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on Utility Models and Prospect Theory |
| By: | Maroussia Favre (ETH Zürich - Department of Management, Technology, and Economics (D-MTEC)); Amrei Wittwer (University of Zurich - Collegium Helveticum); Hans Rudolf Heinimann (ETH Zurich); Vyacheslav I. Yukalov (Joint Institute for Nuclear Research; D-MTEC, ETH Zurich); Didier Sornette (Swiss Finance Institute; ETH Zürich - Department of Management, Technology, and Economics (D-MTEC)) |
| Abstract: | Quantum decision theory (QDT) is a novel theory of decision making based on the mathematics of Hilbert spaces, a framework known in physics for its application to quantum mechanics. This framework formalizes the concept of uncertainty and other effects that are particularly manifest in cognitive processes, which makes it well suited for the study of decision making. QDT describes a decision maker's choice as a stochastic event occurring with a probability that is the sum of an objective utility factor and a subjective attraction factor. This article offers a practical guide to researchers who are interested in applying QDT to a data set of binary lotteries in the domain of gains. We find that our results are in good agreement with the quarter law, a quantitative prediction of QDT. We examine gender differences in our sample in order to illustrate how QDT can be used to differentiate between different groups. We find that women in our sample are on average more risk-averse than men, but stress that our sample is too small to generalize this result to the population outside our sample. |
| Keywords: | decision making, quantum decision theory, risk, uncertainty, how-to guide |
| JEL: | C44 D03 D71 D81 D83 |
| URL: | http://d.repec.org/n?u=RePEc:chf:rpseri:rp1609&r=upt |
| By: | Sabine Vincent (ETH Zurich); Tatyana Kovalenko (ETH Zurich); Vyacheslav I. Yukalov (Joint Institute for Nuclear Research; D-MTEC, ETH Zurich); Didier Sornette (Swiss Finance Institute; ETH Zürich - Department of Management, Technology, and Economics (D-MTEC)) |
| Abstract: | We present the first calibration of quantum decision theory (QDT) to an empirical data set. The data comprise 91 choices between two lotteries (two "prospects") presented in 91 random pairs made by 142 subjects offered at two separated times. First, we quantitatively account for the fraction of choice reversals between the two repetitions of the decisions, using a probabilistic choice formulation in the simplest possible form with no model assumption and no adjustable parameter. The prediction of choice reversal is then refined by introducing heterogeneity between decision makers through a differentiation of the population into two similar sized groups in terms of "over-confident" and "contrarian" decision makers. This supports the first fundamental tenet of QDT, which models the choice of an option as an inherent probabilistic process, such that the probability of a choice can be expressed as the sum of its utility and attraction factors. We propose to model (a) the utility factor with a stochastic version of cumulative prospect theory (logit-CPT), and (b) the attraction factor with a constant absolute risk aversion (CARA) function. This makes logit-CPT nested in our proposed parameterisation of QDT, allowing for a precise quantitative comparison between the two theories. For this data set, the QDT model is found to perform better at both the aggregate and individual levels, and for all considered fit criteria both for the first iteration of the experiment and for predictions (second iteration). The QDT effect associated with the attraction factor is mostly appreciable for prospects with big losses. Our quantitative analysis of the experiment results supports the existence of an intrinsic limit of predictability, which is associated with the inherent probabilistic nature of choice. |
| Keywords: | Quantum decision theory, QDT, prospect probability, utility factor, attraction factor, interference, parametrization, hierarchical estimation method, calibration, empirical data, simple gambles, stochastic Cumulative prospect theory, logit-CPT, probabilistic decision making, limits of predictability |
| JEL: | C44 D81 |
| URL: | http://d.repec.org/n?u=RePEc:chf:rpseri:rp1631&r=upt |
| By: | Kasper LARSEN (Carnegie Mellon University); Mete SONER (ETH Zurich and Swiss Finance Institute); Gordan ZITKOVIC (University of Texas) |
| Abstract: | We establish the existence and characterization of a primal and a dual facelift - discontinuity of the value function at the terminal time - for utility maximization in incomplete semimartingale-driven financial markets. Unlike in the lower- and upper-hedging problems, and somewhat unexpectedly, a facelift turns out to exist in utility-maximization despite strict convexity in the objective function. In addition to discussing our results in their natural, Markovian environment, we also use them to show that the dual optimizer cannot be found in the set of countably-additive (martingale) measures in a wide variety of situations. |
| Keywords: | Boundary layer, convex analysis, convex duality, facelift, financial mathematics, incomplete markets, Markov processes, utility-maximization, unspanned endowment |
| JEL: | C61 G11 |
| URL: | http://d.repec.org/n?u=RePEc:chf:rpseri:rp1461&r=upt |
| By: | Semyon MALAMUD (Ecole Polytechnique Fédérale de Lausanne and Swiss Finance Institute) |
| Abstract: | Carroll and Kimball (1996) show that the consumption function for an agent with time-separable, isoelastic preferences is concave in the presence of income uncertainty. In this paper I show that concavity breaks down if we abandon time-separability. Namely, if an agent maximizing an isoelastic recursive utility has preferences for early resolution of uncertainty, there always exists a distribution of income risk such that consumption function is not concave in wealth. I also derive sufficient conditions guaranteeing that the consumption function is concave if the agent has preferences for late resolution of uncertainty. |
| Keywords: | consumption, saving, marginal propensity to consume, recursive preferences, resolution of uncertainty |
| JEL: | D14 D91 E21 |
| URL: | http://d.repec.org/n?u=RePEc:chf:rpseri:rp1437&r=upt |
| By: | Bruno BOUCHARD (Université de Paris Dauphine and CREST-ENSAE); Ludovic MOREAU (ETH Zurich); Mete SONER (ETH Zurich and Swiss Finance Institute) |
| Abstract: | We consider the problem of option hedging in a market with proportional transaction costs. Since super-replication is very costly in such markets, we replace perfect hedging with an expected loss constraint. Asymptotic analysis for small transaction costs is used to obtain a tractable model. A general expansion theory is developed using the dynamic programming approach. Explicit formulae are obtained in the special cases of exponential and power utility functions. As a corollary, we retrieve the asymptotics for the exponential utility indifference price. |
| Keywords: | Expected loss constraint, hedging, transaction cost, asymptotic expansion |
| JEL: | C61 G11 |
| URL: | http://d.repec.org/n?u=RePEc:chf:rpseri:rp1460&r=upt |
| By: | Hugo Bois |
| Abstract: | This paper discusses about a new framework to explain the decision-making process of modal choice. A specific approach, based on the behavioral framework developed by Ben-Akiva & Boccara (1987), is adopted to understand and analyze the decision processes of individuals. Precisely, we use the Analytic Hierarchy Process (AHP) to build the hierarchy of preferences from attitudes and perceptions. Through the preferences hierarchy, we can apply three different methods to better explain the decision processes; namely a standard compensatory model, a non-compensatory model based on the decision rules, and different possible weightings of the AHP method. The random utility maximization is predominantly used in the transportation literature because of its strong theoretical background, its success in predicting many types of human behavior, and the simplicity of mathematical and statistical analyses and model estimation it offers. Despite that, we believe that non-compensatory approaches are better suited to understand both travel behaviors and decision processes for transportation modes when taking active modes into account. These approaches allow us to better explain the impacts of each modal attributes on the one hand and to build psychological profiles with respect to decision rules on the other hand. Thus, it is possible to simulate shocks all things being equal. |
| Keywords: | Modal choice; Preferences; Decision rules; Hierarchical model; AHP. |
| Date: | 2016 |
| URL: | http://d.repec.org/n?u=RePEc:drm:wpaper:2016-25&r=upt |
| By: | Hummels, David (Purdue University); Munch, Jakob R. (University of Copenhagen); Xiang, Chong (Purdue University) |
| Abstract: | Increased job effort can raise productivity and income but put workers at increased risk of illness and injury. We combine Danish data on individuals' health with Danish matched worker-firm data to understand how rising exports affect individual workers' effort, injury, and illness. We find that when firm exports rise for exogenous reasons: 1. Workers work longer hours and take fewer sick-leave days; 2. Workers have higher rates of injury, both overall and correcting for hours worked; and 3. Women have higher sickness rates. For example, a 10% exogenous increase in exports increases women's rates of injury by 6.4%, and hospitalizations due to heart attacks or strokes by 15%. Finally, we develop a novel framework to calculate the marginal dis-utility of any non-fatal disease, such as heart attacks, and to aggregate across multiple types of sickness conditions and injury to compute the total utility loss. While the ex-ante utility loss for the average worker is small relative to the wage gain from rising exports, the ex-post utility loss is much larger for those who actually get injured or sick. |
| Keywords: | demand shocks, worker effort, health |
| JEL: | I1 F1 J2 F6 |
| Date: | 2016–07 |
| URL: | http://d.repec.org/n?u=RePEc:iza:izadps:dp10036&r=upt |
| By: | Luis A. Alcal\'a |
| Abstract: | A model of collective decisions made by a finite number of agents with constant but heterogeneous discount factors is developed. Collective utility is obtained as the weighted sum of all individual utilities with time-varying weights. It is shown that under standard separability assumptions, collective preferences may be nonstationary but still satisfy time consistency. |
| Date: | 2016–07 |
| URL: | http://d.repec.org/n?u=RePEc:arx:papers:1607.02688&r=upt |
| By: | Suen, Richard M. H. |
| Abstract: | This paper analyses the optimal saving behaviour of a risk-averse and prudent consumer who faces two sources of income risk: risk as described by a given probability distribution and risk in the distribution itself. The latter is captured by the randomness in the parameters underlying the probability distribution and is referred to as distributional risk. Stochastic volatility, which generally refers to the randomness in the variance, can be viewed as a form of distributional risk. Necessary and sufficient conditions by which an increase in distributional risk will induce the consumer to save more are derived under two specifications of preferences: expected utility preferences and Selden/Kreps-Porteus preferences. The connection (or lack of) between these conditions and stochastic volatility is addressed. The additional conditions under which a prudent consumer will save more under greater volatility risk are identified. |
| Keywords: | Stochastic volatility, stochastic convexity, precautionary saving. |
| JEL: | D81 D91 E21 |
| Date: | 2016–07–19 |
| URL: | http://d.repec.org/n?u=RePEc:pra:mprapa:72732&r=upt |
| By: | Carlo Sala (Swiss Finance Institute); Giovanni Barone-Adesi (University of Lugano and Swiss Finance Institute) |
| Abstract: | This paper proposes a theoretical analysis on the impacts of using a suboptimal information set on the three main components used in asset pricing, namely the risk physical and neutral measures and the relative pricing kernel. The analysis is carried out by means of a portfolio optimization problem for a small and rational investor. Solving for the maximal expected utility of terminal wealth, we prove the existence of an information premium between what is required by the theory, a complete information set thus a fully conditional measure, and what is instead achievable by en econometrician. Searching for the best bounds, we then study the impact of the premium on the pricing kernel. Finally, exploiting the strong interconnection between the pricing kernel and its densities, the extension to the risk-neutral measure follows naturally. |
| Keywords: | Portfolio optimization problem, Levy-Ito mixed model, Pricing kernel, Information premium, Optimal bounds |
| JEL: | G10 G11 G12 G14 C61 |
| URL: | http://d.repec.org/n?u=RePEc:chf:rpseri:rp1550&r=upt |
| By: | Pierre-Andre Chiappori (Dept. of Economics, Columbia University); Monica Costa Dias (Institute for Fiscal Studies); Costas Meghir (Cowles Foundation, Yale University) |
| Abstract: | We develop an equilibrium lifecycle model of education, marriage and labor supply and consumption in a transferable utility context. Individuals start by choosing their investmentsin education anticipating returns in the marriage market and the labor market. They then match based on the economic value of marriage and on preferences. Equilibrium in the marriage market determines intrahousehold allocation of resources. Following marriage households (married or single) save, supply labor and consume private and public commodities under uncertainty. Marriage thus has the dual role of providing public goods and offering risk sharing. The model is estimated using the British HPS. |
| Keywords: | Marriage market, Human capital, Labor supply, Life cycle models, Intrahousehold allocations, Collective model, Education choice, Returns to education |
| JEL: | J12 J16 J22 J24 H31 I24 I26 |
| Date: | 2015–03 |
| URL: | http://d.repec.org/n?u=RePEc:cwl:cwldpp:1994r&r=upt |
| By: | Paul Schneider (University of Lugano, EPFL, Swiss Finance Institute, and Boston University) |
| Abstract: | This paper introduces a decomposition of the market return in terms of higher-order realized, and option-implied risk aversion, connecting it to level, slope, and curvature of the implied volatility surface. Empirically, second-order risk aversion -- loss aversion -- explains most of the market return. Signals revealed by this risk anatomy provide predictive power out-of-sample for realized returns in particular for longer maturities. The decomposition also shows that compensation for disaster risk is not prominently featured in the market return. Furthermore it highlights that models with identically and independently distributed state variables are not suited to represent in particular longer-maturity returns. |
| Keywords: | equity premium, model-free, risk aversion, skewness |
| JEL: | C02 C23 C52 C61 G11 G12 |
| URL: | http://d.repec.org/n?u=RePEc:chf:rpseri:rp1561&r=upt |
| By: | Mathieu CAMBOU (Ecole Polytechnique Fédérale de Lausanne); Damir FILIPOVIC (Ecole Polytechnique Fédérale de Lausanne and Swiss Finance Institute) |
| Abstract: | This paper provides a coherent method for scenario aggregation addressing model uncertainty. It is based on divergence minimization from a reference probability measure subject to scenario constraints. An example from regulatory practice motivates the definition of five fundamental criteria that serve as a basis for our method. Standard risk measures, such as value-at-risk and expected shortfall, are shown to be robust with respect to minimum divergence scenario aggregation. Various examples illustrate the tractability of our method. |
| Keywords: | model uncertainty, scenario aggregation, expected shortfall, value-at-risk, statistical divergence, Swiss Solvency Test |
| URL: | http://d.repec.org/n?u=RePEc:chf:rpseri:rp1438&r=upt |
| By: | Per Krusell (Stockholm University); Timo Boppart (IIES, Stockholm University) |
| Abstract: | We argue that a stable utility function of consumption and hours worked for which income effects are slightly stronger than substitution effects can rationalize the long-run data for the main macroeconomic quantities. In these long-run data, in the U.S. as well as in other countries, as productivity grows at a steady rate, hours worked fall slowly and at an approximately constant rate. We narrow down the set of preferences consistent with balanced growth under constant (negative) hours growth. The resulting class amounts to a slight enlargement of the well-known “balanced-growth preferences†that dominate the macro literature and are based on requiring constant hours worked. Thus, hours falling at a constant rate is not inconsistent with the remaining balanced-growth facts but merely requires a slight broadening of the preference class considered. The broadening of the preference class introduces some well-known cases not previously thought to be consistent with balanced growth. From our perspective, we interpret the recent decades of stationary hours worked in the U.S. as a temporary departure from a long-run pattern, and to the extent productivity will keep growing, we predict that hours will fall further. |
| Date: | 2016 |
| URL: | http://d.repec.org/n?u=RePEc:red:sed016:261&r=upt |
| By: | Luisa Herbst |
| Abstract: | In contest experiments, overbidding is a widely observed phenomenon. One common explanation for overbidding is that winning in itself yields utility, termed the joy of winning. However, the joy of winning is difficult to observe and to quantify. This paper develops a novel, incentivized way to measure the individual-specific joy of winning as well as the frustrationof losing in a Tullock lottery contest. We find that the willingness to pay for a restart of the contest differs between winners and losers. Compared to a theoretical benchmark, winners are more satisfied and overbid forrestart of the contest, while losers are less satisfied and underbid. Further, effort levels are higher in the second contest, which can be explained by selection of high effort types with a high joy of winning into the restarted contest. |
| Keywords: | Bidding, contest, desire to win, effort, experiment, emotions, joy of winning, love of winning, overexpenditure |
| JEL: | C78 C91 D72 D74 |
| Date: | 2016–04 |
| URL: | http://d.repec.org/n?u=RePEc:mpi:wpaper:tax-mpg-rps-2016-06&r=upt |
| By: | Aart Gerritsen |
| Abstract: | The usual method of solving for an optimal nonlinear tax schedule is that of the primal approach – ï¬ rst solving for the optimal allocation, and subsequently determining which tax system decentralizes this allocation. While this method is mathematically rigorous, it lacks intuitive appeal. I propose a different method based on the dual approach – directly solving for the optimal tax system – which is equally rigorous, while being much closer in spirit to actual tax policy. I show that this approach can easily incorporate preference heterogeneity, as well as individual behavior that is not fully consistent with utility maximization. Over and above solving for the optimum, the dual approach allows one to obtain new insights into the welfare effects of small nonlinear tax reforms outside the optimum. |
| Keywords: | Optimal taxation, dual approach, preference heterogeneity, individual misoptimization, tax reforms |
| JEL: | H21 H23 H24 |
| Date: | 2016–01 |
| URL: | http://d.repec.org/n?u=RePEc:mpi:wpaper:tax-mpg-rps-2016-02&r=upt |
| By: | Martin Herdegen (ETH Zurich); Johannes Muhle-Karbe (ETH Zurich and Swiss Finance Institute) |
| Abstract: | We study the sensitivity of optimal consumption streams with respect to perturbations of the random endowment. At the leading order, the consumption adjustment does not matter: any choice that matches the budget constraint simply shifts the original utility by the marginal value of the perturbation. Nontrivial results obtain at the next-to-leading order. Here, one first solves the problem for a deterministic perturbation, which leads to a "prognosis measure". The desired consumption adjustment for a general endowment perturbation is in turn given by the conditional expectation of the latter, computed under this measure and appropriately weighted with the conditional expectations of the remaining risk-tolerance. |
| Keywords: | optimal consumption, random endowment, asymptotic analysis |
| JEL: | G11 D91 E21 |
| URL: | http://d.repec.org/n?u=RePEc:chf:rpseri:rp1527&r=upt |
| By: | Semyon MALAMUD (Ecole Polytechnique Fédérale de Lausanne and Swiss Finance Institute) |
| Abstract: | How does private information get incorporated into option prices? To study this question, I develop a non-linear, noisy rational expectations equilibrium model with asymmetric information and a full menu of call and put options available for trading. The model allows for an arbitrary distribution of the underlying payoff and general trader preferences. All equilibrium prices and portfolios are obtained in closed form. Learning from option prices is characterized explicitly in terms of a novel object, the second derivative of the signal-to-noise ratio, whose sign determines whether particular shapes of option prices are "good news". I show that there is a major difference in equilibrium behaviour for constant absolute risk aversion (CARA) and non-CARA preferences due to feedback effects between wealth, price discovery, and private information: First, if informed traders have non-CARA preferences, strong-form efficiency is always obtained, independent of the amount of noise trading. Second, when informed traders have CARA preferences, the nature of equilibrium changes drastically with wealth effects of uninformed trading: If the latter have non-CARA preferences, weak-form efficiency fails and option prices are not sufficient to recover the information contained in the aggregate demand. |
| Keywords: | options, asymmetric information, learning, informational efficiency |
| JEL: | G14 D53 D82 D84 |
| URL: | http://d.repec.org/n?u=RePEc:chf:rpseri:rp1436&r=upt |
| By: | Agata M. Lozinskaia (National Research University Higher School); Evgeniy M. Ozhegov (National Research University Higher School); Alexander M. Karminsky (National Research University Higher School) |
| Abstract: | This paper investigates the distribution of relative credit losses given mortgage default for loans provided by a major government-sponsored creditor in a local area. We use borrower’s individual and loan-level data on residential mortgages originated in the period 2008–2012. Our numerical analysis indicates that mortgages bunching at certain Loan-to-Value ratios (LTV) led to a discontinuity in relative credit loss given mortgage default. Through regression analysis, we demonstrate discrete jumps in the approximated historical credit losses generated by loans with a high LTV ratios and find thresholds allowing the segmentation of loans according their credit risk. In addition, our results suggest that mortgage insurance is a potentially valuable instrument for compensation for expected loss in certain risk segments. |
| Keywords: | discontinuity; credit risk; mortgage default; government mortgage lending programs; loss evaluation. |
| JEL: | C21 G21 G32 R20 R58 |
| Date: | 2016 |
| URL: | http://d.repec.org/n?u=RePEc:hig:wpaper:55/fe/2016&r=upt |
| By: | Jason A. Aimone; Sheryl Ball; Brooks King-Casas |
| Abstract: | Recent studies have suggested that there may be a correlation between the order in which decision-makers collect information about risky gambles and their tendency to make expected value maximizing decisions. In this work we present results from an experiment designed to establish causality by exogenously manipulating the order in which participants view information about probabilities and payoffs. We find that there is a relationship between information presentation and the amount of risk participants take. This suggests that the choice architecture for real world risky decision making may have both intended and unintended consequences. |
| Keywords: | Risky Decision Making, Expected Value Maximization, Nudging |
| Date: | 2016 |
| URL: | http://d.repec.org/n?u=RePEc:vpi:wpaper:e07-52&r=upt |
| By: | Aart Gerritsen |
| Abstract: | The social welfare implications of income tax policy are shown to critically depend on whether or not labor markets are rationed – i.e., on the existence of involuntary unemployment. With rationed labor markets, raising taxes on the employed and transfers towards the unemployed might improve both equity and efficiency. It improves equity by redistributing income from the employed to the unemployed; it improves efficiency as it encourages people with a small utility surplus of employment to exit the labor market, leaving their jobs for people with a higher utility surplus. I derive conditions under which this result continues to hold when only part of the labor market is rationed. I also show that conventional tax incidence results break down in rationed labor markets. |
| Keywords: | Involuntary unemployment, inefficient rationing, optimal taxation |
| JEL: | H21 J21 E24 |
| Date: | 2016–03 |
| URL: | http://d.repec.org/n?u=RePEc:mpi:wpaper:tax-mpg-rps-2016-04&r=upt |
| By: | Paul SCHNEIDER (University of Lugano and Swiss Finance Institute) |
| Abstract: | This paper develops an optimal trading strategy explicitly linked to an agent's preferences and assessment of the distribution of asset returns. The price of this strategy is a portfolio of implied moments, and its expected excess returns naturally accommodate compensation for higher-order moment risk. Variance risk and the equity premium approximate it to first order and it nests cross-sectional asset pricing models such as the CAPM. An empirical study in the US index market compares the investment behavior of an agent with recursive long-run risk preferences to one who merely uses an i.i.d. time series model and takes market prices as given. The two agents exhibit very similar behavior during crises and can be distinguished mostly during calm periods. |
| Keywords: | Predictability, pricing kernel, model risk, trading strategy, model-free, variance premium, skew premium, kurtosispremium OR from SSRN: Preference trading, pricing kernel, model risk, trading strategy, model-free, variance premium, equity premium, skew premium, kurtosis premium |
| JEL: | C02 C23 C52 C61 G11 G12 |
| URL: | http://d.repec.org/n?u=RePEc:chf:rpseri:rp1429&r=upt |
| By: | Ibrahim Abada; Andreas Ehrenmann |
| Abstract: | In resource based economies, regulating the production and export activities have always been an important challenge. Examples in oil and gas show that different behaviors have been adopted ranging from the export monopoly to the complete opening of the export market. This paper tries to explain this multitude of solutions via strategic interactions. When modeling imperfect competition, players are separated in two categories: those who exert market power and those who are competitive and propose the good at their marginal supply cost. Letting a player freely choose whether it wants to exert market power or not when it optimizes its utility is not discussed in the literature. This paper addresses this issue by letting the players choose the level of competition they want to exert in the market. To do so, we analyze the behavior of two countries competing to supply a market with a homogeneous good in an imperfect competition setting. Each country decides the number of firms it authorizes to sell in the market. The interaction between the firms is of a Nash-Cournot type, where each one exerts market power and is in competition with all other firms allowed to sell, whether they belong to the same country or not. Each country optimizes its utility, that is the sum of the profits of its firms. We have studied four kinds of interaction between the countries. The first calculates the closed loop Nash equilibrium of the game between the countries. The second setup analyzes the cartel when the countries collude. The third focuses on the open loop Nash equilibrium and the fourth models a bi-level Stackelberg interaction where one country plays before the other. We demonstrate that in the closed loop Nash equilibrium, our setting leads to the prisoner’s dilemma: the equilibrium occurs when both countries authorize all their firms to sell in the market. In other words, countries willingly chose not to exert market power. This result is at first sight similar to the Allaz & Vila (1993) result but is driven by a completely different economic reasoning. In the Stackelberg and coordinated solutions, the market is on the contrary very concentrated and the countries strongly reduce the number of firms that enter the market in order to fully exert market power and increase the price. The open loop result lies in between: the countries let all their firms sell but market power remains strong. These results suggest that the prisoner’s dilemma outcome is due to the conjectural inconsistency of the Nash equilibrium. Finally, in the Stackelberg setting, we give countries the choice of being leader or follower and demonstrate that the counter-intuitive competitive outcome is very unlikely to occur in the market. |
| Keywords: | Imperfect competition, export oligopoly, open and closed loop Nash equilibrium |
| JEL: | L13 L7 |
| Date: | 2016–07–21 |
| URL: | http://d.repec.org/n?u=RePEc:cam:camdae:1641&r=upt |