Resource depletion under uncertainty: implications for mine depreciation,Hartwick’s Rule and national accounting

Lozada’s equation [Resource and Energy Economics 17 (1995) 137] for the change in value of a non-autonomous dynamic program is generalized to stochastic control and applied to the depreciation of a competitive mine facing price, reserve and discount rate uncertainty. Mine depreciation includes the costs of these risks, as well as an adjustment to the ‘net price’ used to value depletion and revisions. The change in value equation also provides the basis for a stochastic version of Hartwick’s Rule [American Economic Review 67 (1977) 972] for sustainable consumption, that the risk adjusted value of net investment equal zero. The analysis has implications for the Weitzman [Quarterly Journal of Economics 90 (1976) 156] welfare measure and the stochasic Hamiltonian.     

Ranking investment projects

Summary. This paper describes conditions under which one investment project dominates a second project in terms of net present value, irrespective of the choice of the discount rate. The resulting partial ordering of projects has certain similarities to stochastic dominance. However, the structure of the net present value function leads to characterizations that are quite specific to this context. Our theorems use Bernstein's (1915) innovative results on the representation and approximation of polynomials, as well as other general results from the theory of equations, to characterize the partial ordering. We also show how the ranking is altered when the range of discount rates is limited or the rate varies period by period. Received: January 5, 2002; revised version: October 29, 2002 RID="*" ID="*" We thank Robert Driskill, Andrea Maneschi, Roy Radner, and participants of seminars at NYU, Notre Dame, Purdue, and Washington University for helpful comments. The present version of the paper has benefited from comments by a referee and the editor. Foster is grateful for support from the John D. and Catherine T. MacArthur Foundation through its network on Inequality and Poverty in Broader Perspective. Correspondence to: T. Mitra     

Dynamic optimization with a nonsmooth, nonconvex technology: the case of a linear objective function

This paper studies a one-sector optimal growth model with linear utility in which the production function is only required to be increasing and upper semicontinuous. The model also allows for a general form of irreversible investment. We show that every optimal capital path is strictly monotone until it reaches a steady state; further, it either converges to zero, or reaches a positive steady state in finite time and possibly jumps among different steady states afterwards. We establish conditions for extinction (convergence to zero), survival (boundedness away from zero), and the existence of a critical capital stock below which extinction is possible and above which survival is ensured. These conditions generalize those known for the case of S-shaped production functions. We also show that as the discount factor approaches one, optimal paths converge to a small neighborhood of the capital stock that maximizes sustainable consumption.This paper is dedicated to Professor Mukul Majumdar on his 60th birthday. His research with various co-authors in the late 70s and the 80s pioneered innovative techniques for the analysis of nonconvex dynamic optimization models – both deterministic and stochastic. Roy considers himself particularly fortunate for having had the opportunity to learn economic theory and mathematical economics from Professor Majumdar. This paper has benefited from helpful comments and suggestions by an anonymous referee. Financial support from the 21st Century COE Program at GSE and RIEB, Kobe University, is gratefully acknowledged.     

The Ethics of Intertemporal Distribution in a Warming Planet

This paper evaluates, from the ethical viewpoint, current work by economists on intergenerational resource allocation in the presence of global warming. We begin by attempting to elucidate the debate that has recently occurred on the appropriate choice of the discount rate. We offer three justifications for maximizing the discounted sum of generational utilities, and find only one of these to be a satisfactory justification of that practice: the possibility that the human species may become extinct. This implies that a very small discount rate (large discount factor) should be used. We argue that the justification for discounting, inherent in the approaches taken by many economists, is that of ‘the present generation of hegemon,’ which is unacceptable. The role of the Ramsey equation in deducing the discount rate in these theories is explained. As an alternative to discounted utilitarianism, we propose a principle of sustainability; we describe optimal paths that have been calculated for the sustainabilitiarian (Rawlsian) objective function, and paths that will sustain growth in welfare, at a positive rate. We report results concerning optimal paths when the uncertainty of existence of future generations is taken into account. In sharp contrast to the utilitarian model, it turns out that under some conditions, the ‘sustainabilitarian’ can ignore the uncertainty regarding the date at which humans become extinct. There is a striking difference between the solutions of the discounted utilitarian program and the sustainabilitarian program under uncertainty.     

Spatial discounting of ecosystem services

The impact of conservation efforts targeted at preserving ecosystem services largely depend on the welfare implications associated with spatial variations in the provision of ecosystem services. While there is ample empirical evidence of spatial discounting or decay of the valuation of ecosystem services, there are still few underpinnings based on welfare economic theory. We establish a theory of spatial discounting that closely follows the concept of time discounting pertaining to climate change, and show spatial discount rates in the consumption, ecosystem service, and willingness to pay (WTP) numeraires. We consider the role of key parameters such as pure rate of spatial preference, consumption change, ecosystem services change, population density, and elasticity of marginal utility. We find that the spatial discount rate of WTP for ecosystem services that frequently appears in the empirical literature is the difference between the ecosystem service discount rate and consumption discount rate, where the ecosystem service discount rate includes both physical distance decay and welfare effects. Finally, we use numerical simulations to illustrate how the three different spatial discount rates vary with the spatial distance from the source of ecosystem services and with consumption patterns, implying many more possible spatial variations of WTP.     

Cooperation in the commons

This paper delineates circumstances in which a first-best cooperative solution can be supported as a subgame perfect equilibrium in a dynamic common property renewable resource game. In a game with nonlinear resource stock effects on cost, we characterize a worst perfect equilibrium that supports cooperation for the widest range of parameter values for the discount rate, resource growth rate, harvest price, and the number of resource exploiters. The strategy profile that we propose is consistent with human behavior observed in experiments and common property resource case studies.We thank seminar participants at the University of Minnesota, the Heartland Environmental and Resource Economics Conference at Iowa State University, Keio University, and two anonymous referees for helpful comments.     

Optimal random monetary policy with nominal rigidity

This paper studies a money-in-the-utility function model with imperfect competition and one-period ahead nominal price setting. Under standard assumptions on preferences, Friedman's rule—setting the money growth rate equal to the household time discount factor—generates an equilibrium that is optimal within the class of deterministic policies. We then provide conditions under which a random monetary policy increases ex ante expected welfare relative to Friedman's rule. The result obtains because random policy can reduce the distortion associated with imperfect competition. Our result exhibits original features relative to existing cases of welfare-improving random monetary policy, such as Polemarchakis and Weiss (J. Econom. Theory 15 (1977) 345).     

On discounted dynamic programming with unbounded returns

In this paper, we apply the idea of k-local contraction of Rincón-Zapatero and Rodriguez-Palmero (Econometrica 71:1519–1555, 2003; Econ Theory 33:381–391, 2007) to study discounted stochastic dynamic programming models with unbounded returns. Our main results concern the existence of a unique solution to the Bellman equation and are applied to the theory of stochastic optimal growth. Also a discussion of some subtle issues concerning k-local and global contractions is included.     

The Ramsey Discounting Formula for a Hidden-State Stochastic Growth Process

The long term discount rate is critically dependent upon projections of future growth rates that are fuzzier in proportion to the remoteness of the time horizon. This paper models such increasing fuzziness as an evolving hidden-state stochastic process. The underlying trend growth rate is an unobservable random walk hidden by noisy transitory shocks and recoverable only as a probability distribution via Bayesian updating. A simple expression is derived for the time-declining Ramsey discount rate. The components of this hidden-state Ramsey discounting formula are then analyzed, followed by a few remarks about possible implications and applications.     

On capital accumulation paths in a neoclassical stochastic growth model

Summary. Boldrin and Montrucchio [2] showed that any twice continuously differentiable function could be obtained as the optimal policy function for some value of the discount parameter in a deterministic neoclassical growth model. I extend their result to the stochastic growth model with non-degenerate shocks to preferences or technology. This indicates that one can obtain complex dynamics endogenously in a wide variety of economic models, both under certainty and uncertainty. Further, this result motivates the analysis of convergence of adaptive learning mechanisms to rational expectations in economic models with (potentially) complicated dynamics. Received: June 21, 1996; revised version: October 31, 1996     

Entrepreneurship,entry and performance of new business compared in two growth regimes: East and West Germany

The paper provides an outline of the concept of regional growth regimes and empirically illustrates the relevance of the concept. The empirical examples are entrepreneurship, entry and the performance of new businesses in East and West Germany. The differences of the factors determining the formation of new businesses as well as their development between these two growth regimes are immense and clearly demonstrate the relevance of region specific factors.JEL Classification: O11, O18, P25, R11Michael Fritsch: I amindebted to two anonymous referees for helpful comments on an earlier version and to Pamela Mueller and Antje Weyh for energetic support in preparing the data. Correspondence to Technical University of Freiberg, Faculty of Economics and Business Administration, Lessingstraße 45, 09596 Freiberg, Germany     

Discounting and welfare evaluation of policies

We start with the premise that if policy discounting is to have any welfare relevance, one has to accept it being a derivative of a social welfare function (SWF). We show that if that derivative is to have a net present value (NPV) form, then the baseline allocation must be stationary. In addition, we show that at a stationary baseline in an overlapping generations growth economy, the intergenerationally fair discount rate equals the growth rate of per‐capita consumption, which is, roughly, 2% for the United States. This differs from the interest rate, even in the golden rule equilibrium, unless population growth is null. The last result is based on the main theorem in Mertens and Rubinchik (2012) and is demonstrated for a policy space that might naturally arise in applications.     

Information, Incentives, and Option Value: The Silicon Valley Model

This paper analyzes the Silicon Valley model as a novel economic institution in the domain of technological product system innovation. We analyze the informational relationship as well as governance relationships between venture capitalists and a cluster of entrepreneurial firms. The informational conditions that make the Silicon Valley model efficient are identified, leading to an understanding of the significance of standardized interfaces, modularization, and information encapsulation. We then examine the governance/incentive aspect by integrating the models of Aoki (Towards a Comparative Institutional Analysis, Cambridge, MA: MIT Press, 2001) and Baldwin and Clark (Design Rules—Vol. 1: The Power of Modularity, Cambridge, MA: MIT Press, 2000). The paper concludes by evaluating the applicability of the model to other localities and industries. J. Comp. Econ., December 2002, 30(4), pp. 759–786. Department of Economics, Stanford University, Stanford, California 94305, and RIETI, 1-3-1 Kasumigaseki, Chiyoda-ku, Tokyo 100-0013, Japan; and Faculty of Economics, Toyo University, 5-28-20 Hakusan, Bunkyo-ku, Tokyo 112-8606, Japan. © 2002 Association for Comparative Economic Studies. Published by Elsevier Science (USA). All rights reserved.Journal of Economic Literature Classification Numbers: D21, L23, O32, P51, P52.     

Harvest and extinction in multi-species ecosystems

A potential cost of harvesting in multi-species ecosystems is the extinction of nonharvested species that are at the same trophic level as the harvested species. Existing analytical models are not well-suited for studying this harvest externality because they focus on species interactions across trophic levels instead of within them. We identify the conditions under which the harvesting of a single species causes at least one extinction of nonharvested species at the same trophic level. We compare two harvest regimes: uniform management, in which a privately optimal harvest rate is applied to the entire ecosystem; and specialized management, in which a portion of the ecosystem is intensively managed for the harvested species and the rest is left unharvested. Which regime is more likely to result in extinction depends on the discount rate and on the harvested species' competitive ability and colonization rate compared to those of the other species.     

Growth and income inequality: a canonical model

Summary. We develop an endogenous growth model with elastic labor supply, in which agents differ in their initial endowments of physical capital. In this framework, the growth rate and the distribution of income are jointly determined. The key equilibrating variable is the equilibrium labor supply. It determines the rate of return to capital, which in turn affects both the rate of capital accumulation and the distribution of income across agents. We then examine the impact of various structural shocks on growth and distribution. We find that faster growth is associated with a more unequal, contemporaneous distribution of income, consistent with recent empirical findings.Received: 7 October 2004, Revised: 18 February 2005, JEL Classification Numbers: O17, O40.Cecilia García-Peñalosa: Correspondence toGarcía-Peñalosa would like to acknowledge the financial support received from the Institut d’Economie Publique (IDEP), Marseille. Turnovsky’s research was supported in part by the Castor endowment at the University of Washington. The paper has benefited from seminar presentations at the University of California, Riverside, and the University of Kansas, as well as from the comments of an anonymous referee.     

A Note on Discounting an Increasingly Uncertain Future

This note clarifies the roles played by the wealth and precautionary effects in determining the socially efficient discount rate for public investment projects and how the rate should vary over time. We first give a general characterization of the effects of stochastic shifts in the consumption growth rate on the magnitude of the socially efficient discount rate. We then show that increasing uncertainty in the consumption growth rate provides a natural and compelling rationale for discounting more distant future consumption at a lower rate.     

Justifying social discounting: The rank-discounted utilitarian approach

The discounted utilitarian criterion for infinite horizon social choice has been criticized for treating generations unequally. We propose an extended rank-discounted utilitarian (ERDU) criterion instead. The criterion amounts to discounted utilitarianism on non-decreasing streams, but it treats all generations impartially: discounting becomes the mere expression of intergenerational inequality aversion. We show that more inequality averse ERDU societies have higher social discount rates when future generations are better off. We apply the ERDU approach in two benchmark economic growth models and prove that it promotes sustainable policies that maximize discounted utilitarian welfare.     

Philosophy of Economics Rules: introduction to the symposium

ABSTRACT

Economists have long been criticized for their use of highly idealized models. In Economics rules: Why economics works, when it fails, and how to tell the difference [Oxford: Oxford University Press, 2015] Dani Rodrik responds to this criticism by offering an account of models that emphasizes the diversity of models in economics. Rodrik’s account presents a rare opportunity for economists and philosophers of economics to engage in a mutually beneficial exchange that could improve our understanding of the power and limits of economics, and the rights and wrongs of the dismal science. The symposium on Rodrik’s Economics Rules is the first attempt to seize this opportunity.     

Conditions for Sustainable Optimal Economic Development

This paper shows that, for dynamic optimizing economies with different types of natural resource, environmental, and human‐made capital stocks, a necessary and sufficient condition for permanently sustaining an optimal utility/consumption level is the stationarity of the current‐value Hamiltonian. For economies whose development is not exogenously and directly affected by time (i.e., time‐autonomous economies), this stationarity condition generalizes Dixit et al.’s (1980 ) “zero‐net‐aggregate‐investment” rule of sustain‐ability, which in turn generalizes Solow‐Hartwick’s sustainability rule. For non‐autonomous economies, the stationarity condition is not generally fulfilled, and the current‐value Hamiltonian under (over) estimates the true welfare level by an amount equal to the discounted value of the net “pure time effect.” For the non‐autonomous case of a time‐dependent utility discount rate, a general condition on the discount rate function (of which the hyperbolic discount rate function is a special case) upholds the results obtained for autonomous cases. The paper concludes with a discussion of policies that promote both optimality and sustainability objectives.     

The diversity of models as a means to better explanations in economics

ABSTRACT

In Economics Rules, Rodrik [(2015). Economics rules: Why economics works, when it fails, and how to tell the difference. Oxford: Oxford University Press] argues that what makes economics powerful despite the limitations of each and every model is its diversity of models. Rodrik suggests that the diversity of models in economics improves its explanatory capacities, but he does not fully explain how. I offer a clearer picture of how models relate to explanations of particular economic facts or events, and suggest that the diversity of models is a means to better economic explanations.