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Parameter-estimation uncertainty, risk aversion, and "sticky" output decisions.

I. Introduction

A thirty-year onslaught from a variety of launching pads has led to the well-known result that a risk-averse, profit-seeking, quantity-setting, single-product seller faced with uncertain demand will opt for a lower output rate than that which is optimal under risk neutrality.[1] What is neither well known nor even strongly suspected is the extent to which the two optima will differ. That is, from the practical standpoint of a real-world decision maker, is the issue worth worrying about in the first place?

To attempt to answer that question with something beyond an all-purpose "It depends," this paper focuses on a particularly pervasive form of real-world uncertainty, notably, the parameter-estimation uncertainty that accompanies regression-based estimates of the demand curve. In this case, as more data come in over time and previous estimates are revised, uncertainty as to the "true" demand curve and its parameters is reduced and thus, one might suspect a priori, so too would be any discrepancy between the risk-neutral and risk-averse optima. The suspicion, however, turns out to be invalid. Rather, as will be shown, whether the discrepancy tends to decrease or increase with experience and additional information depends directly upon whether the risk-averse sellers' initial output sales are relatively low or high, because this will determine whether their subsequent output rates ultimately approach a long-term equilibrium level from either below or above. The risk-neutral seller follows no such can impact on output rates that are established through regression-based estimates of demand. This is done by first erecting a parsimonious theoretical structure that yields some interesting insights that are subsequently illustrated through some experimental results. A discussion of these results and the conclusions follow.

II. The Seller's Problem

Consider a profit-seeking firm whose management has specified the von Neumann-Morgenstern risk-preference function V = v(W), W = [Pi] + w, where [Pi] is profit and w is wealth. Management is assumed to be non-risk preferring, so that dV/dW > O and [d.sup.2] V/[dW.sup.2] [is less than or equal to] 0.

The firm's profit will be derived from sales of a new product whose demand is unknown. Management's intention is to operate as a quantity setter for whom the market determines price via the firm's unknown demand P = d (Q, [Epsilon]), where P is price, Q is output, and [Epsilon] is an all-encompassing random-error term whose variance is [[Sigma].sup.2] and expectation is [Epsilon] [bar].

Output is obtained at a total cost of C = c(Q), so that [Pi] = PQ - c(Q). Once management specifies d(Q, [Epsilon]) and the probabilistic process that generates [Epsilon], it can determine the optimal Q by solving max (E[V]), where E is the expectations operator. It will be assumed that c(Q) is convex and d(Q, [Epsilon]) is concave in Q, so that the risk-neutral optimum occurs at the [Mathematical Expression Omitted] that satisfies [Delta] E [[Pi]]/[Delta] Q = 0.

For present expositional purposes it will suffice for management to (a) take a second-order Taylor expansion of v(W) around E[[Pi]] + w, and (b) assume that [Mathematical Expression Omitted] are small (where R is remainder term in the Taylor expansion) in order to (c0 determine that the optimum Q = Q* must satisfy (1) [Mathematical Expression Omitted] where [Pi] [bar] = E [Pi], r is Arrow-Hart measure of absolute risk aversion evaluated at [Pi] [bar] + w, and [Mathematical Expression Omitted] is the profit variance.(2) For the risk-neutral firm, r = 0.

Management believes the "true" demand curve to be linear and described by P = [B.sub.0] + [[Beta].sub.1] Q + [Epsilon], where [[Beta].sub.0] and [[Beta].sub.1 are parameters to which management has assigned an "information-less" normal prior density with variance [[Sigma].sup.2]/[n.sub.0] and [n.sub.0], which reflects the extent of management's prior information about the parameters, is equal to zero. To ease our computational burdens it will be assumed that because of its experience with other products, management is confident that it knows, with certainty, [[Sigma].sup.2], the variance of the normally-distributed random-error term; and, [Epsilon] [bar] = E[[Epsilon]]. The assumption that [[Sigma].sup.2] is known, as discussed in footnote 4 below, is nondistortive and easily relaxed. It will also be assumed, solely for computational convenience, that dc (Q)/dQ = c; or, marginal cost = average variable cost = c.

Management's very vague ideas about demand are only partially crystallized by pre-production consumer surveys that have encouraged it to produce initial outputs of [Q.sub.1], . . . ,[Q.sub.k] through the first k production periods. The market-clearing prices realized during these periods are [P.sub.1], . . . ,[P.sub.k]. Then, in Bayesian fashion, management's informationless prior over [Beta] = ([[Beta].sub.0] > 0, [[Beta].sub.1] < 0) is revised, given the sample data ([P.sub.1], . . . ,[P.sub.k]; [Q.sub.1], . . . ,[Q.sub.k]. The posterior density will have E [[Beta].sub.0] = [b.sub.0] and E[[Beta].sub.1] = [b.sub.1], where [b.sub.0] and [b.sub.1] are the ordinary-least-squares estimates of [[Beta].sub.0] and [[Beta].sub.1] based upon the k data points.(3) More critically, [Mathematical Expression Omitted] and [Mathematical Expression Omitted], where [Mathematical Expression Omitted] and [Mathematical Expression Omitted]

To determine [Mathematical Expression Omitted] management must now solve equation (1) with [Mathematical Expression Omitted], and [Mathematical Expression Omitted], which requires solving a cubic equation--given that management specifies a value for r, as we shall assume it does. Specifically, [Mathematical Expression Omitted] must satisfy (2) [Mathematical Expression Omitted]

As successive outputs and random errors result in newly-observed prices, [b.sub.0] and [b.sub.1] are revised along with the historical mean, [Mathematical Expression Omitted]. As the total number of observations, n increases, n replaces k in equation (2), [Mathematical Expression Omitted] and (3) [Mathematical Expression Omitted] where [Mathematical Expression Omitted] is the long-term equilibrium output rate. For the risk-neutral firm, [Mathematical Expression Omitted]

III. Some Initial Insights

1. Equation (1) is a typical mean-variance formulation in which management is trading off "risk" for "return." The maximum expected profit occurs at [Mathematical Expression Omitted] on the strictly concave expected profit function of Figure 1. The profit variance, however, is neither concave nor convex; it takes on its global minimum at Q = 0 and, at least for small n, it has a local minimum at [Mathematical Expression Omitted] and a local maximum at [Mathematical Expression Omitted]. There are, therefore, two possibilities: either [Mathematical Expression Omitted]. As n increase, so too does [Mathematical Expression Omitted] [Q.sub.m] and [Q.sub.M] become imaginary roots, and the profit variance becomes a strictly nondecreasing function of Q.

2. From equation (2), the degree of absolute risk aversion, r, and the process variance, [[Sigma].sub.2], play quantitatively equivalent roles in determining output, in the sense that an x percent change in one of them will have the same impact on output as an x percent change in the other.

3. Equations (2) and (3) suggest that the importance of getting a good estimate, [b.sub.1], of [[Beta].sub.1] is directly related to the size of [[Beta].sub.1] relative to r [[Sigma].sup.2]. That is, if ~[[Beta].sub.1]~ is small relative to r [[Sigma].sup.2], then a failure to estimate it accurately will not divert the seller's optimal output computation of [Mathematical Expression Omitted] from the long-term equilibrium rate of [Mathematical Expression Omitted] to the same extent as would be the case if ~[[Beta].sub.1]~ is large relative to r [[Sigma].sup.2].

4. The conflicting pulls exerted by Q = 0 and Q = [Mathematical Expression Omitted], the variance-minimizing and expected-profit-maximizing points, respectively, together with the fact that the optimal solution, [Mathematical Expression Omitted], cannot fall in the [Q.sub.m] to [Q.sub.M] interval, suggests that, ceteris paribus, solutions "in the neighborhood" of either [Mathematical Expression Omitted], will be particularly attractive to the risk avoider. The latter further implies that the seller's initial outputs can be critical in determining the firm's future path of action, above and beyond their other ancillary real-world impacts on, for example, employee and consumer behavior.

Moreover, if [Mathematical Expression Omitted] fall near [Mathematical Expression Omitted] then in period n + 2 the gap between [Q.sub.m] and [Q.sub.M]--if both are real roots--will be reduced, whereas if [Mathematical Expression Omitted] falls near [Mathematical Expression Omitted] then there is no reason for the next period's output not to fall near [Mathematical Expression Omitted], which will be close to [Mathematical Expression Omitted]. And, as n [right arrow] [infinity], [Mathematical Expression Omitted].

All of the above implies that there will be a strong tendency for the risk-averse management to look to the past when making current decisions, in the sense that there will be a strong tendency to do what was done, on average, in the past, no matter where one will eventually end up. What remains unclear is whether this short-term "stickiness factor" will be sufficiently strong as to dominate the long-term pull of he equilibrium solution. The experimental results that follow shed some light on this issue.

IV. Experimental Results

Suppose the firm's demand curve is P = 100 - .2Q + [Epsilon], where E [[Epsilon] and [[Sigma].sup.2] = .25. As assumed above, management knows [[Sigma].sup.2] but not the fixed [[Beta].sub.0] and [[Beta].sub.1], over which it assesses a normal informationless prior.

Suppose, too, that management initially sets output rates for k = r periods, observers four prices, and in typical Bayesian fashion revises its prior density over [Beta], and in period n = 5 computes E [[Beta].sub.0] = [b.sub.0] and E [[Beta].sub.1] = [b.sub.1], solving equation (2) for [Mathematical Expression Omitted]. Once [P.sub.5] is observed, the process is repeated.

Table I presents various results for this exercise, for the case of r = 1 and c = 0, with the initial four outputs alternatively set at: [Q.sub.1] = 20, [Q.sub.2] = 25, [Q.sub.23] = 30 and [Q.sub.1] = 35 [Q.sub.1] = 25, [Q.sub.2] = 30, [Q.sub.3] = 35, and [Q.sub.4] = 40; and so forth up to [Q.sub.1] = 185, [Q.sub.2] = 190, [Q.sub.3] = 195, and [Q.sub.4] = 200. From equation (3), [Mathematical Expression Omitted] 250. That is, the risk-neutral long-term optimum is 2.25 times as large as the risk-averse equilibrium rate. And should either [[Sigma].sub.2] r increased to .50 or r increase to 4, [Mathematical Expression Omitted] = 100/(1 + .4) = 71.43 or [Mathematical Expression Omitted] is 3.5 times as large as the risk-averse equilibrium rate. Insofar as such differences are credible, the consequences of being risk averse--if management is really conscientious about it and the same time knows what is doing--can be serious indeed. Insofar as such differences are not credible, they call into question the relevance of much of the uncertainty-in-microeconomics literature.

What these results also call to the fore is the potential for conflict that exists when a non-homogeneous group of managers attempts to reach a consensus on how much to produce, in the absence of a group risk-preference function. Reasonable people can reasonably disagree, but the order of magnitude of the disagreement in the present instance is such as to suggest that, in stable real-world organizations, either: (a) managers modify their decisions through some time-honored, rule-of-thumb, caution-invoking process (and even when they can agree on all the objective inputs they eschew the expected-utility-maximizing calculus that can lead to radically different optima); (b) managers tend to hire their clones, thereby minimizing potential sources of conflict, and one manifestation of the cloning process is that management shares common risk attitudes (and, in particular, historically risk-neutral managers and/or shareholders will tend to associate themselves with decision-making colleagues whose "r's" are closer to zero than to two); or (c) both of the above.

To help interpret Table I and assure understanding of how the numbers were generated, consider Case VIII. Here, the firm's initial four outputs are [Q.sub.1] = 55, [Q.sub.2] = 60 [Q.sub.3] = 65, and [Q.sub.4] = 70, whereupon [Mathematical Expression Omitted] = 62.500. Four random values for [Epsilon] were generated from a normal density with zero mean and a variance of .25. Then, prices of [P.sub.i] = 100 - 2 [Q.sub.i] + [[Epsilon].sub.i] (i = l, . . . ,4) were determined. Given the informationless normal prior, the resulting four observation pairs, ([P.sub.i], [Q.sub.i]), were used as inputs to compute the estimated regression line P [caret] = 101.0681 - .22213Q, where P [caret] is the regression-based estimate of P, given Q. The estimated regression coefficients, [b.sub.0] = 101.0681 and [b.sub.1] = -.22213, together with [Mathematical Expression Omitted] = 125.000, k = 4, r - 2, c = 0, and [[Sigma].sup.2] = .25, are then substituted into equation (2), a process that yields two real roots for Q. To determine which root maximizes E[V], the following "firing rules" were employed. [Tabular Data I Omitted]

Let [Q.sub.L] and [Q.sub.H] denote the lower and the higher of two roots. At each stage n + 1, substitute [Q.sub.i] (i = L,H) into [Mathematical Expression Omitted]

Rule 1: If [K.sub.H] [is less than or equal to] [K.sub.L], then [Mathematical Expression Omitted]

Rule 2: If [K.sub.H] > [K.sub.L] then compute [Mathematical Expression Omitted]

If [M.sub.L] [is less than or equal to] [M.sub.h] then [Mathematical Expression Omitted] =

[Q.sub.H]. Otherwise, [Mathematical Expression Omitted] = [Q.sub.L].

The sense of these rules is a follows. First, if [K.sub.H] [is less than or equal to] [K.sub.L] then the higher root does not give a higher profit variance than the lower root. But, we know that the higher root must give a higher expected profit. Thus, [Q.sub.H] is preferred to [Q.sub.L]. Second, suppose we approximately V by a constantly-risk-averse preference function, V [caret] = [-e.sup.-r [Pi]]. Then, it is easily shown that [M.sub.L] [is less than or equal to] [M.sub.H] signals that, [Q.sub.H] is preferred to [Q.sub.L]; otherwise, [Q.sub.L] is preferred to [Q.sub.H]. In no instance was this second firing rule called into play; and in Case VIII, [Mathematical Expression Omitted] = 64.388. Indeed, in Case VIII, with fourteen observations the estimated demand curve is now P [caret] = 99.6794 - .19585Q, and there is only one real root in the solution to Equation (2): notably, [Mathematical Expression Omitted] = 67.679. The risk-neutral optima with k = 4 and n = 14 observations, respectively, are [Mathematical Expression Omitted] = 101.068/.444 = 228 and 99.679/.196 = 254.

The Case VIII results prompt the following suggestions.

1. Equilibrium, like nirvana, may be an interesting intellectual concept, but the harsh reality is that is difficult to attain. Even after seventy-five periods the firm must still increases output by 111.11 - 92.87 = 18.24, or twenty percent to get there--and progress is slow.

2. The past is important, and the tendency to be drawn towards the average of the past--the "stickiness factor"--is strong. Nonetheless, this tendency may dampen as past experience increases. The reason, as seen in equation (2), is that [Mathematical Expression Omitted] increases with increasing n, the third term in the parentheses becomes less of a factor in the determination of Q.

3. For the risk avoider the determination of the optimal output and the rate of progress towards equilibrium does not depend on how well or poorly the estimated demand curved approximates the true demand curve. The only things that really matter are where you have been [Mathematical Expression Omitted], and where you will eventually be going, [Mathematical Expression Omitted]. Under, risk neutrality, however, the current parameter estimates are all that matters, and these will improve, if irregularly, as n [right arrow] [infinity].

Moreover, these results are robust, subject to the following qualifications.(4)

1. When the initial outputs are small relative to [Mathematical Expression Omitted] may actually more rapid than otherwise, because [Mathematical Expression Omitted] will be greater than otherwise (and similarly for n > k + 1), so that [Mathematical Expression Omitted] will also get bigger faster.

2. When the previous condition takes effect, which suggests that [Mathematical Expression Omitted] the risk-neutral optimum, is a potent attractor, and when initial outputs are not small relative to [Mathematical Expression Omitted], then progress towards [Mathematical Expression Omitted] will be at a veritable snail's pace, even if [Q [bar].sub.k] is close to [Mathematical Expression Omitted]

The reason for this result is the stickiness factor, or the pulling power of the historical mean. Once [Mathematical Expression Omitted] is drawn towards [Q [bar].sub.n], then the estimated demand curve is determined almost entirely by the preceding set of observations, say the initial k observations, because there is not longer very much variance in the independent variable.(5) Thus, none of the components in equation (2) change by very much k + 1 to n > k + 1--and neither does the solution to that equation.

3. Table II presents the averages (standard deviations in parentheses) for ten experiments for Cases XVII through XXXIV. These results show that even though there are considerable differences in the parameter estimates in any particular case, even after fifty periods the differences in the optimal output rates, for that case, are negligible. That is, the effect of the initial output rates and stickiness factor dominates all else. To see if things would look much different at considerably higher values of n, the Case XVII experiment was carried out a couple of times with n = 500; in one instance [Mathematical Expression Omitted] and [Mathematical Expression Omitted] and in the other [Mathematical Expression Omitted] and [Mathematical Expression Omitted]. [Tabular Data II Omitted]

4. As seen in the latter result and in Table I, Cases XVI and XVII, the march to equilibrium will not necessarily be monotonic. The problem in each of these cases (and that noted at the end of the preceding paragraph), is that [Mathematical Expression Omitted], and (c) [Mathematical Expression Omitted]. Thus, [Mathematical Expression Omitted] never changes very much but when it does it is drawn towards [b.sub.0] / 2 ([[Sigma].sup.2] - [b.sub.i], which also does not change very much, in the process reducing the historical mean. In principle, this course will ultimately be reversed. That is, as n [right arrow] [infinity], even with minor variations in Q the firm will eventually learn the true value of [Beta]. In practice, however, n = 500 may not be nearly enough observations.

5. In all of the computations it was assumed that r is held constant at r = 2. But, r is a function of [Pi] [bar] + w. Over time, then, as profits are earned the firm's wealth might be presumed to be increasing, and we might also anticipate r to be decreasing; that is, the seller is decreasingly risk averse.(6) Although the individual effects of this are speculative, one certain effect is that it would imply a ceteris paribus preference for higher values of [Mathematical Expression Omitted] as n increases. One particularly positive consequence of the latter preference would be to bring with it greater variability in the observed values of the independent variable and therefore tend to result in an improved estimate of [Beta] at an earlier stage in the process.

6. In the latter regard, on the one hand, risk neutrality or at least relatively weak risk aversion, would seem to carry with it a bonus in that it tends to effect greater variability in early-stage output and, as a direct consequence, improved early-stage estimates of the true demand curve.

On the other hand, output volatility is not necessarily cost free in the real world. There is, in fact, something to be said for an unplanned but anticipatable steady increase or decrease in output over time. Stickiness, too, brings with it some bonus features.

7. Finally, Table III gives values for [Mathematical Expression Omitted] and [Mathematical Expression Omitted] for alternative experiments in which r is permitted to vary between .1 and 5, the values employed by Levy and Markovitz [3, 313-13] in their exponential, constantly-risk-averse, risk-preference function. As they observe, decision makers with much higher values of r, say r = 10, "would have some very strange preferences among probabilities of return" [3, 313]. [Tabular Data III Omitted]

The results are quite consistent, even when r - .1, which approaches perilously close to risk neutrality, and reinforce the previous conclusions for the case of r = 2. Indeed, as a direct result of the stickiness factor, risk preferences can to all intents and purposes become irrelevant in determining optimal outputs. For example, when the seller's initial outputs are as in Cases XIX through XXXIV, the differences between [Mathematical Expression Omitted] and [Mathematical Expression Omitted] for r = 1 and r = 5 are less than five percent (for Case XX, when n + = 75), although the r = 1 equilibrium solution is two-and-one-half times greater than that for r = 5. Thus, an interesting byproduct of these results is the caution against attempting to infer risk preferences from sellers' output decisions.

V. Conclusions

Even a casual glance of random helpings of tables of contents of economics journals over the past quarter of a century will affirm that the explicit incorporation of elements of uncertainly into economic models is one of the important theoretical developments of that period. One of the strongest attacks on this literature stems from the position that people do not really behave in accordance with one or more of the various axioms that underlie the von Neumann-Morgenstern framework upon which most of the formal economic models are built.(7) A second front stems from the view that the von Neumann-Morgenstern approach and its variants lead only to the vacuous conclusion that decision makers behaving in accordance with the underlying axioms will do whatever they think best [1].

The present paper opens a third front, one that suggests that even when decision makers do attempt to follow the prescripts of a formal theoretical model when making real-world decisions, the attempt itself might bring into effect a mode of behavior that the modelers did not anticipate: notably, "sticky" decision making in which the risk avoider's optimal current decision is biased toward a historical average of past decisions.(8) The latter in turn implies, and our experimental results suggest, that the earliest in a sequence of decisions, those that are the least informed, can very easily, if not frequently and indeed ordinarily, take prime of place in dictating the entire sequence, irrespective of its length an the path.

Our results also suggest that the magnitude of the difference between the risk avoider's optimal solutions and those under risk neutrality can be vast indeed and that this magnitude can either increase or decrease over time--and not necessarily with regulatory. A long-term byproduct of this would be the not considerable overall loss in actual profits realized by the firms that survive long enough to enjoy the long term. That is, the costs of caution may be much greater than they are typically judged to be by risk avoiders.

Finally, our results point up the need to consider the dynamic aspects of decision making in real world firms that, in the normal course of events, are getting more and more information about their environments. In a world of "sticky" decision making, random errors, and parameter-estimation uncertainty, this need cannot be satisfied by one- or two-period models. (1)For a discussion of this result and the related literature, see Horowitz [2]. (2)For the algebraic details, see Horowitz [2, 354]. We could also work with a higher Taylor expansion, as in Horowitz, but this would only add what for present purposes are unnecessary algebraic complications. In addition, under subsequent assumptions about the shapes of the demand and cost functions and presumed knowledge of the variance of [Epsilon], for a Bayesian management [Pi] is normally distributed so that its odd moments are zero and all even moments are functions of [Mathematical Expression Omitted]. Thus, using a higher-order expansion would only result in the (r/2) term in Equation (1) being replaced by a more complex expression, one of whose separable components would be r/2. (3)See, e.g., Judge et al. [3, 94-100]. The estimation process assumes that, despite how one might feel about other things, one is prepared to accept a point estimate of [Beta] under quadratic loss. (4)The results are robust in these respects. First, different values of [[Sigma].sup.2] (or r, as discussed below) produced the same general pattern of results. Second, when in more realistic fashion we substitute the variance of the residuals, [S.sup.2], for [[Sigma].sup.2, the same patterns emerge. In the latter case, however, the informationless normal prior would have to be replaced by an informationless normal-gamma prior in order to give regression estimates a Bayesian interpretation, and [Pi] itself would no longer be normally distributed. Third, the pattern of results was impervious to a change in [Beta]. (5)For example, [Mathematical Expression Omitted]. But, [Mathematical Expression Omitted], where [Mathematical Expression Omitted] and, similarly for the numerator. Hence, for values of n in the range considered here, [Mathematical Expression Omitted] and [b.sub.0] [approximately equal to] [P [bar].sub.k] - [b.sub.1] [Q [bar].sub.k]. (6)This property would become explicit and could be measured through [d.sup.3] V / [dW.sup.3] > 0 (see Menzies, Geiss, and Tressler [6]), a term that "drops out" in the Taylor expansion when [Pi] is normally distributed, as discussed in footnote 2, since it is multiplied by the third moment about [Pi] [bar]. (7)For a comprehensive discussion of the issues involved, see Machina [5]. (8)The choice of the term "a historical average" reflects the fact that different model specifications can alter the relevant average. For example, with d[Q, [Epsilon]) linear, [Mathematical Expression Omitted]--the period-(n + 1) price variance--is minimized at [Q = [Q [bar].sub.n]; but, with d(Q, [Epsilon]) hyperbolic, the period-(n + 1) log-price variance is minimized where log(Q) equals the mean of the logarithms of [Q.sub.1], . . . , [Q.sub.n]. While the Q that minimizes either [Mathematical Expression Omitted] or the log-price variance will not also minimize [Mathematical Expression Omitted], which is what is relevant here, since [Mathematical Expression Omitted] the nontrivial solution will be influenced in the direction of that Q.


[1]Harsanyi, John C., "Subjective Probability and the Theory of Games: Comments on Kadane and Larkey's Paper." Management Science, February 1982-120-24. [2]Horowitz, Ira, "Decision Making and Estimation-Induced Uncertainty." Managerial and Decision Economics, December 1990, 349-58. [3]Judge, George G., William E. Griffiths, R. Carter Hill, and Tsoung-Chao Lee. The Theory and Practice of Econometrics. New York: John Wiley and Sons, 1980. [4]Levy, H., and H. M. Markowitz, "Approximating Expected Utility by a Function of Mean and Variance." American Economic Review, June 1979, 308-17. [5]Machina, Mark J., "Choice under Uncertainty: Problems Solved and Unsolved." Journal of Economic Perspectives, Summer 1987, 121-54. [6]Menezes, C., C. Geiss, and J. Tressler, "Increasing Downside Risk." American Economic Review, December 1980, 921-32.
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Author:Horowitz, Ira
Publication:Southern Economic Journal
Date:Apr 1, 1992
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