Kenneth L. Judd

Projection methods for solving aggregate growth models

Abstract We describe a general numerical approach, the projection method, to solve operator equations which arise in economic models. Principles from numerical analysis are then used to develop efficient implementations of the projection method for solving aggregate growth models. Since any numerical approach will involve error, we derive error measures which are related to optimization errors by agents and argue that the numerical approximations can be viewed as equilibria with boundedly rational agents. The results are programs which run hundreds of times faster than competing methods in the literature while achieving high accuracy.

The law of large numbers with a continuum of IID random variables

Abstract There are two problems with the common argument that a continuum of independent and identically distributed random variables sum to a nonrandom quantity in “large economies”. First, it may be unintelligible in that it may call for the measure of a nonmeasurable set. However, there is a probability measure, consistent with the finite-dimensional distributions, which assigns zero measure to the set of realizations having that difficulty. A second difficulty is that the “law of large numbers” may not hold even when there is no measurability problem.

Credible spatial preemption

It is often argued that incumbent firms may deter entry by preemptive investment in new goods. We show that these conclusions are reserved when multiproduct incumbent firms may exit in response to entry. Once entrants are in an industry, an incumbent will often want to withdraw some goods to prevent competition with the entrant from reducing profits on other goods. Such a reaction makes entry more attractive to a potential entrant. The equilibrium industry structure is less likely to be monopolistic as the goods are better substitutes, as exit costs are low, and as the competition between producers of the same good is more intense.

The Welfare Cost of Factor Taxation in a Perfect-Foresight Model

This paper examines the marginal efficiency cost of various factor taxes in a dynamic general-equilibrium model. The au thor solves for the excess burden of anticipated and unanticipated, t emporary and permanent, tax policies. Using a range of estimates for taste and technology parameters, he finds excess burdens larger than those indicated by previous studies that use static models. These exc ess burdens are sensitive to parameter values and timing. However, th e rankings of alternative tax policies turn out to be insensitive to structural parameter values. Copyright 1987 by University of Chicago Press.

Constrained Optimization Approaches to Estimation of Structural Models

To run the code, users need to install the following software: AMPL, KNITRO, and Matlab. AMPL is an algebraic modeling language. Users should contact AMPL Optimization LLC (www.ampl.com) to purchase a license for AMPL. KNITRO is a state-of-the-art constrained optimization solver, initially developed at Northwestern University, now developed and distributed by Ziena Optimization, Inc. (www.ziena.com). Users should contact Ziena Optimization Inc. to purchase a license for KNITRO.

Observable Contracts: Strategic Delegation and Cooperation

The role of commitments in noncooperative games is well acknowledged and documented. One way to achieve commitments is by letting delegates represent the players of a game. In this paper, the authors study a delegation game in which the players can use agents strategically to play on their behalf and the contracts they sign with them are common knowledge. They show that, in such cases, every Pareto optimal outcome of the game can become the unique subgame perfect Nash equilibrium of the delegation game. The authors demonstrate this result by discussing the Cournot-type duopolistic game. Copyright 1991 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.

Optimal taxation and spending in general competitive growth models

Abstract We find that the optimal long-run tax on capital income is zero even if the capital stock does not converge to a steady state nor to a steady state growth rate. The optimal tax on human capital is also zero if human capital is not a final good, but the long-run wage tax is not generally zero. We argue that “consumption” tax proposals, such as the Flat Tax, are not consumption taxes, and are biased against human capital.

Short-Run Analysis of Fiscal Policy in a Simple Perfect Foresight Model

This paper examines the short-run impact of current and future changes in fiscal policy on current investment in a simple representative-agent, perfect foresight model. We show that anticipated investment tax credits may depress current investment, as may an immediate income tax cut financed by f uture cuts in government expenditure. These impacts do result when we parameterize the model with current empirical estimates of the relevant parameters.

Asymptotic methods for aggregate growth models

Abstract We use perturbation methods to compute optimal policy functions in simple continuous- and discrete-time aggregate growth models. We demonstrate that computing the kth degree Taylor expansion of the policy function around the steady state involves solving one quadratic equation and k − 1 linear equations. We also compute Pade expansions, and show that both Taylor and Pade expansions can provide excellent solutions far from the steady state.

Solving Large Scale Rational Expectations Models

We explore alternative approaches to numerical solutions of large rational expectations models. We discuss and compare several current alternatives, focussing on the tradeoffs in accuracy, space, and speed. The models range from representative agent models with many goods and capital stocks, to models of heterogeneous agents with complete or incomplete asset markets. The methods discussed include perturbation and projection methods. We show that these methods are capable of analyzing moderately large models even when we use only elementary, general purpose numerical methods.