Avoiding the Curse of Dimensionality in Dynamic Stochastic Games

Discrete-time stochastic games with a finite number of states have been widely ap- plied to study the strategic interactions among forward-looking players in dynamic en- vironments. However, these games suffer from a "curse of dimensionality" since the cost of computing players' expectations over all possible future states increases exponentially in the number of state variables. We explore the alternative of continuous-time stochas- tic games with a finite number of states, and show that continuous time has substantial computational and conceptual advantages. Most important, continuous time avoids the curse of dimensionality, thereby speeding up the computations by orders of magnitude in games with more than a few state variables. Overall, the continuous-time approach opens the way to analyze more complex and realistic stochastic games than currently feasible.

Avoiding the Curse of Dimensionality in Dynamic Stochastic Games

Continuous-time stochastic games with a finite number of states have substantial computational and conceptual advantages over the more common discrete-time model. In particular, continuous time avoids a curse of dimensionality and speeds up computations by orders of magnitude in games with more than a few state variables. The continuous-time approach opens the way to analyze more complex and realistic stochastic games than is feasible in discrete-time models.

Finite Lifetimes, Borrowing Constraints, and Short-Run Fiscal Policy

Recent developments in public finance in the analysis of dynamic government debt policies have emphasized effects on the distribution of real resources across generations. At the same time, macroeconomists have emphasized the importance of the length of the time horizon over which agents optimize their decisions about consumption for judging the effects of fiscal policy on aggregate demand. Much of the discussion of these issues has focused on whether linkages among generations are sufficient to give consumers infinite horizons. To the extent that horizons are finite, debt burdens can be shifted to future generations, and substitutions of debt for taxes have real effects. This paper argues that, as a matter of quantitative significance, theoretical and empirical emphasis on the importance of finite horizons for the analysis of many fiscal policies is misplaced. Studies of the role of finite horizons in determining the effects of short-run fiscal policies on consumption have been conducted largely under the assumption of perfect capital markets. We show that while the marginal propensity to consume (MPC) out of temporary tax changes is nonzero in finite- horizon models, it is not very large. We demonstrate that the MPC is, however, quite sensitive to the importance of restrictions on borrowing in the economy. The clear implication is that shifting emphasis from the length of the planning horizon to the structure of capital markets is an important step for empirical research.

Optimal Rules for Patent Races

There are two important rules in a patent race: what an innovator must accomplish to receive the patent and the allocation of the benefits that flow from the innovation. Most patent races end before R&D is completed and the prize to the innovator is often less than the social benefit of the innovation. We study the optimal combination of prize and minimal accomplishment necessary to obtain a patent in a dynamic multistage innovation race. A planner, who cannot distinguish between competing firms, chooses the innovation stage at which the patent is awarded and the magnitude of the prize to the winner. We examine both social surplus and consumer surplus maximizing patent race rules. We show that a key consideration is the efficiency costs of transfers and of monopoly power to the patentholder. We show that races are undesirable only when efficiency costs are low, firms have similar technologies, and the planner maximizes social surplus. However, in all other circumstances, the optimal policy spurs innovative effort through a race of nontrivial duration. Races are also used to filter out inferior innovators.

Asymptotic Methods for Asset Market Equilibrium Analysis

General equilibrium analysis is difficult when asset markets are incomplete. We make the simplifying assumption that uncertainty is small and use bifurcation methods to compute Taylor series approximations for asset demand and asset market equilibrium. A computer must be used to derive these approximations since they involve large amounts of algebraic manipulation. To illustrate this method, we apply it to analyzing the allocative, price, and welfare effects of introducing a new derivative security. We find that the introduction of any derivative will raise the value of the risky asset relative to bonds.

Social Security and Individual Welfare: Precautionary Saving, LiquidityConstraints, and the Payroll Tax

Recent advances in the examination of efficiency gains from dynamic tax reforms have used simulation models to isolate intragenerational and/or intergenerational effects. Important considerations having to do with uncertainty or capital market imperfections are frequently missing from such a framework. In this paper, we focus on the welfare gains from introducing social security retirement annuities, given lifetime uncertainty and borrowing restrictions.Our principal findings are four. First, given the considerations mentioned above, "precautionary saving" exceeds life-cycle saving (that would have taken place in the absence of lifetime uncertainty), lending further support to the notion that the perfect-certainty version of the life-cycle model provides an inadequate explanation of observed saving behavior. Second, the introduction of an actuarially fair social security system leads to a significant partial equilibrium increase in lifetime consumption and welfare, accompanied by a reduction in the capital stock.The increase in lifetime welfare is reduced, however,and in many cases eliminated, when borrowing restrictions are imposed.Third, extending the model to general equilibrium, we find that the partial equilibrium gains in lifetime welfare from participation in social security are offset by the interaction of higher steady-state interest rates and binding liquidity constraints. Finally, replacing the proportional payroll tax with a progressive tax (essentially a linear tax with an exemption), we show that age-specific tax schemes can restore much of the potential gain from introducing social security.

Mergers and Dynamic Oligopoly

Static oligopoly theories disagree on whether mergers are profitable. The Cournot model says that many potential mergers would be unprofitable whereas the Bertrand model says that all mergers are profitable. We show that, for economically sensible parameter values, mergers are profitable for merging firms when firms choose both price and output, using inventories to absorb differences between output and sales. Furthermore, substantial cost advantages are necessary for a merger to benefit consumers. The merger predictions of our dynamic model are most similarto predictions of static Bertrand analyses of differentiated products even though our model often behaves like the Cournot model in the long run.oligopoly, dynamic games, mergers

Liquidity Constraints, Fiscal Policy, and Consumption

macroeconomics, liquidity constraints, fiscal policy, and consumption

Optimal control and money targets: should the Fed look at "everything"?

Monetary policy - United States ; Money supply

The Optimal Tax Rate for Capital Income is Negative

We examine the problem of optimal taxation in a dynamic economy with imperfectly competitive markets. We find that the optimal tax system will tend to provide subsidies for the purchase of capital goods to offset gaps between price and marginal cost. The average tax on capital income will be negative, even if pure profits are not taxed away and even if the alternative distortionary taxes have an infinite efficiency cost. These arguments hold even if it is necessary to tax consumption goods which also sell above marginal cost; the difference is that capital goods are intermediate goods and consumption goods are final goods. Since observed markups are greater for equipment than for construction, this analysis justifies the Investment Tax Credit's discrimination in favor of equipment over structures.