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Mathematics and Financial Economics

Erschienen in:

01.06.2015

Pay for performance under hierarchical contracting

verfasst von: Jaeyoung Sung

Erschienen in: Mathematics and Financial Economics | Ausgabe 3/2015

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Abstract

We consider a simple hierarchical contracting problem where the principal contracts the top manager who in turn subcontracts many middle managers. We show that that the top managerial contract is lower-powered and middle managerial contracts are higher-powered in incentives than predicted by the standard agency model. Consequently, at optimum, middle managers work harder, and the CEO (top manager) works less than implied by their standard second-best contracts. Moreover, the CEO contract sensitivity decreases as the number of middle managers increases.

Vorheriger Artikel Dynamic contracts and learning by doing

Nächster Artikel A note on utility-based pricing

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Hall and Liebman [8], Aggarwal and Samwick [1], and Baker and Hall [2] document higher sensitivities than Jensen and Murphy. However, the magnitude of these sensitivities are generally small.

For ease of exposition and without loss of generality, we formulate a discrete-time model which is analogous to its continuous-time counterpart. For the continuous-time methods, readers are referred to Schättler and Sung [19] and Cvitanić et al. [5] for Holmstrom–Milgrom-type models, and [18] for an alternative continuous-time model.

See also Radner [4], Bolton and Dewatripont [17], and Mookherjee [16] for information cost motivations. See Baliga and Sjöström [3], Faure-Grimaud et al. [6], and Mookherjee [16] for collusion motivations.

As shall be seen in our hierarchical contracting problem, the top manager turns out to choose not only the mean of the outcome of his own effort but, in effect, the volatility of the total profit of the firm as he chooses middle managerial contracts. Thus, our problem turns out to be similar to the unobservable project choice problem in Sung [21].

Even without invoking high monitoring costs, the hierarchical contracting may be rationalized if investors are uncertain about middle managers’ abilities, but after the top managerial contract is signed, the top manager can identify their abilities with higher precisions than investors can. However we do not model adverse selection problems here.

Alternatively, one may want to consider allowing $S^i$ to depend also on other managerial effort outcomes such that $S^i = A^i + \sum _{k=0}^N B^{ik} Y^k$. However, under this algebraically more complicated contracting scheme, one can show that our results (Theorems 1 and 2) in this paper will remain qualitatively unaffected.

Note that $\varphi (\mu ):=1 - c_{\mu }(\mu )- R c_{\mu }(\mu ) c_{\mu \mu }(\mu )$ is decreasing in $\mu $, and thus $\varphi (\mu ) > (=, <) 0$ for $\mu < (=, >) \mu _{2nd}$.

The extent of top managerial risk-sharing is based not on the whole outcome $X$, but on a fraction of the aggregate outcome $c_{\mu }^0(\mu ^0) X$, because the top manager is concerned with sharing only his own share of the outcome risk with middle managers.

As soon as positive incentives are given to the top manager, however, he will drive middle managers to work harder than desired by investors in the second best. To see this, assume quadratic cost functions as in (11). Differentiating Eq. (15) with respect to $\mu ^0$, we have

$$\begin{aligned} K \frac{\partial \mu }{\partial \mu ^0}&= \frac{R^0 K^0c_{\mu \mu }(\mu ) \left( 1 - c_{\mu }(\mu ) \right) \bar{\sigma }^2 }{1 + R c_{\mu \mu }(\mu )\bar{\sigma }^2 + R^0 {c_{\mu }^0}(\mu ^0)c_{\mu \mu }(\mu ) \bar{\sigma }^2} > 0. \end{aligned}$$

That is, if the top managerial sensitivity $c_{\mu }^0(\mu ^0)$ is increased, the the top manager responds to it by increasing middle managerial sensitivities $c_{\mu }(\mu ^i)$’s as well.

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Titel: Pay for performance under hierarchical contracting
verfasst von: Jaeyoung Sung
Publikationsdatum: 01.06.2015
Verlag: Springer Berlin Heidelberg
Erschienen in: Mathematics and Financial Economics / Ausgabe 3/2015
Print ISSN: 1862-9679
Elektronische ISSN: 1862-9660
DOI: https://doi.org/10.1007/s11579-014-0138-9