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.

N this paper, I will generalize the modified Newton method previously applied in Chow (1968) to the computation of full-information maximum likelihood estimates of parameters of a system of linear structural equations to the case of a system of nonlinear structural equations. The success of that method for linear systems 1 has stimulated my present attempt to generalize it for nonlinear systems. The subject of maximum likelihood estimation of nonlinear simultaneous equation systems has been studied by Eisenpress and Greenstadt (1966). There are three main differences between their approach and ours. First, their basic formulation is more general, assuming that all parameters in the system may appear in every equation,2 whereas we assume as the basic setup that there is a distinct set of parameters belonging to each equation. Second, partly because of the first, we are able to obtain simpler and more explicit expressions for the derivatives of likelihood function required in the calculations. Third, and also partly because of the first, we can conveniently deal with the important problem of linear restrictions on the parameters in the same equation or in different equations. A fourth feature of this paper, and a feature which has partly motivated it, is the contrast of the linear with the nonlinear case. As it will be shown, there are many similarities in the computations of both. This demonstration can enhance our understanding of the nature of the estimation equations. Two additional features of this paper are the treatments of identities in the system and of residuals which may follow an autoregressive scheme. We will derive in section II the estimation equations for nonlinear systems, under the assumptions that each structural equation contains a distinct set of parameters, that the parameters are not subject to any linear restrictions, and that the (additive) residuals are serially uncorrelated. Section III treats the special case when some equations are linear, and contrasts this case with the nonlinear case. Section IV deals with identities and linear restrictions on the parameters. Section V is concerned with the problem of autoregressive residuals.

They indicate that purchases of durable goods (DUR) are the third most volatile component behind inventory investment (INV I) and federal government expenditures (F GOV). The correlation coefficients between GNP and its components listed in the second row of table 1 show that durable goods purchases have the third hiighest covariance with GNP after inventory investment and nonresidential investment (NRI). Theoretically, these purchases represent either changes in the size of the household portfolio through changes in the flow of savings, or a reallocation of accumulated wealth among assets in response to changes in rates of return. Empirically, some effort has been given to examining the separate influences of rates of return and income. Hamburger (1967) found that interest rates, the price of durable goods relative to other prices faced by the consumer, and disposable personal income all have a significant impact on purchases of durable goods. portance of these variables as sources of fluctuation in purchases. Motley (1970) included a user cost of real assets variable in addition to the rate on savings deposits and expected income, and found in sharp contrast to Hamburger that none of them had a significant influence on the demand for the sum of durables and housing. The analysis of fluctuations in durable good purchases presented here differs from its prede-

Robert C. Blattberg, Evaluation of the Power of the Durbin-Watson Statistic for Non-First Order Serial Correlation Alternatives, The Review of Economics and Statistics, Vol. 55, No. 4 (Nov., 1973), pp. 508-515

[1] Ezekiel, H., and J. Adekunle, Secular Behavior of Income Velocity: An Intermational Cross Section Study, IMF Staff Papers, vol. XVI, no. 2, July 1969, 224-239. [2] Goldsmith, R. W., The Determinants of Financial Structure, Organization for Economic Cooperation and Development, Paris, 1966, 27-30. [3] Khazzoom, J. D., The Currency Ratio in Developing Countries (New York: Frederick A. Praeger, 1966). [4] Melitz, J. and H. Correa, International Differences in Income Velocity, this REviEW LII (Feb. 1970), 12-17. [5] Wallich, H. C., Theory and Quantity Policy, Ten Economic Studies in the Tradition of Irving Fisher (New York: John Wiley & Sons, Inc., 1967), 257-280.

T O measure a nation's wealth, an industry's productive potential or the consumption of a durable stock, we must be able to add machines with different characteristics and different vintages to form an aggregate. Ideally, such a measure would change with machinery deterioration and obsolescence but not with pure price level changes which leave the use of the machinery the same. The aggregation task would be easier if we had information about the nature of depreciation. For example, if depreciation is a constant rate, and that rate remains the same over time, then the aggregate is a simple weighted average of the component machines, the weights being derived from the known depreciation rate. This model is not uncommon, yet its assumptions are clearly restrictive. It would be very useful if there were sufficient empirical evidence pertaining to the nature of depreciation patterns to either confirm or reject such a simple model. It is the intent of this paper to provide some of that evidence. One natural approach to studying decay of capital is to study the in-use cost of machines as they age. Depreciation values could be estimated from changes in rental prices throughout a machine's life. In the absence of welldeveloped rental markets, however, resale values would yield approximations of the remaining value of machinery after a period of use. This paper constructs actual depreciation figures for automobiles from purchase prices, and studies assumptions and hypotheses about the relationship between new and used machinery. In particular, three assumptions are common. First, it is often assumed that depreciation patterns remain fixed over time. For this assumption to be valid, any technological change must be either nonexistent or smooth. There can be no sudden, dramatic innovations, since these would change the nature of depreciation schemes. Similarly, it is often assumed that machinery of the same type depreciates in the same fashion. This assumption will also be studied. The third and most common assumption is that equipment depreciates at a constant rate.' This is a very useful assumption, since it greatly simplifies the relationship between new and used pieces of equipment. These assumptions will be tested for automobiles using figures for nineteen different makes from 1950 to 1969.