The ratchet principle: A diagrammatic interpretation

JOURNAL

OF COMPARATIVE

ECONOMICS

6, 75-80

(1982)

NOTE The Ratchet Principle: A Diagrammatic

Interpretation’

CHAO-NAN LIU Department of Economics, Trenton State College, Trenton, New Jersey 08625 Received

May

1980;

revised

July

1981

Recent study on the Soviet incentive system has shifted from the analysis of static incentives to dynamic, multiperiod structures. Snowberger (1977), for example, investigated the firm’s reactions to uncertainty in case of risk aversion, and introduced multiple performance indicators in his model. In a stochastic optimization model, Weitzman (1980) explored the ratchet principle’s effects on managerial performance. He demonstrated that the manager should simply equate the marginal cost of output to the “ratchet price” to attain bonus maximization over time. In contrast to Snowberger and Weitzman, Peter Murrell ( 1979) concentrated his analysis on managerial time preference, and showed that it virtually dictates the fulfillment of production targets in the multiperiod framework. As a result, the static incentives generally proposed in the literature (Bonin, 1976; Fan, 1975; Miller and Thornton, 1978; Weitzman, 1976) seem unable to induce managers to report honestly their actual accomplishments. Murrell assumes that the incentive scheme remains unchanged over time so that there are no countervailing forces to managerial time preference. The purpose of this note, as an extension of Murrell’s analysis of managerial time preference, is to explore the implications of a flexible reward structure whereby planners may adjust bonuses upward or downward.2 It will be shown that in a two-period situation planners can induce honest reporting of current performance by reducing the incentive coefficient for future reward (provided that future income is not sufficiently preferred to present income). In the tradition of Fisher-Hirshleifer (F-H) models of intertemporal utility maximization, we assume that managers optimize by equating marginal rates of substitution of the utilities available in consecutive periods ’ I am indebted to John M. Montias and an anonymous referee for helpful comments. 2 Sam Gindin (1970) considered flexible incentives but the role of managerial time preference was neglected in managerial decision making. 75

0147-5967/82/010075-06SO2.00/0 Copyright 0 1982 by Academic Press, Inc. All rights of reproduction in any form reserved.

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with marginal rates of transformation of the next.3 Suppose the manager of a socialist firm system of the type analyzed by Fan (1975) properties R=W = W+B+k(m-M)

income from one period to was subject to an incentive and Bonin (1976), with the if

m
if

ma A4,

(1)

where R = manager’s total reward or income, W = manager’s fixed wage, B = lump-sum bonus for meeting the profit target M, k = share of abovetarget profits accruing to the manager, and m = reported profits. Under this reward scheme, the manager will receive only a wage if the profit target is not met. On the other hand, if the profit target is fulfilled the manager will be awarded with an additional lump-sum bonus together with a percentage of the excess profits. A manager seeking a material bonus has to take into consideration the “ratchet principle” whereby the fulfillment or overfulfillment of the current plan usually leads to the imposition of a higher quota in a subsequent period. The state revises targets upward when plans are fulfilled because, under great uncertainty about the capacities of the firms, this would seem to be the best way of coaxing out reserves in production capacities and in profits. Thus, the new profit target in the second period M, can be formally specified as the sum of current reported profits m. plus a fixed percentage of mo, or M, =(l +d)mo, O
model subject

assumes that to his budget

the consumer desires to maximize the level of his lifetime constraint. Fisher (1954). Hirshleifer (1966).

RATCHET

PRINCIPLE

II

control. (The manager’s only problem, given C,, is to decide how much of it he should report.) When the manager maximizes his utility function subject to the constraints specified so far, the completed model can be reformulated as follows: Max U = U(R,) (3) subject to R, = W, if ml < M (4) = W, + B, + kt(m, - M,) mt B ti,,

where

if

m, 2 W ,

M, = (1 + d)mt-, (t = 1, . . . , T).

(5) (6)

There are two sets of equilibrium conditions, depending on whether reported profits are below or above target. If reported profits are below target, the manager receives neither bonus nor extra returns but a straight salary. However, the utility associated with below-target performance, as will be discussed later, is not necessarily lower than that associated with abovetarget performance. In case of above-target performance the equilibrium condition can be obtained as follows. First we substitute (6) into (5) to obtain R, = W, + B, + krm, - k,( 1 + d)m,-, . After substituting Eq. (7) into (3), the differentiation to m,-, yields the first-order condition --=-=dRt dR,-,

MU,-, MU,

k(l

(7)

of (3) with respect

+ 4 k,-, *

According to Eq. (8), the marginal rate of substitution between present and future rewards-provided R,-, and R, both exceed zero-must equal the ratio of the opportunity costs of current and future overfulfillment. Given the background parameters, a set of optimal values can be attained by the manager-viz., mfr.,, m:, RfL,, and RF. Let 1 + r equal the equilibrium marginal rate of substitution in the manager’s preference function between utility of income in times t - 1 and t:

d& --=-=

dR,-,

MU,-1 MU,

1 +r.

(9)

Thus r is the rate at which the reward in period t must be increased to compensate for a reduction in period t - 1 in order to leave the manager’s utility index unchanged. Substituting from (8) and (9) yields the new equilibrium condition: 1 +r=ktl +4 (10) k-, .

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-b-c

s WO

60

FIGURE

1

Equilibrium condition (10) is illustrated in Fig. l4 on the assumption that the utility function U (I?,) can be inferred from observed two-period choices. OS measures the first period’s income on the assumption that m. exceeds MO and that all profits in that period are reported (m. = 6). Thus & at s equals W. + B. + k. (m. - MO). This is clearly a maximum income for the period. og measures the income of the second period that the manager would receive if he reported his actual profits and his target was determined by the decision in the first period to report all profits (i.e., R, ’ The graph was suggested to me by John Montias.

RATCHET

PRINCIPLE

19

at g equals W, + B, + k,m, - k, (1 + d) mo). Here, of course, nzl = fi,, since T = 1. The budget line hf, which has a slope equal to (k,/k,) (1 + d), shows how potential income available in the first period can be translated into income available in the second. The length oc measures the income of the first period that will accrue to the manager if he just fulfills his target (it is equal to IV, + &). ob equals IV,, the salary that he will obtain if he receives no bonus whatsoever. There is a discontinuity between b and c indicated by the dots in the line h$ It may pay the manager to underfulfill his target in the first period, for example, if an indifference curve between RI and RO were tangent to hf at point a, whose first coordinate is smaller than ob. This indifference curve would indicate the reverse of impatiencean actual preference for income in the second period.’ The relative sizes of k,, kO, r and d will determine the location of the equilibrium point on the line h. A. Consider the situation where planners award the same percentage of above-plan profits for both periods, i.e., kl = kO. The slope of the budget line hfbecomes ( 1 + d), and equilibrium condition ( 10) reduces to a simple equality between the marginal rate of substitution between R, and R,, and the rate of target revision, i.e., r = d. Utility maximization will occur at point f in Fig. 1, provided that the marginal rate of substitution between R, and & at fis algebraically smaller than the slope of hJ6 At this corner solution, it will pay the manager to report his actual profits. Alternatively, if r were less than d for any mix of incomes between h andf, the manager would value present reward less than the cost of target revision. He would report only a portion of his actual profits. The concealment of current profits would lead to a new target that would be conservative relative to the production capacities of the firm. The above conclusion is basically consistent with the findings in Murrell’s paper. B. Suppose the planners were to raise the size of k in the second period so that k, > k,. If other things remain unchanged, the budget line would shift upward from hfto vy. As long as r < d for any mix of incomes between h andf, the manager would maximize his two-period utility at a point such as e. The substitution effect of raising k accounts for the movement from a to n and the income effect for the movement from n to e (assuming that neither R0 nor R, is an inferior good for the manager). The substitution effect would induce the manager to conceal a greater portion of his profits in the first period, the income effect to reveal more. s This would he so, for example, if the utility function of the Cobb-Douglas type and the exponent of R, were larger than that of R,,. 6 Consider the utility function U = &I5 Rf/‘. The marginal rate of substitution (MRS) between the periods equals 3R,/2Ro. At RI = &. MRS = 312. For any d less than 0.5, equilibrium will be at f provided og = OS (i.e., RI = &).

80

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C. In general, as long as the manager prefers present to future income, there will always be a k, smaller than kO that will induce the manager to report all his profits in the first period. Alternatively, if future income is sufficiently preferred to present income, the manager will tend to conceal some of his profits. REFERENCES Bonin, John P., “On the Design of Managerial Incentive Structures in a Decentralized Planning Environment.” Amer. Econ. Rev. 66, 4:682-687, Sept. 1976. Fan, L-S., “On the Reward System.” Amer. Econ. Rev. 65, 1:226-269, March 1975. Fisher, Irving, The Theory of Interest. New York: Kelley and Millman, 1954. Gindin, Sam, “A Model of the Soviet Firm.” Econ. Planning 10, 3:145-147, 1970. Hirshleifer, Jack, “On the Theory of Optimal Investment Decision.” J. Polit. Econ. 66,4:329352, Aug. 1958. Miller, Jeffrey B., and Thornton, James R., “Effort, Uncertainty, and the New Soviet Incentive System.” Southern Econ. J. 45, 2:432-446, Oct. 1978. Murrell, Peter, “The Performance of Multiperiod Managerial Incentive Schemes.” Amer. Econ Rev. 69, 5:934-940, Dec. 1979. Snowberger, Vinson, “The New Soviet Incentive Model: Comment.” Bell J. Econ. 8, 2:591600, Autumn 1977. Weitzman, Martin L., “The New Soviet Incentive Model.” Bell J. Econ. 8, 1:25 l-257, Spring 1976. Weitzman, Martin L., “The Ratchet Principle and Performance Incentives.” Bell J. Econ. 11, 1:302-308, Spring 1980.