このたびジョン・ベイツ・クラーク賞を受賞したユリ・サニコフの業績を、A Fine Theoremブログが表題のエントリ(原題は「Yuliy Sannikov and the Continuous Time Approach to Dynamic Contracting」)で取り上げている。以下はそこからの引用。

Sannikov’s most famous work is in the pure theory of dynamic contracting, which I will spend most of this post discussing, but the methods he has developed turn out to have interesting uses in corporate finance and in macroeconomic models that wish to incorporate a financial sector without using linearization techniques that rob such models of much of their richness.

Solving for the optimal contract essentially involves solving two embedded dynamic optimization problems. The agent optimizes effort over time given the contract the principal committed to, and hence the agent chooses an optimal dynamic history-dependent contract given what the agent will do in response. The space of possible history-dependent contracts is enormous. Sannikov shows that you can massively simplify, and solve analytically, for the optimal contract using a four step argument.


  1. 継続的価値がマルチンゲール(過去を条件として、明日の継続的価値の期待値が今日の継続的価値そのものとなる)であることから、マルチンゲール表現定理を使って、継続的価値が正確に過去の約束を踏襲するという制約の下で、同価値の最適な確率過程が解析的に求まる。その解析解はエージェントの行動に依存し、特性も自然(例:フローの効用を今日支払えば、明日の支払いは少なくて済む)である。
  2. 上で求めた継続的価値を決定する方程式に、エージェントのインセンティブとの両立、という制約を放り込む。
  3. エージェントへの継続的な支払いを決定する確率過程の下で、プリンシパルの利益を最大化する。
  4. このプリンシパル問題からハミルトン=ヤコビ=ベルマン方程式が導かれ、それは伊藤のルールと、境界条件の確認によって解ける。結果的に、時間を通じてエージェントの継続的価値をコントロールする最適な手法に関する解析的な表現が得られ、その方程式であらゆる比較静学が実行できる。

その意義についてブログ主のKevin Bryanは以下のように書いている。

What does this method give us? Because the continuation value and the flow payoffs can be constructed analytically even for positive discount rates, we can actually answer questions like: should you use long-term incentives (continuation value) or short-term incentives (flow payoffs) more when, e.g., your workers have a good outside option? What happens as the discount rate increases? What happens if the uncertainty in the mapping between the agent’s actions and output increases? Answering questions of these types is very challenging, if not impossible, in a discrete time setting.
この手法から我々が得るものは何か? 継続的価値とフローの支払いが、正の割引率についても解析的に導出できるため、次のような質問にも答えることができるようになる:例えばあなたの労働者が外部に良い選択肢を有している時、長期のインセンティブ(継続的価値)を使うべきか、それとも短期のインセンティブ(フローの支払い)を使うべきか? 割引率が上昇すると何が起きるか? エージェントの行動と生産の増大の間のマッピングにおける不確実性が増大すると何が起きるか? 離散的時間の枠組みでは、こうした類の質問に答えるのは不可能と言わないまでも非常に難しい。


Though I’ve presented the basic Sannikov method in terms of incentives for workers, dynamic moral hazard – that certain unobservable actions control prices, or output, or other economic parameters, and hence how various institutions or contracts affect those unobservable actions – is a widespread problem. Brunnermeier and Sannikov have a nice recent AER which builds on the intuition of Kiyotaki-Moore models of the macroeconomy with financial acceleration. The essential idea is that small shocks in the financial sector may cause bigger real economy shocks due to deleveraging. Brunnermeier and Sannikov use the continuous-time approach to show important nonlinearities: minor financial shocks don’t do very much since investors and firms rely on their existing wealth, but major shocks off the steady state require capital sales which further depress asset prices and lead to further fire sales. A particularly interesting result is that exogenous risk is low – the economy isn’t very volatile – then there isn’t much precautionary savings, and so a shock that hits the economy will cause major harmful deleveraging and hence endogenous risk. That is, the very calmness of the world economy since 1983 may have made the eventual recession in 2008 worse due to endogenous choices of cash versus asset holdings. Further, capital requirements may actually be harmful if they aren’t reduced following shocks, since those very capital requirements will force banks to deleverage, accelerating the downturn started by the shock.