Time Inconsistency Problem Research Articles

On the Gradient Method in One Portfolio Management Problem

This study refines the methodology for solving stochastic optimal control problems with quality criteria that include the sum of the quality functional of the classical formulation and an extremal measure. A two-level optimization solution of these kinds of problems is presented already for the case where the quality functional consists only of the extremal measure. Our study shows the possibility of solving the original time inconsistency problem through solving a two-level optimization problem, where the outer problem is solved by gradient methods since the value function is convex and the inner problem is solved by classical methods. Some experiments were carried out and confirmed the validity of the theory. The results of the study can be applied to the case of portfolio management with quality criteria containing the Conditional Value-at-Risk (CVaR) metric.

Managing strategic inventories in a three-echelon supply chain of durable goods

Firms can hold strategic inventory to bargain with their suppliers about wholesale prices. In this paper, we model a three-echelon durable goods supply chain consisting of a supplier, a wholesaler, and a retailer, where the latter two firms can choose to hold strategic inventory. Our major findings are as follows. First, contrary to the literature, our study shows that the retailer's use of strategic inventory benefits all supply chain members, irrespective of whether the wholesaler holds strategic inventory. Second, the strategic inventories of the wholesaler and the retailer suppress each other when the holding cost is low. Otherwise, the retail strategic inventory can stimulate the wholesale strategic inventory. Third, after the retailer employs strategic inventory, introducing wholesale strategic inventory will harm each supply chain member. However, after the wholesaler employs strategic inventory, introducing retail strategic inventory can benefit all members. For the entire supply chain, holding strategic inventory exclusively at the retailer is optimal. Fourth, when product durability is high, the supplier may suffer from the wholesaler's strategic inventory. Additionally, the retailer's strategic inventory can solve the time inconsistency problem associated with durable goods. Finally, we extend our main model to consider multiple retailers and different holding costs. Our analysis confirms the robustness of our major findings and managerial insights.

In Defense of Central Bank Independence against its Enthusiasts

The problem of time inconsistency provides the basic economic rationale behind the independence of central banks, understood as the most suitable institutional mechanism for ensuring price stability. According to the consensus that emerged in the 1990s and the 2000s, endowing the central bank with independent decision-making authority over the use of monetary policy instruments, with a mandate for price stability, enhances the effectiveness of monetary policy and reduces the inflation volatility. It may be surprising that criticisms of the once almost sacrosanct independence arise after the Great Financial Crisis, when it is widely recognized that central banks “saved the world” from repeating the Great Depression of the 1930s. The paper is an attempt to explain the rationale why these cracks in the consensus have arisen and offers suggestions what to do about it. Independence has an economic basis that is as solid as its political foundation is fragile, because central banks are unelected power. And in the realm of economics and politics, the political economy of central banking may help us understand the evolution of the degree and the specific form of independence because of both internal and external political and economic constraints. The paper examines these issues in some detail, to contextualize the economic and non-economic threats to independence and discusses possible mitigants. Properly assessing the challenges is a prerequisite for preserving the value of this institution, which is perhaps the most important and effective institutional innovation of our time in macroeconomic policy.

Counterfeit Competition With Strategic Consumers

This article investigates competition between a branded firm selling a durable good over two periods and a deceptive counterfeiter entering the market in the second period. The two firms engage in a price signaling game in which the branded firm designs its price strategy over two periods, and strategic consumers decide whether to buy the authentic product upfront or wait until the second period. We find that the branded firm may benefit from the counterfeit competition if the quality gap between the two products is sufficiently large. The intuition is that the branded firm would charge a high second-period price to signal its authenticity, inducing more consumers to buy the genuine product upfront. This strategy allows the branded firm to increase its first-period price and demand simultaneously, thus effectively mitigating the time-inconsistency problem. Otherwise, when the quality gap is small, counterfeit competition leads to reduced profits for the authentic product. These results remain robust throughout several extensions of the base model, including partially informed or naive consumers, asymmetric retail channels, post-purchase regret, and endogenized counterfeit.

LVI-Fusion: A Robust Lidar-Visual-Inertial SLAM Scheme

With the development of simultaneous positioning and mapping technology in the field of automatic driving, the current simultaneous localization and mapping scheme is no longer limited to a single sensor and is developing in the direction of multi-sensor fusion to enhance the robustness and accuracy. In this study, a localization and mapping scheme named LVI-fusion based on multi-sensor fusion of camera, lidar and IMU is proposed. Different sensors have different data acquisition frequencies. To solve the problem of time inconsistency in heterogeneous sensor data tight coupling, the time alignment module is used to align the time stamp between the lidar, camera and IMU. The image segmentation algorithm is used to segment the dynamic target of the image and extract the static key points. At the same time, the optical flow tracking based on the static key points are carried out and a robust feature point depth recovery model is proposed to realize the robust estimation of feature point depth. Finally, lidar constraint factor, IMU pre-integral constraint factor and visual constraint factor together construct the error equation that is processed with a sliding window-based optimization module. Experimental results show that the proposed algorithm has competitive accuracy and robustness.

Towards a public sustainable finance paradigm for the green transition

AbstractSustainable finance is often discussed as a solution to the climate crisis, but its impacts are limited and its discourse focuses on mobilising private investments through public de-risking, without considering direct government action. We argue that this is due to an implicit reference to mainstream economic theory assuming that an active state leads to time inconsistency problems and crowding-out effects. However, these assumptions have been sufficiently refuted as public investments may actually crowd-in private capital. We therefore propose a paradigm shift towards what we call Public Sustainable Finance, aimed at empowering the role of the state in the green transition on the discursive, policy, and political economy levels. Studying the case of Germany, we show how Public Sustainable Finance can be introduced despite tight fiscal regimes. To this end, we propose that the Climate- and Transformation Fund be given its own borrowing powers. By borrowing an average of 23 billion euros annually from 2024 to 2030, the existing financing gap that has been exacerbated following the November 2023 constitutional court ruling can be closed, enabling a more rapid and effective green transition.

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.

Inattention and the Taxation Bias

Abstract This paper shows that inattention to taxes generates a time-inconsistency problem in the choice of tax policy, leading to higher taxes in equilibrium. These discretionary tax increases are inefficient as they are deviations from the socially optimal commitment policy. We call these deviations a taxation bias. Combining sufficient statistics and structural approaches, we quantify the magnitude of this policy distortion for the U.S. redistributive tax-transfer system. We find that the taxation bias ranges between 3 and 8 percentage points, alters tax-transfer progressivity, and has significant welfare effects. Overall, our findings shed new light on the implications of inattention and misperceptions.

A Household’s Preferences Vary Depending on Whether Incomes Are Permanent or Temporary: A Solution to the Time-Inconsistency Problem and Equity-Premium Puzzle

A household’s preferences are usually assumed not to vary temporally or depending on the objects to which they are applied, but this assumption is often inconsistent with empirical estimates, for example, with the time-inconsistency problem of the time preference rate and the equity-premium puzzle. I show that these inconsistencies are generated because a household’s preferences vary depending on whether they are applied to permanent or temporary incomes. Preferences applied to permanent incomes are anchored to the steady state or a balanced growth path, but those for temporary incomes are not. Hence, the former is fixed and unchanged, but the latter can take various values depending on conditions. Copyright© 2023 The Author(s). This article is distributed under the terms of the license CC-BY 4.0., which permits any further distribution in any medium, provided the original work is properly cited.

Nonlocal fully nonlinear parabolic differential equations arising in time-inconsistent problems

We prove the local well-posedness results, i.e. existence, uniqueness, and stability, of the solutions to a class of nonlocal fully nonlinear parabolic partial differential equations (PDEs), where there is an external time parameter t on top of the temporal and spatial variables (s,y) and thus the problem could be considered as a flow of equations. The nonlocality comes from the dependence on the unknown function and its first- and second-order derivatives evaluated at not only the local point (t,s,y) but also at the diagonal line of the time domain (s,s,y). Such equations arise from time-inconsistent problems in game theory or behavioral economics, where the observations and preferences are (reference-)time-dependent. We first study the linearized version of the nonlocal PDEs with an innovative construction of appropriate norms and Banach spaces and contraction mappings over which. With fixed-point arguments, we obtain the solvability of nonlocal linear PDEs and establish a Schauder-type prior estimate for the solutions. Then, by the linearization method, we establish the well-posedness under the fully nonlinear case. Moreover, we reveal that the solution of a nonlocal fully nonlinear parabolic PDE is an adapted solution to a flow of second-order forward-backward stochastic differential equations.

Open-loop equilibriums for a general class of time-inconsistent stochastic optimal control problems

<p style='text-indent:20px;'>This paper studies open-loop equilibriums for a general class of time-inconsistent stochastic control problems under jump-diffusion SDEs with deterministic coefficients. Inspired by the idea of Four-Step-Scheme for forward-backward stochastic differential equations with jumps (FBSDEJs, for short), we derive two systems of integro-partial differential equations (IPDEs, for short). Then, we rigorously prove a verification theorem which provides a sufficient condition for open-loop equilibrium strategies. As special cases, a mean-variance portfolio selection problem and a time-inconsistent problem under non-exponential discounting are discussed.</p>

A Stackelberg reinsurance-investment game under Heston's stochastic volatility model

<p style='text-indent:20px;'>This paper studies the optimal reinsurance-investment problem under Heston's stochastic volatility (SV) in the framework of Stackelberg stochastic differential game, in which an insurer and a reinsurer are the two players. The aim of the insurer as the follower in the game is to find the optimal reinsurance strategy and investment strategy such that its mean-variance cost functional is maximized, while the aim of the reinsurer as the leader is to maximize its mean-variance cost functional through finding its optimal premium pricing strategy and investment strategy. To overcome the time-inconsistency problem in the game, we derive the optimization problems of the two players as embedded games and obtain the corresponding extended Hamilton–Jacobi–Bellman (HJB) equations. Then we present the verification theorem and get the equilibrium reinsurance-investment strategies and the corresponding value functions of both the insurer and the reinsurer. Finally, we provide some numerical examples to draw the findings.</p>

Implementing the commitment solution via discretionary policy-making

AbstractThis paper demonstrates that, in a large class of linear-quadratic models with rational expectations, losses due to time-inconsistency problems can be avoided, as the commitment solution can be implemented by a policy-maker who acts under discretion. We focus on two approaches. First, we show that a non-Markovian, reputational equilibrium that implements the commitment solution always exists. Second, we show how delegation to a policy-maker with an additional objective for the policy instrument can be used to implement the commitment solution via a standard discretionary Markov equilibrium. Implementation is facilitated by the fact that the commitment outcome can be attained irrespective of the weight that the policy-maker assigns to the additional target. Using the standard new Keynesian model as an example, we study the dynamics of the economy under optimal additional output targets as well as optimal interest-rate targets for central banks.

Commitment versus flexibility and sticky prices: Evidence from life insurance

Life insurance premiums display significant rigidity in the data, on average adjusting once every 3 years by more than 10%. This contrasts with the underlying marginal cost which exhibits considerable volatility due to the movements in interest and mortality rates. We build a dynamic model where policyholders are held-up by long-term insurance contracts, resulting in a time inconsistency problem for the insurer. The optimal contract balances commitment and flexibility and takes the form of a simple cutoff rule: premiums are rigid for cost realizations smaller than the threshold, while adjustments must be large and are only possible when cost realizations exceed it. We use a calibrated version of the model to show that it matches the data and captures several aspects of premium rigidity in the cross-section and over time.

Endogenizing Discretion in Disclosures

A firm’s publicly available information reflects ex ante committed information releases and ex post discretionary disclosures. In a setting wherein a firm’s CEO is concerned with stock market valuation and confrontations with the employee union, this paper endogenizes when the CEO retains ex post disclosure discretion and when the CEO curtails discretion, opting instead to ex ante commit to not gathering information or publicly disclosing information. The underlying economic forces entail a trade-off between the time-inconsistency problem (the CEO’s ex post preferences can diverge from the CEO’s ex ante goals) and how silence (absence of information) under the ex ante and ex post paths is valued differentially by the union. In effect, silence under different systems “speaks” differently, and this influences both the labor union’s incentives to engage in bargaining with the firm and the stock market’s pricing of the firm’s equity.This paper was accepted by Ranjani Krishnan, accounting.

Time-consistent consumption-portfolio control problems with regime-switching-modulated habit formation: an essentially cooperative approach

This paper is devoted to consumption-portfolio control problems with regime-switching-modulated habit formation. The formation of a habit depends on the current regime, which leads to time inconsistency in optimal control. General utility functions are used to evaluate the preference for the consumption net of the habit. To derive an analytical solution, we consider an enlarged financial market with Markov jump assets. Then, the time-inconsistent problem for pre-commitment controls is reduced to solving a regime-switching Hamilton–Jacobi–Bellman (HJB) equation. In seeking a time-consistent control, we investigate an alternate problem by means of multi-person sequential games with an essentially cooperative approach. The analytical expression of the time-consistent control is derived via a straightforward application of the pre-commitment control results. Moreover, the limit case in which the mesh size of time approaches zero is studied.

Debt Spillovers in a Monetary Union: A Novel Rationale for Central Bank Independence

Abstract Central bank independence has been championed on the grounds that it avoids political business cycles, the time-inconsistency problem of discretionary monetary policy, and political conflicts. However, after the financial crisis, central banks have resorted to unconventional monetary policies and embraced additional tasks, making monetary authorities more exposed to political interference. This new reality has put into question the long-lasting consensus on the desirability of central bank independence. We add to this debate a new argument in support of that independence, namely, it internalizes the fiscal spillovers that arise in a monetary union, which is not a full fiscal union.

Do We Need to Implement Multi-Interval Real-Time Markets?

Many market-based power systems have implemented a form of ‘look-ahead dispatch’ which simultaneously solves for the optimal dispatch and prices over several intervals into the future. A few papers have pointed out that the dispatch outcomes which emerge from look-ahead dispatch may not be time consistent. We emphasise that this time inconsistency is not inherent in look-ahead dispatch but is a consequence of the assumption of linear cost and utility functions, which is arguably a special case. Various augmentations to the dispatch process to resolve the time inconsistency problem have been proposed, but these augmentations suffer from the drawback that they do not allow the power system to efficiently adjust to new information. We query whether it is necessary to implement multi-interval real-time markets. We show how under certain assumptions, a sequence of one-shot dispatch processes will achieve the efficient outcome.

A Risk Extended Version of Merton’s Optimal Consumption and Portfolio Selection

A risk management version of the classical investment-consumption problem known as Merton's problem in the finance literature is proposed. Risk is measured by variance, which introduces a nonlinear function of the expected value into the control problem. Standard stochastic theory cannot properly handle this type of nonlinear stochastic optimization problem. Therefore, we study this time-inconsistent problem within the mean field-type control framework. We derive the sufficient condition of optimality and solve the problem completely. Numerical results illustrating the effect of risk on optimal policies are also presented. Applications can be numerous, including all kinds of investment decisions or operations decisions.

Distributionally Robust Inventory Control When Demand Is a Martingale

Demand forecasting plays an important role in many inventory control problems. To mitigate the potential harms of model misspecification in this context, various forms of distributionally robust optimization have been applied. Although many of these methodologies suffer from the problem of time inconsistency, the work of Klabjan et al. established a general time-consistent framework for such problems by connecting to the literature on robust Markov decision processes. Motivated by the fact that many forecasting models exhibit very special structure as well as a desire to understand the impact of positing different dependency structures in distributionally robust multistage optimization, we formulate and solve a time-consistent distributionally robust multistage newsvendor model, which naturally robustifies some of the simplest inventory models with demand forecasting. In particular, in some of the simplest such models, demand evolves as a martingale (i.e., expected demand tomorrow equals realized demand today). We consider a robust variant of such models in which the sequence of future demands may be any martingale with given mean and support. Under such a model, past realizations of demand are naturally incorporated into the structure of the uncertainty set going forward. We explicitly compute the minimax optimal policy (and worst-case distribution) in closed form by combining ideas from convex analysis, probability, and dynamic programming. We prove that, at optimality, the worst-case demand distribution corresponds to the setting in which inventory may become obsolete at a random time. To gain further insight, we prove weak convergence (as the time horizon grows large) to a simple and intuitive process. We also compare with the analogous setting in which demand is independent across periods (analyzed previously by Shapiro) and identify several differences between these models in the spirit of the price of correlations studied by Agrawal et al. Finally, we complement our results by providing both numerical experiments that illustrate the potential benefits and limitations of our approach as well as additional theoretical analyses exploring what happens when our modeling assumptions do not hold.