Discounting long run average growth in stochastic dynamic programs

Summary. Finding solutions to the Bellman equation often relies on restrictive boundedness assumptions. In this paper we develop a method of proof that allows to dispense with the assumption that returns are bounded from above. In applications our assumptions only imply that long run average (expected) growth is sufficiently discounted, in sharp contrast with classical assumptions either absolutely bounding growth or bounding each period (instead of long run) maximum (instead of average) growth. We discuss our work in relation to the literature and provide several examples. Received: July 26, 2000; revised version: July 10, 2002 RID="*" ID="*" I am specially grateful to Cuong Le Van and to anonymous referee for detecting an error in a previous version of this paper and for suggestions that sensibly improved the paper. Comments and suggestions are also acknowledged to Michele Boldrin, Raouf Boucekkine, Fabrice Collard, Tim Kehoe, Omar Licandro, and Luis Puch. I am also indebted to participants to the III Summer School on Economic Theory held at the Universidade de Vigo, the Macroeconomics Workshop at the Universitat Autò}noma de Barcelona, and the Econometrics Seminar at Tilburg University. Financial support from the Belgian government, under project PAI P4/01, at the IRES-UCL, from a European Marie Curie fellowship, Grant HPMF-CT-1999-00410, at the CEPREMAP, and from IVIE and Spanish Ministerio de Ciencia y Tecnología and FEDER, under project BEC2001-0535, at the Universidad de Alicante, is gratefully acknowledged.     

On an optimal consumption problem for p-integrable consumption plans

A generalization is presented of the existence results for an optimal consumption problem of Aumann and Perles [4] and Cox and Huang [10]. In addition, we present a very general optimality principle. Received: July 14, 1999; revised version: September 9, 1999     

Regional-incentive programs and the problem of time-inconsistent plans

Capital grants are the mainstay of most countries' regional-incentive packages. From an economic point of view it is hard to understand why governments choose to give subsidies to reduce the cost of capital. For if the objective is to reduce the unemployment problem in a specific district, it seems reasonable to encourage labor-intensive firms to establish production in that region. In this article, I discuss this discrepancy between the policy usually recommended by economists and the policy carried out by politicians. I show that giving capital grants can be the best regional policy if important dynamic aspects of the policy problem are considered.I have benefited from discussions with Geir Asheim, Kjell Erik Lommerud, Torsten Persson, James Poterba, and Agnar Sandmo.     

Infinite-horizon competitive programs are optimal

For infinite-horizon optimal-growth problems the standard result in the literature says that a program is optimal if and only if associated with it is a sequence of present-value prices at which the program satisfies (i) a set of myopic competitive conditions, and (ii) an asymptotic transversality condition. The principal result of this paper points out the interesting and surprising fact that at least for a class of multisector models where the production side is described by a simple linear model, and there are some limiting primary factors, the competitive conditions alone characterize an optimal program.     

Designing optimal gain sharing plans to promote energy conservation

We analyze the optimal design of gain sharing plans to promote energy conservation. We show how the optimal plan varies as industry conditions and the regulator??s information change. We demonstrate the importance of allowing the energy supplier a choice among plans, some of which offer the prospect of both pronounced financial gains for superior performance and substantial losses for inferior performance.     

Computing uniformly optimal strategies in two-player stochastic games

We provide a computable algorithm to calculate uniform ε-optimal strategies in two-player zero-sum stochastic games. Our approach can be used to construct algorithms that calculate uniform ε-equilibria and uniform correlated ε-equilibria in various classes of multi-player non-zero-sum stochastic games.     

On the solution of a stochastic optimal growth model

This note discusses a stochastic optimal growth model in which the optimal paths can be obtained by a simple direct argument. The structural characteristics of the model are the infinite horizon, the form of the instantaneous utility function, and uncertainty as a Wiener process in a linear production constraint. The note explains that, for optimality, at each point in time a formally identical problem must be solved. This implies that the optimal saving ratio must be constant.A proof, employing the rules of stochastic calculus, that the ensuing paths are the unique globally optimal paths is also given.We are very grateful to two referees of this journal for their invaluable comments and suggestions.     

Self-regulatory strength and dynamic optimal purchase

I study a single-agent dynamic purchase problem by using a Gul and Pesendorfer’s [Gul, F., Pesendorfer, W., 2007. Harmful addiction. Review of Economic Studies 74 (1), 147-172] dynamic preference: extreme self-control cannot be sustained and leads to addiction. This type of agent exhibits a behavioral pattern of self-regulatory depletion.     

Stability of stochastic optimal growth models: a new approach

The paper proposes an Euler equation technique for analyzing the stability of differentiable stochastic programs. The main innovation is to use marginal reward directly as a Foster-Lyapunov function. This allows us to extend known stability results for stochastic optimal growth models, both weakening hypotheses and strengthening conclusions.     

Computation of non-linear continuous optimal growth models: experiments with optimal control algorithms and computer programs

In this paper we discuss the use of optimal control methods for computing non-linear continuous optimal growth models. We have discussed various recently developed algorithms for computing optimal control, involving step-function approximations, Runge–Kutta solutions of differential equations, and we suggest that the discretization approach is preferable to methods which solve first-order optimality conditions. We have surveyed some powerful computer programs by : , and for computing such models numerically. These programs have no substantial optimal growth modelling applications yet, although they have numerous engineering and scientific applications. A computer program named by is developed in this study. Results are reported for computing the Kendrick–Taylor optimal growth model using and programs based on the discretization approach. References are made to the computational experiments with and . The results are used to compare and evaluate mathematical and economic properties, and computing criteria. While several computer packages are available for optimal control problems, they are not always suitable for particular classes of control problems, including some economic growth models. The -based and , however, offer good opportunities for computing continuous optimal growth models. It is argued in this paper, that optimal growth modellers may find that these recently developed algorithms and computer programs are relatively preferable for a large variety of optimal growth modelling studies.     

A turnpike property of optimal programs for a class of simple linear models of production

Summary. This paper establishes a ‘turnpike theorem’ for a closed linear model of production with a primitive input requirement matrix. Optimal programs of resource allocation have a ‘turnpike property’ if the growth factor of every sector in the economy converges, in the long run, to a common value. The usefulness of such a theorem is due to the fact that the input requirement matrix for an economy with a large number of goods may be primitive (some power of the matrix is strictly positive). Received: April 19, 1998; revised version: July 15, 1998     

Comparative statics of the optimal dynamic incentive contract

Each period a firm must carry out an indivisible project, the cost of which depends on the firm's constant efficiency and its (variable) level of ‘effort’. Only the cost is observed by the regulator who has to choose the incentive scheme. The optimal dynamic mechanism is derived under possible commitment of the regulator over time. Then, under non-commitment, it is shown that there are four types of continuation equilibria. Also the profile of effort levels is compared to the commitment case. Finally, some numerical comparative statics is performed and it is shown that all types of equilibria can be optimal.     

The crisis,Fed, Quants and stochastic optimal control

The Dodd–Frank (D–F) Financial Reform Bill authorizes the Federal Reserve to monitor the financial services marketplace to identify potential threats to the stability of the US financial system. Alan Greenspan's retrospective indicates what he has learned from the crisis. He argues that the crisis, the housing price bubble, was unpredictable and unavoidable. Greenspan now focuses on desirable capital requirements, or leverage, for banks and financial intermediaries. I explain why the Fed's and Greenspan's views stem from a lack of the appropriate tools of analysis of what is an excessive debt or leverage. The Quants who devised the highly leveraged financial derivatives ignored systemic risk.My theme is that the application of stochastic optimal control (SOC) is an effective approach to implement what the D–F bill is authorizing. I explain: first, what is the optimal capital requirement/leverage that balances expected return against risk. Second, what is a theoretically derived early warning signal of a crisis. Third, I derive an excess debt ratio, equal to the difference between the actual and optimal ratios. The probability of a debt crisis is directly related to the excess debt ratio. The excess debt ratio starting from 2004–05 indicated that a crisis was most likely. The Fed should use this SOC analysis in implementing the Dodd–Frank bill.     

Optimal disaster-preventive expenditure in a dynamic and stochastic model

The purpose of this study is to present an analytical framework for publicly optimal disaster-preventive expenditure. We examine the optimal policy combination of tax rate, disaster-preventive expenditure, and productive government expenditure in a neoclassical growth model, in which natural disasters occur stochastically and partially destroy existing capital. Based on this model, we can decompose the welfare effect of raising preventive expenditure into three effects: the damage reduction, crowding out, and precautionary effects. By identifying these marginal benefits and costs, we obtain the policy conditions that maximize household welfare. Furthermore, we show that optimal prevention is increasing in disaster probability, and by using a numerical example, we show that there is an inverse U-shaped relationship between the expected growth rate and disaster probability.     

A stochastic estimated version of the Italian dynamic General Equilibrium Model

This paper aims at identifying the main drivers of the Italian economic cycle. To this end, we estimate a small-open economy model based on a dual labor market, which captures the main features of the Italian economy. Our results indicate that labor market rigidities are important structural features of the Italian economy, but they provide a limited contribution in explaining the business cycle fluctuations. Long-term dynamics are mostly driven by supply factors (productivity and markups). However, demand factors, including monetary and fiscal policies, play a sizeable role in the short run. Policy experiments show that expansionary fiscal policies crowd out private consumption and investment. The paper also contributes to the recent debate on fiscal consolidation. Estimated fiscal multipliers support the view that plans aimed at reducing the public debt should be based on tax increases rather than expenditure cuts.     

Performance of the Norwegian dairy farms: A dynamic stochastic approach

From a theoretical perspective, it is well stated that the farm's decision on the use of inputs depends on the farmer's ability to make an efficient decision over time. The existing literature in performance analysis of the dairy farms based on static modeling and thus ignores the inter-temporal nature of production decisions. This paper aims to construct a dynamic stochastic production frontier incorporating the sluggish adjustment of inputs, to measure the performance of dairy farms in Norway. The empirical application focused on the farm-level analysis of the Norwegian dairy sector for 2000- 2018. The dynamic frontier estimated using the system Generalized Method of Moments estimator. The analysis shows that the static model in the previous studies underestimates the performance of the dairy farms.     

Stackelberg leadership in a dynamic duopoly with stochastic capital accumulation

I propose a dynamic duopoly model where firms enter simultaneously but compete hierarchically á la Stackelberg at each instant over time. They accumulate capacity through costly investment, with capital accumulation dynamics being affected by an additive shock the mean and variance of which are known. The main findings are the following. First, the Stackelberg game is uncontrollable by the leader; hence, it is time consistent. Second, the leader invests more than the follower; as a result, in the steady state, the leader’s capacity and profits are larger than the follower’s. Therefore, the present analysis does not confirm Gibrat’s Law, since the individual growth rate is determined by the timing of moves.JEL Classification: C61, C73, D43, D92, L13Financial support within the project The post-entry performance of firms: technology, growth and survival lead by Enrico Santarelli, co-financed by the University of Bologna and MIUR, is gratefully acknowledged. I thank Uwe Cantner, Roberto Cellini, Roberto Golinelli, Helen Louri, Enrico Santarelli, Antonello Scurcu, Peter Thompson, two anonymous referees and the audience at the final workshop of the project (Bologna, November 22-23, 2002) for useful comments and discussion. The usual disclaimer applies.     

Endogenous innovation under New Keynesian dynamic stochastic general equilibrium model

This paper constructs an endogenous growth model using the framework of New Keynesian dynamic stochastic general equilibrium models. We incorporate the Schumpeterian approach that generates seemingly sticky prices and reinterpret the Calvo mechanism from the perspective of Bertrand competition and successful entrepreneurs. Our results demonstrate that both positive productivity shocks and endogenous innovation have a negative effect on subsequent endogenous innovation. These self-destructive effects of endogenous innovation might account for the IT productivity paradox and productivity slowdown seen in advanced countries. Furthermore, it is shown that there are both neutral and non-neutral properties of monetary policy shocks. They are neutral in terms of the growth effect, but non-neutral in terms of the level effect. In particular, expansionist monetary policies are desirable to facilitate endogenous innovation.