First-order approach

In microeconomics and contract theory, the first-order approach is a simplifying assumption used to solve models with a principal-agent problem.[1] It suggests that, instead of following the usual assumption that the agent will take an action that is  utility-maximizing, the modeller use a weaker constraint, and look only for actions which satisfy the  first-order conditions of the agent's problem. This makes the model mathematically more tractable (usually resulting in  closed-form solutions), but it may not always give a valid solution to the agent's problem. [2]

## History[edit]

Historically,[1] the first-order approach was the main tool used to solve the first formal moral hazard models, such as those of Richard Zeckhauser, [3] Michael Spence,[4] and Joseph Stiglitz.[5] Not long after these models were published, James Mirrlees was the first to point out that the approach was not generally valid, and sometimes imposed even stronger necessary conditions than those of the original problem.[2] Following this realization, he[6] and other economists such as Bengt Holmström,[7] William P. Rogerson[2] and Ian Jewitt[8] gave both sufficient conditions for cases where the first-order approach gives a valid solution to the problem, and also different techniques that could be applied to solve general  principal-agent models.

## Mathematical formulation[edit]

In mathematical terms, the first-order approach relaxes the more general incentive compatibility constraint in the  principal's problem. The principal decides on an action  a {\displaystyle a} and proposes a contract  w {\displaystyle w} to the agent by solving the following program:

max w , a {\displaystyle {\underset {w,a}{\operatorname {max} }}} U ( w , a ) {\displaystyle U(w,a)}
subject to
( 1 ) {\displaystyle (1)} V ( w , a ) ≥ V ¯ {\displaystyle V(w,a)\geq {\bar {V}}}
( 2 ) {\displaystyle (2)} a ∈ argmax a ^ ∈ A V ( w , a ^ ) {\displaystyle a\in {\underset {{\hat {a}}\in A}{\operatorname {argmax} }}V(w,{\hat {a}})}

where  U ( w , a ) {\displaystyle U(w,a)} and  V ( w , a ) {\displaystyle V(w,a)} are the principal's and the agent's expected utilities, respectively. Constraint  ( 1 ) {\displaystyle (1)} is usually called the participation constraint (where  V ¯ {\displaystyle {\bar {V}}} is the agent's reservation utility), and constraint  ( 2 ) {\displaystyle (2)} is the incentive compatibility constraint.

Constraint  ( 2 ) {\displaystyle (2)} states that the action  a {\displaystyle a} that the principal wants the agent to take must be utility-maximizing for the agent – that is, it must be compatible with her incentives. The first-order approach relaxes this constraint with the  first-order condition

( 3 ) {\displaystyle (3)} V a ( w , a ) = 0 {\displaystyle V_{a}(w,a)=0}

Equation  ( 3 ) {\displaystyle (3)} is oftentimes much simpler and easier to work with than constraint  ( 2 ) {\displaystyle (2)} , which justifies the attractiveness of the first-order approach. Nonetheless, it is only a necessary condition, and not equivalent to the more general incentive compatibility constraint.

## References[edit]

1. ^ a b Mirrlees, James (1999). "The Theory of Moral Hazard and Unobservable Behaviour: Part I". The Review of Economic Studies. 66 (1): 3–21.
2. ^ a b c Rogerson, WIlliam P. (1985). "The First-Order Approach to Principal Agent Problems". Econometrica. 53 (6): 1357–1367. doi:10.2307/1913212.
3. ^ Zeckhauser, Richard (1970). "Medical insurance: A case study of the tradeoff between risk spreading and appropriate incentives". Journal of Economic Theory. 2 (1): 10–26.
4. ^ Spence, Michael; Zeckhauser, Richard (1971). "Isurance, Information, and Individual Action". American Economic Review. 61 (2): 380–387.
5. ^ Stiglitz, Joseph (1974). "Incentives and Risk Sharing in Sharecropping". The Review of Economic Studies. 41 (2): 219–255. doi:10.2307/2296714.
6. ^ Mirrlees, James (1976). "The Optimal Structure of Incentives and Authority within an Organization". The Bell Journal of Economics. 7 (1): 105–131. doi:10.2307/3003192.
7. ^ Holmström, Bengt (1979). "Moral Hazard and Observability". The Bell Journal of Economics. 10 (1): 74–91. doi:10.2307/3003320.
8. ^ Jewitt, Ian (1988). "Justifying the First-Order Approach to Principal-Agent Problems". Econometrica. 56 (5): 1177–1190. doi:10.2307/1911363.