Equilibrium Investment Research Articles

Greenness transformation, environmental awareness, and green product policy in spatial markets

AbstractThis paper examines the transformation towards greenness, focusing on consumer environmental awareness within a circular city model characterized by elastic demand. Our findings reveal that the equilibrium investment in environmentally friendly initiatives falls short of the socially optimal level for a given market structure. The degree of excess or insufficient market entry is influenced by the level of environmental awareness. Further, we delve into the role of environmental delegation, demonstrating that heightened environmental awareness can mitigate underinvestment in the environment and reduce inefficiency in market entry. From a policy perspective, we expand our analysis to include the “Green Product Policy.” This extension aims to explore the impact of such a policy on environmental awareness and assess the applicability of the “excess entry theorem” in this context.

Equilibrium multi-period investment strategy for a DC pension plan with incomplete information: Hidden Markov model

In the financial market with both observable and unobservable market states, this article explores the equilibrium investment strategy for a DC pension plan with the mean-variance criterion in a discrete-time setting. The dynamics of the partially observed market state are described by a discrete-time finite-state hidden Markov chain. There is a riskless asset and a risky asset in the financial market, where the return rate of the risky asset depends both on the observable and unobservable market states. Meanwhile, the stochastic salary process is also modulated by the observable and unobservable market states. Under the framework of non cooperative game, we first define the equilibrium investment strategy for the multi-period mean-variance DC pension plan. By adopting the sufficient statistics method, the investment problem for the mean-variance DC pension plan with incomplete information is transformed into the one with complete information. The closed-form equilibrium investment strategy is derived by solving the extended Bellman equation. Finally, numerical results show that the incomplete information has significant impacts on the equilibrium investment strategy and the equilibrium efficient frontier. Neglecting the reality of incomplete information in the financial market will reduce the investment benefit of the DC pension plan. The longer the investment horizon, the greater the investment benefit loss for the DC pension plan.

Robust asset-liability management games for n players under multivariate stochastic covariance models

This paper investigates a non-zero-sum stochastic differential game among n competitive CARA asset-liability managers, who are concerned about the potential model ambiguity and aim to seek the robust investment strategies. The ambiguity-averse managers are subject to uncontrollable and idiosyncratic random liabilities driven by generalized drifted Brownian motions and have access to an incomplete financial market consisting of a risk-free asset, a market index and a stock under a multivariate stochastic covariance model. The market dynamics permit not only stochastic correlation between the risky assets but also path-dependent and time-varying risk premium and volatility, depending on two affine-diffusion factor processes. The objective of each manager is to maximize the expected exponential utility of his terminal surplus relative to the average among his competitors under the worst-case scenario of the alternative measures. We manage to solve this robust non-Markovian stochastic differential game by using a backward stochastic differential equation approach. Explicit expressions for the robust Nash equilibrium investment policies, the density generator processes under the well-defined worst-case probability measures and the corresponding value functions are derived. Conditions for the admissibility of the robust equilibrium strategies are provided. Finally, we perform some numerical examples to illustrate the influence of model parameters on the equilibrium investment strategies and draw some economic interpretations from these results.

Dynamic competition and market structure for platform‐based products: roles of product quality and indirect network effect

AbstractWe consider dynamic competition between two platform‐based products that exhibit two‐sided network effects, such as game consoles and intelligent hardware. Firms compete in terms of pricing and Research & Development investment, which are driven by consumers’ marginal utility of quality, indirect network effects, and the differentiation between consumers’ preference for the products. We find that with a small marginal utility of consumers, the equilibrium investment of one firm increases first and decreases later with its product quality but is not obviously sensitive to the rival's. In contrast, with a large marginal utility, if the quality of the two products is comparable, the equilibrium investment increases substantially. However, if one product pulls ahead, the other firm will stop investing. Besides, we find that when the intrinsic value and the network value coexist, the market becomes more concentrated, even when consumers’ marginal utility is small. The market concentration increases in both the consumers’ marginal utility and the indirect network effect but decreases in the degree of differentiation between consumers’ preference toward the two products. Interestingly, we show that differentiation plays an important role in shaping the market structure only when the network effect is weak. Counterintuitively, we show that the differentiation corrodes firms’ total profits, which demonstrates that less fierce competition resulting from low substitution leads to a lower profit. Additionally, we reveal that dynamic interaction between quality and installed base motivates firms to improve from a low quality, the first‐mover advantage does not ensure market dominance.

Nash equilibria for relative investors with (non)linear price impact

We consider the strategic interaction of n investors who are able to influence a stock price process and at the same time measure their utilities relative to the other investors. Our main aim is to find Nash equilibrium investment strategies in this setting in a financial market driven by a Brownian motion and investigate the influence the price impact has on the equilibrium. We consider both CRRA and CARA utility functions. Our findings show that the problem is well-posed as long as the price impact is at most linear. Moreover, numerical results reveal that the investors behave very aggressively when the price impact is close to a critical parameter.

Equilibrium investment–reinsurance strategy under information asymmetry and random horizon

This paper investigates an investment–reinsurance problem incorporating information asymmetry and random horizon. At the beginning of the transaction, the insurer owns some inside information related to the risky asset price and the insurance claims. Simultaneously, the return rate and the driving noise for the risky asset process cannot be observed directly. Moreover, the trading horizon is random due to some exogenous factors. Under the dynamic mean–variance criterion, the equilibrium feedback strategy and the associated equilibrium value function are derived in closed form by solving the extended Hamilton–Jacobi–Bellman (HJB) equations. Several special cases are further discussed. Finally, numerical simulations are conducted to analyze the influences of some important parameters on the equilibrium strategy and the efficient frontier.

Wind power investment equilibrium guided by reliability option considering electricity-carbon emission trading-trading green certificate joint market

Reliability option (RO) is a capacity adequacy mechanism and it is coupled with electricity market, carbon emission trading market and trading green certificate market. This article explores the impact of implementing the RO mechanism in a multi-market trading context on wind power investment equilibrium and user side cost, under different demand elasticities and the degrees of competition on the generation side. We introduce a novel multi-agent deep reinforcement learning algorithm, leveraging the twin delayed deep deterministic policy gradient, aimed at obtaining equilibrium. Initially, we propose a method for calculating the comprehensive income of power producers, integrating risk preferences using Conditional Value at Risk. Subsequently, we employ a decentralized partially observable Markov decision process to emulate multi-agent investment patterns. During each exploration phase, wind power output scenarios are meticulously crafted via Latin hypercube sampling. Evaluations of our algorithm underscore its superior convergence and discerning rationality. Numerical simulations reveal that in areas with high demand elasticity, the RO mechanism primarily reduces user side risk without much impact on equilibrium. Yet, in regions with low demand elasticity, its role in influencing wind power investment and user side risk hinges on the degree of market competition and the designated wind power RO sales ratio.

A non-zero-sum investment and reinsurance game between two mean–variance insurers with dynamic CVaR constraints

This paper is devoted to investigating a non-zero-sum game between two competing insurers. The insurers can diversify their insurance risks by purchasing proportional reinsurance and investing their collected premiums into a financial market composed of one risk-free asset and one stock. The reinsurance premiums charged by the reinsurer follow the generalized mean–variance premium principle. Moreover, the dynamic CVaR constraints are incorporated in the game problem to control risks. With the dynamic mean–variance objective, we introduce two forward deterministic auxiliary processes to represent the expectations of the insurers’ wealth processes and transform the original time inconsistent game problem into a standard time consistent game problem with two state variables for each insurer. By adopting the dynamic programming principle and the Lagrange duality method, we derive the Nash equilibrium investment–reinsurance strategies for the two insurers. Finally, the effects of several important model parameters on the optimal policies are analyzed by numerical examples.

Optimal investments of port authorities facing ambiguity on uncertain market demands

This study explores how two competing port authorities facing ambiguity about uncertain market demands determine their optimal investments. Using the Choquet expected utility function, the port authorities are categorized into four types: ambiguity-averse and risk-averse, ambiguity-neutral and risk-averse, ambiguity-averse and risk-neutral, and ambiguity-neutral and risk-neutral. Each type can reach equilibria through no-port investing, only-one-port investing, or both-port investing. Our findings indicate that considering ambiguity does not alter the types of equilibrium investments for port authorities, regardless of their risk aversion or neutrality. However, the emergence of equilibrium investments for each type is affected by the consideration of ambiguity. Specifically, when risk-neutral port authorities transition from ambiguity-neutral to ambiguity-averse, they will invest more (less) if they are optimistic (pessimistic) enough. Nevertheless, this relationship may not apply to risk-averse port authorities, as the magnitudes of the first to the fourth moments of probability distributions, which maximize and minimize the expected utility of ports, also impact their investment decisions. Importantly, our results hold true even when ports exhibit ambiguity-loving characteristics, possess distinct ambiguity or risk attitudes, consider Knight uncertainty, or compete in prices.

Robust reinsurance contract and investment with delay under mean-variance framework

We consider a (non) zero-sum differential game between two insurers and determine a robust reinsurance contract from joint interests of the insurers and the reinsurer under the framework of Stackelberg differential game, that is, a hybrid stochastic differential game framework is introduced in this article. We assume that each party can invest in a financial market consisting of a risk-free asset and a risky asset whose price evolution follows a stochastic volatility model, and the reinsurer charges the premium according to both the expected value premium principle and the variance premium principle. Since the performance-related capital inflow or outflow feature is introduced, the insurers and reinsurer all aim to maximize their own time-inconsistent mean-variance criterion with delay. Using the dynamic programming approach, we derive the equilibrium investment strategy, the robust equilibrium reinsurance contract, and the corresponding equilibrium value functions under two premium principles with certain conditions, respectively. Some interesting properties of the equilibrium investment strategy are proved explicitly, and several numerical examples and sensitivity analysis are presented to demonstrate the effects of model parameters on the equilibrium reinsurance contract.

A fiscal theory of money and bank liquidity provision

Fiscal-backed money can provide additional liquidity to consumers and mitigate the liquidity shortage problem in an economy with banks where agents face idiosyncratic liquidity shocks without being fully insured. The government issues fiat money and creates real value for money by levying a tax and accepting money for tax payments. Tax reallocates the distribution of liquidity in the economy. An increase in tax, by increasing fiscal surplus and the real value of money, reduces the equilibrium investment. Additionally, imposing taxes influences the incentive of private information production, which may impose a constraint on optimal fiscal policy.

Two-Sided Beneficial Value-Added Service Investment and Pricing Strategies in Asymmetric/Symmetric Investment Scenarios

We explore media platforms’ investment strategies for two-sided beneficial value-added services, which can directly benefit both consumers and advertisers, and ad pricing strategies by using game theory. We consider an asymmetric investment scenario, scenario A, and a symmetric investment scenario, scenario S, and develop game models under each scenario. First, we obtain the equilibrium investments, prices and profits and analyze the influence of the three important parameters—marginal investment costs, positive consumer effects and negative advertising effects, on the equilibrium outcomes in each scenario. Then we compare these equilibrium outcomes between both scenarios. Finally, we conduct numerical simulations to verify the conclusions obtained in both scenarios. We show that in scenario A, the value-added service levels and ad prices of the investment platforms remain constant and then decrease with marginal investment costs. The ad prices and profits of the investment platforms increase (decrease) with positive consumer effects (negative advertising effects). The same change is true for the value-added service levels only under certain conditions. In scenario S, the value-added service levels of the investment platforms change with positive consumer effects or negative advertising effects only when marginal investment costs are high. The ad prices of the platforms always increase with positive consumer effects but increase with negative advertising effects only when marginal costs are low. The profits of the platforms vary monotonically with negative advertising effects, but not necessarily with positive consumer effects. Compared to scenario S, the ad prices of the investment platforms in scenario A are higher, but that is not always true for the value-added service levels.

Contagion risks and security investment in directed networks.

We develop a model for contagion risks and optimal security investment in a directed network of interconnected agents with heterogeneous degrees, loss functions, and security profiles. Our model generalizes several contagion models in the literature, particularly the independent cascade model and the linear threshold model. We state various limit theorems on the final size of infected agents in the case of random networks with given vertex degrees for finite and infinite-variance degree distributions. The results allow us to derive a resilience condition for the network in response to the infection of a large group of agents and quantify how contagion amplifies small shocks to the network. We show that when the degree distribution has infinite variance and highly correlated in- and out-degrees, even when agents have high thresholds, a sub-linear fraction of initially infected agents is enough to trigger the infection of a positive fraction of nodes. We also demonstrate how these results are sensitive to vertex and edge percolation (intervention). We then study the asymptotic Nash equilibrium and socially optimal security investment. In the asymptotic limit, agents' risk depends on all other agents' investments through an aggregate quantity that we call network vulnerability. The limit theorems enable us to capture the impact of one class of agents' decisions on the overall network vulnerability. Based on our results, the vulnerability is semi-analytic, allowing for a tractable Nash equilibrium. We provide sufficient conditions for investment in equilibrium to be monotone in network vulnerability. When investment is monotone, we demonstrate that the (asymptotic) Nash equilibrium is unique. In the specific example of two types of core-periphery agents, we illustrate the strong effect of cost heterogeneity on network vulnerability and the non-monotonous investment as a function of costs.

Analysis of generation investments under price controls in cross-border trade of electricity

Price controls are commonly employed in Latin America and other regions to prevent price increases in exporting regions during cross-border trade of electricity. This paper introduces a methodology to evaluate the economic impact of price controls on generation investments, dispatch decisions, and carbon emissions in the context of cross-border electricity trade. We propose a non-linear adjusting function to represent a price-control rule, which can be incorporated into equilibrium models for generation firms and price-sensitive consumers. By employing fixed-point iterations between the adjusting function and an optimization problem that determines equilibrium investments and dispatch decisions for generation firms in a competitive environment, we obtain an equilibrium solution. We apply the proposed methodology to analyze the current price controls governing cross-border electricity trade between Chile and Argentina, with a focus on the exporting country. Our findings reveal that existing price controls can impede incentives for renewable energy investments, particularly in wind and solar, while stimulating investments in fossil fuel-based generation capacity. Moreover, these controls lead to increased CO2 emissions compared to a scenario without price controls. Additionally, we observe that the inefficiency resulting from price controls is proportionate to the volume of exports relative to local demand and the price elasticity of demand in the importing region.

A Stackelberg reinsurance-investment game under α-maxmin mean-variance criterion and stochastic volatility

ABSTRACT This paper investigates a Stackelberg game between an insurer and a reinsurer under the α-maxmin mean-variance criterion. The insurer can purchase per-loss reinsurance from the reinsurer. With the insurer's feedback reinsurance strategy, the reinsurer optimizes the reinsurance premium in the Stackelberg game. The financial market consists of cash and stock with Heston's stochastic volatility. Both the insurer and reinsurer maximize their respective α-maxmin mean-variance preferences in the market. The criterion is time-inconsistent and we derive the equilibrium strategies by the extended Hamilton-Jacobi-Bellman equations. Similar to the non-robust case in [Li, D. & Young, V. R. (2022). Stackelberg differential game for reinsurance: mean-variance framework and random horizon. Insurance: Mathematics and Economics 102, 42–55.], excess-of-loss reinsurance is the optimal form of reinsurance strategy for the insurer. The equilibrium investment strategy is determined by a system of Riccati differential equations. Besides, the equations determining the equilibrium reinsurance strategy and reinsurance premium rate are given semi-explicitly, which is simplified to an algebraic equation in a specific example. Numerical examples illustrate that the game between the insurer and reinsurer makes the insurance more radical when the agents become more ambiguity averse or risk averse. Furthermore, the level of ambiguity, ambiguity attitude, and risk attitude of the insurer (reinsurer) have similar effects on the equilibrium reinsurance strategy, reinsurance premium, and investment strategy.

Equilibrium investment with random risk aversion.

We solve the problem of an investor who maximizes utility but faces random preferences. We propose a problem formulation based on expected certainty equivalents. We tackle the time-consistency issues arising from that formulation by applying the equilibrium theory approach. To this end, we provide the proper definitions and prove a rigorous verification theorem. We complete the calculations for the cases of power and exponential utility. For power utility, we illustrate in a numerical example that the equilibrium stock proportion is independent of wealth, but decreasing in time, which we also supplement by a theoretical discussion. For exponential utility, the usual constant absolute risk aversion is replaced by itsexpectation.

A Stackelberg reinsurance-investment game with derivatives trading

This paper studies a stochastic Stackelberg differential reinsurance-investment game with derivatives trading under a stochastic volatility model. The reinsurer who occupies a monopoly position can price a reinsurance premium and invest her wealth in the financial market consisting of a riskless asset and a stock and derivatives tied to the stock. The insurer, the follower of the Stackelberg game, purchases proportional reinsurance from the reinsurer and invests in the same financial market. The main target of the reinsurer and the insurer is to seek their own optimal strategy to maximize the CARA utility of the relative performance. An explicit equilibrium strategy with derivatives trading is deduced by solving Hamilton-Jacobi-Bellman (HJB) equations sequentially. The equilibrium investment strategy demonstrates that the insurer and the reinsurer imitate each other’s investment strategies, showing a herd effect. In numerical experiments, the sensitivity of the equilibrium strategy to model parameters is analyzed. For the optimal investment strategy, we find that a short position in the derivative may switch to a long position with parameters changing, which provides investors with important decision-making information.

Career Concerns, Investment, and Management Forecasts

ABSTRACTA firm manager is concerned about both the firm value and the market assessments of his abilities. When investing in a project, he has private knowledge of his project-related ability that interacts with the project investment, and his general ability that produces a cash flow independent of the project cash flow. The concerns about the general (project-related) ability assessment create a signaling incentive to decrease (increase) investment. In the presence of underinvestment (overinvestment), higher-quality earnings information reduces (improves) equilibrium efficiency. When the manager issues an earnings forecast as an additional signaling device, the forecast is upwardly biased, and the equilibrium investment is smaller than that without a forecast. The latter is because the signaling incentive to decrease investment is strengthened. When the manager’s concerns about the general ability assessment are relatively large, he is better off by committing to no forecast. Novel empirical predictions about investment and earnings forecast emerge.JEL Classifications: D61; G14; M41.

Equilibrium investment strategy for multi-period DC pension funds with stochastic interest rate and regime switching

This paper studies the equilibrium investment strategy for multi-period defined contribution (DC) pension funds under the mean-variance criterion. We assume that the financial market consists of one risk-free asset and multiple risky assets, the interest rate is stochastic and characterized by the discrete-time Ho-Lee model, and the interest rate as well as the returns of the risky assets both depend on the market states, the evolution of the market states is described by a Markov chain. We regard this problem as a non-cooperative game whose equilibrium strategy is a time-consistent strategy, and derive the analytical expressions for the equilibrium strategy, the equilibrium value function and the equilibrium efficient frontier by using the extended Bellman equation and the matrix representation technique. Finally, a numerical example based on real data from the UK market is provided to shed light on the theoretical results established in this paper.

Equilibrium characterizations of multi-resource Lotto games

The allocation of heterogeneous resources plays an increasingly important role in the completion of system objectives and tasks. In this paper, we focus on deriving optimal strategies for the allocation of heterogeneous resources in a well-known adversarial model known as the General Lotto game. In standard formulations, outcomes are determined solely by the players’ allocation strategies of a single type of resource across multiple contests. In particular, a player wins a contest if it sends more resources than the opponent. Here, we propose a multi-resource extension where the winner of a contest is now determined not only by the amount of resources allocated, but also by the composition of resource types that are allocated. We completely characterize the equilibrium payoffs and strategies for two distinct formulations. The first consists of a weakest-link/best-shot winning rule, and the second considers a winning rule based on a linear combination of the allocated resources. We then provide equilibrium investment strategies in scenarios where the resource types are costly to purchase, and players each have a limited monetary budget.