Economics

We consider the model of economic growth with time delayed investment function. Assuming the investment is time distributed we can use the linear chain trick technique to transform delay differential equation system to equivalent system of ordinary differential system (ODE). The time delay parameter is a mean time delay of gamma distribution. We reduce the system with distribution delay to both three and four-dimensional ODEs. We study the Hopf bifurcation in these systems with respect to two parameters: the time delay parameter and the rate of growth parameter. We derive the results from the analytical as well as numerical investigations. From the former we obtain the sufficient criteria on the existence and stability of a limit cycle solution through the Hopf bifurcation. In numerical studies with the Dana and Malgrange investment function we found two Hopf bifurcations with respect to the rate growth parameter and detect the existence of stable long-period cycles in the economy. We find that depending on the time delay and adjustment speed parameters the range of admissible values of the rate of growth parameter breaks down into three intervals. First we have stable focus, then the limit cycle and again the stable solution with two Hopf bifurcations. Such behaviour appears for some middle interval of admissible range of values of the rate of growth parameter.

This paper explores the gain maximization problem of two nations engaging in non-cooperative bilateral trade. Probabilistic model of an exchange of commodities under different price systems is considered. Volume of commodities exchanged determines the demand each nation has over the counter party's currency. However, each nation can manipulate this quantity by imposing a tariff on imported commodities. As long as the gain from trade is determined by the balance between imported and exported commodities, such a scenario results in a two party game where Nash equilibrium tariffs are determined for various foreign currency demand functions and ultimately, the exchange rate based on optimal tariffs is obtained.

This chapter examines how positivity and order play out in two important questions in mathematical economics, and in so doing, subjects the postulates of continuity, additivity and monotonicity to closer scrutiny. Two sets of results are offered: the first departs from Eilenberg's (1941) necessary and sufficient conditions on the topology under which an anti-symmetric, complete, transitive and continuous binary relation exists on a topologically connected space; and the second, from DeGroot's (1970) result concerning an additivity postulate that ensures a complete binary relation on a {\sigma}-algebra to be transitive. These results are framed in the registers of order, topology, algebra and measure-theory; and also beyond mathematics in economics: the exploitation of Villegas' notion of monotonic continuity by Arrow-Chichilnisky in the context of Savage's theorem in decision theory, and the extension of Diamond's impossibility result in social choice theory by Basu-Mitra. As such, this chapter has a synthetic and expository motivation, and can be read as a plea for inter-disciplinary conversations, connections and collaboration.

The betweenness property of preference relations states that a probability mixture of two lotteries should lie between them in preference. It is a weakened form of the independence property and hence satisfied in expected utility theory (EUT). Experimental violations of betweenness are well-documented and several preference theories, notably cumulative prospect theory (CPT), do not satisfy betweenness. We prove that CPT preferences satisfy betweenness if and only if they conform with EUT preferences. In game theory, lack of betweenness in the players' preference relations makes it essential to distinguish between the two interpretations of a mixed action by a player - conscious randomizations by the player and the uncertainty in the beliefs of the opponents. We elaborate on this distinction and study its implication for the definition of Nash equilibrium. This results in four different notions of equilibrium, with pure and mixed action Nash equilibrium being two of them. We dub the other two pure and mixed black-box strategy Nash equilibrium respectively. We resolve the issue of existence of such equilibria and examine how these different notions of equilibrium compare with each other.

We present tight bounds and heuristics for personalized, multi-product pricing problems. Under mild conditions we show that the best price in the direction of a positive vector results in profits that are guaranteed to be at least as large as a fraction of the profits from optimal personalized pricing. For unconstrained problems, the fraction depends on the factor and on optimal price vectors for the different customer types. For constrained problems the factor depends on the factor and a ratio of the constraints. Using a factor vector with equal components results in uniform pricing and has exceedingly mild sufficient conditions for the bound to hold. A robust factor is presented that achieves the best possible performance guarantee. As an application, our model yields a tight lower-bound on the performance of linear pricing relative to optimal personalized non-linear pricing, and suggests effective non-linear price heuristics relative to personalized solutions. Additionally, our model provides guarantees for simple strategies such as bundle-size pricing and component-pricing with respect to optimal personalized mixed bundle pricing. Heuristics to cluster customer types are also developed with the goal of improving performance by allowing each cluster to price along its own factor. Numerical results are presented for a variety of demand models that illustrate the tradeoffs between using the economic factor and the robust factor for each cluster, as well as the tradeoffs between using a clustering heuristic with a worst case performance of two and a machine learning clustering algorithm. In our experiments economically motivated factors coupled with machine learning clustering heuristics performed best.

This survey explores the literature on game-theoretic models of network formation under the hypothesis of mutual consent in link formation. The introduction of consent in link formation imposes a coordination problem in the network formation process. This survey explores the conclusions from this theory and the various methodologies to avoid the main pitfalls. The main insight originates from Myerson's work on mutual consent in link formation and his main conclusion that the empty network (the network without any links) always emerges as a strong Nash equilibrium in any game-theoretic model of network formation under mutual consent and positive link formation costs. Jackson and Wolinsky introduced a cooperative framework to avoid this main pitfall. They devised the notion of a pairwise stable network to arrive at equilibrium networks that are mainly non-trivial. Unfortunately, this notion of pairwise stability requires coordinated action by pairs of decision makers in link formation. I survey the possible solutions in a purely non-cooperative framework of network formation under mutual consent by exploring potential refinements of the standard Nash equilibrium concept to explain the emergence of non-trivial networks. This includes the notions of unilateral and monadic stability. The first one is founded on advanced rational reasoning of individuals about how others would respond to one's efforts to modify the network. The latter incorporates trusting, boundedly rational behaviour into the network formation process. The survey is concluded with an initial exploration of external correlation devices as an alternative framework to address mutual consent in network formation.

We present a directed variant of Salop (1979) model to analyze bus transport dynamics. The players are operators competing in cooperative and non-cooperative games. Utility, like in most bus concession schemes in emerging countries, is proportional to the total fare collection. Competition for picking up passengers leads to well documented and dangerous driving practices that cause road accidents, traffic congestion and pollution. We obtain theoretical results that support the existence and implementation of such practices, and give a qualitative description of how they come to occur. In addition, our results allow to compare the current or base transport system with a more cooperative one.

I develop a novel macroeconomic epidemiological agent-based model to study the impact of the COVID-19 pandemic under varying policy scenarios. Agents differ with regard to their profession, family status and age and interact with other agents at home, work or during leisure activities. The model allows to implement and test actually used or counterfactual policies such as closing schools or the leisure industry explicitly in the model in order to explore their impact on the spread of the virus, and their economic consequences. The model is calibrated with German statistical data on time use, demography, households, firm demography, employment, company profits and wages. I set up a baseline scenario based on the German containment policies and fit the epidemiological parameters of the simulation to the observed German death curve and an estimated infection curve of the first COVID-19 wave. My model suggests that by acting one week later, the death toll of the first wave in Germany would have been 180% higher, whereas it would have been 60% lower, if the policies had been enacted a week earlier. I finally discuss two stylized fiscal policy scenarios: procyclical (zero-deficit) and anticyclical fiscal policy. In the zero-deficit scenario a vicious circle emerges, in which the economic recession spreads from the high-interaction leisure industry to the rest of the economy. Even after eliminating the virus and lifting the restrictions, the economic recovery is incomplete. Anticyclical fiscal policy on the other hand limits the economic losses and allows for a V-shaped recovery, but does not increase the number of deaths. These results suggest that an optimal response to the pandemic aiming at containment or holding out for a vaccine combines early introduction of containment measures to keep the number of infected low with expansionary fiscal policy to keep output in lower risk sectors high.

This note considers cartel stability when the cartelized products are vertically differentiated. If market shares are maintained at pre-collusive levels, then the firm with the lowest competitive price-cost margin has the strongest incentive to deviate from the collusive agreement. The lowest-quality supplier has the tightest incentive constraint when the difference in unit production costs is sufficiently small.

We propose a decision-theoretic model akin to Savage (1972) that is useful for defining causal effects. Within this framework, we define what it means for a decision maker (DM) to act as if the relation between the two variables is causal. Next, we provide axioms on preferences and show that these axioms are equivalent to the existence of a (unique) Directed Acyclic Graph (DAG) that represents the DM's preference. The notion of representation has two components: the graph factorizes the conditional independence properties of the DM's subjective beliefs, and arrows point from cause to effect. Finally, we explore the connection between our representation and models used in the statistical causality literature (for example, Pearl (1995)).