nach oben

Mathematics and Financial Economics

Erschienen in:

01.06.2015

Dynamic contracts and learning by doing

verfasst von: Julien Prat

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

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

This paper studies the design of optimal contracts in dynamic environments where agents learn by doing. We derive a condition under which contracts are fully incentive compatible. A closed-form solution is obtained when agents have CARA utility. It shows that human capital accumulation strengthens the power of incentives and allows the principal to provide the agent with better insurance against transitory risks.

Nächster Artikel Pay for performance under hierarchical contracting

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Nur mit Berechtigung zugänglich

See the seminal work of [1].

Fernandes and Phelan [4] propose a framework where the state space remains manageable. They assume that output is only correlated from one period to the next, and that agents can take one of only two actions. Then the relevant deviation is unique, and it is sufficient to keep track of a single “threat keeping constraint”.

He et al. [7] extends Jovanovic and Prat (2013) by introducing hidden savings and effort costs which are convex instead of linear.

The companion paper [13] studies a dynamic moral hazard problem where the agent has access to states which the principal cannot observe. As an illustration of his general results, [13] solves a model where the hidden states corresponds to the amount of savings accumulated by the agent.

By contrast, our closed form example uses a utility function that is not separable in consumption and leisure.

It is straightforward to generalize the production function by adding a multiplicative term $\alpha \in \left[ 0,1\right] $ in front of effort, i.e., $Y_{t}=\int _{0}^{t}(\alpha a_{s}+h_{s})ds+\int _{0}^{t}\sigma dZ_{s}.\ $ All the derivations presented in this paper go through under this more general specification. Notice that [10] focuses on cases where $ \alpha =0$ so that effort affects output solely through its impact on the stock variable $h$.

Our incentive constraint is also closely related to the one in [10] which reads $-u_{a}\left( \cdot \right) =E_{t}^{a}\left[ \int _{t}^{T}e^{-\rho \left( s-t\right) }f\left( s-t\right) \gamma _{s}ds \right] \!.\,$The function $f\left( s-t\right) $ captures the effect of past action $a_{s}$ on the value at time $t$ of the stock variable. It is therefore equal to $\exp (-\delta \left( s-t\right) )$ in our model. More importantly, the volatility coefficient $\gamma _{t}$ does not anymore appear on the right hand side of the incentive constraint. This is because, as discussed in footnote 6, [10] assumes that effort has no direct effect on output, i.e., $Y_{t}=\int _{0}^{t}h_{s}ds+\int _{0}^{t}\sigma dZ_{s}.$

The argument is mostly heuristic because both wages and effort are endogenous. Yet, we will see that its intuition is valid when we solve for the optimal contract under CARA utility.

This constructive approach shows that the solution in Proposition is not invalidated by the lack of mutually absolutely continuous measure in the infinite horizon limit.

In Williams’ [12] model, the variable $p$ corresponds to the co-state associated to the variable $z$ capturing the covariation between reports and the stock of lies.

The sensitivity coefficient $\gamma $ is unambiguously positive because $v$ is negative and, as shown in the proof of Corollary 1, $\rho >k.$

We show in Corollary 1 that $\rho >k$, whereas $v<0$ follows from the specification (11) of the utility function.

This will not necessarily be true if we allowed effort to have a cumulative effect, i.e., if we let $\delta $ be negative. Then persistence may even lead to backloaded transfers.

To see that this conclusion holds true in our set-up, set $\delta $ equal to infinity. It follows that $\lim _{\delta \rightarrow \infty }p_{t}=\lim _{\delta \rightarrow \infty }E_{t}\left[ \int _{t}^{T}e^{-\left( \rho +\delta \right) \left( s-t\right) }\gamma _{s}ds\right] =0.$ Thus the volatility coefficient $\vartheta _{t}=0$, which implies in turn that the sufficient condition is satisfied since $2u_{aa}\left( w_{t},a_{t}\right) \le 0=\vartheta _{t}.$

We cannot directly apply the standard Martingale Representation theorem because we are considering weak solutions, so that $\left\{ Z_{t}^{a}\right\} $ does not necessarily generate $\left\{ \mathcal {F} _{t}^{Y}\right\} $.

The additional expectation term vanishes because both$\ \nabla A_{s}$ and $ \Delta a_{s}$ are bounded and so

$$\begin{aligned} \left( \int _{0}^{t}U\left( \tau ,Y_{\cdot },a_{\tau }\right) d\tau \right) E_{t}^{a}\left[ \int _{t}^{T}\left( \nabla A_{s}+\Delta a_{s}\right) dZ_{s}^{a}\right] =0. \end{aligned}$$

Square integrability of $\Gamma _{s}^{2}$ can be established for any $ \varepsilon \in \left[ 0,\varepsilon _{0}\right) $ following the same steps as in Lemma 7.3 of [2].

$\chi ^{*}$ is predictable since both $ \xi ^{*}$ and $h^{*}$ are $\mathbb {F}^{Y}-$predictable.

In our particular problem, switching from a weak formulation of the agent’s problem, to a strong formulation of the principal’s problem, does not raise measurability issues because the agent’s action is constant over time, and so does not directly depends on the Brownian motion.

To establish this claim analytically, one can differentiate $L\left( k\right) $ and $R\left( k\right) $ to obtain: $L^{\prime \prime }\left( k\right) =\left( \theta \lambda \sigma \right) ^{2}$ and $R^{\prime \prime }\left( k\right) =2\left[ \theta \lambda \sigma \left( \delta +1\right) \right] ^{2}\left( k+\delta +1\right) ^{-3}$. Hence there exists a unique $ \hat{k}$ such that $L^{\prime \prime }\left( k\right) \gtrless R^{\prime \prime }\left( k\right) $ for all $k\gtrless \hat{k},$ which rules out the possibility that $R\left( k\right) $ intersects twice $L\left( k\right) $ from above.

The second equality below follows from the definition of $k$ as

$$\begin{aligned} k\left( \frac{k+\delta }{k+\delta +1}\right) =\frac{\rho -k}{\left( \theta \lambda \sigma \right) ^{2}k}. \end{aligned}$$

Becker, G.: Human Capital: A Theoretical and Empirical Analysis, with Special Reference to Education. University of Chicago Press, Chicago (1964)

Cvitanić, J., Wan, X., Zhang, J.: Optimal compensation with hidden action and lump-sum payment in a continuous-time model. Appl. Math. Optim. 59, 99–146 (2009)MATHMathSciNet

DeMarzo, P.M., Sannikov, Y.: Learning in Dynamic Incentive Contracts. Princeton University University, Unpublished manuscript (2011)

Fernandes, A., Phelan, C.: A recursive formulation for repeated agency with history dependence. J. Econ. Theory 91, 223–247 (2000)MATH

Fujisaki, M., Kallianpur, G., Kunita, H.: Stochastic differential equation for the non linear filtering problem. Osaka J. Math. 9, 19–40 (1972)MATHMathSciNet

Gibbons, R., Murphy, K.J.: Optimal incentive contracts in the presence of career concerns: theory and evidence. J. Polit. Econ. 100(3), 468–505 (1992)

He, Z., Wei, B., Yu, J.: Optimal Long-term Contracting with Learning. University of Chicago, Unpublished manuscript (2012)

Jovanovic, B., Prat, J.: Dynamic Contracts when Agent’s Quality is Unknown. Forthcoming in Theoretical Economics (2013)

Sannikov, Y.: A continuous-time version of the principal-agent problem. Rev. Econ. Stud. 75(3), 957–984 (2008)MATHMathSciNet

10.

Sannikov, Y.: Moral Hazard and Long-Run Incentives. Princeton University, Unpublished manuscript (2012)

11.

Schättler, H., Sung, J.: The first-order approach to the continuous-time principal-agent problem with exponential utility. J. Econ. Theory 61, 331–371 (1993)MATH

12.

Williams, N.: Persistent private information. Econometrica 79, 1233–1274 (2011)

13.

Williams, N.: On Dynamic Principal-Agent Problems in Continuous Time. University of Wisconsin - Madison, Unpublished manuscript (2013)

Titel: Dynamic contracts and learning by doing
verfasst von: Julien Prat
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/s11579-014-0120-6