Sequential optimal contracting in continuous time

Guillermo Alonso Alvarez; Erhan Bayraktar; Ibrahim Ekren; Liwei Huang

doi:10.3934/fmf.2025001

Article Contents

March 2025, Volume 4, 114-139. Doi: 10.3934/fmf.2025001 Free Access

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Sequential optimal contracting in continuous time

Department of Mathematics, University of Michigan, USA

^*Corresponding author: Guillermo Alonso Alvarez
^*Corresponding author: Guillermo Alonso Alvarez

Received: November 2024

Revised: January 2025

Early access: February 8, 2025

Published online: February 8, 2025

The second author was funded in part by the NSF through DMS-2106556 and by the Susan M. Smith Chair. The third author was funded in part by the NSF through DMS-2406240.

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Abstract

This paper studies a principal-agent problem in continuous time with multiple lump-sum payments (contracts) paid at different deterministic times. We reduce the non-zero-sum Stackelberg game between the principal and agent to a standard stochastic optimal control problem. We apply our result to a benchmark model to investigate how different inputs (payment frequencies, payment distribution, discounting factors, agent's reservation utility) affect the principal's value and agent's optimal compensations.

Keywords:

Principal-agent,
contract theory,
stochastic control,
backward stochastic differential equations,
viscosity solutions of partial differential equations.

Mathematics Subject Classification: 91B41, 93E03.

Citation:

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Figure 1. Principal's value and optimal utilities $ \eta_i^* = U_a(\xi_i^*) $, when $ k_a = 0 $

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Figure 2. Principal's value, when $ k_a>0 $

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Figure 3. Optimal utilities $ \eta_i^* = U_a(\xi_i^*) $, when $ k_a>0 $

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Figure 4. Principal's value with $ N = 2 $ payments for different $ T_1 $

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Figure 5. Principal's value for different payment frequencies

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Figure 6. Initial Negotiation vs. Renegotiation, when $ k_a = 0 $

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Figure 7. Initial Negotiation vs. Renegotiation, when $ k_a = 0.4 $

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Figure 8. Initial Negotiation vs. Renegotiation, when $ k_a = 0.4 $

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