Kevin Clinton has provided an interesting counterexample to the verbal argument that the omission of cross-adjustment coefficients necessarily misspecifies the system of asset adjustment proposed by William Brainard and James Tobin.I However, Clinton's specification is not a counterexample to any of the formal propositions developed in my paper. As Clinton observes in a footnote, I dealt with two alternative sets of sufficient conditions for consistency of that system. He has presented a third alternative. All three of these alternatives are special cases of the complete set of necessary and sufficient conditions for consistency. I did not develop these latter conditions in my paper since, at the time I wrote it, I did not recognize their economic interpretation. This interpretation can now be provided. I therefore welcome the opportunity to develop the complete set of conditions. I will derive as one of these conditions the proposition that all columns of the adjustment coefficient matrix, A, have the same sum (i.e., Ei aij is constant for all j). This is a proposition which Clinton needs in order to rearrange his equation (4) into his equation (5), and which he simply assumes. More important, however, will be our discussion of the meaning of the conditions. We begin with equation (11') of mv 1971 paper in this Review:
and of alternative regulatory strategies. Fluorocarbons are widely believed to threaten the stratospheric ozone layer and to influence climate. In fact, all the pertinent effects are highly uncertain, and could well be entirely different from what currently popular hypotheses suggest. The eventual ozone-depleting effect of fluorocarbons has been reduced approximately to zero in recent trials of the going models of stratospheric chemistry, due to new experimental data on the pertinent chemical reactions. The climatic effects could imply either net benefits or costs. Subject to these wide uncertainties, the recent data revisions lead my assessment of benefits of costs to a net benefit expected from continued unregulated emissions of fluorocarbons, due mainly to energy-related benefits of climatic warming. That is, it suggests that regulation would do more harm than good. However, there is a nontrivial possibility of extreme damage, and we might wish to insure against this threat by restricting emissions. Analysis of the buildup of risk over time suggests that the optimum time for a decision on regulation will be ten years or so in the future. In the meanwhile, further research will have narrowed down the uncertainties. At present they are so wide that a wrong decision either way could be very costly to the U.S. economy. The worldwide character of the problem adds both to the uncertainties and to the
Elhanan Helpman, David Pines, and Eli Borukhov (H-P-B) provide an interesting generalization of my earlier paper (1974b). Their less specific characterization of local public services permits them to consider both completely congestible club goods,.I which are the types of services implicitly considered in my earlier model,2 and pure local public services. H-P-B also draw a useful distinction between public goods that act as substitutes for increased land occupancy (complements for increased population density) and can be viewed as people-enhancing services, and those public goods that are complementary to large land holdings a-nd can be considered as property-enhancing services.3 A case can be made that the specific assumptions in my paper are realistic approximations for most urban areas in the United States, and that none of H-P-B's results contradict my conclusions within that framework of restricted optimality; nevertheless, the power of what I would call the planninganalytic approach4 in dealing with very generalized functional relationships is obvious. What is equally obvious is that H-P-B have not considered the parallel case to my analysis where income levels vary according to some known distribution. It may be appropriate to argue that ultimate urban policy prescriptions should not be tied to specific assumptions about functional forms, but similarly, a useful analysis should also incorporate the reality of tolerated differences in the income (or utility) levels and at the very least demonstrate their effect on solutions.5 H-P-B have substantially overcome the first difficulty, and the present paper adopts their technique to account for the effects of mandated dispersions in household utility levels. The interesting conclusions show how changes in the mandated distribution of utility level alter the optimal urban form. I begin by establishing the importance of a city's income distribution in determining its shape. Previously developed spatial equilibrium market solutions are used to numerically simulate the sizable differences in a city's density gradient6 under different' assumed distributions of income. Next, assumptions of specific functional forms are relaxed and the planning-analytic approach is Ipursued under several assumed distributions of household utility. In particular, sufficient conditions for a rising density gradient in the city are established, and the possibility ofhaving an optimal allocation of public *Assistant professor of economics and of environmental engineering, Cornell University. I wish to thank Yoshitsugu Kanemoto for his helpful comments. 'See James Buchanan. 2Both Yoshitsugu Kanemoto and John Wile have also pointed out the restrictions that are inherent in my previous local public goods formulation. 'A Cobb-Douglas or log-linear utility function such as I used in my paper specifies the public good as the borderline case, neither substitute nor complement. 4The distinction is between a straightforward optimal solution to the urban spatial problem, as analyzed by H-P-B, versus a two-step procedure in which equilibrium market solutions are developed and then a social optimum is computed based upon this restricted market framework. Robert Solow used the latter procedure, and I used it in my paper, and it may or may not lead to a global optimum. One advantage of theplanning-analytic approach is the ease with which externalities may be identified and therefore second best solutions in a market analysis can be recognized. 5The prior works of Martin Beckmann and Aldo Montesano have also pursued this problem within a market framework. John Riley, James Mirrlees, and Avinash Dixit have explored slightly more limited cases using the planning-analytic approach. As an. example Dixit allows unequal utilities to exist, weighted by a negative exponential welfare function. Dixit's model also assumes specific Cobb-Douglas utility and production functions., 'The density gradient is the relationship between population density,and distance from the city's central business district (CBD).
The stability of the demand function for money has received extensive attention over the past two decades. However, there is no precise meaning of the term stability in the literature. The issue is most often discussed in reference to time-series estimates of the function and generally based on three characteristics: 1) The demand for money can be explained by a small set of variables as determined by various statistical tests; 2) the function does not exhibit marked shifts over time; 3) the function is capable of generating reasonable forecasts outside of the interval of estimation. Stephen Goldfeld (1973) and John Boorman in exhaustive surveys conclude that relatively simple formulations of the demand for money yield stable shortand long-run functions. Despite some negative evidence (see William Poole), stability of the demand function has been fairly well accepted, at least up to the last few years (Goldfeld, 1976). The overwhelming majority of evidence is based on time-series models using constant coefficient estimation procedures. Yet, arguments can be developed to show that estimating a demand function for money via constant coefficient methods amounts to misspecification. The time varying characteristics of the demand function should be explicitly recognized in the estimation procedure to properly investigate the stability issue. This study is organized around two objectives: First, to use a theoretical model of risk preferences to develop the opportunity cost aspect of the demand function for money implying time varying coefficients and second, to provide within and outside sample comparisons of constant and variable coefficient estimates of various demand function specifications.
has shown that in an international economy satisfying the conditions of the factor price equalization theorem, the same equilibrium which is sustained by both factor mobility and free trade can be sustained by either free trade alone or mobility of the one factor capital alone. Thus it would seem to follow that if free trade is impeded by tariffs, capital mobility will replace trade (p. 325) and maintain undiminished the efficiency of the international economy. This result is noteworthy because it indicates a remarkable resilience in the international economy.
In this study we consider the stability conditions and equilibrium properties of a long-run model of economic growth in which population and labor force are endogenous. Our way of formulating the stability conditions of this model in terms of deviations from a trend solution enables us to relate the model to a variety of historical data and to begin to explore the disequilibrium properties of the model. An additional purpose of this study is to show how sensitive the model is to specification changes. The prime example of this is our finding that the stability of the model with endogenous population and labor force requires the rates of growth of the birth rate and per capita income to be negatively related.