The ‘curse of dimensionality’ resolved: The effects of climate change and trade barriers in large dimensional modelling

Solving large scale optimisation problems over space and time quickly generates a computational impasse, termed the ‘curse of dimensionality’. This severely limits the practical use of economic models, especially for determining the effects of climate change and protectionist trade policies. In this paper, we employ an innovative approach to solving (otherwise unsolvable) large scale systems through the use of parallel processing methods and a proper ordering of variables and equations in a ‘Nested Doubly Bordered Block Diagonal’ form. We illustrate how the approach can be used to solve an intertemporal CGE model with more than 500 million equations. Using existing damage functions, the framework allows us to determine the impact of climate change on long-run economic growth for 112 countries as a result of the effect of sea-level rise on land endowments, variation in crop yields and productivity and shifts in the demand for energy and transportation. We also compare our solution to more common (and smaller dimensional) recursive methods, in terms of both the economic effects of climate change and potential increases in trade barriers, showing the power and efficiency of our computational approach and parallel processing routine.     

On dynamic programming with unbounded returns

Summary. Some economic models like those of endogenous growth motivate the analysis of a class of recursive models sharing the property that the return function is not bounded along feasible paths. We consider a strategy of proof which allows to deal with many unbounded recursive models exploiting bounds to the rates of growth rather than to the levels.     

Computing DSGE models with recursive preferences and stochastic volatility

This paper compares different solution methods for computing the equilibrium of dynamic stochastic general equilibrium (DSGE) models with recursive preferences such as those in Epstein and Zin, 1989, Epstein and Zin, 1991 and stochastic volatility. Models with these two features have recently become popular, but we know little about the best ways to implement them numerically. To fill this gap, we solve the stochastic neoclassical growth model with recursive preferences and stochastic volatility using four different approaches: second- and third-order perturbation, Chebyshev polynomials, and value function iteration. We document the performance of the methods in terms of computing time, implementation complexity, and accuracy. Our main finding is that perturbations are competitive in terms of accuracy with Chebyshev polynomials and value function iteration while being several orders of magnitude faster to run. Therefore, we conclude that perturbation methods are an attractive approach for computing this class of problems.     

The recursive approach to time inconsistency

We introduce a new class of finite horizon stochastic decision problems in which preferences change over time, and provide a proof of the existence of a recursively optimal strategy. Recursive optimization techniques dominate many areas of economic dynamics. However, in decision problems in which tastes change over time, the solution technique most commonly applied is not recursive, but rather strategic (subgame perfection). In this paper we argue in favor of the recursive approach, and we take the necessary theoretical steps to make the recursive methodology applicable.     

Modeling the dynamics of regulated resource systems: a fishery example

In this paper, a discrete-time model of regulated fisheries is developed. This class of models is interesting because most modern real-world fisheries are under some kind of regulation. The regulatory part of the fishery in this paper is partitioned into two stages. In the first stage, which is the main focus, total allowable catch quotas (TACs) based on biological and economic considerations are determined in a way that guarantees the safety of the stock from a conservation viewpoint. In addition, we assume that a target biomass level is set by the management authorities to be achieved over a given time horizon to satisfy an economic objective. Since we assume here that the main goal is to rebuild the stock, we propose a gradual approach to the target biomass level via a simple recursive rule.     

Economic Dynamics with Learning: New Stability Results

Drawing upon recent contributions in the statistical literature, we present new results on the convergence of recursive, stochastic algorithms which can be applied to economic models with learning and which generalize previous results. The formal results provide probability bounds for convergence which can be used to describe the local stability under learning of rational expectations equilibria in stochastic models. Economic examples include local stability in a multivariate linear model with multiple equilibria and global convergence in a model with a unique equilibrium.     

A 4-Stated DICE: Quantitatively Addressing Uncertainty Effects in Climate Change

We introduce a version of the DICE-2007 model designed for uncertainty analysis. DICE is a wide-spread deterministic integrated assessment model of climate change. Climate change, long-term economic development, and their interactions are highly uncertain. The quantitative analysis of optimal mitigation policy under uncertainty requires a recursive dynamic programming implementation of integrated assessment models. Such implementations are subject to the curse of dimensionality. Every increase in the dimension of the state space is paid for by a combination of (exponentially) increasing processor time, lower quality of the value or policy function approximations, and reductions of the uncertainty domain. The paper promotes a state-reduced, recursive dynamic programming implementation of the DICE-2007 model. We achieve the reduction by simplifying the carbon cycle and the temperature delay equations. We compare our model’s performance and that of the DICE model to the scientific AOGCM models emulated by MAGICC 6.0 and find that our simplified model performs equally well as the original DICE model. Our implementation solves the infinite planning horizon problem in an arbitrary time step. The paper is the first to carefully analyze the quality of the value function approximation using two different types of basis functions and systematically varying the dimension of the basis. We present the closed form, continuous time approximation to the exogenous (discretely and inductively defined) processes in DICE, and we present a numerically more efficient re-normalized Bellman equation that, in addition, can disentangle risk attitude from the propensity to smooth consumption over time.     

Uncertainty and Learning in Stochastic Macro Models

Limits on information have deep economic impact and affect the conduct of economic policy. In the present paper we explore the effect of substantive uncertainty in a macro model, from both an analytical and methodological point of view. Agents are boundedly rational and make their forecasts according to different techniques and try to learn the values of the various parameters. In this context, a Markov regime switching rule, a VAR system, and recursive least square are considered and compared. As a result, we obtain a model which is mostly keynesian in nature that can be compared with the new neoclassical synthesis models. Simulations are carried out and show the possible appearence of endogenous and persistent fluctuations.     

RECURSIVE EQUILIBRIA IN AN AIYAGARI‐STYLE ECONOMY WITH PERMANENT INCOME SHOCKS

We prove existence of a recursive competitive equilibrium (RCE) for an Aiyagari‐style economy with permanent income shocks and derive important economic implications. We show that there exist equilibria where borrowing constraints are never binding and establish a nontrivial lower bound on the equilibrium interest rate. These results imply distinct consumption dynamics compared to existing studies. We present a new approach to solve the agent's problem that uses lattices of consumption functions to deal with permanent income shocks and an unbounded utility function. The approach provides a theoretical foundation for convergence of the time iteration algorithm widely used in applied work.     

On Impatience, Economic Growth and the Environmental Kuznets Curve: A Dynamic Analysis of Resource Management

The paper presents a dynamic analysis of natural resource management and investigates some key factors that affect optimal management and resource conservation.Using a recursive specification of time preferences, we show how endogenous discounting and impatience can affect the motivation for both capital investments and environmental preservation. We examine the relationships between economic growth and environmental quality. Endogenous discounting provides new insights in the economic dynamics underlying the environmental Kuznets curve. By treating growth as endogenous, we examine how externalities and economic growth interact with each other. We investigate how economic development can contribute to an increased demand for environmental preservation. As an important new result, we also show how poverty can contribute to environmental degradation.     

Existence and Uniqueness of Equilibrium in Distorted Dynamic Economies with Capital and Labor

In this paper, we provide a set of sufficient conditions under which recursive competitive equilibrium exists and is unique for a large class of distorted dynamic equilibrium models with capital and elastic labor supply. We develop a lattice based approach to the problem. The class of economies for which we are able to obtain our existence and uniqueness result is considerably larger than those considered in previous work. We conclude by applying the new results to some important examples of monetary economies often used in applied work. Journal of Economic Literature Classification Numbers: C62, D51, D90, E10.     

Initial beliefs and the global stability of least squares learning

This article provides an operational framework of global stability analysis when Ljung's (1977) ordinary differential equation (ODE) approach is applied to the recursive stochastic system. We first establish the notion of stable set under which sufficient conditions for the equivalence between them can be translated to restrictions on initial beliefs in agents' forecasts. The maximum ODE-stable set is simply the largest range of initial beliefs. We then demonstrate how to implement this operational framework for the analysis of stability beyond the local sense in some well-known models in the learning literature.     

NUMERICAL SIMULATION OF NONOPTIMAL DYNAMIC EQUILIBRIUM MODELS

In this article, we propose a recursive equilibrium algorithm for the numerical simulation of nonoptimal dynamic economies. This algorithm builds upon a convergent operator over an expanded set of state variables. The fixed point of this operator defines the set of all Markovian equilibria. We study approximation properties of the operator. We also apply our recursive equilibrium algorithm to various models with heterogeneous agents, incomplete financial markets, endogenous and exogenous borrowing constraints, taxes, and money.     

Modeling input–output impacts with substitutions in the household sector: A numerical example

Although there have been many elaborations of the basic input–output approach, including multi-regional models, dynamic models, models with variable coefficients, supply-side models, etc., these approaches all have the same limitation. The fixed-coefficients production function assumptions ignore substitutions in response to price changes that can be expected to accompany most shocks— skipping over the heart and soul of market economics. This research note suggests a simple approach to estimating new technical coefficients matrices after a shock so that the consequences of short-term substitution effects can be studied. Given a reduction in income (as reflected in the value added row), households are likely to make substitutions, reducing their final demand by less than the application of base-year I–O coefficients would indicate. But if ex post changed income and consumption can be observed, the application of RAS procedures can generate an appropriately modified A matrix. The resulting set of interdependent substitutions that occurred can be identified. Due to some well known limits in applying the traditional RAS approach, we reformatted it and suggest a new economic model that can link coefficient adjustments to degrees of a priori substitutability and complementarity. Based on this resolution, we look forward to detailed studies of specific coefficients and how they evolve over the short term.     

How diverse are national banking systems? An analysis on banking business models,profitability, and riskiness

In this paper, we apply an algorithm developed by Martin Weitzman to quantify the extent of diversity among the business models of financial intermediaries at an international level. In particular, we investigate the relationship between the diversity of the business models of EU national banking systems and their profitability and riskiness. We show how Weitzman's approach can be generally applied to the issue at hand; as a by-product, the analysis allows us to assess whether the diversity among banking systems was affected by the imported systemic financial crises of 2007/8 as well as its domestic sequel centered on the sovereign debt crisis that occurred in 2010–12. The motivation for this paper is twofold. First, we provide an operational measure of the diversity of business models among banking sectors. Second, we enrich the economic literature relating to banking business models by providing a macro-founded analysis. To this end, we highlight the range of diversity of national banking business models correlated with high performances in terms of profitability and riskiness.     

Moving beyond the iceberg model: The role of trade relations in endogenizing transportation costs in computable general equilibrium models

Modeling trade and transportation costs is an essential part of multiregional or spatial computable general equilibrium models where interregional trade plays an important rolein shaping economic activity. The majority of such models use the iceberg trade cost approach where part of the produced output (representing the material costs of transportation) is assumed to melt away during transportation. There are a few models which employ a more refined approach with an explicit transportation sector providing transportation services which are then used to ship goods between locations. In this paper we show that this approach, although much more convenient than the iceberg approach, still lacks full usability due to the fact that markets, hence prices are defined at the regional level and as a result, transportation costs can not be endogenous at the trade relation level. Moreover, under regional level market clearing the iceberg and the more detailed approach are equivalent. We propose to refine the definition of market equilibrium and move it to the trade relation level. Using this approach we can gain full advantage of the explicit transport sector in the model with respect to trade cost evolution. We show through simulations that refining the way trade costs are modelled indeed gains new insights, and that moving the market definition to the trade relational level leads to qualitative changes in the effect of labor supply shocks on main model variables. The paper also presents a method to estimate a SAM by reallocating data from standard industries to a transportation sector which is then consistent with the model setup. This SAM can be used to calibrate the refined model with a detailed transportation sector.     

Specifications of Social Welfare in Economic Studies of Climate Policy: Overview of Criteria and Related Policy Insights

An important assumption in economic studies of climate policy is the social welfare function. This paper shows that applying distinct decision or social welfare criteria can result in different optimal policies of climate control, notably if climate change impacts are uncertain. First, decision criteria in current climate-economic studies are reviewed. Next, the most important alternatives are discussed, including their (mathematical) formalization and incorporation in economy-climate models. Most of these alternative criteria suggest more stringent climate policies to be optimal than the standard discounted utilitarianism approach. However, several important welfare criteria have not or hardly seen any application in the economic analysis of climate policy. We conclude that there is clear need for systematic research on this theme, for which the current review provides a solid basis.     

On the indeterminacy of capital accumulation paths

In neoclassical optimal growth models the stability of the accumulation paths depends on the discount parameter. We prove that, for discount factors small enough, the policy function which describes an optimal path can be of any type. The result is achieved using the notion of α-concavity. We adopt a constructive approach. Given any twice differentiable map we show how to construct an optimal growth problem which produces that map as the optimal policy function. A consequence is that “chaos” can appear in these models. We also provide bounds on the values of the discount parameter for which “indeterminacy” is possible.     

The measurement of environmental economic inefficiency with pollution-generating technologies

This study introduces the measurement of environmental inefficiency from an economic perspective. We develop our proposal using the latest by-production models that consider two separate and parallel technologies: a standard technology generating good outputs, and a polluting technology for the by-production of bad outputs. While research into environmental inefficiency incorporating undesirable or bad outputs from a technological perspective is well established, no significant attempts have been made to extend it to the economic sphere. Based on the definition of net profits, we develop an economic inefficiency measure that accounts for suboptimal behavior in the form of foregone private revenue and environmental cost excess. We show that economic inefficiency can be consistently decomposed according to technical and allocative criteria, considering the two separate technologies and market prices, respectively. We illustrate the empirical implementation of our approach using a dataset on agriculture at the level of US states.     

Non-self-averaging and the statistical mechanics of endogenous macroeconomic fluctuations

Strong fluctuation phenomena are an endogenous feature of economic systems if they are non-self-averaging. We show that an important consequence of non-self-averaging is that current forms of economic policy can be rendered useless. We also find non-self-averaging both to exist in microeconomic models of cluster development within economies and to be consistent with observed economic power laws. These results suggest the need for straightforward identification of non-self-averaging in economic systems and to this end we present a sufficient condition for non-self-averaging in terms familiar to financial risk management.