This expression is the same as in our basic model, except that now the initial ability functions depend on the composition of the lower level, and further the wage differential generates a direct effect on profits from the lower level composition, —qiW.

A key quantity for characterizing the equilibrium is the “mentoring differential” at firm i, which we define as

The mentoring differential of a firm is increasing in the representation of A types in management m» and in the importance of mentoring.

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# Monthly Archives: August 2014

# MENTORING AND DIVERSITY: A Two-Firm Model

Unless otherwise stated, we retain the earlier assumptions from our single-firm model. The main change is the addition of a second firm that competes for entry level employees in a competitive labor market. We simplify the analysis by limiting ourselves to a static model, where each firm makes hiring and promotion decisions only once, and by assuming that the initial ability function is linear there.

Formally, there are two firms indexed by i = 1,2. Each firm hires a unit measure of lower level employees at the beginning of the period. Let be the proportion of lower level employees at firm i of type A and let w be the wage differential between A’s and S’s. In the labor market there is a unit measure of each type of employee, and the labor market must clear. That is, qi(w) + <72(w) = 1> where qi(w) is firm Vs demand for entry level A’s at a given wage differential. More generally, we could include an upward-sloping supply curve for both types of labor.

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# MENTORING AND DIVERSITY: Market Equilibrium and Segregation 2

Of course, in practice, we do sometimes observe segregated outcomes. For example, in the U.S., several industries such as law and banking historically included some firms with a large Jewish representation. Construction companies in some U.S. cities are dominated by a particular ethnic group. Some academic disciplines have large representations from particular nationalities, and gender integration is surprisingly heterogeneous across disciplines, even within the sciences. In other examples, the software company T/Maker in Cupertino, CA was founded by a woman and the firm remained over two-thirds women after 15 years in business; as discussed in the introduction, the fashion company Esprit de Corps has approximately two-thirds women in management. However, more often firms are not strongly segregated.

How do we reconcile the lack of strongly segregated outcomes with the existence of type-based mentoring? First, there are a host of significant costs to homogeneity of a firm’s lower levels. For example, it is rare that firms are truly identical from the perspective of workers. Their locations, cultures, and the skills required will differ; it may be efficient for a diverse group of workers to remain in the same firm, despite mentoring disadvantages.

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# MENTORING AND DIVERSITY: Market Equilibrium and Segregation

As with a, there are two effects. There is direct effect, captured in the term —(2m—1)7, which favors diversity. As the retirement rate goes up the firm must dig deeper into its applicant pool, and the importance of mentoring relative to initial ability falls. However, there is also an indirect effect: a firm is more willing to bias promotions as r increases since current promotions have an immediate and large effect on diversity. If the bias is opposed to the minority, the indirect effect opposes the direct effect more.

However, if SCV holds everywhere, we are able to make a prediction. To see this, observe that by Proposition 3 and optimality of the stable promotion policy, SCV everywhere implies that rm3 < zUB(ms). But then, the second term in -^F0C3{m\ r) must be less than \-§^UB (m), which in turn is negative by SCV everywhere.

There are other potential comparative statics as well: for example, if the initial ability function for one type of worker increased relative to the other, the steady states would entail more workers of that type. If such asymmetries were important, far sighted firms might be willing to change their composition so that the better type is in the majority.

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# MENTORING AND DIVERSITY: Comparative Statics 5

The comparative statics on SUB for a and r do not extend to the case with optimal promotions. Consider first the effect of a, where /x(m) = It might seem that when mentoring becomes more important, the firm would always want to increase homogeneity in order to improve the mentoring of the majority candidates. While this intuition is correct for large enough a, there is a competing effect for intermediate levels. This can be seen in reference to the first order conditions:

The importance of mentoring has two effects on steady states. First, it increases the mentoring advantage of majority candidates, which directly increases their promotion rates (as reflected by the term £(m) — /2(1 — m) > 0), a force which favors increased homogeneity. Second, as mentoring becomes more important, the firm becomes more willing to bias promotions to optimize mentoring. Since this bias may be for or against the minority, the effect of a change in the importance of mentoring is in general ambiguous for intermediate levels of a.

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# MENTORING AND DIVERSITY: Comparative Statics 4

Further, as usual, if This result identifies the nature of the inefficiency which arises if the agents responsible for promotions were to discount the future more than the firm’s owners. In practice, long-term compensation schemes are rarely used for agents in the middle levels of a firm’s hierarchy, and workers take into account the fact that they will leave the firm (or at least their current job) with positive probability in the future.

Such a divergence of discount factors would lead to inefficiencies, according to our analysis, and firms whose mentoring functions are globally SCV might move towards diversity too slowly. In such a case, firms have an incentive to create internal rules or policies about hiring and promotion which help correct the agency problem. This is consistent with the existence of internal affirmative action policies within many large firms.

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# MENTORING AND DIVERSITY: Comparative Statics 3

The First Order Condition illustrates that the return to promoting more majority type workers today is balanced against to the (weighted) return of beginning a new period with a higher level of diversity. Given uniqueness of the optimal promotion policy, the implicit function theorem together with differentiability of the objective function implies that a small increase in FOCs(ms) must lead to an increase in the stable steady state. Reading here We exploit this fact in our comparative statics analysis.

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# MENTORING AND DIVERSITY: Comparative Statics 2

Now we turn to analyze how the set of stable steady states changes with exogenous parameters when optimal promotion policies are used. A natural first approach to this problem would be to derive comparative statics results on the optimal bias, and use these results to analyze changes in steady states. Since the bias increases exactly when the value function gets steeper, this approach is equivalent to deriving comparative statics on the slope of the value function.

In Proposition 3, we showed that when the one-period value function is nonincreasing (nondecreasing) in m, so is the infinite horizon value function. Unfortunately, similar connections do not in general hold for other properties of the value function: making the one-period value function steeper everywhere does not necessarily make the infinite-horizon value function steeper everywhere, without additional restrictions on the curvature of the one-period payoff function.

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# MENTORING AND DIVERSITY: Comparative Statics

Next, we turn to the comparative statics on the set of steady states. We introduced the parameters of interest in Section 3.2.2; our analysis in this section derives predictions about how the level of diversity varies across different economic environments.

The effect of any parameter on the long-run steady states of profit-maximizing firms can be decomposed into two parts, the effect on the unbiased dynamics and the effect on the optimal bias. Reading here We start by characterizing the first effect.

Proposition 7 The diversity of SUB is decreasing in a, increasing in 7 , and is independent of 6. If dzUB/dr < 1/2, then diversity is increasing in r. Proof: Recall that m € SUB satisfies zUB(m) — rm = 0. Consider m > Then the diversity of SUB is increasing (decreasing) in a parameter if zUB(m) — rm is decreasing (increasing) in that parameter. It is straightforward to show that dzVB/да > 0, dzUB/d7 < 0 and dzUB/d6 = 0. Finally, if dzUB/dr < 1/2.

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# MENTORING AND DIVERSITY: The Set of Stable Steady States 5

When SCV holds everywhere and there are multiple steady states, Proposition 3 ranks the steady states in terms of profits: the less diverse steady state mB represents an “inertial trap” with lower profits. Thus, we see that firms employing the same technologies and facing identical worker pools can have different levels of profitability depending on historical conditions which affect their initial levels of diversity. While the less profitable firm could imitate the organization of the other firm, the existence of multiple steady states implies that the cost of the transition is more than the less diverse firm is willing to bear.

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