This paper studies the effect of lifetime uncertainty on optimal consumption decisions. It is shown that for risk averters changing the distribution of lifetime uncertainty decreases consumption due to the higher probability of having a longer life and increases consumption due to the desire for sure consumption in the present. The stronger of these effects determines the effect of lifetime uncertainty on optimal consumption decisions. The major result is that if the utility function is Cobb-Douglas and the rate of return is not too large relative to the amount of future discounting then lifetime uncertainty will always increase consumption.

Kenneth J. Arrow, David Levhari, Eytan Sheshinski; A Production Function for the Repairman Problem12, The Review of Economic Studies, Volume 39, Issue 3, 1 July

Journal Article On the Sensitivity of the Level of Output to Savings: Embodiment and Disembodiment: A Clarificatory Note Get access Franklin M. Fisher, Franklin M. Fisher Massachusetts Institute of Technology Search for other works by this author on: Oxford Academic Google Scholar David Levhari, David Levhari Hebrew University Search for other works by this author on: Oxford Academic Google Scholar Eytan Sheshinski Eytan Sheshinski Hebrew University Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Economics, Volume 83, Issue 2, May 1969, Pages 347–348, https://doi.org/10.2307/1883092 Published: 01 May 1969

It is natural to consider the stock of any renewable resource as a capital stock and treat the exploitation of that resource in much the same way as one would treat accumulation of a capital stock. This has been done to some extent by Cohn Clark and Gordon Monro and by Ngo Van Long, whose papers contain a discussion of this point of view. However, the analysis is much simpler than it appears in the literature especially since the interaction between markets and the natural biological dynamics has not been made clear. Three issues which have been raised only tangentially are important for the understanding of the economics of renewable resources. The first arises from the fact that the fish population might not be able to sustain the market determined amount of fish and hence should be allocated optimally over time. Second, there is the cost externality. Here costs are affected by the size of the stock of fish which again affects the allocation over time. The third issue is how central (typically used in capital theory) can be employed as a tool in analyzing a decentralized economy with perfect futures markets and property rights. In this paper a simple framework, using only elementary mathematical techniques, will be provided in which many important results new to the literature, as well as many familiar results, can be derived. One bonus of this approach is that much of the previous literature can be organized and rationalized. The simplicity of the mathematics has the effect of revealing the underlying economic intuition of the subject. Throughout the paper, as a matter of convenience, the selfrenewing resource will be referred to as However, it should be clear that the analysis is perfectly general in that it can be applied to any self-renewing resource. In fact, by specializing the production function, an elementary exposition of the theory of exhaustible resources is implicit. There are two main sections to this work plus an Appendix. The first section analyzes the time path of prices, output, and resource stock under the assumption of free-entry competition, and the second section studies optimal planning (or perfect competition). In the first part of the discussion of each market structure, extraction costs are assumed to be independent of the size of the fish stock. In the second part of each section this assumption is relaxed. Finally the Appendix contains several of the results used in the body of the paper. The case of free-entry competition can be treated in a very simple manner. Each period's output is determined by the intersection of the demand and marginal harvesting cost curves (i.e., supply equals demand). Subtracting this output from the natural dynamics yields the dynamics of free-entry competition. Using this technique, it is easy to see the conditions under which free-entry competition leads to the exhaustion of the stock of fish. Furthermore, the free-entry case serves as a benchmark by which to evaluate the effects of the institution of property rights on the exploitation of the resource. The case of a centrally planned economy is studied next. This case is shown to be identical to a competitive outcome in which property rights are clearly defined and prices and outputs are endogenously determined so as to be consistent with demand conditions. One result, coincidentally, is that it is possible that the solution might be obtained without property rights. The tech*The Hebrew University; University of Virginia; University of Illinois and Northwestern University, respectively. Research support from the National Science Foundation under grants SOC 77-27340 and SOC 7905900 and the US-Israel BSF under grant 1828-79 is gratefully acknowledged.

TN following the mean-variance analysis developed by Markowitz (1952) and Tobin (1958), Sharpe (1964), Lintner (1965a, b) and Treynor (1961) have developed the theory for determination of asset prices under conditions of uncertainty. The equilibrium asset pricing model, and its implication for measuring ex post performance of individual securities, have been empirically tested by Lintner,1 Jensen (1968, 1972), Miller and Scholes (1972), Douglas (1969), Roll (1969) and others. The empirical results obtained by both Douglas and Lintner deviated from the theory of the model. Moreover, Miller and Scholes have run empirical tests similar to those of Douglas and Lintner and found a significant disparity between the theoretical model and the empirical evidence. They maintain that part of this discrepancy can be explained by possible statistical biases, measurement errors, and consideration of the skewness of the distribution of returns. Black, Jensen, and Scholes (1972) (hereafter B-J-S) confirm the systematic bias. Using monthly data covering a 35 year period, they discovered that, on average, high risk securities earned less than the amount predicted by the model. Similarly, earnings on low risk securities exceeded the amount predicted.2 Although these disparities clearly suggest some systematic empirical bias, we are not examining a possible statistical bias but a mathematical bias stemming from one of the assumptions underlying the capital asset pricing model (CAPM). To be more specific, the model assumes that all investors are single period, expected utility of terminal wealth maximizers. There is no particular restriction on the length of this period as long as it is identical for all investors. Clearly, the length of the true investment horizon affects asset prices under conditions of uncertainty. We claim that the disparities noted above may result from using data calculated for an investment horizon that differs from the true investment horizon. In the various empirical tests, the investment horizon has been selected arbitrarily. For example, Lintner and Miller and Scholes use annual data (i.e., they implicitly assume a one-year horizon), Douglas uses quarterly and annual data; Black, Jensen and Scholes, as well as Friend and Blume (1970), use monthly rates of return in their empirical tests, while Roll uses weekly data. It has been shown elsewhere by Levy (1972) that the Reward to Variability index (developed by Sharpe, 1966) is a function of the investment horizon assumed. Hence, there exists a systematic mathematical bias that is a function of the horizon assumed. The above theoretical findings are related to the theory of pricing capital assets, but deal only with efficient portfolios and not individual stocks. In this paper we illustrate that the assumed horizon plays a crucial role in empirical testing. Any deviation from the true horizon causes a systematic bias in the regression coefficient (i.e., in the security systematic risk). This in turn causes a systematic bias in the performance measures of each security, and hence the deviation between the theoretical model and the empirical evidence. The results of this paper are not limited to the theory of pricing capital assets; they are applicable to any econometric study in which the variables have multiplicative rather than additive properties. In such a case the regression coefficients will have a matheReceived for publication February 27, 1975. Revision accepted for publication March 8, 1976. The authors acknowledge the technical assistance of Moshe Smith and two anonymous referees. The first author has been partially financed by the Maurice Falk Foundation, and the second author has been financed by the Ford Foundation. 1 Lintner's paper Security Prices and Risk: The Theory and a Comparative Analysis of A.T.&T. and Leading Industrials was presented at the Conference on Economics of Regulated Public Utilities, June 24, 1965, Chicago. 2 Miller and Scholes show that the presence of certain biases could have accounted for the Douglas and Lintner findings. BJ-S maintain that the assumption of borrowing at riskless interest rates could have accounted for these deviations.

David Levhari, Jacob Paroush, Bezalel Peleg; Efficiency Analysis for Multivariate Distributions1, The Review of Economic Studies, Volume 42, Issue 1, 1 Jan