliant hydrogen production poses a supply uncertainty risk to hydrogen developers. As both such project developers seek ways to mitigate these risks, this paper reviews the existing literature on modelling strategies for hedging wind power production risks. The review, based upon a structured literature review following vom Brocke et al. (2009), provides a comprehensive overview of arbitrage models incorporating seasonal elements and stochastic jump risks, as well as equilibrium pricing models. The variations and conclusions of these models are analyzed in the context of the altered market conditions in 2024. This analysis offers insights into the applicability of current models for pricing risk premia and identifies gaps under the evolving market realities.

Competitive Price Discrimination in Asymmetric Oligopoly

AbstractSegmentation seldom results in symmetric market demands in the real world. We allow the demands in the segmented markets to exhibit asymmetric market characteristics across firms as well as markets, and study the effects of asymmetries on equilibrium prices and profits under price discrimination. Using a standard linear demand system for differentiated oligopolies, we show that both best-response symmetry and best-response asymmetry can arise—dependent on the parameter values. Distinguishing between inter-market and inter-firm parameter asymmetries, we find that the former is consequential in determining equilibrium price rankings as well as best-response (a)symmetries, while the latter controls the overall competitiveness of a segmented market and thus influences equilibrium profits. We also find that profits tend to increase with strong asymmetries, which results in enhanced profits with discrimination under the environments of best-response asymmetry. This result highlights the importance of a market expansion effect that is often overlooked in the best-response asymmetry literature that employs a unit-demand Hotelling structure.

The Antitrust Regulation Harms Consumers

This article examines the effects of antitrust regulation. It is always assumed that this regulation has the purpose and function of protecting the interests of consumers. The other function of regulation is to protect competition, and it is assumed that this also indirectly protects consumer interests, because the dogma ‚the more competition the better‘ is accepted without reservation. We will demonstrate that antitrust regulation can harm consumer interests and will consistently prove the following theses in general: Antitrust regulation leads to a new, forced market equilibrium from which no participant has an interest in deviating. Regulation results in an equilibrium that is less profitable for consumers but more profitable for players that are not subject to antitrust regulation. In maximising their profit, players will offer quantities to the market such that the price on the market will increase and the total quantity offered will decrease. The efficiency of production decreases as a result of regulation. Although anti-monopoly regulation leads to a redistribution of market shares and profits, it generally leads to an increase in the equilibrium price for consumers and thus harms their interests.

Dual-Channel Supply Chain Pricing Coordination and Channel Selection with Reference Quality Effect Under Blockchain Traceability

This paper uses symmetry to develop a dual-channel supply chain network model, both with and without blockchain technology, ensuring that the solutions for pricing and channel selection are symmetric in both modes. And, in the model that does not employ blockchain technology, mechanisms are used to make the benefits of centralized and decentralized decision-making symmetrical. Consumers often have varied psychological expectations when choosing products. This paper considers the reference quality effect, where consumers establish a reference quality before making a purchase decision. However, the lack of transparency in product sales across supply chain channels can lead to a disparity between actual and expected product quality, impacting market demand and brand reputation. Therefore, manufacturers must balance increasing profits with meeting consumer demand for product traceability. This paper focuses on a dual-channel supply chain model involving manufacturers, retailers, and consumers. Firstly, the pricing issues and channel selections under centralized decision-making and a Stackelberg game with decentralized decision-making are comparatively analyzed in the mode of without adopting blockchain technology and adopt two-part pricing contract to coordinate. Secondly, the impact of blockchain technology on pricing is examined when it is adopted. Thirdly, pricing issues and channel selection strategies based on actual product quality in the dual-channel supply chain are compared under both models. Finally, numerical simulations validate the findings. The study found that: (1) the reference quality effect has a cross-positive influence on the equilibrium price and profit of a dual-channel supply chain; (2) the two-part pricing contract can mitigate the double-marginal effect and enhance profits; and (3) the adoption of the blockchain technology leads to increased equilibrium prices and total profits in the dual-channel supply chain compared to not using it. This paper offers a theoretical foundation for manufacturers and retailers to develop effective pricing and channel selection strategies in dual-channel operations to achieve higher returns.

A Big Data Discriminatory Pricing Strategy for Takeaway Platforms Considering Different Attributional Behaviors of Users

To address the big data discriminatory pricing problem of two takeout platforms with different attribution behaviors of platform users, the equilibrium pricing of the two platforms under different attribution behaviors is analyzed and compared based on the Hotelling model, and the effects of switching costs and network effects on pricing strategies are examined. The results show that: in the single-attribution-single-attribution model, the network effect affects the user scale of the platform, and then the pricing of the platform is affected by the switching costs and the network effect, and the pricing of the platform in the first stage affects the pricing decision in the second stage; in the single-attribution-partial-multi-attribution model, the pricing of the platform is affected not only by the switching cost and the network effect but also by the switching costs and the network effect of the user base. In the single-attribution-partial-multi-attribution model, platform pricing is not only affected by switching costs and network effects but also by users' base utility, and the platform's pricing in the first stage partially affects the pricing decision in the second stage.

China and G7 in the Current Context of the World Trading

The paper analyses trade between the most developed economies of the world. The analysis is based on the previously proposed model of international trade. This model of international trade is based on the theory of general economic equilibrium. The demand for goods in this model is built on the import of goods by each of the countries participating in the trade. The structure of supply of goods in this model is determined by the structure of exports of each country. It is proved that in such a model, given a certain structure of supply and demand, there exists a so-called ideal equilibrium state in which the trade balance of each country is zero. Under certain conditions on the structure of supply and demand, there is an equilibrium state in which each country have a strictly positive trade balance. Among the equilibrium states under a certain structure of supply and demand, there are some that differ from the ones described above. Such states are characterized by the fact that there is an inequitable distribution of income between the participants in the trade. Such states are called degenerate. In this paper, based on the previously proposed model of international trade, an analysis of the dynamics of international trade of 8 of the world's most developed economies is made. It is shown that trade between these countries was not in a state of economic equilibrium. The found relative equilibrium price vector turned out to be very degenerate, which indicates the unequal exchange of goods on the market of the 8 studied countries. An analysis of the dynamics of supply to the market of the world's most developed economies showed an increase in China's share. The same applies to the share of demand.

Expectations and Learning from Prices

Abstract We study mislearning from equilibrium prices, and contrast this with mislearning from exogenous fundamentals. We micro-found mislearning from prices with a psychologically founded theory of “Partial Equilibrium Thinking” (PET), where traders learn fundamental information from prices, but fail to realize others do so too. PET leads to over-reaction, and upward sloping demand curves, thus contributing to more inelastic markets. The degree of individual-level over-reaction and the extent of inelasticity vary with the composition of traders, and with the informativeness of new information. More generally, unlike mislearning from fundamentals, mislearning from prices (i) generates a two-way feedback between prices and beliefs that can provide an arbitrarily large amount of amplification and (ii) can rationalize both over-reaction and more inelastic markets. The two classes of biases are not mutually exclusive. Instead, they interact in very natural ways, and mislearning from prices can vastly amplify mislearning from fundamentals.

Insider trading at a random deadline with correlation between dynamic asset and stochastic liquidity

We propose a generalized continuous-time insider trading model, building upon the frameworks of Caldentey and Stacchetti (2010) and Collin-Dufresne and Fos (2016), with a correlation between the value of a risky asset following an Ornstein-Uhlenbeck-type process and the noise trading volume with volatility characterized by a general stochastic process. And a closed form of the market equilibrium is established, consisting of the insider's trading strategy and the market makers' pricing rule. It shows that at the equilibrium: (i) all of the insider's private information is released at the end of the transaction; (ii) market depth and market liquidity evolve as semi-martingales, respectively; and (iii) the equilibrium price is driven by a bridge process that solves an Ornstein-Uhlenbeck-type SDE. Numerical simulations show that as the correlation coefficient increases, the equilibrium price becomes more informative, leading to a decrease in both the trading intensity and the expected payoff for the insider.

Dynamic spatial price equilibrium, dynamic user equilibrium, and freight transportation in continuous time: A differential variational inequality perspective

In this paper we provide a statement of dynamic spatial price equilibrium (DSPE) in continuous time as a basis for modeling freight flows in a network economy. The model presented describes a spatial price equilibrium due to its reliance on the notion that freight movements occur in response to differences between the local and distant prices of goods for which there is excess demand; moreover, local and distant delivered prices are equated at equilibrium. We propose and analyze a differential variational inequality (DVI) associated with dynamic spatial price equilibrium to study the Nash-like aggregate game at the heart of DSPE using the calculus of variations and optimal control theory. Our formulation explicitly considers inventory and the time lag between shipping and demand fulfillment. We stress that such a time lag cannot be readily accommodated in a discrete-time formulation. We provide an in-depth analysis of the DVI's necessary conditions that reveals the dynamic user equilibrium nature of freight flows obtained from the DVI, alongside the role played by freight transport in maintaining equilibrium commodity prices and the delivered-price-equals-local-price property of spatial price equilibrium. By intent, our contribution is wholly theoretical in nature, focusing on a mathematical statement of the defining equations and inequalities for dynamic spatial price equilibrium (DSPE), while also showing there is an associated differential variational inequality (DVI), any solution of which is a DSPE. The model of spatial price equilibrium we present integrates the theory of spatial price equilibrium in a dynamic setting with the path delay operator notion used in the theory of dynamic user equilibrium. It should be noted that the path delay operator used herein is based on LWR theory and fully vetted in the published dynamic user equilibrium literature. This integration is new and constitutes a significant addition to the spatial price equilibrium and freight network equilibrium modeling literatures. Among other things, it points the way for researchers interested in dynamic traffic assignment to become involved in dynamic freight modeling using the technical knowledge they already possess. In particular, it suggests that algorithms developed for dynamic user equilibrium may be adapted to the study of urban freight modelled as a dynamic spatial price equilibrium. As such, our work provides direction for future DSPE algorithmic research and application. However, no computational experiments are reported herein; instead, the computing of dynamic spatial price equilibria is the subject of a separate manuscript.

Dynamics of a Price Adjustment Model with Distributed Delay

This paper deals with the stability and occurrence of Hopf bifurcation of a distributed delay differential cobweb model using the chain trick technique. This is a generalized form of the fixed delay cobweb model to which it is compared using the same parameter values. The results from the delay distribution showed that whenever less weight (γ=0.146) is put on past prices, the current equilibrium price is adjusted upwards while the reverse is observed when a higher weight (γ=0.186) is put on the previous price. It is also observed that if the initial price is set below/above the equilibrium price, the price adjustment either affects the consumers or benefits the suppliers. However, the fixed delay cobweb model does not display the consumers or suppliers benefits of the price dynamics in either direction. These are unique, underlying patterns in price dynamics discovered when using a distributed delay model compared to traditional fixed delay cobweb models. Furthermore, our model challenges the traditional cobweb model’s requirement for divergence, as it is based on the weight assigned to past prices rather than the relationship between the elasticities of supply and demand, which is the determining factor in the classical model. Based on these insights, we recommend that future price adjustment models incorporate distributed delays, as they reveal more intricate price dynamics and provide a more comprehensive understanding of market behavior than fixed delay models.

Higher order expectations, learning, and sentiment pricing dynamics

We present a dynamic asset pricing model combining individual investor sentiment, higher order expectations with learning. In the basic model, the forward-looking expectation of individual investors is distorted by individual sentiment and higher order expectations, so prices react more sluggishly to changes in fundamentals of the asset. We find that investor sentiment plays a significant role in the effect of higher order expectations on asset pricing. Investor sentiment not only makes the price tightly anchor to the initial price, but also increases the sentiment drift of the price. Higher order expectations exhibit inertia, therefore aggravating the anchor to the initial price. With the increase of the order, more and more investor sentiment is integrated into the prices, amplifying the bias of pubic signal relative to fundamentals. When individual sentiment investors learn valuable public information through price system in the long term, the information component of the equilibrium price increases, thus drawing the asset price back toward the rational expected value. The model could offer a partial explanation to the inertia and drift in the price path.

Globally solving a class of bilevel programs with spatial price equilibrium constraints

Globally solving a class of bilevel programs with spatial price equilibrium constraints

Direct Channel or Distribution Channel? The Role of Internet-Based Targeted Advertising With Vertical Differentiation in Distinct Channels

The selection of optimal channels is considered the critical marketing strategy for the firm. Effective vertical channel selection could induce distinct effects on the firm’s profit and social warfare by selecting a specific type of advertising. Otherwise, as the targeted population is quite different for the firm using mass or targeted advertising, this differentiation is becoming more significant for the population in the final marketing channel. Within the vertical differentiation model, the role of Internet-based targeted advertising on direct sales and distribution channels is studied. Compared to mass advertising, the firm using Internet-based targeted advertising in distinct channels can benefit the firm’s equilibrium profit by reducing advertising waste and enhancing the product’s price. The manufacturer might use Internet-based targeted advertising to control the total profit of the channel. Meanwhile, Internet-based targeted advertising might reduce the retailer’s and the entire channel’s profit. Otherwise, the targeting accuracy of the firm shows a bidirectional role in the firm’s equilibrium price in the direct channel. The firm’s equilibrium advertising level and profit might be reduced with the increasing targeting accuracy. The public department of the government needs to consider the potential risk of targeted advertising that might reduce the real benefits of distribution channels and harm the development of the market.

Pricing and bundling decisions for complementary products in an agency selling platform with risk‐averse suppliers and stochastic demand

AbstractThe exponential rise of e‐commerce has prompted many companies to utilize online platforms (e‐tailers) to sell their products. Agency selling is a prevalent policy among e‐tailers where suppliers determine retail prices of their products, and the e‐tailer charges an agency rate. This paper investigates the selling format and pricing problem of two risk‐averse suppliers who sell complementary products on the e‐tailer's platform. The e‐tailer can offer products separately and as a bundle. A game‐theoretic analysis reveals that the equilibrium prices of individual products and the bundle uniquely exist. The study explores that the e‐tailer does not always benefit from a higher agency rate. Furthermore, the agency rate can influence the superiority of mixed bundling over separate selling. Moreover, a mixed‐bundling strategy may hurt the supplier's profitability which is sensitive to marginal costs and the degree of risk aversion. Finally, when demand correlation increases, the supplier sets a lower price to mitigate the risk cost.

An equilibrium model of reinsurance pricing

Reinsurance is the insurance of insurance companies. We develop a dynamic equilibrium model for reinsurance pricing based on the supply and demand of reinsurance. In the model, an insurance company and a reinsurance company make decisions on the optimal reinsurance policy as a demander and a supplier of reinsurance, respectively. The reinsurance price is determined when the demand matches the supply. We incorporate the proportional reinsurance policy as well as investment opportunities into the framework. The high flexibility of this equilibrium pricing model paves a new way for studies of optimal reinsurance and reinsurance pricing.

Carbon market design and market sentiment

Concerns about systematic price distortions in the EU Emission Trading System (ETS) have risen in recent years. This paper shows how carbon-market design affects the impact of market sentiment, i.e. systematic deviations of price expectations from fundamentals by at least some market participants, on equilibrium prices. The Market Stability Reserve (MSR) of the EU ETS that adjusts supply based on past allowance banking undermines self-stabilization of the carbon market by discouraging rational intertemporal arbitrage. The MSR increases vulnerability of the EU ETS to market sentiments. Initially, the allowance price responds more to distorted expectations while the MSR (partially) prevents the distortion to spread across periods. Making the MSR more responsive to banking decisions increases the likelihood that distorted expectations turn out to be correct ex post. In contrast, a mechanism that adjust the cap based on the current allowance price does not face a trade off between stabilizing prices in the present and the future.

Existence of equilibrium in a dynamic supply chain game with vertical coordination, horizontal competition, and complementary goods

We consider supply chain competition and vertical coordination in a linear–quadratic differential game setting. In this setting, supply chains produce complementary goods and each of them includes a single manufacturer and a single retailer who coordinate their decisions through a revenue-sharing contract with a wholesale price and a fixed sales revenue share. We study a multiple leader-follower Stackelberg game where the manufacturers are the leaders and the retailers are the followers. Competition occurs at both levels of the supply chains. Retailers play Nash and compete in price; manufacturers also play Nash but they compete in choosing their production capacities by exploiting the equilibrium price decisions made by the retailers. We show that open-loop Nash equilibria exist when the manufacturers only receive a wholesale price (there are no longer exploiting the equilibrium price decision made by the retailers, however). When the manufacturers receive both a wholesale-price and a share of the retailers’ sales revenues, equilibria generally no longer exist. The non-existence of an equilibrium stems from the fact that the manufacturers’ instant profits are discontinuous functions of their production capacities. This discontinuity leads to a major technical difficulty in that one cannot apply standard optimal control approaches to study the equilibria of the dynamic game. Our results illustrate the possibility that competition between supply chains might not be sustainable when they sell complementary products and rely on a revenue-sharing agreement.

Predictive crypto-asset automated market maker architecture for decentralized finance using deep reinforcement learning

This study proposes a quote-driven predictive automated market maker (AMM) platform with on-chain custody and settlement functions, alongside off-chain predictive reinforcement learning capabilities, to improve the liquidity provision of real-world AMMs. The proposed architecture augments Uniswap V3, a cryptocurrency AMM protocol, by using a novel market equilibrium pricing to reduce divergence and slippage losses. Furthermore, the proposed architecture involves a predictive AMM capability, for which a deep hybrid long short-term memory (LSTM) and Q-learning reinforcement learning framework is used. It seeks to improve market efficiency through obtaining more accurate forecasts of liquidity concentration ranges, where liquidity starts moving to expected concentration ranges prior to asset price movement; thus, liquidity utilization is improved. The augmented protocol framework is expected to have practical real-world implications through (1) reducing divergence loss for liquidity providers; (2) reducing slippage for crypto-asset traders; and (3) improving capital efficiency for liquidity provision for the AMM protocol. The proposed architecture is empirically benchmarked against the well-established Uniswap V3 AMM architecture. The preliminary findings indicate that the novel AMM framework offers enhanced capital efficiency, reduced divergence loss, and diminished slippage, which could potentially address several of the challenges inherent to AMMs.

The ascending auction with flexible reporting

In the assignment problem where multiple heterogeneous indivisible items are assigned to unit-demand bidders, we introduce a novel ascending auction called the Flexible Reporting Ascending Auction to balance the privacy preservation and the speed of the ascending auction. Assuming bidders behave truthfully, it always results in a Walrasian equilibrium, along with the minimum Walrasian equilibrium price vector. If each bidder reports without a contradiction, bidders’ truthful behavior forms a Nash equilibrium. Our auction generalizes some well-known ascending auctions in literature, and we also provide simulation results to compare these auctions.

Optimizing Price Markup: The Impact of Power Purchase Agreements and Energy Production Uncertainty on the Economic Performance of Onshore and Offshore Wind Farms

Wind energy is rapidly expanding its capacity as part of the global energy transition. To ensure economic viability, wind energy projects increasingly rely on risk mitigation strategies. While Power Purchase Agreements (PPAs) manage spot market price variability, wind farms still face energy production uncertainty, directly impacting the quantity of energy generated. This study presents a comprehensive analysis of the price markup for onshore and offshore wind farms, considering energy production uncertainty under PPA and non-PPA scenarios. The study examines key metrics such as equilibrium prices, price markups, Internal Rate of Return (IRR), Net Present Value (NPV), and their correlation to energy production risk. Results demonstrate that PPAs significantly alter the relationship between price markup and variability. Offshore wind farms can potentially benefit more from PPAs compared to onshore, especially at lower levels of wind energy variability. However, as variability increases, the risk mitigation provided by PPAs diminishes, and both onshore and offshore wind farms may require higher price markups for financial viability. These findings highlight the necessity of carefully designed PPA structures and pricing strategies to ensure long-term competitiveness and sustainability of wind energy projects.