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Sequential optimal contracting in continuous time

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

Abstract

In this paper we study 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 for which we investigate how different inputs (payment frequencies, payments' distribution, discounting factors, agent's reservation utility) affect the principal's value and agent's optimal compensations.

Sequential optimal contracting in continuous time

Abstract

In this paper we study 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 for which we investigate how different inputs (payment frequencies, payments' distribution, discounting factors, agent's reservation utility) affect the principal's value and agent's optimal compensations.

Paper Structure

This paper contains 12 sections, 5 theorems, 103 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Set up
  3. The principal-agent problem
  4. The benchmark model
  5. Numerical results
  6. The case with no discounting $\kappa_a=0$
  7. The effect of the discounting factor $k_a>0$
  8. Analysis of the payments' distribution
  9. Analysis of the payment frequency
  10. Initial Negotiation VS Renegotiation
  11. Appendix
  12. Proof of Theorem \ref{['hjb:pp:in']}

Key Result

Proposition 1

Let $\bar{\xi}_{N}:=(\xi_1,\ldots,\xi_N) \in \Sigma_{N}$ be a contract schedule. Then, there exists a pair of processes $(Z,Y)\in \mathbb{H}([0,T])\times \mathbb{D}_{\exp}([0,T])$, solving the following recursive system of BSDEs: for all $i \in \{1,\ldots,N-1\}$.

Figures (8)

  • Figure 1: Initial Negotiation $k_a = 0$
  • Figure 2: Discounting Comparison (Principal's Value Function)
  • Figure 3: Optimal utilities $\eta_i^*=U_a(\xi_i^*)$.
  • Figure 4: Uneven contracting periods.
  • Figure 5: Payment Frequency Comparison
  • ...and 3 more figures

Theorems & Definitions (13)

  • Remark 1
  • Remark 2
  • Proposition 1
  • Proposition 2
  • Remark 3
  • Theorem 1
  • Remark 4
  • Theorem 2
  • Definition 1
  • Lemma 1
  • ...and 3 more