Market-clearing Conditions Research Articles

A Random-Supply Mean Field Game Price Model

We consider a market where a finite number of players trade an asset whose supply is a stochastic process. The price formation problem consists of finding a price process that ensures that when agents act optimally to minimize their trading costs, the market clears, and supply meets demand. This problem arises in market economies, including electricity generation from renewable sources in smart grids. Our model includes noise on the supply side, which is counterbalanced on the consumption side by storing energy or reducing the demand according to a dynamic price process. By solving a constrained minimization problem, we prove that the Lagrange multiplier corresponding to the market-clearing condition defines the solution of the price formation problem. For the linear-quadratic structure, we characterize the price process of a continuum population using optimal control techniques. We include numerical schemes for the price computation in the finite and infinite games, and we illustrate the model using real data.

A Mean Field Game Approach to Equilibrium Pricing with Market Clearing Condition

In this work, we study an equilibrium-based continuous asset pricing problem which seeks to form a price process endogenously by requiring it to balance the flow of sale-and-purchase orders in the exchange market, where a large number of agents $1\leq i\leq N$ are interacting through the market price. Adopting a mean field game approach, we find a special form of forward-backward stochastic differential equations of McKean--Vlasov type with common noise whose solution provides an approximate of the market price. We show the convergence of the net order flow to zero in the large $N$-limit and get the order of convergence in $N$ under some conditions.

Incorporating learning-by-doing into mixed complementarity equilibrium models

Market equilibrium models are often specified and solved as mixed complementarity problems (MCPs). These formulations combine the Karush–Kuhn–Tucker (KKT) optimality conditions of the optimization problems faced by multiple strategic players with market-clearing conditions, such that the solution to this system provides the Nash equilibrium prices and quantities. MCPs are widely applied to energy markets including those for electricity, oil, and natural gas. While researchers have made substantial progress on expanding the model features included in MCPs and on solving these problems, a limitation of existing MCPs is that they treat costs as exogenous input parameters. Therefore, MCPs have not been able to capture learning-by-doing (LBD), the empirically observed phenomenon whereby production costs tend to decline as a function of cumulative production experience. In this paper, we demonstrate the incorporation of LBD into a mixed complementarity equilibrium model. We consider two closely related, but nevertheless distinct, LBD formulations: one with discrete changes in cost from period to period, and another where cost declines continuously. Through theoretical analysis and numerical exploration, we establish the conditions under which these LBD formulations lead to convex optimization problems. Confirming convexity is important because it guarantees that their KKT conditions are sufficient for optimality. Then, we demonstrate the practical application of a mixed complementarity equilibrium model with LBD using the North American natural gas market as an example. When LBD is incorporated into the cost of liquefaction, North America exports more liquefied natural gas, which raises prices and reduces domestic consumption.

Incorporating Learning-by-Doing into Mixed Complementarity Equilibrium Models

Market equilibrium models are often specified and solved as mixed complementarity problems (MCPs). These formulations combine the Karush-Kuhn-Tucker (KKT) optimality conditions of the optimization problems faced by multiple strategic players with market-clearing conditions, such that the solution to this system provides the Nash equilibrium prices and quantities. MCPs are widely applied to energy markets including those for electricity, oil, and natural gas. While researchers have made substantial progress on expanding the model features included in MCPs and on solving these problems, a limitation of existing MCPs is that they treat costs as exogenous input parameters. Therefore, MCPs have not been able to capture learning-by-doing (LBD), the empirically observed phenomenon whereby production costs tend to decline as a function of cumulative production experience. In this paper, we demonstrate the incorporation of LBD into a mixed complementarity equilibrium model. We consider two closely related, but nevertheless distinct, LBD formulations: one with discrete changes in cost from period to period, and another where cost declines continuously. Through theoretical analysis and numerical exploration, we establish the conditions under which these LBD formulations lead to convex optimization problems. Confirming convexity is important because it guarantees that their KKT conditions are sufficient for optimality. Then, we demonstrate the practical application of a mixed complementarity equilibrium model with LBD using the North American natural gas market as an example. When LBD is incorporated into the cost of liquefaction, North America exports more liquefied natural gas, which raises prices and reduces domestic consumption.

Market Clearing Conditions and the CAPM Equation

Market Clearing Conditions and the CAPM Equation

A new approach to the existence and regularity of linear equilibrium in a noisy rational expectations economy

This paper gives a new approach to show the existence and regularity of linear equilibrium established by Lou ⓡ al. (2019) for a noisy rational expectations economy. Different from the existing method which essentially requires to find a fixed point of a system of nonlinear algebraic equations, the new approach is operated directly on an alternative form of market-clearing conditions. One main advantage of the new approach is that besides homogeneous-valuation economies, it can also handle the existence of equilibrium in economies with heterogeneous valuations where the existing method for dealing with homogeneous-valuation economies fails to work.

A Mean Field Game Approach to Equilibrium Pricing with Market Clearing Condition

In this work, we study an equilibrium-based continuous asset pricing problem which seeks to form a price process endogenously by requiring it to balance the flow of sales-and-purchase orders in the exchange market, where a large number of agents are interacting through the market price. Adopting a mean field game (MFG) approach, we find a special form of forward-backward stochastic differential equations of McKean-Vlasov type with common noise whose solution provides a good approximate of the market price. We show the convergence of the net order flow to zero in the large N-limit and get the order of convergence in N under some conditions. We also extend the model to a setup with multiple populations where the agents within each population share the same cost and coefficient functions but they can be different population by population.

Brokered versus dealer markets: Impact of proprietary trading with transaction fees

In this study, we consider a one-period financial market with a dealer/broker and an infinite number of investors. While the dealer who trades on his own account (with proprietary trading) simultaneously sets both the transaction fee and the asset price, the broker who brings investors' orders to the market (with no proprietary trading) sets only the transaction fee, given that the price is determined according to the market-clearing condition among investors. We analyze the impact of proprietary trading on the asset price, transaction fee, trading volume, and the welfare of investors. We find that the bid and ask prices set by the dealer who can engage in proprietary trading are more favorable to average investors. As a result, both the trading volume and the transaction fee increase, and social welfare improves.

Pricing in C2C Sharing Platforms

Sharing platforms such as Zilok.com enable the sharing of durable goods among consumers, and seek to maximize profits by charging transaction-based platform fees. We develop a model where consumers that are heterogeneous in their need to use a durable good decide whether to purchase and share (i.e., be a lender) or borrow (i.e., be a borrower), and a monopoly sharing platform decides on the platform fees. We find first that consumers with a greater need to use a durable good purchase and share, and that consumers with a lesser need borrow. Second, sharing platforms maximize profits only if the supply of a durable good matches demand – that is, the market must clear for platform fees to be profit maximizing. Third, the market-clearing condition requires that lender and borrower fees are classic strategic complements. Fourth, to maintain the market-clearing condition, sharing platforms have to increase their lender fee or decrease their borrower fee in response to increases in the sharing price, increases in usage capacity, and decreases in the purchase price of a durable good, and vice versa. These findings indicate that commonly applied one-sided pricing models in sharing platforms can be improved.

Pricing in C2C Sharing Platforms

Sharing platforms such as Zilok.com enable the sharing of durable goods among consumers, and seek to maximize profits by charging transaction-based platform fees. We develop a model where consumers that are heterogeneous in their need to use a durable good decide whether to purchase and share (i.e., be a lender) or borrow (i.e., be a borrower), and a monopoly sharing platform decides on the platform fees. We find first that consumers with a greater need to use a durable good purchase and share, and that consumers with a lesser need borrow. Second, sharing platforms maximize profits only if the supply of a durable good matches demand – that is, the market must clear for platform fees to be profit maximizing. Third, the market-clearing condition requires that lender and borrower fees are classic strategic complements. Fourth, to maintain the market-clearing condition, sharing platforms have to increase their lender fee or decrease their borrower fee in response to increases in the sharing price, increases in usage capacity, and decreases in the purchase price of a durable good, and vice versa. These findings indicate that commonly applied one-sided pricing models in sharing platforms can be improved.

Monopolistic Dealer versus Broker: The Impact of Proprietary Trading with Transaction Fees

In this study, we consider a one-period financial market with a monopolistic dealer/broker and an infinite number of investors. While the dealer who trades on his own account (with proprietary trading) simultaneously sets both the transaction fee and the asset price, the broker who brings investors' orders to the market (with no proprietary trading) sets only the transaction fee, given that the price is determined according to the market-clearing condition among investors. We analyze the impact of proprietary trading on the asset price, transaction fee, trading volume, and the welfare of investors. Results that proprietary trading increases both the trading volume and the transaction fee, and improves social welfare. Our study effectively demonstrates how proprietary trading affects market equilibrium and welfare of investors.

A Market-Clearing Model of the Uranium and Enrichment Industries

A market-clearing model of the uranium and enrichment industries through 2030 is presented. Built around thorough databases of primary and secondary uranium supplies as well as enrichment facilities, the model derives market-clearing conditions by locating the intersections between the annual supply-and-demand curves for uranium and enrichment services. Considering the effects of secondary supplies including highly enriched and natural uranium inventories along with depleted uranium enrichment, the model solves embedded optimization problems to account for trade-offs between uranium and enrichment requirements. The model can inform policy decisions tied to uranium inventory management and sale and market effects of purchase and disbursement from a uranium bank. This paper documents the methodologies behind the model, describes a stochastic implementation to propagate uncertainties, and contrasts its forecasts to static projections. Further, it is applied to an illustrative reference case featuring moderate (2.6%/yr) demand growth for reactor fuel. The model predicts near-level uranium prices with declining separative work unit prices and enrichment tails assays through the mid-2020s. This behavior is largely driven by the coming online of several new centrifuge enrichment plants and capacity expansions at others, which encourages more aggressive tails assays while suppressing uranium requirements.

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A Theory of Asset Prices Based on Heterogeneous Information

We propose a theory of asset prices that emphasizes heterogeneous information as the main element determining prices of different securities. Our main analytical innovation is in formulating a model of noisy information aggregation through asset prices, which is parsimonious and tractable, yet flexible in the specification of cash flow risks. We show that the noisy aggregation of heterogeneous investor beliefs drives a systematic wedge between the impact of fundamentals on an asset price, and the corresponding impact on cash flow expectations. The key intuition behind the wedge is that the identity of the marginal trader has to shift for different realization of the underlying shocks to satisfy the market-clearing condition. This identity shift amplifies the impact of price on the marginal trader's expectations. We derive tight characterization for both the conditional and the unconditional expected wedges. Our first main theorem shows how the sign of the expected wedge (that is, the difference between the expected price and the dividends) depends on the shape of the dividend payoff function and on the degree of informational frictions. Our second main theorem provides conditions under which the variability of prices exceeds the variability for realized dividends. We conclude with two applications of our theory. First, we highlight how heterogeneous information can lead to systematic departures from the Modigliani-Miller theorem. Second, in a dynamic extension of our model we provide conditions under which bubbles arise.

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On the currency effect to home bias puzzle

We propose an international asset allocation model to investigate whether home currency concern has impacts on home bias puzzle. Domestic investors choosing their international portfolios depend on home currency premium and thus will behave in home bias scenarios. Besides, the optimal portfolio suggests that it is isolated with nontradable and tradable consumptions. After employing the market-clearing condition, we price the equilibrium exchange rate and it tells that Interest Rate Parity (IRP) has more potential influence than Purchasing Power Parity (PPP) on exchange rate pricing in our model.

Single- and Multi-Commodity Models of Spatial Equilibrium in a Linear Programming Framework

This paper uses linear programming to construct a set of single- and multi-commodity models of spatial equilibrium within a competitive market. The models are based on grid linearization techniques which appear in the agricultural economics literature. First, a single-commodity spatial equilibrium model is constructed and is shown to satisfy both the pricing and quantity conditions required to obtain a spatial competitive market equilibrium solution. Next, the model is extended to include multiple commodities. This extension lakes two forms. The first form slates the objective function as the sum of consumers' and producers' surplus, and requires the assumption that the cross-price elasticity coefficients of supply and demand are symmetric. Finally, a model with asymmetric cross-price elasticities is staled as a primal-dual problem in which the constraints are the conditions necessary for a spatial competitive market equilibrium. Cet exposé présente un ensemble de modèles ďéquilibre spatial qui appartiennent à des denrées individuelles el multiples au marché de concours et qui sont construits dans le cadre de la programmation linéaire. La construction des modèles de programmation linéaire est fondée sur des techniques du quadrillage linéaire qui se montrent dans les revues ďéconomie rurale. Ďabord, un modèle ďéquilibre spatial ďune denrée individuelle est construit et on fait preuve que ce modèle remplit les conditions de prix el de quantité qui sont exigées pour qu ‘on peut se procurer des résultats. Ensuite le modèle de la denrée individuelle est étendu pour comprendre des denrées multiples. Ľauteur présente deu x formes du modèle étendu. Ďabord. un modèle est développé qui affirme la fonction économique sous la forme de la somme du surplus du producteur et celui du consommateur. Cette conformation exige que les élasticités prix-croisées de ľoffre et de la demande soient symétriques. Enfin, un modèle dont les élasticités sont asymétriques est exprimé en fonction ďun problème primal-dual, un problème oú les contraintes sont les conditions nécéssaires pour un équilibre spatial au marché de concours. Summary Beginning with Samuelson's work, mathematical programming methods to solve spatial competitive market equilibria problems have appeared extensively throughout the agricultural economics literature. The vast majority of these mathematical programming models have been developed in terms of a quadratic programming framework. Duloy and Norton have shown how a quadratic objective function can be approximated as a linear objective function through the use of grid linearization techniques. A set of spatial competitive market equilibrium models for single and multiple commodities in a linear programming framework were presented in this paper. The linearization techniques used by Duloy and Norton provided a basis for constructing the linear programming whole. First, a single-commodity spatial equilibrium model was constructed and shown to satisfy both the pricing and quantity conditions required for a competitive spatial equilibrium solution to be obtained. The single-commodity model was extended to include multiple commodities. This extension took two forms. First, if the model objective function is stated in the form of Samuelson's “net social payoff‘ function, the commodity demand and supply functions must be assumed to be integrable. This assumption is consistent with the neoclassical assumptions of the theory of production, but creates a number of problems on the demand side. It is implied here that the demand cross-price effects will be equal across all commodity prices. However, this will be true only for very restrictive cases. The second form of the multiple commodity model was designed to circumvent the integrability assumption. This was accomplished by specifying a primal-dual problem in which the constraints are the necessary conditions for a spatial competitive market equilibria. Implementation of the models set out in this paper require a set of numerical initial conditions. However, these do not necessarily need to be market-clearing conditions.

Pension Freezes and Household Saving over the Life Cycle

Defined benefit (DB) pension freezes in large healthy firms such as Verizon and IBM, as well as terminations of plans in the struggling steel and airline industries, highlight the fact that these traditional pensions cannot be viewed as risk-free promises from the employee's perspective. Indeed, the current turmoil in financial markets and difficult economic outlook for many firms suggest that many more pension plans could be frozen soon. In this preliminary paper we develop an empirical dynamic programming framework to investigate household saving decisions in a model economy with freeze-prone DB pensions. The model incorporates important sources of uncertainty facing households, including asset returns, employment, income, and mortality, as well as pension freezes. Applying a compensating variation measure of welfare, we find that pension freezes reduce welfare by a maximum of about $6,000 for individuals with a high school degree and about $2,000 for individuals with a college degree. We close by highlighting a few important issues that are missing from our preliminary analysis, including a labor supply decision and the effects of market-clearing conditions in the labor market. We hope to address these issues in future work.

A complementarity model for the European natural gas market

In this paper, we present a detailed and comprehensive complementarity model for computing market equilibrium values in the European natural gas system. Market players include producers and their marketing arms which we call “traders”, pipeline and storage operators, marketers, LNG liquefiers, regasifiers, tankers, and three end-use consumption sectors. The economic behavior of producers, traders, pipeline and storage operators, liquefiers and regasifiers is modeled via optimization problems whose Karush–Kuhn–Tucker (KKT) optimality conditions in combination with market-clearing conditions form the complementarity system. The LNG tankers, marketers and consumption sectors are modeled implicitly via appropriate cost functions, aggregate demand curves, and ex post calculations, respectively. The model is run on several case studies that highlight its capabilities, including a simulation of a disruption of Russian supplies via Ukraine.

A complementarity model for solving stochastic natural gas market equilibria

This paper presents a stochastic equilibrium model for deregulated natural gas markets. Each market participant (pipeline operators, producers, etc.) solves a stochastic optimization problem whose optimality conditions, when combined with market-clearing conditions give rise to a certain mixed complementarity problem (MiCP). The stochastic aspects are depicted by a recourse problem for each player in which the first-stage decisions relate to long-term contracts and the second-stage decisions relate to spot market activities for three seasons. Besides showing that such a market model is an instance of a MiCP, we provide theoretical results concerning long-term and spot market prices and solve the resulting MiCP for a small yet representative market. We also note an interesting observation for the value of the stochastic solution for non-optimization problems.