Holmström's Informativeness Principle

Holmström's informativeness principle (also known as the sufficient statistic theorem for moral hazard, alongside Holmström's 1982 moral hazard in teams theorem) is a foundational pillar of modern contract theory, organizational economics, and business and economics theory, formulated by Finnish economist and Nobel laureate Bengt Holmström in two landmark Bell Journal of Economics articles in 1979 ("Moral Hazard and Observability") and 1982 ("Moral Hazard in Teams"). Operating within the canonical principal–agent problem in which a principal cannot directly observe an agent's costly effort, Holmström (1979) proved that the optimal compensation contract is governed by the likelihood ratio of observable output, and that any additional verifiable signal y (such as an industry stock index, peer output, or an accounting audit) should be included in the agent's compensation contract if and only if output x is not a statistical sufficient statistic for (x, y) with respect to the agent's effort a. In his companion 1982 paper, Holmström extended this analysis to multi-agent production teams, proving that under unobservable individual effort and deterministic joint output, no budget-balanced sharing rule can sustain a Pareto-efficient Nash equilibrium, establishing the economic rationale for a residual claimant or corporate budget breaker.

Structure of the Hidden-Action Model

Consider a principal–agent problem in which a risk-neutral (or risk-averse) principal hires a risk-averse agent to take an unobserved effort action a ∈ A ⊂ ℝ₊:

• Stochastic Technology: Effort a is not verifiable by a court; instead, both parties observe a verifiable output realization x ∈ ℝ drawn from a conditional probability density function f(x | a) > 0 whose support is independent of a, with cumulative distribution F(x | a) satisfying first-order stochastic dominance (F_a(x | a) ≤ 0).
• Preferences: The principal designs a compensation schedule w(x) paid as a function of realized output x. The principal's utility is V(x − w(x)) with V′ > 0 and V″ ≤ 0 (for a risk-neutral principal, V(x − w(x)) = x − w(x)). The agent has additively separable utility U(w, a) = u(w) − c(a), where u(w) is increasing and concave (u′(w) > 0, u″(w) < 0, reflecting risk aversion) and c(a) is increasing and convex (c′(a) > 0, c″(a) > 0, reflecting disutility of effort).

To implement desired effort a at maximum expected profit, the principal chooses w(x) to maximize ∫ V(x − w(x))f(x | a)*dx* subject to two constraints:

1. Individual Rationality (IR / Participation Constraint) with shadow multiplier λ > 0:

1. Incentive Compatibility (IC Constraint): Given w(x), the agent maximizes expected utility by choosing a. Under the First-Order Approach (validated by Rogerson's 1985 Monotone Likelihood Ratio Property [MLRP] and Convexity of the Distribution Function Condition [CDFC]), the IC constraint is represented by the agent's first-order stationarity condition with shadow multiplier μ > 0:

The Holmström–Mirrlees Likelihood-Ratio Equation

Taking the pointwise variational derivative of the Lagrangian with respect to w(x) at each output realization x yields the Holmström–Mirrlees Characterization Equation (1979):

When the principal is risk-neutral (V′ ≡ 1), this simplifies to the inverse marginal utility of consumption:

If effort a were directly verifiable, μ = 0, so 1 / u′(w(x)) = λ is constant across all x, meaning the risk-neutral principal provides full insurance with a fixed wage w^*. Under moral hazard (μ > 0), the optimal wage w(x) varies with output x through the score function or likelihood ratio f_a(x | a) / f(x | a). Output realizations x are rewarded because a positive likelihood ratio (f_a(x | a) > 0) provides statistical evidence that the agent exerted high effort a rather than shirking. Consequently, the wage schedule w(x) is monotonically non-decreasing in output x if and only if f(x | a) satisfies the Monotone Likelihood Ratio Property (MLRP): ∂[f_a(x | a) / f(x | a)] / ∂x ≥ 0.

Suppose that in addition to primary output x, a second verifiable signal y (such as an industry peer index or an operational audit) is available with joint conditional density f(x, y | a). Pointwise optimization of the two-signal Lagrangian yields:

Thus, the optimal contract w(x, y) is independent of y if and only if the joint likelihood ratio f_a(x, y | a) / f(x, y | a) does not depend on y—which, by the Fisher–Neyman Factorization Theorem of mathematical statistics, occurs if and only if the joint density factors as:

Theorem (Holmström's Informativeness Principle / Sufficient Statistic Theorem, 1979): Let x be the agent's primary performance measure and y be an additional verifiable signal. Conditioning the agent's contract on y fails to yield a Pareto improvement (so w(x, y) = w(x)) if and only if x is a sufficient statistic for the pair (x, y) with respect to unobserved effort a. Equivalently, y is contractually valuable if and only if y contains incremental statistical information about a or about the exogenous noise contaminating x conditional on x.

Application to Relative Performance Evaluation (RPE)

An important implication of the Informativeness Principle is that a signal y should be included in a CEO's compensation contract even if the CEO has zero control over y (∂f(y | a) / ∂a = 0), provided y is correlated with the exogenous shock affecting x. In the linear normal benchmark where firm output is x = a + θ + ε (with common industry shock θ ~ N(0, σ_θ²) and idiosyncratic firm noise ε ~ N(0, σ_ε²)) and peer firms generate average output y = ā + θ + η (with peer noise η ~ N(0, σ_η²)), x alone is not a sufficient statistic for (x, y) because y is informative about θ. Instead, the linear combination:

is a sufficient statistic for (x, y) with respect to a. Basing compensation on x − β^*y (Relative Performance Evaluation [RPE]) filters out common industry luck θ, lowering compensation variance and reducing the risk premium required to motivate effort.

In his 1982 article "Moral Hazard in Teams," Holmström analyzed free-riding in joint production when n ≥ 2 agents exert unobservable efforts a = (a₁, …, a_n) ∈ ℝ₊ⁿ that jointly determine verifiable team output y = F(a₁, …, a_n), and each agent i incurs private effort cost c_i(a_i). Maximizing aggregate team surplus F(a) − ∑_i=1ⁿ c_i(a_i) requires the Pareto-efficient effort profile a^* where ∂F(a^*) / ∂a_i = c_i′(a_i^*) for all i.

Theorem 1 (Holmström's Team Impossibility Theorem, 1982): Under budget balance (∑_i=1ⁿ s_i(y) = y for all y) with differentiable sharing rules, no Nash equilibrium can achieve the Pareto-efficient effort profile a^*.

• Proof: For a^* to be a Nash equilibrium, each agent i's first-order condition requires s_i′(F(a^*))[∂F(a^*) / ∂a_i] = c_i′(a_i^*). Matching Pareto efficiency requires s_i′(y^*) = 1 for every agent i ∈ {1, …, n}, which sums to ∑_i=1ⁿ s_i′(y^*) = n ≥ 2, contradicting the differentiated budget-balance identity ∑_i=1ⁿ s_i′(y) = 1.

Theorem 2 (Role of the Principal as a Budget Breaker, 1982): Relaxing budget balance to ∑_i=1ⁿ s_i(y) ≤ y by introducing an external residual claimant or budget breaker (a non-producing owner or corporate principal who absorbs any forfeited output when y < F(a^*)) implements a^* as a Nash equilibrium via the forcing contract s_i(y) = b_i if y ≥ F(a^*) and s_i(y) = 0 if y < F(a^*).

Does Holmström's Informativeness Principle imply that contracts should be as complex as possible?

In a frictionless single-task model where additional signals are costless and cannot be gamed, the Informativeness Principle states that every non-redundant signal correlated with effort or noise should enter the contract. However, in two later papers with Paul Milgrom (1987, 1991), Holmström explained why real-world contracts are often simple and linear: first, when an agent controls a dynamic sequence of actions over time (1987), non-linear bonus thresholds invite intertemporal gaming, making a linear commission contract (w = α + βx) optimal; second, in multi-task environments (1991) where quantity is easily measured but long-term quality is not, high-powered incentives on the measurable metric cause agents to neglect unmeasured quality.

Why do many CEO compensation contracts appear to "pay for luck" despite Relative Performance Evaluation (RPE) theory?

Marianne Bertrand and Sendhil Mullainathan (2001) documented empirically that CEO pay often rises when industry-wide oil prices or exchange rates boost firm profits. Economists have reconciled this observation with Holmström (1979) through two channels: (1) managerial skimming / governance capture when corporate boards are weak, and (2) changing outside options (Oyer, 2004)—when an entire industry booms, the marginal value of executive talent across rival firms rises simultaneously, so indexing away industry booms would violate the manager's retention constraint during expansions.

Why can't a worker-owned partnership agree to burn or throw away output if the team misses its target F(a^*)?

In principle, destroying output when y < F(a^*) would satisfy weak budget balance (∑_i s_i(y) = 0 < y). However, once y < F(a^*) has been realized, destroying valuable output is not renegotiation-proof for a closed partnership: the partners have a unanimous ex-post incentive to renegotiate and divide y among themselves rather than burn it. Introducing a legally distinct third party (a residual-claimant owner) makes the penalty credible because the third party prefers keeping the forfeited output to returning it to the team.

1. Holmström, B. (1979). "Moral Hazard and Observability." The Bell Journal of Economics, 10(1), 74–91. https://doi.org/10.2307/3003320
2. Holmström, B. (1982). "Moral Hazard in Teams." The Bell Journal of Economics, 13(2), 324–340. https://doi.org/10.2307/3003457
3. Holmström, B., & Milgrom, P. (1987). "Aggregation and Linearity in the Provision of Intertemporal Incentives." Econometrica, 55(2), 303–328. https://doi.org/10.2307/1913238
4. Holmström, B., & Milgrom, P. (1991). "Multitask Principal–Agent Analyses: Incentive Contracts, Asset Ownership, and Job Design." Journal of Law, Economics, & Organization, 7(Special Issue), 24–52. https://doi.org/10.1093/jleo/7.special_issue.24
5. Rogerson, W. P. (1985). "The First-Order Approach to Principal–Agent Problems." Econometrica, 53(6), 1357–1367. https://doi.org/10.2307/1913212