注释

1.F.P.Ramsey，“A Mathematical Theory of Saving,”Economic Journal,Vol.38,（December 1928）,pp.543-559.

2.G.Debreu,“Topological Methods in Cardinal Utility Theory,”in K.J.Arrow et al.（eds.）,Mathematical Methods in the Social Sciences 1959（Stanford:Stanford University Press,1960）.

3.T.C.Koopmans,“Stationary Ordinal Utility and Impatience,”Econometrica,Vol.28（April 1960）,pp.287-309.

4.P.A.Samuelson and R.M.Solow,“A Complete Capital Model Involving Heterogeneous Capital Goods,”Quarterly Journal of Economics,Vol.70,（November 1956）,pp.537-562.

5.C.C.von Weizsacker,“Existence of Optimal Programs of Accumulation for an Infinite Time Horizon,”Review of Economic Studies,Vol.32,（April 1965）,pp.85-104.

6.H.Atsumi,“Neoclassical Growth and the Efficient Program of Capital Accumulation,”Review of Economic Studies,Vol.32（April 1965）,pp.127-136.

7.T.C.Koopmans,“On the Concept of Optimal Economic Growth,”in Le Role de L’analyse Econometrique dans la Formulation de Plans de Développement,Vol.28,Part 1 in the series Scripta Varia（Pontificia Academia Scientarium,Vatican City,1965）,pp.225-287.

8.如果C**（t）是唯一的最大化路径，那么对其他所有路径都有

这里，在某些t上C≠C**（t）。因此，

即

如果下面的积分在T上是连续的，那么对某个T⁰有

所以C**（t）一定是唯一的最优解。

9.J.E.Meade,Trade and Welfare:Mathematical Supplement（London:Oxford University Press,1955）.

10.实际上，餍足点只可能渐进地达到而不可能完全达到。所以凯恩斯的论证需要做一些改进才能成立。

11.R.E.Bellman，Dynamic Programming（Princeton:Princeton University Press,1957）,pp.249-250.

12.Samuelson and Solow,op.cit.

13.E.S.Phelps,“The Accumulation of Risky Capital,”Econometrica,Vol.30,（October 1962）,pp.729-743.

14.I.F.Pearce,“The End of the Golden Age in Solovia,”American Economic Review,Vol.52（December 1962）,pp.1088-1097.

15.T.N.Srinivasan,“Optimal Savings in a Two-Sector Model of Growth,”Econometrica,Vol.32（July 1964）,pp.358-373.

16.H.Uzawa,“Optimal Growth in a Two-Sector Model,”Review of Economic Studies,Vol.31（January 1964）,pp.1-24.

17.Koopmans,op.cit.

18.Weizsacker,op.cit.

19.M.Inagaki,“The Golden Utility Path,”Memorandum,Netherlands Economic Institute,Rotterdam（November 1963）.

20.Atsumi,op.cit.

21.P.A.Samuelson,“A Catenary Turnpike Theorem Involving Consumption and the Golden Rule,”American Economic Review,Vol.51（June 1965）,pp.486-496.

22.D.Cass,“Optimum Growth in an Aggregative Model of Capital Accumulation,”Review of Economic Studies,Vol.32（July 1965）,pp.233-240.

23.Cass,op.cit.

24.Koopmans,op.cit.

25.Srinivasan,op.cit.

26.Uzawa,op.cit.

27.R.Radner,Notes on the Theory of Economic Planning,（Athens:Center of Economic Research,1963）and“Optimal Growth in a Linear-Logarithmic Economy,”International Economic Review,Vol.7（January 1966）,pp.1-33.

28.Inagaki,op.cit.

29.库普曼斯的这个证明有些“过强”。因为只要证明N=0时结论成立就足够了。

30.Samuelson,op.cit.

31.D.Cass,Studies in the Theory of Optimum Economic Growth,（Ph.D.Dissertation,Stanford University,1965）.

32.Pearce,op.cit.

33.为什么利率r不会跌到零呢？这是因为当r=γ时，当前的一单位人均消费的减少只能换取未来一单位人均消费的增加。这样，进一步减少当前消费的代价并不低于未来消费增加的收益。既然资本存量不会再提高，利率也就不会被推到低于γ的水平。

34.R.M.索洛在1963年耶鲁大学的一次报告上给出了ρ=0时对数函数型效用函数的这个结果。

35.J.Tobin,“Economic Growth as an Objective of Government Policy,”American Economic Review,Vol.54,（May 1964）,pp.1-20,especially,Figure 2,p.8 and pp.15-16.

36.Weizsacker,op.cit.

37.Inagaki,op.cit.

38.Koopmans,op.cit.,p.226.

39.E.S.Phelps,Fiscal Neutrality Toward Economic Growth（New York:McGraw-Hill,1965）.