Isoelastic Demand Function Research Articles

Dynamics of a Cournot game with bounded rational firms and various scale effects.

In this paper, we focus on a dynamic Cournot game in the market with a nonlinear (isoelastic) demand function. In our model, there are N competing firms featured by nonlinear cost functions, which enhances our model's resemblance to real-world scenarios. Firstly, we focus on the homogeneous case where firms' marginal costs change at equal rates. We establish analytical expressions of the market supply at equilibrium and perform comparative static analysis. In addition, we investigate the local stability under different economies of scale and show that there could be multiple stable equilibria if firms face economies of scale. The heterogeneous case where firms' marginal costs change at distinct rates is much more complex, thus we investigate the duopoly game with only two firms involved. Methods of symbolic computations such as triangular decomposition and partial cylindrical algebraic decomposition are employed in the analytical investigations of the equilibrium, which is nearly impossible by using the pencil-and-paper approach since the closed-form equilibrium is quite complicated. According to the computational results, we derive that two stable positive equilibria may coexist if both firms face economies of scale. Additionally, we conduct preliminary numerical simulations and find two different types of complex dynamics of the model considered in this paper: complex trajectories such as periodic and chaotic orbits may appear; the topological structure of the basins of attraction may be complex.

Comparison between Simultaneous and Sequential Movements in a Duopoly Game

In general, a Stackelberg competition benefits the leader and the consumer due to making the first move in a sequential game. However, in this paper, sequential movement does not offer an advantage over simultaneous movement in a static duopoly under quantity competition with an isoelastic demand function. By introducing dynamics into the duopoly model through numerical simulations of simultaneous and sequential movements towards the Cournot equilibrium, there is evidence of first-mover advantage. However, simultaneous movement is advantageous over sequential movement in terms of better profits for both firms and lower prices for the consumer.

Big or small? A new economic geography model with an endogenous switch in the market structure

We present a new economic geography (NEG) model with a linear demand function where firms may change the perception of their relative dimension with respect to the local market. If they perceive themselves as big, they behave as Cournot oligopolists, otherwise they behave as monopolistic competitors. We first compare the pure cases in which only one market form prevails in the two-region economy. Comparing these pure cases of monopolistic and oligopolistic competition only a quantitative difference emerges. Subsequently, we assume that firms switch behavior and start to interact strategically when the number of local firms is below a threshold; in that case, the market form evolves endogenously. Results change substantially. ‘Break’ and ‘sustain’ points are separated as in standard NEG models with an isoelastic demand function leading to a co-existence of equilibria. Stable partial agglomeration and oscillations with small amplitude are possible. The dynamics are described by a 1D piecewise smooth map, which can be continuous (in the pure monopolistic or oligopolistic cases) or discontinuous (when the market structure evolves endogenously). We analyze the bifurcation structure of the parameter space of the map comparing these cases. We show that the continuous maps have rather standard dynamics, while the discontinuous map is characterised by border collision bifurcations of fixed points and cycles, which lead to rich and complex bifurcation structures.

Stability, multi-stability and instability in Cournot duopoly game with knowledge spillover effects and relative profit maximization

A dynamic Cournot duopoly model with isoelastic demand function and knowledge spillover effects is established, in which the goals of two enterprises are relative profit maximization. Since these two enterprises only have incomplete information of the market, then it is assumed that the enterprises are boundedly rational, and they need to adjust their production decision in next period according to the gradient adjustment mechanism. Sufficient conditions for the existence and stable properties of unique Nash equilibrium can be judged with the help of the Jury criterion. The size of stability region is related to the values of parameters and the system has two different paths to chaos, one is via flip bifurcation and another is via Neimark-Sacker bifurcation. Finally, the complex dynamic characteristics also can be further studied by using numerical simulation, including the structures of multi-layer Arnold's tongues and coexistence of attractors.

Local Stability of Cournot Equilibrium as the Number of Firms Increases

In this paper, a Cournot oligopoly with isoelastic demand function and constant marginal cost is considered. The local stability conditions of the Cournot equilibrium are determined for four models with different decision mechanisms. In the first model, firms adjust their outputs using the best reply response with naive expectations. The second model is a generalization of the first one, where firms have adaptive expectations. Meanwhile, the third and fourth models adopt the bounded rationality and local monopolistic approximation, respectively. The results show that, in the case of identical firms, the Cournot equilibrium is always stable when the firms adopt the local monopolistic approximation mechanism.

Decision Analytic Pricing with Constant Price Elasticities

This article introduces a new pricing model which uses decision analysis and isoelastic demand functions. Using the methodologies discussed here will enable companies to choose prices that will maximize the profit of a given product based on the state of the economy. An exponential utility function is used to incorporate the risk attitude of the company. The use of the model is demonstrated through a case study on 2018 Chevrolet Malibus. The proper use of the model will “take the guesswork out” of the pricing decision and help companies make pricing decisions using the best available data. Keywords: Pricing, Decision Analysis, Elasticity, Utility.

Quantum Cournot duopoly game with isoelastic demand function

This paper studies the quantum Cournot duopoly games with isoelastic demand function and unequal marginal costs by using the Li–Du–Massar and the Frąckiewicz quantum schemes. The influences of relative marginal cost and degree of quantum entanglement on the optimal profits of the two players are analyzed theoretically and illustrated numerically. The results show that the profit of one player increase, but the profit of the other player decreases with increasing the relative marginal cost for any fixed degree of quantum entanglement. The profits of two players both increase with increasing the degree of quantum entanglement as the relative marginal cost is in a certain range.

Quantum Stackelberg duopoly game with isoelastic demand function

We apply Frąckiewicz's scheme to investigate a quantum Stackelberg duopoly game with isoelastic demand function and unequal marginal costs. We analyze the existence conditions of quantum equilibrium, and the influences of relative marginal cost (m) and degree of quantum entanglement (γ) on the optimal profits and the first-mover advantage in theory. We demonstrate the theoretical analysis by numerical simulations. The results show that an increase of m will increase one player's profit, but will decrease the other player's profit for any fixed γ. An increase of γ will increase both players' profits for any fixed m. An increase of γ will enhance the first-mover advantage for 0<m<1, but will suppress it for 1<m<2.

A Dynamic Duopoly Model: When a Firm Shares the Market with Certain Profit

The current paper analyzes a competition of the Cournot duopoly game whose players (firms) are heterogeneous in a market with isoelastic demand functions and linear costs. The first firm adopts a rationally-based gradient mechanism while the second one chooses to share the market with certain profit in order to update its production. It trades off between profit and market share maximization. The equilibrium point of the proposed game is calculated and its stability conditions are investigated. Our studies show that the equilibrium point becomes unstable through period doubling and Neimark–Sacker bifurcation. Furthermore, the map describing the proposed game is nonlinear and noninvertible which lead to several stable attractors. As in literature, we have provided an analytical investigation of the map’s basins of attraction that includes lobes regions.

Distribution of profit in a smart phone supply chain under isoelastic demand

This study investigates a novel supply chain (smart phone) where end consumers purchase handsets (smart phones) and service either separately (free channel) or together as a package (bundled channel). Our study considers three different power structures for the bundled channel – the handset manufacturer is the Stackelberg leader, the service operator is the Stackelberg leader and finally both the players (handset manufacturer and service provider) decide simultaneously – Vertical Nash. Previous studies have used a linear deterministic demand function to show that the supply chain player possessing superior channel power gains superior profit. We use an isoelastic deterministic demand function to point out anomalies or contradictions in already existing smart phone supply chain literature. Under isoelastic deterministic function that represents demand for smart phones we show that the supply chain player possessing superior channel power need not earn superior profits.

Monopoly with differentiated final goods and heterogeneous markets

Abstract This work attempts to characterise the dynamic properties of a nonlinear model in which a monopolist produces a fixed amount of an intermediate good. The firm employs this good in producing two vertically differentiated final commodities, sold in two distinct markets. Consumers’ preferences are described by quasi-linear and quadratic utility functions respectively yielding isoelastic and linear demand functions. In addition, we assume that the monopolist adopts a gradient adjustment rule based on profit variation to adjust its choice about the amount of intermediate good to employ in the production of each final good. Two alternative scenarios emerge: in the first, there exists coexistence of two markets; in the second, the monopolist specialises on the production of a single final good. The dynamics show that the elasticities of market demands play an ambiguous role in affecting the stability of the equilibrium, whereas the speed of adjustment unambiguously destabilises the system. Moreover, the article investigates the role of the productive capacity and the gap in marginal costs in defining the different scenarios. In particular, economic dynamics may be chaotic and/or multiple attractors may appear.

Emission reduction and profit-neutral permit allocations

The present paper addresses two policy objectives: to implement a market for pollution permits and to make regulation acceptable for businesses. Profit-neutral permit allocations are defined as the number of permits that the regulator should give for free so that post-regulation profits (i.e. a firm's profits in the products market plus the value of the allowances granted for free) are equal to pre-regulation profits. The proposed model is developed by assuming that firms use polluting technologies and compete “à la Cournot”. The paper demonstrates that a low number of free allowances is sufficient to meet these two goals. Moreover, the regulator can fully offset losses, even when the reduction in emissions is high, provided that the sectors concerned are not monopolies, both for isoelastic and linear demand functions.

Chaotic dynamics in a three-dimensional map with separate third iterate: The case of Cournot duopoly with delayed expectations

We consider a Cournot duopoly with isoelastic demand function and constant marginal costs. We assume that both producers have naive expectations but one of them reacts with delay to the move of its competitors, due to a “less efficient” production process of a competitor with respect to its opponent. The model is described by a 3D map having the so-called “cube separate property”, that is its third iterate has separate components. We show that many cycles may coexist and, through global analysis, we characterize their basins of attraction. We also study the chaotic dynamics generated by the model, showing that the attracting set is either a parallelepiped or the union of coexisting parallelepipeds. We also prove that such attracting sets coexist with chaotic surfaces, having the shape of generalized cylinders, and with different chaotic curves.

Neimark–Sacker bifurcation analysis and complex nonlinear dynamics in a heterogeneous quadropoly game with an isoelastic demand function

In this paper, a nonlinear quadropoly game based on Cournot model with fully heterogeneous players is established. This game extends the model introduced by Tramontana and Elsadany (Nonlinear Dyn 68:187–193, 2012) who considered a heterogeneous triopoly game with an isoelastic demand function. Here, four different types of players and potentially different marginal costs are considered. Moreover, the assumption of an isoelastic demand function increases the nonlinearity of the final four-dimensional map. The stability of the resulting discrete-time dynamical system is analyzed. The existence of Neimark–Sacker bifurcation near the Nash equilibrium point of the game is shown. Also, based on the Kuznetsov’s normal form technique for discrete-time system, the stability of the Neimark–Sacker bifurcation is also discussed which indicates that the bifurcation is supercritical. Moreover, it is shown that the Nash equilibrium point of the game undergoes period-doubling (flip) bifurcation. Furthermore, the double route to chaotic dynamics in this model, via flip bifurcations and via Neimark–Sacker bifurcation of the Nash equilibrium point, is illustrated. Coexistence of multi-chaotic attractors of the model is illustrated. Simulation tools like bifurcation diagrams, stability regions of parameters, Lyapunov exponent spectrum, phase plots and basins of attraction are used to verify the complex dynamics of the game.

Nonlinear oligopolistic game with isoelastic demand function: Rationality and local monopolistic approximation

Isoelastic demand function have been used in literature to study the dynamic features of systems constructed based on economic market structure. In this paper, we adopt the so-called Cobb–Douglas production function and study its impact on the steady state of an oligopolistic game that consists of four oligopolistic competitors or firms. Briefly, the paper handles three different scenarios. The first scenario introduces four oligopolistic firms who plays rational against each other in market. The firms use the myopic mechanism (or bounded rational) to update their production in the next time unit. The steady state of the obtained system in this scenario, which is the Nash equilibrium, is unique and its characteristics are investigated. Based on a local monopolistic approximation (LMA) strategy, one competitor prefers to play against the three rational firms and this is illustrated in the second scenario. The last scenario discusses the case when three competitors use the LMA strategy against a rational one. For all scenarios discrete dynamical systems are used to describe the game introduced in all scenarios. The stability analysis of the Nash equilibrium is investigated analytically and some numerical simulations are used to confirm the obtained analytical results.

Optimal rebate strategies in a two-echelon supply chain with nonlinear and linear multiplicative demands

We examine the pure rebate strategies in a two-echelon supply chain under stochastic demand with multiplicative error. Given exogenous wholesale price and retail price, we characterize the unique Nash equilibrium when both manufacturer and retailer provide rebate policy to consumers under nonlinear and linear price-dependent demand functions, including iso-elastic multiplicative demand function (EMDF) and linear multiplicative demand function (LMDF). Based on a game theoretical framework, we prove that there still exists a unique equilibrium when the price elasticity is rather small with constraint conditions in the former case. We also find that in this case the retailer(manufacturer) may increase its rebate value in reaction to the manufacturer's(retailer's) rebate value in order to stimulate sales, which is contrary to the conventional wisdom that the retailer(manufacturer) will shrink its rebate value to gain an ``extra advantage unfairly. As a result, both parties share the same profit at equilibrium. Further, we compare the expected profit outcomes at equilibrium among joint-rebate game, single-party rebate game and no-rebate game by using numerical examples. It is shown that the joint-rebate policy is not always dominates the others unless the price elasticity is sufficiently flexible.

Heterogeneity and the (de)stabilizing role of rationality

In this paper we study oligopolies of generic size consisting of heterogeneous firms, which adopt best response adjustment mechanisms with either perfect foresight (rational firms) or static expectations (naive firms). Assuming an isoelastic demand function and possibly different marginal costs for the two groups of firms, we focus on the local stability of the Nash equilibrium. We show that, with respect to the oligopoly composition, described in terms of the fraction of rational firms, different scenarios are possible. We find that a high rationality degree may not always guarantee stability, in particular when rational firms have sufficiently larger marginal costs. In fact, in this situation, increasing the fraction of rational firms can even introduce instability. Besides the usual scenarios in which replacing some naive firms with rational ones leads to a stabilization of (or at least keeps unchanged) the dynamics, we provide a family of situations, characterized by costs ratio favorable to naive firms, in which equilibrium loses its stability when naive firms are replaced by rational ones. The results we present are both analytical and simulative.

Two different routes to complex dynamics in an heterogeneous triopoly game

We study a triopoly game with heterogeneous players. The market is characterized by a nonlinear (isoelastic) demand function and three competitors. The main novelty is the double route to complex dynamics that we find and is quite rare in heterogeneous triopoly models. We show that the two routes have important implications for the economic interpretation of the dynamics emerging when the Cournot–Nash equilibrium becomes locally unstable. Moreover the model displays multistability of different attractors, requiring a global analysis of the dynamical system.

Nonlinear dynamics and convergence speed of heterogeneous Cournot duopolies involving best response mechanisms with different degrees of rationality

In this paper, we propose and compare three heterogeneous Cournotian duopolies, in which players adopt best response mechanisms based on different degrees of rationality. The economic setting we assume is described by an isoelastic demand function with constant marginal costs. In particular, we study the effect of the rationality degree on stability and convergence speed to the equilibrium output. We study conditions required to converge to the Nash equilibrium and the possible route to destabilization when such conditions are violated, showing that a more elevated degree of rationality of a single player does not always guarantee an improved stability. We show that the considered duopolies exhibit either a flip or a Neimark–Sacker bifurcation. In particular, in heterogeneous oligopolies models, the Neimark–Sacker bifurcation usually arises in the presence of a player adopting gradient-like decisional mechanisms, and not best response heuristic, as shown in the present case. Moreover, we show that the cost ratio crucially influences not only the size of the stability region, but also the speed of convergence toward the equilibrium.

Nonlinear dynamics and global analysis of a heterogeneous Cournot duopoly with a local monopolistic approach versus a gradient rule with endogenous reactivity

We study a heterogeneous duopolistic Cournotian game, in which the firms, producing a homogeneous good, have reduced rationality and respectively adopt a “Local Monopolistic Approximation” (LMA) and a gradient-based approach with endogenous reactivity, in an economy characterized by isoelastic demand function and linear total costs. We give conditions on reactivity and marginal costs under which the solution converges to the Cournot–Nash equilibrium. Moreover, we compare the stability regions of the proposed oligopoly to a similar one, in which the LMA firm is replaced by a best response firm, which is more rational than the LMA firm. We show that, depending on costs ratio, the equilibrium can lose its stability in two different ways, through both a flip and a Neimark–Sacker bifurcation. We show that the nonlinear, noninvertible map describing the model can give rise to several coexisting stable attractors (multistability). We analytically investigate the shape of the basins of attractions, in particular proving the existence of regions known in the literature as lobes.