Quadratic Utility Function Research Articles

Optimal insurance design under mean-variance preference with narrow framing

In this paper, we study an optimal insurance design problem under mean-variance criterion by considering the local gain-loss utility of the net payoff of insurance, namely, narrow framing. We extend the existing results in the literature to the case where the decision maker has mean-variance preference with a constraint on the expected utility of the net payoff of insurance, where the premium is determined by the mean-variance premium principle. We first show the existence and uniqueness of the optimal solution to the main problem studied in the paper. We find that the optimal indemnity function involves a deductible provided that the safety loading imposed on the “mean part” of the premium principle is strictly positive. Our main result shows that narrow framing indeed reduces the demand for insurance. The explicit optimal indemnity functions are derived under two special local gain-loss utility functions – the quadratic utility function and the piecewise linear utility function. As a spin-off result, the Bowley solution is also derived for a Stackelberg game between the decision maker and the insurer under the quadratic local gain-loss utility function. Several numerical examples are presented to further analyze the effects of narrow framing on the optimal indemnity function as well as the interests of both parties.

Research on Energy Management of Fuel-Cell Electric Tractor Based on Quadratic Utility Function

Research on Energy Management of Fuel-Cell Electric Tractor Based on Quadratic Utility Function

A comparative study to find the optimal ordering quantity of the risk-neutral and risk-averse newsvendor

In the classical newsvendor problem, it is considered that the newsvendor is risk-neutral and the Optimal Ordering Quantity (OOQ) was found which maximizes the newsvendor expected profit. In the real world, different investors have different attitudes towards risk. Accordingly, this paper considers the utility function which is commonly being used to model the attitude of the investor who makes the investment decision to maximize his expected utility instead of expected profit. This study considers the quadratic utility function and demonstrates that it can be used to describe the riskaverse as well as a risk-neutral investor with some conditions. Finally, by considering the risk profile of the investor at different levels of investor initial wealth, we have developed a method to determine the OOQ which maximizes the expected utility. From the numerical examples, it is clear that the OOQ and hence the expected utility varies depending on an investor?s attitude toward the risk and investment decision. At different levels of initial wealth, the attitudes toward the risk of the investors are different and the sensitivity analysis demonstrates how an investor can choose the initial wealth and OOQ to maximize his expected utility.

Optimization of day-ahead pricing electricity markets based on a simplified methodology for stochastic utility function estimation

In the context of smart markets’ popularization, day-ahead pricing of electricity has been deployed more commonly as an attempt to improve the grid’s operation and maximize the socioeconomic welfare. However, estimating the utility function is necessary to maximize the socioeconomic welfare, which is not a trivial task. The utility function models the quality of life added by electricity consumption and depends on several factors, such as the consumer’s class (residential, commercial, industrial, etc.), income, climate conditions, and individual preferences. This paper presents a simplified methodology to estimate the utility function in the context of day-ahead pricing. The main benefit of the proposed method is that its application is straightforward since the modeling is mainly based on simple matrix equations. The demand elasticities (self- and cross-elasticities), which are general concepts in economics, are addressed as input parameters. Due to the inherent uncertainties related to human behavior, a stochastic utility function is proposed to enhance the model’s accuracy/robustness. The proposed model particularly thrives when the effects of cross-elasticities are significant but not massive (although procedures can be applied to mitigate the error if higher cross-elasticities are verified). By applying the proposed model to a hypothetical smart microgrid with 100 consumers and 24 daily periods (hourly prices), results demonstrate that it successfully provides the rates that maximize the socioeconomic welfare created by the market. In the case study, the conventional quadratic utility function implies an error of 73%, whereas the proposed model an error of 10%, highlighting the usefulness of the proposed model.

MONOPOLISTIC COMPETITION MODEL WITH ENTRANCE FEE

We study the monopolistic competition model with producer-retailer-consumers two-level interaction. The industry is organized according to the Dixit–Stiglitz model. The retailer is the only monopolist. A quadratic utility function represents consumer preferences. We consider the case of the retailer's leadership; namely, we study two types of behavior: with and without the free entry condition. Earlier, we obtained the result: to increase social welfare and/or consumer surplus, the government needs to subsidize (not tax!) retailers. In the presented paper, we develop these results for the situation when the producer imposes an entrance fee for retailers.

On the dynamics of Cournot duopoly game with private firms: Investigations and analysis

In this paper, we introduce a Cournot duopoly game whose players’ inverse prices are isoelastic functions and are derived from a simple quadratic utility function. The proposed game consists of two competing firms seeking the optimum of their productions by maximizing weighted objectives that are profit and social welfare under the same marginal costs. Based on those objectives, the bounded rationality is adopted to construct the model describing the game’s evolution. A unique fixed point which is Nash equilibrium point is obtained for the game. The stability conditions for this point are discussed and shown that the equilibrium point can be unstable through flip bifurcation. In our discussion, we show that the model’s map is noninvertible and belongs to Z3−Z1 type. Numerical experiments for the dynamics of the map are performed and shown that the map possesses several stable attractors. Furthermore, the shapes of some attractive basins for the map are analyzed providing the existence of the so-called lobes.

General equilibrium cryptocurrency pricing in an OLG model

We study the valorization of cryptocurrency in an overlapping generations model where agents have the choice between crypto- and central bank currency. We pay attention to three features of cryptocurrency: it is a risky asset (because of frauds and hacks), which one acquires by bearing transaction costs, and its supply reaches its upper limit in a finite time (e.g., Bitcoin). We consider both the standard case with linear–quadratic utility functions and the more general case where utility functions are not specified. We find that the effect of an increase in cryptocurrency on the currency prices depends on the transaction costs that individuals bear to get cryptocurrency. In particular, when these costs are convex non-increasing in the quantity of cryptocurrency, the cryptocurrency price may increase with this quantity. Otherwise, when the transaction costs are either increasing or concave non-increasing, the cryptocurrency price decreases with the quantity of cryptocurrency. We also show that the cryptocurrency and the central bank currency prices always move in opposite directions.

Quadratic utility function-based energy management strategy for electric vehicle considering degradation of fuel cells and batteries

The vehicle economy and durability of power sources are strongly affected by the energy management strategy (EMS). In order to balance the economy and durability, this paper proposes a novel EMS based on the quadratic utility function (QUF). Degradation models of fuel cells and batteries are first established. The QUF and finite state machine (FSM) are then used to decompose the demand current into the output current of fuel cells and batteries. In order to improve the power sources durability and the economy of the vehicle, the actual output characteristics and current change of fuel cells are considered in the QUF, in which the key coefficients are determined and optimized using the Multi-Objective Particle Swarm Optimization (MOPSO) algorithm and the pareto solution set. The simulation results show that the proposed control strategy can efficiently improve the durability of fuel cells and batteries, reduce the hydrogen consumption and extend the driving range. Compared with the FSM control strategy, the proposed strategy can reduce the degradation of the batteries by 19.4%, and extend the driving range by 1.2% under the WLTC driving cycle. Finally, compared with the fuzzy control strategy, the proposed strategy can reduce the degradation of the fuel cells by 52.1%, and extend the driving range by 3.0% under the WLTC driving cycle.

On the Dynamics of Cournot Duopoly Game with Governmental Taxes

A quadratic utility function is introduced in this paper to study the dynamic characteristics of Cournot duopoly game. Based on the bounded rationality mechanism, a discrete dynamical map that describes the game’s dynamic is obtained. The map possesses only one equilibrium point which is Nash point. The stability conditions for this point are analyzed. These conditions show that the point becomes unstable due to two bifurcation types that are flip and Neimark–Sacker. The synchronization property for that map is studied. Through local and global analysis, some dynamics of attracting sets are investigated. This analysis gives some insights on the basis of those sets and the shape of the critical curves. It also shows some lobes found in those attracting sets which are constructed due to the origin focal point.

Forecasting the European Monetary Union equity risk premium with regression trees

This paper investigates whether classification and regression trees ensemble algorithms such as bagging, random forests and boosting improve on traditional parametric models for forecasting the equity risk premium. In particular, we work with European Monetary Union (EMU) data for the period from its foundation in 2000 to 2020. The paper first compares the monthly out-of-sample forecasting ability of multiple economic and technical variables using univariate linear regression models and regression tree techniques. The results obtained suggest that regression trees do not show better forecasting ability than a first-order autoregressive benchmark model and univariate linear regressions. The paper then analyses asset allocation strategies with regression trees and checks whether these can select the best economic predictors to form dynamic portfolios composed of two assets: a risk-free asset and an equity index. The results indicate that trading strategies built with two or three economic predictors selected with boosting and random forest algorithms can generate economic value for a risk-averse investor with a quadratic utility function.

On Dynamic Investigations of Cournot Duopoly Game: When Firms Want to Maximize Their Relative Profits

This paper studies a Cournot duopoly game in which firms produce homogeneous goods and adopt a bounded rationality rule for updating productions. The firms are characterized by an isoelastic demand that is derived from a simple quadratic utility function with linear total costs. The two competing firms in this game seek the optimal quantities of their production by maximizing their relative profits. The model describing the game’s evolution is a two-dimensional nonlinear discrete map and has only one equilibrium point, which is a Nash point. The stability of this point is discussed and it is found that it loses its stability by two different ways, through flip and Neimark–Sacker bifurcations. Because of the asymmetric structure of the map due to different parameters, we show by means of global analysis and numerical simulation that the nonlinear, noninvertible map describing the game’s evolution can give rise to many important coexisting stable attractors (multistability). Analytically, some investigations are performed and prove the existence of areas known in literature with lobes.

Complex Investigations of a Piecewise-Smooth Remanufacturing Bertrand Duopoly Game

This paper considers a Bertrand competition between two firms whose decision variables are derived from a quadratic utility function. The first firm produces new products with their own prices while the second firm re-manufactures returned products and sells them in the market at prices that may be less than or equal to the price of the first firm. Dynamically, this competition is constructed on which boundedly rational firms apply a gradient adjustment mechanism to update their prices in each period. According to this mechanism and the nature of the competition, a two-dimensional piecewise smooth discrete dynamic map was constructed in order to study the complex dynamic characteristics of the game. The phase plane of the map was divided into two different regions, separated by border curve. The equilibrium points of the map, in each region on where they are defined, were calculated, and their stability conditions were investigated. Furthermore, we conducted a global analysis to investigate the complex structure of the map, such as closed invariant curves, periodic cycles, and chaotic attractors and their basins, which cause qualitative changes as some parameters are allowed to vary.

The (Un)identification of Continuous-Action Rational Inattention Decision Models

In this paper, we discuss some properties of optimal consideration set in binary state continuous-action rational inattention decision problems. First, we show that it is the form of the utility function that determines the form of posterior distribution, i.e. whether it is a binary distribution or a triple or quadruple distribution, etc. Besides, we will also show that the posterior distribution may not be resulted after information acquisition in some circumstances. Second, we show that the optimal consideration set is essentially the optimal action strategies. Therefore, without further conditions, the rational inattention decision model is unidentifiable. Third, in order to identify the rational inattention decision model, we introduce another condition to the model. The discrete actions after information acquisition that satisfy the endogenously-determined probability (optimal action strategies) should satisfy the properties of probabilities, and therefore the discrete actions should also follow an exogenous probability distribution. This idea is tested by an example with quadratic utility function. Through this example, we find that under this condition, the optimal consideration set can be identified. The elements in the optimal consideration set, or equivalently the discrete signals after information acquisition, exhibit a curve or curves.

Multivehicle Task Assignment Based on Collaborative Neurodynamic Optimization With Discrete Hopfield Networks.

This article presents a collaborative neurodynamic optimization (CNO) approach to multivehicle task assignments (TAs). The original combinatorial quadratic optimization problem for TA is reformulated as a quadratic unconstrained binary optimization (QUBO) problem with a quadratic utility function and a penalty function for handling load capacity and cooperation constraints. In the framework of CNO with a population of discrete Hopfield networks (DHNs), a TA algorithm is proposed for solving the formulated QUBO problem. Superior experimental results in four typical multivehicle operation scenarios are reported to substantiate the efficacy of the proposed neurodynamics-based TA approach.

Who gains and who loses on stock markets? Risk preferences and timing matter

AbstractThis paper uses an agent‐based multi‐asset model to examine the effect of risk preferences and optimal rebalancing frequency on performance measures while tracking profit and risk‐adjusted return. We focus on the evolution of portfolios managed by heterogeneous mean‐variance optimizers with a quadratic utility function under different market conditions. We show that patient and risk‐averse agents are able to outperform aggressive risk‐takers in the long‐run. Our findings also suggest that the trading frequency determined by the optimal tolerance for the deviation from portfolio targets should be derived from a tradeoff between rebalancing benefits and rebalancing costs. In a relatively calm market, the absolute range of 6% to 8% and the complete‐way back rebalancing technique outperforms others. During particular turbulent periods, however, none of the existing rebalancing techniques improves tax‐adjusted profits and risk‐adjusted returns simultaneously.

Analysis of Risk Measures in Portfolio Optimization for the Uganda Securities Exchange

For the most recent years, risk has become one of the essential parameters in portfolio optimization problems. Today most practitioners and researchers in portfolio optimization have used variance as a standard risk measure. This approach has been found subjective. The Markowitz (1952) mean-variance model considered variance as an adequate portfolio risk measure, and asset returns are multivariate normally distributed and that investors have a quadratic utility function which is subjective too. Other risk measures have been suggested to overcome the limitations of the mean-variance model. This paper analyzes which portfolio optimization models can better explain the optimal portfolio performance (high return, low risk) for the Uganda Security Exchange (USE). We compare Mean-Variance (MV), Mean Absolute Deviation (MAD), Robust Portfolios and Covariance Estimation Models (The Shrinked Mean-Variance (SMV) Models & Alternative Covariance Estimator (ACE) Models) and Mean-Conditional Value-at-Risk (Mean-CVaR) models in terms of the risk and performance. For the computed monthly returns and price data (February 2010 to January 2021) for USE selected stocks, we considered the results to show that Mean-CVaR and ACE portfolios have the highest performance ratio compared to other models. We find that VaR is the best risk measure for portfolio optimization for the USE since it has lower values across all models than other risk measures. It is vital to consider all the available risk measures for a regulator or practitioner to make a good decision since using one can be subjective; as seen in our results, different risk measures yield different results.

Factors that Influence Farmer’s Behavior Towards Risk

The research was carried out with the aim to find out the behavior of farmers towards risk and the factors that influence it. The research sample was 100 paddy farmers in flood-prone area paddy fields in Pangandaran District, West Java Province, Indonesia. Farmer’s behavior towards risk was analyzed using quadratic utility functions, while the factors that influence farmer’s behavior towards risk were analyzed using logistic regression. The results showed farmers 87 was risk neutral, while 13 farmer risk takers were farmers. Education, familys size and income significantly influence farmer’s behavior towards risk; while age, experience, land area, production risk, price risk, income risk and group did not significantly influence farmer’s behavior towards risk.

PENDAPATAN DAN RISIKO USAHATANI UBI KAYU (Manihot utilisima) DI DESABUMI AGUNG MARGA KECAMATAN ABUNG TIMUR KABUPATEN LAMPUNG UTARA

This study aims to analyze income, price and production risks, and farmer behavior on the risk of cassava farming. This research is a survey research, carried out in Bumi Agung Marga Village, Abung Timur Distric of North Lampung Regency in February 2018. There are 46 samples of cassava farmers who were chosen randomly. Farm income is analyzed by income analysis, whereas price risk and production risk are analyzed by coefficient of variation analysis. To determine the behavior of farmers against risk is used quadratic utility function analysis. The research results showed that cassava farming was profitable in which income value was Rp15, 146,206.10/ha and R/C was 3.30. The production risk had a CV value of 0.057 and L 20.061; it meant that the production risk was low, because the result of CV <0.5 and L>1. The price risk had a CV value of 0.135 and L 644.695; it meant that the risk was low because the CV <0.5 and L>1. All cassava farmers behave neutral in facing the risk of cassava farming.Key words: cassava, farming, income,risk

Deep-Learning Solution to Portfolio Selection with Serially-Dependent Returns

This paper investigates a deep-learning solution to high-dimensional multi-period portfolio optimization problems with bounding constraints on the control. We propose a deep neural network (DNN) architecture to describe the underlying control process. The DNN consists of $K$ subnetworks, where $K$ is the total number of decision steps. The feedback control function is determined solely by the network parameters. In this way, the multi-period portfolio optimization problem is linked to a training problem of the DNN, that can be efficiently computed by the standard optimization techniques for network training. We offer a sufficient condition for the algorithm to converge for a general utility function and general asset return dynamics including serially-dependent returns. Specifically, under the condition that the global minimum of the DNN training problem is attained, we prove that the algorithm converges with the quadratic utility function when the risky asset returns jointly follow multivariate AR(1) models and/or multivariate GARCH(1,1) models. Numerical examples demonstrate the superior performance of the DNN algorithm in various return dynamics for a high-dimensional portfolio (up to 100 dimensions).

Formalization of Cost and Utility in Microeconomics

Cost and utility modeling of economics agents based on the differential theory is fundamental to the analysis of the microeconomics models. In particular, the first and second-order derivative tests are used to specify the desired properties of the cost and utility models. Traditionally, paper-and-pencil proof methods and computer-based tools are used to investigate the mathematical properties of these models. However, these techniques do not provide an accurate analysis due to their inability to exhaustively specify and verify the mathematical properties of the cost and utility models. Additionally, these techniques cannot accurately model and analyze pure continuous behaviors of the economic agents due to the utilization of computer arithmetic. On the other hand, an accurate analysis is direly needed in many safety and cost-critical microeconomics applications, such as agriculture and smart grids. To overcome the issues pertaining to the above-mentioned techniques, in this paper, we propose a theorem proving based methodology to formally analyze and specify the mathematical properties of functions used in microeconomics modeling. The proposed methodology is primarily based on a formalization of the derivative tests and root analysis of the polynomial functions, within the sound core of the HOL-Light theorem prover. We also provide a formalization of the first-order condition, which is used to analyze the maximum of the profit function in a higher-order-logic theorem prover. We then present the formal analysis of the utility, cost and first-order condition based on the polynomial functions. To illustrate the usefulness of proposed formalization, the proposed formalization is used to formally analyze and verify the quadratic cost and utility functions, which have been used in an optimal power flow problem and demand response (DR) program, respectively.