Knowledge that Transforms

I have entirely avoided in this comment the important question of whether the empirical difficulties, especially correlation between pollution and unmeasured neighborhood characteristics, are so overwhelming as to render the entire method useless. I hope that with the air cleared of theoretical misconceptions, future work can proceed to solving these practical problems. For example, recent improvements in pollution measurements and in county assessors' data on individual homes might be combined to permit regressions within a more uniform area. The degree of attention devoted to this kind of detail is what will really determine whether the method stands or falls, once it is recognized that its theoretical underpinnings are sound.

N the study of optimal economic policy using a linear econometric model and a quadratic welfare function the parameters of the model are often assumed to be known for certain. Under this assumption the solution in the form of an optimal feedback equation can be obtained easily. Although one recognizes that in a realistic situation the parameters of an econometric model are never known for certain, he might still apply the above solution, using a set of estimates of the parameters as if they were the true values, if he believes that it is a good approximation to optimal policy. Such a procedure is well known to be a certainty equivalence solution. In a recent paper (Chow, 1973b), I have presented a method of obtaining the optimal feedback equations and the associated welfare costs by allowing for uncertainty in the parameters as expressed in the posterior density function computed from data available at the time of the current decision but not for possible future revision of this posterior density in the derivation of the current policy. Because future learning about the model is not explicitly taken into account in the design of the current policy, the above method is not truly optimal. However if the sample period is long as compared with the planning period, this method will probably be close to being optimal. The purposes of this paper are to present an approximate solution to optimal when learning is taken into account and to contrast this solution with the first two solutions. In the literature, the term control is used for a problem having the dual purpose of improving the system performance and of learning more about the system for the sake of future control. Numerous approximate solutions to this problem have been suggested.' The solution of this paper appears to be the simplest in conception, and yet it incorporates all theoretical elements in the calculations. It contains a logical structure which brings out clearly the effect of learning on the optimization process and enables the effect to be measured numerically. It provides useful contrasts to the certainty equivalence solution and the solution for unknown parameters without learning, being a natural generalization of these two solutions. We will set up the problem and describe the method of solution in section II. This method will be compared in section III with the two other methods just mentioned, both in conceptual terms and in terms of computations. Two simpler, modified versions of the method will also be briefly described. They are simpler to compute but they still take learning partially into account. Some numerical results using a simple one-equation model will be presented in section IV to bring out the effects of learning on the optimal solution. This paper is confined mainly to presenting the method and providing some illustrative calculations. A comprehensive study of the effect of learning on optimal policies using the method of this paper remains to be undertaken. From the viewpoint of economics in general, other than the study of quantitative economic policy using econometric models, the content of this paper may also be relevant. Maximization is in the heart of economics. Most of economic theory assumes maximization to take place in Received for publication December 26, 1973. Revision accented for publication July 8, 1974. * I am much indebted to Andrew Abel for extremely able research assistance, to Edison Tse, Ray C. Fair and several members of the Econometric Research Program seminar at Princeton for valuable suggestions and discussions, to a referee for comments on an early draft, and to the National Science Foundation for financial support through Grant GS32003X. 1 The references in the literature are too numerous to cite. In the economics literature, Prescott (1972) deals with the problem of learning using a very simple model but provides no new method of solution; its results were computed by complete enumeration. MacRae (1972) and Tse (1974) provide interesting approximations to the optimal solution and are highly recommended to the reader.

ln Q -0.1988 + 0.6922 ln L (2.8) (4.7) + 0.5896 In K 0.0223 In MF (4.3) (0.2) + 0.2106InMc + 0.0018T. (3.7) (0.4) R = 0.986 D.W. = 1.03 The results indicate that the real money balances held by firms seem to have no significant effect on output. The level of money held by consumers retains its significance. These results are again consistent with induced innovation approach but not with the production factor approach.

A survey of the empirical work involving interest rates indicates an almost universal use of seasonally unadjusted rates, apparently due to a conviction that seasonal factors are not important in interest rates. Studies by V. Kerry Smith and Richard Marcis (1972), Stanley Diller (1969) and William Gibson (1970), however, provide evidence showing that interest rates exhibit seasonal variation. Seasonality in interest rates merits careful consideration for at least two reasons. First, as noted by Gibson, An aim of the Federal Reserve System is to accommodate seasonal swings in the financial needs of trade, and the system tries to do this by removing seasonal fluctuations from interest rates. The seasonal variations remaining in interest rates suggest that the system is not wholly successful in these efforts. . (Gibson, 1970, p. 442). Second, from an econometric point of view, any model, such as a money demand model, which contains an interest rate as an independent variable should also contain seasonal dummy variables; otherwise, the coefficient of the interest rate variable may be both biased and inconsistent.' These issues underscore the importance of determining whether there is seasonality in interest rates. In this regard, it is interesting to note that the Federal Reserve does not report seasonally adjusted rates, apparently because the Board does not recognize the existence of a seasonal component. The purpose of this paper is to provide additional information regarding seasonality in interest rates. Unlike earlier studies, both daily and monthly data are analyzed. And, whereas Smith and Marcis employed spectral analysis to detect seasonality, ordinary least squares techniques with seasonal dummies are employed here. Using this technique, we find, in contrast to the findings reported by other investigators, no evidence of a significant seasonal component in monthly rates. Seasonality is, however, present in the daily rate. The plan of the remainder of the paper is as follows. In section II, the magnitude and the importance of seasonality in three interest rates are assessed. Section III contains a discussion of seasonal variation in a short-term rate employing daily as opposed to monthly data. A summary and the conclusions are presented in the last section.

The transformation of Puerto Rico from a sugar-monoculture began in the early 1950's and by 1963 created a diversified manufacturing export economy. During these years, gross domestic product more than doubled, while gross investment and exports nearly quadrupled (Table 1). The magnitude of the industrialization, as seen in the rise of the industrial share of GDP by 50% and the decline of the agricultural share by about the same extent, resulted in the doubling of per capita income while employment remained stable.

Alterman, J., Interindustry Employment Requirements, Monthly Labour Review (July 1965), 841850. Carter, A. P., Structural Change in the American Economy, Harvard Studies in Technology and Society (Cambridge, Harvard University Press, 1970), 34. Economic Commission for Europe (ECE) Standardised Tables of ECE countries for Years Around 1959, (1970), 1-31. (mimeo). Lancaster, K., Mathematical Economics (New York: Macmillan Company, 1968), 86. Leontief, W. W., The Structure of American Economy 1919-39 (Oxford University Press, 1960), 160. U.K., Central Statistical Office, Input-Output Tables for the United Kingdom, Studies in Official Statistics, no. 16 (H.M.S.O. 1970), 1-89. U.S. Bureau of Commerce, Office of Business Economics, Input-Output Structure of the U.S. Economy: 1963, Survey of Current Business (Nov. 1969), 16-47.

THE pathbreaking introduction of the linear expenditure model by Klein and Rubin (1947-1948) opened the way to an econometric implementation of the theory of the true cost-of-living index. The hypothesis of a given static utility function, however, raises doubts about its relevance to a world of changing preferences. This paper aims at developing an econometric application of the theory of the true cost-of-living index for the case of taste changes. It is our hope that we will thus contribute to the improvement of the official indices measuring the cost of living. In a static context, the relevant indifference class is mostly chosen with reference to a price income vector. (For example, prices and income may be those prevailing during the base period or those prevailing during the period of comparison.) But the indifference class can also be made subject to the choice of a commodity vector, there being a one-to-one correspondence between the two vectors. In a dynamic world with continuous and systematic taste changes, on the contrary, the two vectors will not lead to the identification of the same indifference class, as was pointed out by de Souza (1974). Furthermore, the indifference class chosen is indicated by a corresponding utility level in a static model, while the utility function is time dependent in a dynamic model. A reference year is then needed to fix the time dependent parameters of the dynamic utility function. This opens the way to using the utility level not only as an indicator representative of an indifference class but also as determining of itself a level of satisfaction in a cardinalist sense. A correspondence has then to be established between indifference curves of one map at one moment of time and those of a map at another moment (i.e., after a change in tastes). And this correspondence has to be interpreted as representing equal welfare. In this paper, we develop an algorithm for the computation of two dynamic indexes. One index implements the theory developed by Fisher and Shell (F-S for short) in their wellknown 1969 paper on taste and quality change in the pure theory of the true cost-of-living index and is called the (ordinal) F-S index. This index belongs to the class of simultaneous indices, the comparison being based on current tastes while the indifference class is chosen with reference to the base period price-income vector. The other index, called the index, belongs to the class of temporal indices. It takes the base year utility level as a reference point and determines the income that together with comparison period prices and tastes will allow the consumer to attain the base-year utility level. The base period utility level is obtained by maximizing the base period utility function subject to the base period price-income vector constraint. Before proceeding, it is worth noticing that the adjective characterizes the choice of the base-year utility level as the reference point. As a result, the cardinal index is invariant only under monotonic transformations of the utility function that are not time dependent. In particular, this index is based on the assumption that there is no change over time in the of the consumer as a pleasure machine. (This assumption corresponds to the hypothesis of the absence of neutral technical progress in the theory of production.) A change in efficiency does not affect the preference ordering nor the demand functions, but does affect the utility level and therefore the index.' Received for publication May 10, 1973. Revision accepted for publication August 1, 1974.

PREVIOUS studies of residential and commercial energy use, exclusive of electricity, have generally found that price is not a significant determinant of demand.' This paper analyzes the demand for space heating energy using cross-sectional data by state for 1971. We find that price is a significant determinant of demand with an elasticity of approximately -0.3. In deriving this estimate, explicit attention is given to: (1) discrimination among alternative functional forms since average price data rather than marginal prices are used; and (2) specification of relationships that attempt to capture the trade-off that exists between energy inputs and housing construction and insulation materials.