I. Introduction, 705.—II. The model, 707.—III. Tax incidence, 710.—IV. Conclusion, 715.

The notion of stability in the sense of Lyapunov is applied to economic dynamic processes of the Champsaur-Dreze-Henry type. Our purpose in this note is to fill a small gap in the literature concerning dynamic processes in economic theory, of the type presented by Champsaur, Dreze, and Henry [3]. Indeed, these authors do not discuss stability in the sense of Lyapunov [7]. However, a recent result of Maschler and Peleg [9] on this kind of stability (presented in a discrete model) can easily be applied to both continuous and discrete processes used in economics. We shall present this result for a very general class of such processes and conclude with references to a few economic applications. For our purpose a (set valued) dynamic system is simply a pair 〈X,φ〉, where X is a compact subset of R and φ a correspondence from X to its nonempty subsets. Let T be a subset of [0,∞) containing 0 and x0 an element of X. Then a φ-process starting at x0 is a pair of functions: x(·) : T → X, ẋ(·) : T → R, such that: x(0) = x0 and, ∀ t ∈ T , ẋ(t) ∈ φ(x(t)). If T = {0, 1, 2, · · · , } and ẋ(t) = x(t + 1) then the process 〈x(·), ẋ)(·)〉 is called discrete. If T = [0,∞), if x(·) is absolutely continuous on any interval [0, τ ] in T , and ẋ(t) = dx(t)/dt for almost every t in T , then the process 〈x(·), ẋ(·)〉 is called continuous. In the first case the Econometrica, 47(3), 733-737, 1979. As pointed out by Negishi [10], this is the same as Samuelson’s stability of the second kind [11]. The term “stability in the sense of Lyapunov” is used by Arrow and Hahn [1]. Heal [5] and Hori [6] also use this concept of stability. See Champsaur, Dreze, and Henry [3, Section 5].

The composite measures of investment performance: the reward-to-variability index, by Sharpe ([29], [30]) and Lintner [23], and the reward-to-volatility index, by Treynor [33], were developed after Markowitz ([24], [25]) and Tobin [32] popularized the mean-variance framework of analyzing the problems of certain investments. Since these are ex ante measures they are not directly applicable to the evaluation of ex post performance. A theoretical basis for doing so has been provided by Jensen ([17], [18]) who also developed another composite performance measure, the predictability index. In practice, these composite measures have been found to have problems. Foremost, they have been observed to exhibit systematic biases. Various causes of the biases have been proposed. These are: the existence of unequal lending and borrowing rates, the failure to consider higher moments of return distributions, and the elusive “true” holding period.

This paper is concerned with costs of capital in mathematical programming formulations of the problem of capital budgeting under capital rationing. It shows that there is a serious error in the method outlined by previous authors for converting the shadow prices from the solution of the dual into measures of the firm's marginal internal opportunity rates. In addition to demonstrating that the traditional approach leads to erroneous and nonsensical results, this paper presents a correct procedure for determining these rates.