Unique Steady-state Equilibrium Research Articles

Price and product innovation competition with network effects and consumers’ adaptive learning: A differential game approach

For many high-tech and Internet related products, the utility of consumers depends partly on the scale of user groups, which is called network effects. In this paper, we formulate a differential game model to investigate the joint price and product innovation competition in a market exhibiting network effects. In the spirit of behavior economics, the consumers’ adaptive learning about network size is considered. The unique steady-state equilibrium is analytically characterized under both committed and feedback information structures. Our analysis mainly shows that (i) the network effects have negative influence on the result of price competition but have nothing to do with the result of product innovation competition under the premise of low market density; (ii) the marginal impact of network effects on the steady-state price is decreasing with the consumers’ learning speed; (iii) in the long-run, the product price converges to the level either higher or lower than the stationary price, depending on the strength of network effects; (iv) in a market where the consumers are extremely sensitive to the product price, the quality provision is insufficient compared with the first-best optimal level regardless of the choice of solution concept. The dynamic evolution patterns of the price and product innovation competition are illustrated by numerical experiments.

Unobservable investments, trade efficiency and search frictions

AbstractInformation structure can affect market efficiency by influencing participants' investment incentives. In this case, the entrants' type distribution and the market trade efficiency are both affected by the information structure, so they are jointly determined and correlated. In order to investigate the effects of asymmetric information, this paper uses a random search model of ex ante investments and trade efficiency, assuming that the amount of investment remains the investor's private information. I show that the ex ante payoffs are equivalent to the equilibrium payoffs when investments are observable. In the unique steady state equilibrium, non‐degenerate investment distribution and price distribution emerge simultaneously with ex ante identical agents. The investments motivated by unobservability fail to improve social welfare due to the mismatches caused by unobservability. Allocating positive bargaining power to investors alleviates the mismatch problem and improves social welfare.

The uniqueness of steady-state equilibrium under state-dependent pricing: The case of deflation

John and Wolman (2004, 2008) show that steady-state equilibrium is likely to be unique under inflation (i.e., a positive net inflation rate) in a New Keynesian state-dependent pricing model based on the Dotsey et al. (1999) model. I show that the uniqueness of equilibrium holds in a stronger way under deflation (i.e., a negative net inflation rate).

Unobservable Investments, Trade Efficiency and Search Frictions

Market participants can invest to increase their gains from trade before searching for partners. The entrants' type distribution and trade efficiency are jointly determined and correlated. This paper explores a random search model of ex ante investments and trade efficiency, assuming that the investment amount remains the investor's private information. We show that the ex ante payoffs are equivalent to the equilibrium payoffs when investments are observable. In the unique steady state equilibrium, non-degenerate investment distribution and price distribution emerge simultaneously with ex-ante identical agents. In the basic model, where investing agents have no bargaining power, the extra investments fail to improve social welfare even when the search friction vanishes and the investments become efficient due to the mismatches caused by unobservability. Allocating positive bargaining power to investors alleviates the mismatch problem and improves social welfare.

Uniqueness and Multiple Trajectories for the Case of Lucas Model

The main aim of this paper is to prove that the model introduced by Lucas and further analyzed by Caballe and Santos, Mulligand and Sala-I-Martin, Benhabib and Perli and finally by Boucekkine and Ruiz-Tamarit, has two interesting properties. If the externality parameter in the production of human capital is greater than the elasticity of output with respect to physical capital, then the system is characterized by multiple transitional paths, indexed by the starting value of the fraction of labor allocated to the production of physical capital, leading to different steady-states equilibrium. Alternatively, if the externality parameter in the production of human capital is smaller than the elasticity of output with respect to physical capital, then the system is characterized by an unique transitional path, convergent to the unique steady-state equilibrium. For the special case where the inverse of the elasticity of intertemporal substitution equals the elasticity of output with respect to physical capital, we obtain closed-form solutions and thus, our approach contains as particular cases some other results, as those obtained by Boucekkine and Ruiz-Tamarit. More than this, differently to Boucekkine and Ruiz-Tamarit we obtain closed-form solutions for all variables of the model and thus we are able to verify the two transitional conditions.

From club convergence of per capita industrial pollutant emissions to industrial transfer effects: An empirical study across 285 cities in China

The process of industrialization has led to an increase in air pollutant emissions in China. At the regional level, industrial restructuring and industrial transfer from eastern China to western China have caused a significant difference in pollutant emissions among various cities. This paper analyzes per capita industrial pollutant emissions across 285 prefecture-level cities from 2003 to 2015, aiming to reveal how industrial transfer affects the formation of convergence clubs. Whether industrial pollutant emissions across heterogeneous cities converge to a unique steady-state equilibrium is first identified based on the concept of club convergence. Logit regression analysis is then applied to assess the effects of industrial transfer on the observed clubs. The log t-test highlights four convergence clubs for industrial SO2 emissions and three clubs for industrial soot emissions. The regression analysis results reveal that the effects of industrial transfer can lead to multiple steady-state equilibria, suggesting region-specific environmental policies and execution strategies. In addition, accelerating the development of clean energy technologies in emission-intense regions should be further emphasized.

Secular stagnation: Theory and remedies

To investigate secular stagnation, I add two features to a standard Ramsey model with money: (i) Households have a preference for wealth; (ii) Wages are downward rigid. In this framework, there exists a frictionless neoclassical steady state equilibrium characterized by a low natural real interest rate. In addition, if wages are sufficiently rigid and the natural real interest rate sufficiently low, then there also exists a Keynesian secular stagnation steady state characterized by under-employment, low inflation, and a binding zero lower bound on the nominal interest rate. As wages become more flexible, the Keynesian steady state diverges away from the neoclassical steady state, until wages are so flexible that it ceases to exist. If monetary policy is excessively restrictive, then the secular stagnation steady state is the unique steady state equilibrium of the economy. The optimal policy response to secular stagnation is to move the economy to the neoclassical steady state. This can either be achieved by raising the central bank's inflation ceiling or by taxing wealth and subsidizing investment in physical capital. This optimal tax policy is revenue-neutral.

Club convergence and clustering of U.S. energy-related CO2 emissions

This study examines the convergence of energy-related carbon dioxide emissions among a panel of U.S. states between the period 1960–2010. This examination is carried out by means of a two-stage procedure. In the first stage, we conduct an endogenous grouping, regression-based convergence test. Unlike previous studies, this methodology endogenously identifies groups of states with emissions that are converging to a similar steady state growth path over time. In the second stage, we evaluate the conditional rate of convergence for the whole sample and for each club using panel data, fixed effects models that control for unobserved, time-invariant heterogeneous effects. More specifically, we examine the rates of convergence conditional on certain structural and non-structural characteristics of the state economy. Results from stage one and stage two suggest that one group of twenty-six states is converging to a unique steady-state equilibrium, and otherwise, the remaining states are diverging. Finally, we discuss different policy approaches to mitigating carbon dioxide emissions based on the club convergence hypothesis.

Distressed Sales in OTC Markets

We present a stylized model of the over-the-counter markets in the tradition of Duffie Gârleanu and Pedersen (2005) with two distinctive features: (i) buyers have heterogenous preferences and their willingness to pay is private information and (ii) sellers become financially distressed if they cannot sell for too long. A unique steady-state equilibrium exists and it is characterized by predatory buying. Specifically, during periods where sellers are more likely to become distressed (e.g. during economic crises, financial turmoils etc.) buyers become more selective and hold off purchasing despite the abundance of distressed sales and low prices. This reluctance triggers the number of distressed sellers to grow even further and forces them for additional price cuts.

THE RICH AND THE POOR IN A SIMPLE MODEL OF GROWTH AND DISTRIBUTION

We consider a model of economic growth with altruistic agents who care about their consumption and the disposable income of their offspring. The agents' consumption and the offspring's disposable income are subject to positional concerns. We show that, if the measure of consumption-related positional concerns is sufficiently low and/or the measure of offspring-related positional concerns is sufficiently high, then there is a unique steady-state equilibrium, which is characterized by perfect income and wealth equality, and all intertemporal equilibira converge to it. Otherwise, in steady-state equilibria, the population splits into two classes, the rich and the poor; under this scenario, in any intertemporal equilibrium, all capital is eventually owned by the households that were the wealthiest from the outset and all other households become poor.

When Ricardo Meets Chamberlin: A Simple Dynamic Model of Monopolistic Competition

We develop a dynamic model of monopolistic competition which sheds light on how the interplay between the degree of product differentiation and intertemporal elasticity of substitution affects the steady-state equilibrium. Consumers love variety and split their labor endowment between wage labor, which brings immediate income, and producing capital, which yields a rent in the future. The impact of the elasticity of substitution across varieties on the market outcome depends crucially on whether consumption today and consumption tomorrow are gross substitutes or gross complements. The case of Cobb-Douglas intertemporal utility is a borderline situation, when the market outcome is invariant to the degree of product differentiation. We also fully characterize the unique steady-state equilibrium path and show that the key dynamic properties of the model, such as local stability and determinacy of equilibrium, also hinge mainly on the interplay between the intra- and intertemporal elasticities of substitution.

How urbanization affect employment and social interactions

We develop a model where the unemployed workers in the city can find a job either directly or through weak or strong ties. We show that, in denser areas, individuals choose to interact with more people and meet more random encounters (weak ties) than in sparsely populated areas. We also demonstrate that, for a low urbanization level, there is a unique steady-state equilibrium where workers do not interact with weak ties, while, for a high level of urbanization, there is a unique steady-state equilibrium with full social interactions. We show that these equilibria are usually not socially efficient when the urban population has an intermediate size because there are too few social interactions compared to the social optimum. Finally, even when social interactions are optimal, we show that there is over-urbanization in equilibrium.

Some Properties of The Solution of The Ramsey Model

Abstract The main aim of this paper is to improve some results obtained by Smith. We provide here a closed-form solution for the Ramsey model. We prove that if the inverse of the constant elasticity of intertemporal substitution is different to the elasticity of output with respect to physical capital, then the economy described by this model reaches the unique steady-state equilibrium, for any starting values of per-capita consumption. The advantage of the paper over the existing paper is that the arguments here are not local, that is, our hypotheses are most generally possible. Consequently, we can obtain almost all other results as particular cases of our solution. Our approach clarifies why different starting values of per-capita consumption generates the same steady-state equilibrium, but of different periods of time. We also clarifiy some of previous result obtained by other authors as for example Luca Guerrini.

Unemployment and Innovation

This paper analyzes equilibrium, dynamics, and optimal decisions on the factor bias of innovation in a model of induced innovation. In a model with full employment, we show that (a) if the elasticity of substitution is always less than or greater than unity, there is a unique steady state equilibrium; (b) if the elasticity of substitution is less than unity, the steady state is stable, but convergence is oscillatory; (c) if the elasticity of substitution is greater than unity, the steady state is a saddle point; and (d) if the elasticity of substitution is less than unity for both high and low effective capital labor ratios but greater than unity for intermediate values, then there can be multiple steady states. In a model where efficiency wages lead to equilibrium unemployment, we show that if the elasticity of substitution is less than unity, there will be a bias towards excessive labor augmenting innovation, resulting in too high unemployment, with convergence to the unique steady state being oscillatory, rather than monotonic. Similarly, if the elasticity of substitution between skilled and unskilled labor is less than unity, and there is efficiency wage unemployment for unskilled labor only, there is will be excessively skill-biased innovation. This paper provides an alternative resolution to the Harrod-Domar conundrum of the disparity between the natural and warranted rate of growth to that of Solow, with strong policy implications, for instance, concerning the effects of income distribution and monetary policy both in the short run and the long.

How Urbanization Affects Employment and Social Interactions

We develop a model where the unemployed workers in the city can find a job either directly or through weak or strong ties. We show that, in denser areas, individuals choose to interact with more people and meet more random encounters (weak ties) than in sparsely populated areas. We also demonstrate that, for a low urbanization level, there is a unique steady-state equilibrium where workers do not interact with weak ties, while, for a high level of urbanization, there is a unique steady-state equilibrium with full social interactions. We show that these equilibria are usually not socially efficient when the urban population has an intermediate size because there are too few social interactions compared to the social optimum. Finally, even when social interactions are optimal, we show that there is over-urbanization in equilibrium.

On Ramsey's conjecture.

Studying a one-sector economy populated by finitely many heterogeneous households that are subject to no-borrowing constraints, we confirm a conjecture by Frank P. Ramsey according to which, in the long run, society would be divided into the set of patient households who own the entire capital stock and impatient ones without any physical wealth. More specifically, we prove (i) that there exists a unique steady state equilibrium that is globally asymptotically stable and (ii) that along every equilibrium the most patient household owns the entire capital of the economy after some finite time. Furthermore, we prove that despite the presence of the no-borrowing constraints all equilibria are efficient. Our results are derived for the continuous-time formulation of the model that was originally used by Ramsey, and they stand in stark contrast to results that – over the last three decades – have been found in the discrete-time version of the model.

Motivated sellers and predation in the housing market

We develop an equilibrium search model of the housing market where sellers may become distressed as they are unable to sell. A unique steady state equilibrium exists where distressed sellers attempt liquidation sales by accepting prices that are substantially below fundamental values. During periods where a large number of sellers are forced to liquidate customers exhibit ‘predation’: they hold off purchasing and strategically slow down the speed of trade, which in turn causes more sellers to become distressed. The model naturally suggests several proxies of liquidity. Interestingly, the average time on the market (TOM), one of the most frequently used statistics in the literature, does a poor job within the context of liquidation sales and predation. Specifically we show that TOM falls during periods of predatory buying, which, if interpreted on face value, indicates that the market becomes more liquid with predation. We propose an alternative proxy – the profit loss in fire sales – which appears to be a more robust measure of liquidity than TOM.

EXISTENCE AND UNIQUENESS OF STEADY-STATE EQUILIBRIUM IN A GENERALIZED OVERLAPPING GENERATIONS MODEL

Galor and Ryder [Journal of Economic Theory 49 (1989), 360–375] establish conditions for the existence of equilibrium in a Diamond-type overlapping-generations (OLG) model. Although theoretically appealing, these conditions are implicit and not convenient to apply. This paper provides explicit and easily applied conditions for the existence and uniqueness of steady-state equilibrium, with which one only needs to check the first derivatives of the production and utility functions and their interactions, with no need to solve the optimization problem. Our theorems on the existence and uniqueness of steady-state equilibrium can be applied to a larger class of OLG models that do not require second-order differentiability of the production and utility functions. We present examples to show how to check the existence and uniqueness of equilibrium.

Sorting by search intensity

In this paper, I characterize matching in an on-the-job search model with endogenous search intensity, heterogeneous workers and firms, and match surplus is shared between workers and firms through bargaining. I provide proof of existence and uniqueness of steady state equilibrium. Given equally efficient matched and unmatched search, the worker skill conditional distribution of firm productivity over matches is stochastically increasing (decreasing) in worker skill if the production function is supermodular (submodular). I also show that this strong notion of sorting does not obtain everywhere for the firm productivity conditional match distribution.

Health, infrastructure, environment and endogenous growth

This paper attempts to develop a model of endogenous growth with special focus on the role of health capital, public infrastructure and environmental pollution. It is an extension of the model of Agenor (2008) who does not consider environmental pollution. We analyze properties of optimal fiscal policy in the steady-state growth equilibrium when the level of production of the final good is the source of pollution. Tax revenue of the government is channelized into three expenditure heads-health expenditure, pollution abatement expenditure and public infrastructure expenditure. It is found that the optimum ratio of public infrastructural expenditure to national income in the balanced growth equilibrium is less than the competitive output share of the public input; and it varies inversely with the magnitude of the emission-output coefficient. There is no conflict between the social welfare maximizing solution and the growth rate maximizing solution in the balanced growth equilibrium. There may exist indeterminacy in the transitional growth path converging to the unique steady-state equilibrium point that never satisfies saddle-point stability. The market economy growth rate is not necessarily less than the socially efficient growth rate; and the relationship between these two depends on the magnitude of the emission-output coefficient.