Luca Lambertini

• Professor
• Professor at University of Bologna

This paper identifies a class of state-redundant games whose open-loop solution is a degenerate feedback one. The structure of the game features an instantaneous payoff which is at least cubic, and state equations linear in states and controls. The model can be used to illustrate games in the field of industrial organization and management, involvi...

Typical problems of negative effects of CO\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{2}$$\end{document} emissions are that (i) they are suffered and generated no...

Perseverance in R&D effort is the first fundamental step towards any kind of innovation. We investigate the beginning of the innovation story, rather than its end, through duration models. Among the drivers of our unconventional IO approach, we focus on heterogeneity, path dependence and market power, measured as elasticity of firm-specific demand...

We analyse a model of environmental regulation where two firms can optimally decide to invest in an emission abatement technology and the regulator taxes firms’ emissions in a time-consistent manner. Depending on the values of the parameters measuring the extent of emission abatement that firms may achieve and the degree of product differentiation,...

We consider a resource extraction model with the possibility of extending the resource stock by drilling efforts and an oligopolistic competition of symmetric firms. Assuming a hyperbolic market demand, we adopt the open-loop Nash equilibrium to analyze and compare the outcomes of a private vs a common resource pool. For both cases, the drilling ef...

Citizens are expected to play a great role in the future global energy transition, being able to give a decisive contribution to limit global warming to 1.5° and avoid the worst consequences. Empowering citizens is crucial and assigning them the role of prosumers in the new energy market is necessary to ensure a sustainable and fair pathway to the...

We participate in the lasting debate about the persistence of monopolies under technological change, by examining two deterministic games modelling innovation auctions. We highlight some novel aspects within such debate. If product innovation is at stake, the joint effect of diseconomies of scope and product differentiation may allow the entrant to...

We address the issue of regulating both polluting emissions through a generic tax and access to a common resource pool in a dynamic oligopoly game. Our analysis shows that once industry structure is regulated so as to induce the industry to harvest the resource in correspondence of the maximum sustainable yield, social welfare is either independent...

As is well known, zero-sum games are appropriate instruments for the analysis of several issues across areas including economics, international relations and engineering, among others. In particular, the Nash equilibria of any two-player finite zero-sum game in mixed-strategies can be found solving a proper linear programming problem. This chapter...

We revisit a well known differential Cournot game with polluting emissions, to propose a version of the model in which environmental taxation is levied on emissions rather than the environmental damage. This allows to attain strong time consistency under open-loop information, and yields two main results which can be summarized as follows: (i) to a...

Is socially efficient taxation conducive to the win‐win solution associated with the strong version of the Porter Hypothesis? Using a Bertrand duopoly yielding a continuum of Nash equilibria, we show that this is true for almost any level of environmental damage and equilibrium pricing strategy. We also prove that the only case in which no conflict...

A well established dynamic model describing the impact of oligopolistic interaction on a renewable resource is revisited here to illustrate its dual interpretation as a waste removal differential game. The regulatory implications are illustrated by assuming that the public agency may control market price and possibly also access to the commons. Two...

Zero-sum games are a well known class of game theoretic models, which are widely used in several economics and engineering applications. It is known that any two-player finite zero-sum game in mixed-strategies can be solved, i.e., one of its Nash equilibria can be found solving a linear programming problem associated to it. The idea of this work is...

The Porter hypothesis and the pollution haven hypothesis seem to predict opposite reactions by firms facing environmental regulation, as the first invokes the arising of a win–win solution while the second envisages the possibility for firms to flee abroad. We illustrate the possibility of designing policies (taking the form of either emission taxa...

We investigate the Cournot–Ramsey differential game using a homogeneous good oligopoly with three different specifications of market demand: convex, linear and concave, allowing us to obtain fully analytical solutions in the presence of linear cost functions. The analysis of the static game predicts that the ranking of equilibrium quantities across...

We determine the emergence of the Porter hypothesis in a large oligopoly setting where the industry-wide adoption of green technologies is endogenously determined as a result of competition among coalitions. We examine a framework where firms decide whether to be “brown” or “green” and compete in quantities. We find that the Porter hypothesis may e...

We investigate the relationship between market concentration and industry innovative effort within a familiar two-stage model of R&D race in which firms compete à la Cournot in the product market. With the help of numerical simulations, we show that such a setting is rich enough to generate Arrovian, Schumpeterian, and inverted U curves. We interpr...

We revisit the discussion about the relationship between price’s cyclical features, implicit collusion and the demand level in an oligopoly supergame where a positive shock may hit demand and disrupt collusion. The novel feature of our model consists in characterising the post-shock noncooperative price and comparing it against the cartel price pla...

A dynamic Hotelling monopoly model is proposed, with the purpose of characterising the effects of informative and persuasive advertising in a regime of partial market coverage. The two models are solved to show the emergence of a unique saddle-point steady-state equilibrium in both cases. Then, equilibrium profits are compared to illustrate that th...

We revisit the spatial duopoly model à la Hotelling (1929), to show that, provided the parameter scaling marginal cost is sufficiently high, quadratic production costs guarantee equilibrium existence in presence of linear transportation costs and a uniform distribution, with minimum product differentiation and no undercutting. We also discuss the c...

This paper analyzes whether consumers’ misperception on the quality of the product influences firms’ investment choices. We examine a setup with horizontal and vertical (green) differentiation, where consumers are heterogeneous in the exogenous perception of environmental quality. Demands, true qualities and profits are increasing in the perception...

In this paper we show that environmental consciousness may act as a substitute for environmental regulation. We consider a vertically differentiated duopoly in which the high quality firm pollutes more than the low quality rival. Consumers attach a positive value to the green firm, while stigmatizing the brown one. For relatively high values of thi...

We revisit the interplay between differentiation and divisionalization in a duopoly version of Ziss (Econ Lett 59:133–138, 1998). We model divisionalization as a discrete problem to prove that (i) firms may choose not to become multidivisional; and (ii) there may arise asymmetric outcomes in mixed strategies, due to the existence of multiple symmet...

We reformulate the Verhulst–Lotka–Volterra model of natural resource extraction under the alternative assumptions of Cournot behaviour and perfect competition, to revisit the tragedy of commons vs the possibility of sustainable harvesting. After a brief layout of the open-loop solution including the Ramsey rule, we rely on the state-redundancy prop...

We show that managerial delegation based upon comparative performance may generate collusive outcomes observationally equivalent to those typically associated with repeated games or cross ownership. This happens when rivals’ profits are positively weighted in the managerial incentive scheme. We also identify the level of time discounting at which a...

We model an industry in which a discrete number of firms choose the output of their differentiated products, deciding whether or not to consider the impact of their decisions on aggregate output. The firm’s choice of ignoring the impact of its production on aggregate output, which is typical of monopolistic competition, is derived as an equilibrium...

A leitmotiv of the analysis of marketing channels’ behaviour is the possibility of designing contractual relations so as to replicate the performance of vertically integrated firms, whenever this is efficient for firms. This is particularly relevant when the vertical externality provokes distortions in the firms’ incentives to invest in R&D or adve...

The optimal design of two-part tariffs is investigated in a dynamic model where two firms belonging to the same supply chain invest in R&D (research and development) activities to increase the perceived quality of the final product. It is shown that the replication of the vertically integrated monopolist’s performance can be attained using a two-pa...

We revisit the adoption of voluntary export restraints (VERS) in the differential Cournot game with sticky price and intraindustry trade by Dockner and Haug (Can J Econ 3:679–685, 1991). The analysis relies on linear and nonlinear feedback strategies, to encompass the special cases considered in Fujiwara (Aust Econ Pap 49:101–110, 2010). We show th...

In a homogeneous oligopoly, under standard regularity conditions, we prove that the Cournot–Nash equilibrium emerges under price competition and Cournot conjectures. We illustrate this result also under exogenous product differentiation.

Cambridge Core - Econometrics and Mathematical Methods - Differential Games in Industrial Economics - by Luca Lambertini

We study a class of games featuring payoff functions where best reply functions are orthogonal and therefore the pure-strategy non-cooperative solution is attained as a Nash equilibrium in dominant strategies. We prove that the resulting threshold of the discount factor above which implicit collusion on the Pareto frontier is stable in the infinite...

Multiplicity of equilibria under supply function competition is a two-layer problem. To prove it, we investigate an extended game with observable delay under duopolistic competition in affine supply functions. Firms use the intercepts of supply functions as their strategic variables. Best replies are downward (upward) sloping if the common slope of...

This paper aims at providing an explanation of the observed espresso price dispersion across major Italian cities. We present some preliminary empirical evidence that suggests a positive relationships between the average espresso price in a city and the number of coffee shops (normalized for the adult population) operating in that city. The finding...

This review summarizes the extant debate on the interplay between oligopolistic behavior and policy stimuli in determining firms’ green R&D efforts. It encompasses models based on the representative consumer and discrete choice approaches, under both Cournot and Bertrand competition. It also draws on the growing literature on environmental quality,...

We examine the relationship between competition and innovation in an industry where production is polluting and R&D has the aim to reduce emissions (“green” innovation). We present an n-firm oligopoly where firms compete in quantities and decide their investment in “green” R&D. We analyse the case where the emission tax is set endogenously by a com...

Firms' innovation portfolios include several dimensions ranging from organizational aspects to cost reduction and product characteristics. All of these efforts take place during the product life cycle, and interact with each other in determining the spectrum of features of the product and its performance on the market. This paper contributes to the...

We show that a frequently used direct demand system with product differentiation in a duopoly market generates unexpected effects of increasing the substitutability of firms’ products on prices, outputs, profits and welfare. Using the original demand system introduced by Bowley (The Mathematical Groundwork of Economics, Oxford, Oxford University Pr...

The separation between ownership and control has become common practice over the last century, in most medium and large firms across the world. Throughout the twentieth century, the theory of the firm and the theory of industrial organization developed parallel and complementary views on managerial firms. This book offers a comprehensive exposition...

I propose a differential oligopoly game of resource extraction under linear and nonlinear feedback strategies, where firms are managerial and delegation contract are based on output levels. The model shows that delegation expands the set of stable nonlinear feedback equilibria as well as the residual steady state resource stock. Additionally, the s...

We show that the standard argument according to which supply function equilibria rank intermediate between Bertrand and Cournot equilibria may be modified. We prove this result within a static oligopolistic game in which both supply function competition and Cournot competition yield a unique Nash equilibrium, whereas price setting yields a continuu...

We investigate positional effects in a vertically differentiated duopoly, evaluated against the first best. Positional concerns distort the allocation of consumers across varieties, as well as the average quality. If the external effect is sufficiently relevant, the resulting welfare loss is increasing in the extent of the externality itself. The b...

We investigate a linear state dfferential game describing an asymmetric Cournot duopoly with capacity accumulation à la Ramsey and a negative environmental externality (pollution), in which one of the firms has adopted corporate social responsibility (CSR) in its statute, and therefore includes consumer surplus and the environmental effects of prod...

We revisit the two-stage duopoly game with strategic delegation and asymmetric technologies of Sen and Stamatopoulos (2015). We show that their conclusions are driven by the restrictive assumption that the extent of delegation to managers is confined to a binary set. Allowing for a continuous set of delegation incentives, we prove that the delegati...

We build up a differential game to investigate the interplay between the quality of health care and the presence of an evolving disease in a duopoly where patients are heterogeneous along the income dimension. We study the Markov perfect equilibrium, and we identify the admissible parameter region wherein price regulation achieves the twofold objec...

Within a simple model of differentiated oligopoly, we show that tacit collusion may be prevented by the threat of nationalising a private firm coupled with the appropriate choice of the weight given to private profits in the maximand of the nationalised company. This happens because the threat of nationalization eliminates the prisoners’ dilemma us...

We revisit Fujiwara’s (2008) linear–quadratic differential duopoly game to show that the degenerate nonlinear feedback identified by the tangency point with the stationary state line is indeed unstable, given the dynamics of the natural resource exploited by firms. To do so, we fully characterise the continuum of nonlinear feedback solution via Row...

The optimal design of two-part tariffs is investigated in a dynamic model where two firms belonging to the same supply chain invest in R&D activities to increase the quality of the final product. It is shown that the replication of the vertically integrated monopolist’s performance can be attained using a TPT in which the fee is a linear function o...

We revisit the discussion about the relationship between price’s cyclical features, implicit collusion and the demand level in an oligopoly supergame where a positive shock may hit demand and disrupt collusion. The novel feature of our model consists in characterising the post-shock noncooperative price and comparing it against the cartel price pla...

Studies about innovation find evidence of a positive relationship between technological advancement and firm performance, in particular when the innovative effort is continuous. This paper aims to further the analysis on the duration of R&D investment at the firm level. The contribution of this study is threefold: first, we extend Manez et al. [201...

This paper investigates how socially responsible behaviour influences firms' profits and social welfare when production entails an environmental externality. We study a Cournot oligopoly with pollution, with one CSR operating in the market. A CSR firm not only takes into account its profits but also internalises its own share of pollution and is se...

We modify the price-setting version of the vertically differentiated duopoly model by Aoki (2003) by introducing an extended game in which firms noncooperatively choose the timing of moves at the quality stage. Our results show that there are multiple equilibria in pure strategies, whereby firms always select sequential play at the quality stage. W...

This background note offers first a synthesis of the main research questions addressed under the three topics of the 5th European Conference on Corporate R&D and Innovation - Industrial Research and Innovation: Evidence for Policy (CONCORDi 2015). Section II positions the selected papers in the current academic literature. It focuses on the extent...

In this note we revisit the result by Menezes and Quiggin (2012), showing that under linear supply function competition, the same Nash equilibrium results when firms choose slopes or intercepts of their supply functions. This is because the first order conditions emerging in the two strategy spaces are not linearly independent.

We propose a model of environmental overcompliance where …rms set the environmental quality of their products and compete in quantities , while the government imposes an environmental standard with the aim to maximise welfare. We show that all …rms overcomply if the environmental impact of production is su¢ ciently low, otherwise unilateral overcom...

We investigate the R&D portfolio of a monopolist investing in cost-reducing and quality enhancing R&D. Incentives along the two directions are inversely related to the size of market demand, and independent of each other. The stability analysis shows the existence of a unique stable steady state equilibrium, which is a saddle point. Finally, we sho...

Within a simple model of homogeneous oligopoly, we show that the traditional ranking between Bertrand and Cournot equilibria may be reversed. For price setting entails a continuum of price equilibria under convex variable costs, departure from marginal cost pricing may be observed. As a consequence, Bertrand-Nash equilibrium profits (welfare) may b...

We characterize the equilibrium in a homogeneous good Cournot duopoly in which firms have the choice to react to a cost-push shock by paying a lump-sum adjustment cost in order to offset the initial rise in marginal cost. Our results show that the size of the shock and the size of the adjustment cost jointly determine the nature and the number of t...

With the rising public and political concern about greener production, there is unrelenting pressure on individual companies to mitigate and abate pollution and adopt cleaner technologies. Governments adopt several types of environmental policies and regulations that are aimed at protecting the environment and encouraging efficient use of natural r...

In a Cournot duopoly, if only one firm hires a manager while the other remains entrepreneurial, the Cournot-Stackelberg equilibrium emerges, with the managerial firm as the leader. This happens under at least three different delegation schemes. We illustrate the different meachanisms driving this outcome through the analysis of the map of best repl...

We show that supply functions cannot be classified as either strategic complements or substitutes according to the twofold criterion advanced by Bulow et al. (1985). This is because while the slope of the best reply is univocally positive, this is not the case with the sign of the cross derivative of marginal profit. We first show this discrepancy...

We study a class of games featuring payoff functions being parabolic cylinders where best reply functions are orthogonal and therefore the pure-strategy non-cooperative solution is attained as a Nash equilibrium in dominant strategies. We prove that the resulting threshold of the discount factor above which implicit collusion on the Pareto frontier...

This paper aims at participating in the long-lasting debate about the analytical foundations of the Cournot equilibrium. In a homogeneous oligopoly, under standard regularity conditions, we prove that Cournot-Nash emerges both under (i) price competition and Cournot conjectures; and (ii) supply function competition with ex post market clearing. We...

We examine the relationship between competition and innovation in an industry where production is polluting and R&D aims to reduce emissions ("green" innovation). We present an n-firm oligopoly where firms compete in quantities and decide their investment in "green" R&D. When environmental taxation is exogenous, aggregate R&D investment always incr...

In this note we revisit the result by Menezes and Quiggin (2012), showing that under linear supply function competition, the same Nash equilibrium results when firms choose slopes or intercepts of their supply functions. This is because the first order conditions emerging in the two strategy spaces are not linearly independent.

We revisit the Cournot–Bertrand debate in the light of Cournot, Edgeworth and Launhardt, tracing back to Launhardt the origin of price competition in duopoly models with constant returns to scale. Then, we discuss the formalisation of consumer utility function for differentiated products, first appearing in Launhardt and then in Bowley. This allows...

A well established belief both in the game-theoretic IO and in policy debates is that market concentration facilitates collusion. We show that this piece of conventional wisdom relies upon the assumption of profit-seeking behaviour, for it may be reversed when firms pursue other plausible goals. To illustrate our intuition, we investigate the incen...

We revisit a recent literature on productive asset exploitation describing a differential oligopoly game of resource extraction under static, linear feedback and nonlinear feedback strategies, where we explicitly allow for the possibility of resource exhaustion. We show that (i) feedback rules entail resource exhaustion for a finite number of firms...