Economics

On taking a non-trivial and semi-transitive bi-relation constituted by two (hard and soft) binary relations, we report a (i) p-continuity assumption that guarantees the completeness and transitivity of its soft part, and a (ii) characterization of a connected topological space in terms of its attendant properties on the space. Our work generalizes antecedent results in applied mathematics, all following Eilenberg (1941), and now framed in the context of a parametrized-topological space. This re-framing is directly inspired by the continuity assumption in Wold (1943-44) and the mixture-space structure proposed in Herstein and Milnor (1953), and the unifying synthesis of these pioneering but neglected papers that it affords may have independent interest.

It is well understood that the structure of a social network is critical to whether or not agents can aggregate information correctly. In this paper, we study social networks that support information aggregation when rational agents act sequentially and irrevocably. Whether or not information is aggregated depends, inter alia, on the order in which agents decide. Thus, to decouple the order and the topology, our model studies a random arrival order. Unlike the case of a fixed arrival order, in our model, the decision of an agent is unlikely to be affected by those who are far from him in the network. This observation allows us to identify a local learning requirement, a natural condition on the agent's neighborhood that guarantees that this agent makes the correct decision (with high probability) no matter how well other agents perform. Roughly speaking, the agent should belong to a multitude of mutually exclusive social circles. We illustrate the power of the local learning requirement by constructing a family of social networks that guarantee information aggregation despite that no agent is a social hub (in other words, there are no opinion leaders). Although the common wisdom of the social learning literature suggests that information aggregation is very fragile, another application of the local learning requirement demonstrates the existence of networks where learning prevails even if a substantial fraction of the agents are not involved in the learning process. On a technical level, the networks we construct rely on the theory of expander graphs, i.e., highly connected sparse graphs with a wide range of applications from pure mathematics to error-correcting codes.

We study the nature (i.e., constructive as opposed to non-constructive) of social welfare orders on infinite utility streams, and their representability by means of real-valued functions. We assume finite anonymity and introduce a new efficiency concept we refer to as asymptotic density-one Pareto. We characterize the existence of representable and constructive social welfare orders (satisfying the above properties) in terms of easily verifiable conditions on the feasible set of one-period utilities.

We present a polynomial-time algorithm that determines, given some choice rule, whether there exists an obviously strategy-proof mechanism for that choice rule.

We consider a platform facilitating trade between sellers and buyers with the objective of maximizing consumer surplus. Even though in many such marketplaces prices are set by revenue-maximizing sellers, platforms can influence prices through (i) price-dependent promotion policies that can increase demand for a product by featuring it in a prominent position on the webpage and (ii) the information revealed to sellers about the value of being promoted. Identifying effective joint information design and promotion policies is a challenging dynamic problem as sellers can sequentially learn the promotion value from sales observations and update prices accordingly. We introduce the notion of confounding promotion policies, which are designed to prevent a Bayesian seller from learning the promotion value (at the expense of the short-run loss of diverting consumers from the best product offering). Leveraging these policies, we characterize the maximum long-run average consumer surplus that is achievable through joint information design and promotion policies when the seller sets prices myopically. We then establish a Bayesian Nash equilibrium by showing that the seller's best response to the platform's optimal policy is to price myopically at every history. Moreover, the equilibrium we identify is platform-optimal within the class of horizon-maximin equilibria, in which strategies are not predicated on precise knowledge of the horizon length, and are designed to maximize payoff over the worst-case horizon. Our analysis allows one to identify practical long-run average optimal platform policies in a broad range of demand models.

We revisit the linear Cournot model with uncertain demand that is studied in Lagerlöf (2006)* and provide sufficient conditions for equilibrium uniqueness that complement the existing results. We show that if the distribution of the demand intercept has the decreasing mean residual demand (DMRD) or the increasing generalized failure rate (IGFR) property, then uniqueness of equilibrium is guaranteed. The DMRD condition implies log-concavity of the expected profits per unit of output without additional assumptions on the existence or the shape of the density of the demand intercept and, hence, answers in the affirmative the conjecture of Lagerlöf (2006) that such conditions may not be necessary. *Johan Lagerlöf, Equilibrium uniqueness in a Cournot model with demand uncertainty. The B.E. Journal in Theoretical Economics, Vol. 6: Iss 1. (Topics), Article 19:1--6, 2006.

This paper identifies the mathematical equivalence between economic networks of Cobb-Douglas agents and Artificial Neural Networks. It explores two implications of this equivalence under general conditions. First, a burgeoning literature has established that network propagation can transform microeconomic perturbations into large aggregate shocks. Neural network equivalence amplifies the magnitude and complexity of this phenomenon. Second, if economic agents adjust their production and utility functions in optimal response to local conditions, market pricing is a sufficient and robust channel for information feedback leading to macro learning.

We study the mathematical and economic structure of the Kolkata (k) index of income inequality. We show that the k-index always exists and is a unique fixed point of the complementary Lorenz function, where the Lorenz function itself gives the fraction of cumulative income possessed by the cumulative fraction of population (when arranged from poorer to richer). We show that the k-index generalizes Pareto's 80/20 rule. Although the k and Pietra indices both split the society into two groups, we show that k-index is a more intensive measure for the poor-rich split. We compare the normalized k-index with the Gini coefficient and the Pietra index and discuss when they coincide. We establish that for any income distribution the value of Gini coefficient is no less than that of the Pietra index and the value of the Pietra index is no less than that of the normalized k-index. While the Gini coefficient and the Pietra index are affected by transfers exclusively among the rich or among the poor, the k-index is only affected by transfers across the two groups.

In a recent paper, Naz and Chaudry provided two solutions for the model of Lucas-Uzawa, via the Partial Hamiltonian Approach. The first one of these solutions coincides exactly with that determined by Chilarescu. For the second one, they claim that this is a new solution, fundamentally different than that obtained by Chilarescu. We will prove in this paper, using the existence and uniqueness theorem of nonlinear differential equations, that this is not at all true.

A fundamental property of choice functions is stability, which, loosely speaking, prescribes that choice sets are invariant under adding and removing unchosen alternatives. We provide several structural insights that improve our understanding of stable choice functions. In particular, (i) we show that every stable choice function is generated by a unique simple choice function, which never excludes more than one alternative, (ii) we completely characterize which simple choice functions give rise to stable choice functions, and (iii) we prove a strong relationship between stability and a new property of tournament solutions called local reversal symmetry. Based on these findings, we provide the first concrete tournament---consisting of 24 alternatives---in which the tournament equilibrium set fails to be stable. Furthermore, we prove that there is no more discriminating stable tournament solution than the bipartisan set and that the bipartisan set is the unique most discriminating tournament solution which satisfies standard properties proposed in the literature.