Yuliy Sannikov and the Continuous Time Approach to Dynamic Contracting

The John Bates Clark Award, given to the best economist in the United States under 40, was given to Princeton’s Yuliy Sannikov today. The JBC has, in recent years, been tilted quite heavily toward applied empirical microeconomics, but the prize for Sannikov breaks that streak in striking fashion. Sannikov, it can be fairly said, is a mathematical genius and a high theorist of the first order. He is one of a very small number of people to win three gold medals at the International Math Olympiad – perhaps only Gabriel Carroll, another excellent young theorist, has an equally impressive mathematical background in his youth. 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. A quick warning: Sannikov’s work is not for the faint of heart, and certainly not for those scared of an equation or two. Economists – and I count myself among this group – are generally scared of differential equations, as they don’t appear in most branches of economic theory (with exceptions, of course: Romer’s 1986 work on endogenous growth, the turnpike theorems, the theory of evolutionary games, etc.). As his work is incredibly technical, I will do my best to provide an overview of his basic technique and its uses without writing down a bunch of equations, but there really is no substitute for going to the mathematics itself if you find these ideas interesting.

The idea of dynamic contracting is an old one. Assume that a risk-neutral principal can commit to a contract that pays an agent on the basis of observed output, with that output being generated this year, next year, and so on. A risk-averse agent takes an unobservable action in every period, which affects output subject to some uncertainty. Payoffs in the future are discounted. Take the simplest possible case: there are two periods, an agent can either work hard or not, output is either 1 or 0, and the probability it is 1 is higher if the agent works hard than otherwise. The first big idea in the dynamic moral hazard of the late 1970s and early 1980s (in particular, Rogerson 1985 Econometrica, Lambert 1983 Bell J. Econ, Lazear and Moore 1984 QJE) is that the optimal contract will condition period 2 payoffs on whether there was a good or bad outcome in period 1; that is, payoffs are history-dependent. The idea is that you can use payoffs in period 2 to induce effort in period 1 (because continuation value increases) and in period 2 (because there is a gap between the payment following good or bad outcomes in that period), getting more bang for your buck. Get your employee to work hard today by dangling a chance at a big promotion opportunity tomorrow, then actually give them the promotion if they work hard tomorrow.

The second big result is that dynamic moral hazard (caveat: at least in cases where saving isn’t possible) isn’t such a problem. In a one-shot moral hazard problem, there is a tradeoff between risk aversion and high powered incentives. I either give you a big bonus when things go well and none if things go poorly (in which case you are induced to work hard, but may be unhappy because much of the bonus is based on things you can’t control), or I give you a fixed salary and hence you have no incentive to work hard. The reason this tradeoff disappears in a dynamic context is that when the agent takes actions over and over and over again, the principle can, using a Law of Large Numbers type argument, figure out exactly the frequency at which the agent has been slacking off. Further, when the agent isn’t slacking off, the uncertainty in output each period is just i.i.d., hence the principal can smooth out the agent’s bad luck, and hence as the discount rate goes to zero there is no tradeoff between providing incentives and the agent’s dislike of risk. Both of these results will hold even in infinite period models, where we just need to realize that all the agent cares about is her expected continuation value following every action, and hence we can analyze infinitely long problems in a very similar way to two period problems (Spear and Srivistava 1987).

Sannikov revisited this literature by solving for optimal or near-to-optimal contracts when agents take actions in continuous rather than discrete time. Note that the older literature generally used dynamic programming arguments and took the discount rate to a limit of zero in order to get interested results. These dynamic programs generally were solved using approximations that formed linear programs, and hence precise intuition of why the model was generating particular results in particular circumstances wasn’t obvious. Comparative statics in particular were tough – I can tell you whether an efficient contract exists, but it is tough to know how that efficient contract changes as the environment changes. Further, situations where discounting is positive are surely of independent interest – workers generally get performance reviews every year, contractors generally do not renegotiate continuously, etc. Sannikov wrote a model where an agent takes actions that control the mean of output continuously over time with Brownian motion drift (a nice analogue of the agent taking an action that each period generates some output that depends on the action and some random term). The agent has the usual decreasing marginal utility of income, so as the agent gets richer over time, it becomes tougher to incentivize the agent with a few extra bucks of payment.

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.

First, as in the discrete time approach, we can simplify things by noting that the agent only cares about their continuous-time continuation value following every action they make. The continuation value turns out to be a martingale (conditioning on history, my expectation of the continuation value tomorrow is just my continuation value today), and is basically just a ledger of my promises that I have made to the agent in the future on the basis of what happened in the past. Therefore, to solve for the optimal contract, I should just solve for the optimal stochastic process that determines the continuation value over time. The Martingale Representation Theorem tells me exactly and uniquely what that stochastic process must look like, under the constraint that the continuation value accurately “tracks” past promises. This stochastic process turns out to have a particular analytic form with natural properties (e.g., if you pay flow utility today, you can pay less tomorrow) that depend on the actions the agents take. Second, plug the agent’s incentive compatibility constraint into our equation for the stochastic process that determines the continuation value over time. Third, we just maximize profits for the principal given the stochastic process determining continuation payoffs that must be given to the agent. The principal’s problem determines an HJB equation which can be solved using Ito’s rule plus some effort checking boundary conditions – I’m afraid these details are far too complex for a blog post. But the basic idea is that we wind up with an analytic expression for the optimal way to control the agent’s continuation value over time, and we can throw all sorts of comparative statics right at that equation.

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.

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.

Sannikov’s entire oeuvre is essentially a graduate course in a new technique, so if you find the results described above interesting, it is worth digging deep into his CV. He is a great choice for the Clark medal, particularly given the deep and rigorous application he has applied his theory to in recent years. There really is no simple version of his results, but his 2012 survey, his recent working paper on moral hazard in labor contracts, and his dissertation work published in Econometrica in 2007 are most relevant. In related work, we’ve previously discussed on this site David Rahman’s model of collusion with continuous-time information flow, a problem very much related to work by Sannikov and his coauthor Andrzej Skrzypacz, as well as Aislinn Bohren’s model of reputation which is related to the single longest theory paper I’ve ever seen, Sannikov and Feingold’s Econometrica on the possibility of “fooling people” by pretending to be a type that you are not. I also like that this year’s JBC makes me look like a good prognosticator: Sannikov is one of a handful of names I’d listed as particularly deserving just two years ago when Gentzkow won!


5 thoughts on “Yuliy Sannikov and the Continuous Time Approach to Dynamic Contracting

  1. CC says:

    “Get your employee to work hard today by dangling a chance at a big promotion opportunity tomorrow, then actually give them the promotion if they work hard tomorrow.”

    At the risk of becoming overly pedantic: you don’t actually give them the promotion if they work hard; you give them the promotion if they happen to produce a good outcome. The problem is interesting precisely because you can’t tell if your employee is working hard. (Though I’m sure you know that — some of your other readers may not, however!)

    • afinetheorem says:

      Indeed! There is always a struggle between precision and accessibility when I write about high theory, I am afraid.

  2. Gabriel Carroll says:

    Hi – very minor comment on the first paragraph of this post, Mihai Manea should get as much “youth mathematical background” credit as I do. Congratulations to Yuliy! An exciting moment for micro theory. – Gabriel

  3. CrispyAndy says:

    Thanks for the informative post, but were you serious about this?
    ‘Economists – and I count myself among this group – are generally scared of differential equations, as they don’t appear in most branches of economic theory’
    You know, there was this dude, I think Bob Solow was his name, with this growth model in the 1950s, or that other guy, Paul Samuelson? With this correspondence principle, somewhere in the 1940s?

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