The aggregate rate of R&D in a competitive economy is compared with the optimal rate. The optimal rate of R&D is shown to be the same for all preferences in a broad family, while the competitive rate is sensitive to the form of substitutability among products and so can vary dramatically within a family. The second-best level of R&D is shown to be also common within a family and equal to the optimal rate. Numerical examples suggest that diminishing returns in the innovation technology is the most important potential source for excessive R&D in a competitive economy.

North-South trade is studied in a model of vertical product differentiation. The South produces a low-quality spectrum of goods and the North a high-quality spectrum. An increase in the South's population lowers its relative wage, expands the spectrum of Southern goods at the top, and shifts the Northern spectrum upward. An increase in Northern labour productivity raises its relative wage. If the increase is neutral or export-biased, then the South's terms of trade improve, the spectrum of Northern products expands, the spectrum of Southern products contracts, and the volume of trade grows. If it is biased against Northern exports, these effects are reversed. Similar results hold for neutral increases in Southern productivity.

Although an income tax is often a government's most important instrument for raising revenue and redistributing income, its potential usefulness for either of these purposes is limited by its negative impact on work incentives. The implications of the incentive effect have been studied by examining optimal tax structures under a variety of assumptions about preferences, the distribution of wage rates, and the form of the social welfare function. Much of this work has been done using models that include interpersonal variation in ability (wage rates), but in which no saving or dissaving occurs, i.e. in which the consumption of each individual is exactly equal to his labour income net of taxes within the time period (Mirrlees (1971), Sheshinski (1972a), (1972b), Atkinson (1973a), (1973b), Phelps (1973), Cooter and Helpman (1974), Itsumi (1974), Sadka (1976)). While the latter assumption might be innocuous if wage rates were approximately constant over an individual's lifetime, so that there was little incentive to borrow or save, or if capital markets were non-existent, so that borrowing and saving were impossible, wage profiles are in fact quite steep and most individuals make use of (admittedly imperfect) capital markets. Multiperiod models incorporating consumption-saving decisions have been used to study the effects on capital accumulation of wage, interest, capital gains, and other taxes, but even those models that include variation in ability have in general assumed that labour is supplied at a constant rate over the individual's working years (Ordover and Phelps (1975), Sheshinski (1976), Feldstein (1974)). A life-cycle model of individual behaviour that includes both labour supply and consumption decisions is used below. Although a general equilibrium framework is used, for simplicity real capital is ignored; labour is the only factor of production and government debt is the only asset available to savers. Only steady states and only linear tax schedules are considered, and the utilitarian social welfare function is used throughout. First, the conditions under which the life-cycle model reduces to the one-period case are derived, as well as the conditions under which the first-best tax policies for the two are identical. Next it is shown that if all individuals are identical, the optimal policy consists of lump-sum taxes together with an interest rate equal to the rate of pure time preference. The non-optimality of the biological interest rate proposed by Samuelson (1958) in his Exact Consumption-Loan Model is discussed. Finally, an upper bound on the optimal marginal tax rate is derived. This bound depends on the elasticity of total labour supply and on the elasticity of demand for debt.

A model of growth is developed in which finite-lived individuals invest in human capital, investments have a positive external effect on the human capital of later cohorts, and labor with more human capital produces higher-quality goods. Stationary growth paths are analyzed, paths along which human capital and the quality of goods grow at a common, constant rate. It is also shown that if a small open economy is either very advanced or very backward relative to the rest of the world, then its rate of investment in human capital is lower under free trade than under autarky.

The determination of wage rates and skill requirements for jobs is studied in a model where workers, who vary in ability, acquire skill through (costly) training. If firms' skill requirements are fixed, there exists a unique Nash equilibrium in wage rates, but the resulting allocation of labor is inefficient. The Nash equilibrium wage rates and allocation of labor approach the competitive equilibrium if and only if the number of job types increases on every skill range and no firm remains a monopsonist on any skill range. If firms choose both skill requirements and wage rates, no Nash equilibrium in pure strategies exists.

I. Introduction, 355.—II. No production costs: "pure" discrimination, 357.—III. Positive production costs, 363.—IV. Conclusion, 367.

Recent work has shown that reputational arguments based on trigger strategies can sometimes be used to deal with the problem of time inconsistency in government policy.' This idea was exploited by Robert Barro and David Gordon (1983) in a model of inflation and by V. V. Chari and Patrick Kehoe (1988) in a model of capital taxation. Here it is shown that reputational arguments of this sort are widely applicable.2 A general policy model is developed, based on an economy that is stationary over time, contains identical households, and has no endogenous state variables. It is shown in general that the infinite-horizon version of this model has equilibria in trigger strategies that yield outcomes strictly better than those attainable as equilibria in the one-period model.

In his work on economic development, Lucas?s goals were to understand the enormous diversity across countries in levels and growth rates of per capita income and to build a framework for studying developing economies with the same kind of precision and rigor that the neoclassical growth model had brought to the study of already industrialized countries. Lucas found knowledge spillovers to be a key mechanism. Thus, human capital, which he viewed broadly, is at the center of much of his work, and external effects are crucial. Lucas stated explicitly that in his view, physical capital plays a distinctly secondary role.