Wage change and the quit behavior of workers: implications for efficiency wage theory.

• Author: Campbell, Carl M., III

1. Introduction

   Over the past decade, economists have developed efficiency wage models to explain the presence of wage rigidity and thus of involuntary unemployment. In these models, workers' productivity depends positively on the wage or firms' costs depend negatively on the wage, giving firms an incentive to pay wages above the market-clearing level. One type of efficiency wage model is the turnover cost model of Stiglitz [26], Schlicht [24], and Salop [23], in which fewer workers quit at a firm paying high wages. Since hiring and training new workers is costly, firms pay high wages to reduce the number of workers who quit. In the turnover cost model, quits depend on the level of the wage.

   This study argues that quits also depend on the change in the wage as well as on its level. Note that a worker's current wage is determined by his initial wage and by the amount the wage changes over his tenure at the firm. A model is developed in section II demonstrating that, under reasonable conditions, a rise in a worker's current wage not matched by a rise in his starting wage will reduce quits to a greater extent than will an equal rise in the worker's current and starting wages. This means that the change in the wage has a negative effect on the quit rate, even controlling for the current level of the wage. In fact, it is possible that quits are more affected by the change in the wage than by the current level of the wage.

   The hypothesis that quits depend on the change in the wage has some interesting implications for efficiency wage theory. In section III it is argued that if quits depend on the change in the wage, efficiency wage theory may be able to explain why the economy tends to return to a fixed natural rate of unemployment in mild recessions but not in times of severe recessions such as experienced by the United States in the 1930s and by Europe in the 1980s, why wages are rigid upward as well as downward, why wage rigidity varies across industries and occupations, and why nominal wage cuts are so rare.

2. A Model of a Worker's Quit Decision

   A worker's current wage depends on two components: the worker's starting wage and the amount the firm changes the worker's wage over her tenure. This section develops a partial equilibrium model demonstrating that, under reasonable conditions, a worker's decision on whether to quit depends more on the amount a firm changes the wage over the worker's tenure than on her initial wage.

   In this study, the wage (w) is defined as the present discounted value of the worker's current and expected future salary over her expected working life, where a worker's expected future salary is likely to depend on the firm's current wage-tenure profile. Thus, a change in the wage refers to deviations in a worker's pay from its expected path. For example, suppose that a worker is hired by a firm in which salaries normally rise by 10% with each year of tenure. If the firm then cuts salaries by 5% at every level of tenure in the following year, the recently hired worker would be considered to have taken a wage cut, since the present discounted value of her lifetime pay would decrease, in spite of the fact that her salary is higher in the second year than in the first year.(1)

   Assume a two period model in which a worker decides whether to accept or reject an offer in the first period and then decides whether to quit or to remain with the firm in the second period.(2) Both a worker's decision on whether to accept a job offer and her decision on whether to quit depend on the wage she could be earning elsewhere, which in turn depends on her characteristics and on a stochastic component. The stochastic component of a worker's alternate wage arises because different finns, making differing wage offers, have openings at different times. Thus, an individual's best alternative wage offer will vary over time, depending on factors such as which firms have a position open for which the individual is suited, whether the individual is the most qualified for that position, and how much each firm is offering. Again, the alternate wage refers to the present discounted value of the salary offer at another firm.

   Let the individual's alternative wage (net of the cost of changing jobs) in periods 1 and 2 be, respectively,

   [[Omega].sub.1] = [Beta][prime]x + [e.sub.1],

   [[Omega].sub.2] = [Beta][prime]x + g - c + [e.sub.2],

   where x represents a vector of worker characteristics, [Beta] represents a vector of coefficients, c represents the cost of changing jobs, and [e.sub.1] and [e.sub.2] represent the stochastic component of the individual's alternate wage in period 1 and period 2. In addition, g represents the normal change over time in the present discounted value of a worker's alternate wage because of his increased age, his increased experience, and general inflation.(3) Note that g will normally be positive since future pay (which will generally be higher than current pay) will be discounted less with each passing year. However, g can be negative if the individual is near retirement.

   The individual will accept the job in period 1 if

   [w.sub.1] [is greater than] [[Omega].sub.1] = [Beta][prime]x + [e.sub.1],

   and will remain with the firm in period 2 if

   [w.sub.2] [is greater than] [[Omega].sub.2] = [Beta][prime]x + g - c + [e.sub.2],

   where [w.sub.1] and [w.sub.2] represent the wage the firm is offering in each period.

   Assume that x is distributed normally, x [is similar to] N([[Mu].sub.x], [[Sigma].sub.x]),(4) that e is distributed normally, [Mathematical Expression Omitted], and that [e.sub.1] and [e.sub.2] are independent of each other (i.e., E([e.sub.1][e.sub.2]) = 0) and of x (i.e., E([e.sub.1]x) = E([e.sub.2]x) = 0). Then,

   [Mathematical Expression Omitted],

   [Mathematical Expression Omitted],

   where [Mathematical Expression Omitted].

   Since a worker accepts a job offer in period 1 if [w.sub.1] [is greater than] [[Omega].sub.1] = [Beta][prime]x + [e.sub.1], the probability of a worker accepting a job (Prob[a]) can be expressed as

   Prob[a] = Prob[[w.sub.1] [is greater than] [[Omega].sub.1]] = [integral of] h([[Omega].sub.1])d[[Omega].sub.1] between limits [w.sub.1] and -[infinity], (1)

   where [Mathematical Expression Omitted].(5)

   Only a worker who has previously accepted a job can quit. The probability of a worker quitting in period 2 (Prob[q]) is thus the conditional probability that a worker finds a job offer in period 2 that exceeds his current wage (net of the cost of changing jobs) given that he previously accepted a job offer in period 1:

   Prob[q] = Prob[[w.sub.2] [is less than] [[Omega].sub.2][where][w.sub.1] [is greater than] [[Omega].sub.1]] = Prob[w.sub.2] [is less than] [w.sub.2] and [w.sub.1] [is greater than] [[Omega].sub.1]]/Prob[[w.sub.1] [is greater than] [[Omega].sub.1]]. (2)

   The numerator of this fraction can be expressed as,

   [Mathematical Expression Omitted],

   where

   [Mathematical Expression Omitted].

   As demonstrated in the Appendix, the correlation ([Rho]) between [[Omega].sub.1] and [[Omega].sub.2] is,

   [Mathematical Expression Omitted].

   Note that [Rho] = 0 if there is no variation in worker characteristics and that [Rho] = 1 if the worker's alternate wage contains no stochastic component.

   Let [Delta]w = [w.sub.2] - [w.sub.1], so that [w.sub.2] = [w.sub.1] + [Delta]w. Then from (1), (2), and (3), the probability of a worker quitting can be expressed as,

   [Mathematical Expression Omitted].

   This model differs from early models in the reservation wage literature.(6) Early reservation wage models did not allow on-the-job search and thus assumed that once a worker accepted a job, she holds it forever. The assumption that workers cannot search while employed means that workers may reject a wage offer to continue searching, and the wage below which a worker will reject a job is termed the reservation wage in this literature.(7) In contrast, on-the-job search plays an important role in the model in this study. Consistent with this model, Holzer [18] and Blau [6] present evidence that many employed workers do, in fact, engage in on-the-job search. Since the model in this study assumes that the worker's expected alternate wage is the same in both periods(8) (except for the normal growth rate, g) a worker will never remain unemployed to look for another job, although a worker may choose to be unemployed to engage in household production. In this case [[Omega].sub.1] or [[Omega].sub.2] would represent the present discounted value of household production. This study makes different assumptions than models in the reservation wage literature since it examines quit behavior rather than search unemployment.

   The important question for this study is how a change in [Delta]w and a change in [w.sub.1] each affects the probability of a quit. First, consider the effect of [Delta]w on quits. Taking the derivative of (4) with respect to [Delta]w and applying Leibnitz's role (see the Appendix for a derivation of (5)) yields,

   [Mathematical Expression Omitted].

   Similarly, in the Appendix it is shown that,

   [Mathematical Expression Omitted].

   From (5) and (6) it can be seen that d Prob[q]/d[Delta]w is greater than d Prob[q]/d[w.sub.1] in absolute value (i.e., more negative) if and only if

   [Mathematical Expression Omitted].

   This expression will be positive if h([[Omega].sub.1])f([w.sub.1], [[Omega].sub.2]) - h([w.sub.1])f([[Omega].sub.1], [[Omega].sub.2]) [is greater than] 0 over the domain in (7).(9) This, in turn, will be positive if...