Uncertainty, Contracting, and Beliefs in Organizations: Dicks & Fulghieri (2025)

What this is. The paper’s core results, the theoretical model with its key equations, and the method: enough to know what it found and how, without reading all 44 pages. To replicate or extend it, read the full source at https://doi.org/10.1093/rfs/hhaf005.

TL;DR

Dicks and Fulghieri study optimal incentive contracts in a two-division firm where headquarters (HQ) and division managers are uncertainty averse in the sense of Gilboa and Schmeidler (1989): they hold a set of admissible priors and evaluate random variables by their worst-case expected utility. The paper extends the classical moral hazard framework of Holmstrom (1979) and the linearity results of Holmstrom and Milgrom (1987) to settings where agents lack a single prior on the probability distribution of cash flows. Uncertainty creates two novel costs. First, an “incentive effect”: conservative beliefs about own division productivity suppress effort, requiring higher pay-performance sensitivity. Second, an “uncertainty discount”: HQ and division managers disagree on the value of compensation contracts because their positions in the hierarchy give them different exposures to uncertainty, making participation constraints more costly. The key insight is that HQ can partly resolve both costs by designing contracts with cross-divisional exposure. Linking pay to the other division’s output hedges division managers’ uncertainty, improves beliefs, and lowers incentive costs. This motive for cross-pay is present even when divisions are uncorrelated, in contrast to the informativeness principle of Holmstrom (1982). When uncertainty is sufficiently high for both HQ and division managers, pure equity is optimal in the risk-neutral-manager case regardless of cash-flow correlation. With uncertainty-neutral HQ, high uncertainty instead produces relative-performance pay for positively correlated divisions.

Core results

Magnitudes and significance are as reported. Locators point into the source PDF.

#	Result	Locator	Magnitude
R1	Cross-division pay is optimal even absent correlation, violating the informativeness principle; uncertainty aversion motivates uncertainty hedging through equity or relative-performance contracts	Theorem 1, p. 2201; Corollary 1, p. 2203	With uncertainty-neutral HQ and risk-neutral division managers, optimal cross-division exposure is $\lvert\gamma_d\rvert = \xi_d \beta_d > 0$ when division managers face uncertainty ($\eta > 0$); $\beta_d = 1/(1 + 3(1 - \hat{q}^d_d/q_d)) < 1$ and effort decrease in $\eta$ (eq. 20). With risk-averse managers, Corollary 1 gives nonzero cross-pay when uncertainty exceeds its risk- and correlation-dependent threshold.
R2	Pure equity replaces relative-performance pay under sufficiently high uncertainty for both HQ and division managers, regardless of cash-flow correlation	Theorem 3, p. 2207	With risk-neutral division managers, if HQ uncertainty is sufficiently high ($\eta^{HQ} > \max_d \ln\hat{H}_d$) and division-manager uncertainty satisfies $\eta > \eta^{HQ} + 2\ln(3/2)$, then $\beta_d = \gamma_d = 1/[1 + 3(1 - \hat{q}^d_d/\hat{q}^{HQ}_d)] < 1$. By contrast, Corollary 1(ii) with uncertainty-neutral HQ sets $\operatorname{sign}(\gamma)=-\operatorname{sign}(\rho)$.
R3	HQ uncertainty aversion makes relative-performance contracts more costly because division managers take a short position in the other division while HQ remains long, widening belief disagreement	Theorem 3, p. 2207; Section 4.1, pp. 2206-2208	Under the theorem’s uncertainty conditions, $\beta_d = \gamma_d = 1/[1+3(1-\hat{q}^d_d/\hat{q}^{HQ}_d)] < 1$; increasing HQ uncertainty raises pay sensitivity while HQ uncertainty remains below division-manager uncertainty (Figure 5).
R4	Internal hedging (cross-division pay) dominates external benchmarks for uncertainty hedging; with large HQ uncertainty, optimal contracts exclude external hedges entirely	Lemma 5, p. 2213; Theorem 5, p. 2213	HQ weakly prefers contract $(\beta, \lvert\psi\rvert, 0)$ over $(\beta, 0, \lvert\psi\rvert)$ when choosing between internal (division B) and external (variable C) hedges; if $\eta^{HQ} > \tilde{\eta}^{HQ}$, optimal contracts set $\psi = 0$ (no external hedge)
R5	Synergies reinforce the optimality of equity-based pay; for any uncertainty level there exists a synergy threshold above which pure equity is optimal	Theorem 7, p. 2215	For any $\eta \geq 0$ there is a threshold $\bar{\zeta}$ such that for all $\zeta > \bar{\zeta}$, the optimal contract has $\gamma = \beta$; at $\eta = 0$, pure equity is optimal only when $\zeta = 1$ (perfect effort substitutes)
R6	Risk aversion changes the optimal uncertainty hedge, but does not eliminate cross-division pay	Theorem 2, p. 2203; eq. 21	$$\beta_d a_d q_d + r\sigma^2\beta_d^2 =
R7	Cross-pay raises equilibrium effort by improving own-division beliefs	Lemma 3, p. 2199	$a_d = \beta_d\theta_d\hat{q}^d_d$; effort increases in $$\beta_d,
R8	With risk-averse managers, the correlation and HQ uncertainty determine whether cross-pay is relative-performance or equity	Theorem 4, pp. 2209-2210	At low uncertainty and low HQ uncertainty, $\gamma = -(\rho-\bar{\rho})\beta$; with high uncertainty, $\gamma=\beta$ when $\rho\leq0$ or when HQ uncertainty exceeds the stated threshold; for positive correlation and lower HQ uncertainty, $\gamma=-\hat{\xi}(\eta^{HQ})\beta$
R9	Rectangular beliefs remove the uncertainty-hedging response, and sufficiently uncertain HQ chooses no cross-pay	Theorem 6, p. 2215	For $\rho>0$ and $\eta^{HQ}>\eta^{HQ}_1$, $\gamma_d=0$; rectangular-belief worst-case assessments do not vary with contract exposure
R10	The rectangular-belief no-cross-pay branch has a closed-form own-pay sensitivity	Theorem 6, p. 2215; eq. 30	$\beta_d=\frac{1}{1+\left(1-\frac{\hat{q}^d_d}{\hat{q}^{HQ}_d}\right)+\frac{r\sigma^2}{\theta\hat{q}^{HQ}_d\hat{q}^d_d}}$
R11	At low uncertainty, risk-hedging pay remains optimal, while the cross-pay threshold depends on risk aversion and cash-flow correlation	Corollary 1(i), p. 2203	$\gamma=-\rho\beta$ and $\beta<\beta^*$ for $\eta\leq\bar{\eta}(r,\rho)$; $\bar{\eta}(0,\rho)=0$, and the threshold increases in $r$ and decreases in $$

Overall (paper’s conclusion). Uncertainty aversion provides a unifying explanation for three otherwise puzzling compensation practices: the prevalence of equity-based pay for lower-level managers (even when risk-bearing arguments do not support it), the rarity of relative-performance contracts especially in high-uncertainty environments such as young and innovative firms, and the optimism gradient whereby senior managers hold systematically more favorable beliefs about firm prospects than rank-and-file employees.

Theory / model

The paper’s model is a one-period, two-division firm. There are two divisions $d \in \{A, B\}$, each run by a division manager supervised by HQ.

Cash flows and effort. Each division’s cash flow is specified on p. 2189:

$$Y_d = \mu_d + \varepsilon_d, \quad \text{where} \quad \mu_d = a_d q_d$$

Division cash flows $(Y_A, Y_B)$ have a joint normal distribution $N(\mu, \Sigma)$ with homoscedastic variance $\sigma^2$ and correlation $\rho$. Effort $a_d \in \mathbb{R}_+$ affects the mean; the cost is $c_d(a_d) = \tfrac{1}{2\theta_d} a_d^2$, where $\theta_d$ is efficiency of effort. Division managers have CARA utility $U(w) = -e^{-r w}$ with coefficient $r$; HQ is risk neutral (in the base model).

Linear incentive contracts. HQ offers linear contracts (p. 2192). Division manager $d$‘s compensation is:

$$w_d(Y) = s_d + \beta_d Y_d + \gamma_d Y_{d'}$$

where $s_d$ is a fixed base pay, $\beta_d$ is pay-performance sensitivity on own division output, and $\gamma_d$ is cross-division (“cross-pay”) exposure. Setting $\gamma_d > 0$ gives an equity component; $\gamma_d < 0$ gives relative-performance pay.

Uncertainty aversion (MEU, Gilboa and Schmeidler (1989)). Both HQ and division managers are uncertainty averse: they evaluate payoffs by minimizing expected utility over a set of admissible priors $\mathcal{P}$: The minimization is eq. 1, p. 2190.

Miao and Rivera (2016) study robust contracts in continuous time as an alternative uncertainty formulation.

$$\mathcal{U} = \min_{p \in \mathcal{P}} E_p[U(w)] \tag{1}$$

The core beliefs set is defined using the relative entropy (Kullback-Leibler divergence) of candidate distribution $\hat{P}(x)$ relative to reference $P(x)$ (Hansen and Sargent (2001); eq. 3, p. 2190):

$$R\!\left(\hat{P}(x) \,\middle|\, P(x)\right) \equiv \int \hat{p}(x) \ln\!\left(\frac{\hat{p}(x)}{p(x)}\right) dx \tag{3}$$

The admissible set is $\mathcal{P}(P) \equiv \{\hat{P}: R(\hat{P}(x)|P(x)) \leq \eta^P\}$, where $\eta^P$ is the uncertainty parameter. A higher $\eta^P$ means greater uncertainty aversion and a larger set of admissible beliefs.

Parametric approximation. For tractability (following Dicks and Fulghieri (2019, 2021)), the core beliefs set for agent $i \in \{HQ, A, B\}$ is approximated as in eq. 11, p. 2195:

$$K^i(q) \equiv \left\{\hat{q}^i \,\middle|\, D(\chi^i_A) + D(\chi^i_B) \leq \eta^i\right\}, \quad \chi^i_d = \frac{\hat{q}^i_d - q_d}{q_d}, \quad D(\chi) = -\ln(1-\chi) \tag{11}$$

where $\hat{q}^i_d$ is agent $i$‘s belief about division $d$‘s productivity and $q_d$ is the reference productivity. This set is strictly convex with smooth boundaries, guaranteeing that beliefs respond to changes in compensation contracts.

Division manager utility. Given beliefs $\hat{q}^d$ and action $a$, division manager $d$‘s certainty-equivalent utility is given in eq. 5, p. 2192:

$$u_d(\hat{q}^d, a) \equiv E\!\left[w_d|\hat{q}^d, a\right] - \frac{r}{2} Var(w_d) - c_d(a_d) \tag{5}$$

where $Var(w_d) = \sigma^2(\beta_d^2 + 2\rho\beta_d\gamma_d + \gamma_d^2)$. The key feature is that the expected wage $E[w_d|\hat{q}^d, a]$ depends on division managers’ beliefs about productivity of both divisions (through own-pay $\beta_d$ and cross-pay $\gamma_d$), while $Var(w_d)$ does not.

The uncertainty-hedging axiom implies the following inequality for acts $y_1,y_2$ and mixture weight $\alpha$ (eq. 2, p. 2190):

$$\alpha\min_{p\in\mathcal{P}}E_p[U(y_1)] +(1-\alpha)\min_{p\in\mathcal{P}}E_p[U(y_2)] \leq \min_{p\in\mathcal{P}}\{\alpha E_p[U(y_1)]+(1-\alpha)E_p[U(y_2)]\} \tag{2}$$

The relative-entropy ball defining admissible priors is eq. 4, p. 2190:

$$\mathcal{P}(P(x)) = \{\hat{P}:R(\hat{P}(x)|P(x))\leq\eta^P\} \tag{4}$$

The contract’s fixed component makes the participation constraint bind (eq. 9, p. 2194):

$$s_d=c_d(a_d)+\frac{r}{2}\operatorname{Var}(w_d)-E[w_d|\hat{q}^d,a] \tag{9}$$

For symmetric divisions the paper assumes condition (S), eq. 12, p. 2195:

$$\theta_A=\theta_B\equiv\theta,\quad q_A=q_B\equiv q,\quad \eta_A=\eta_B\equiv\eta \tag{12}$$

In the no-uncertainty benchmark, risk-neutral HQ’s optimal own-output sensitivity and cross-pay are given by eq. 13, p. 2195:

$$\beta_d^*=\frac{1}{1+\frac{r\sigma^2(1-\rho^2)}{\theta_d q_d^2}},\quad \gamma_d^*=-\rho\beta_d^*,\quad a_d^*=\beta_d^*\theta_dq_d \tag{13}$$

Method

The paper derives analytical solutions to a minimax contracting problem. HQ maximizes expected profits subject to division managers’ incentive constraints (IC) and participation constraints (PC), while both HQ and managers minimize over their worst-case beliefs. The problem and constraints are eqs. 6-8, p. 2193:

$$\max_{\{w_d, a_d\}} \min_{\hat{q}^{HQ} \in K^{HQ}} \pi(\hat{q}^{HQ}) \equiv \sum_{d \in \{A,B\}} E\!\left[Y_d(a_d) - w_d|\hat{q}^{HQ}\right] \tag{6}$$

subject to the division managers’ IC constraints:

$$\max_{a_d} \min_{\hat{q}^d \in K^d} u_d(\hat{q}^d, a) \equiv E\!\left[w_d|\hat{q}^d, a_d, a_{d'}\right] - \frac{r}{2}Var(w_d) - c_d(a_d) \tag{7}$$

and PC constraints:

$$\min_{\hat{q}^d \in K^d} u_d(\hat{q}^d, a_d, a_{d'}) \geq u_0 = 0 \tag{8}$$

The solution strategy is three-step: (1) characterize how contracts determine beliefs via Lemma 2; (2) derive equilibrium effort from beliefs via Lemma 3; (3) characterize optimal contracts by substituting the binding PC into the objective, yielding the reduced-form HQ payoff (eq. 10, p. 2194):

$$\pi = \sum_{d \in \{A,B\}} \left\{E(Y_d(a_d)|\hat{q}^{HQ}_d) - \frac{r}{2}Var(w_d) - c_d(a_d) - \left(E[w_d|\hat{q}^d,a] - E[w_d|\hat{q}^{HQ}_d,a]\right)\right\} \tag{10}$$

The fourth term is the “uncertainty discount” arising from belief disagreement; it is novel and central to the paper’s results.

Key analytical result (Theorem 1, eq. 20, p. 2201). With uncertainty-neutral HQ and uncertainty-averse risk-neutral division managers, optimal contracts set $H_d = 1$ (uncertainty-hedging ratio equal to one), where $H_d \equiv |\gamma_d| a_{d'} q_{d'} / (\beta_d a_d q_d)$. Optimal pay-performance sensitivity is:

$$\beta_d = \frac{1}{1 + 3\!\left(1 - \hat{q}^d_d/q_d\right)} < 1, \quad |\gamma_d| = \xi_d \beta_d \tag{20}$$

with $\xi_d \equiv \frac{a_{d'} q_{d'}}{a_d q_d}$. Both $\beta_d$ and effort $a_d$ are decreasing in uncertainty $\eta$. Under symmetry, pure equity is optimal: $\beta = \gamma < 1$.

Theorem 2 (risk-averse division managers) shows that the optimal contract must satisfy (eq. 21, p. 2203):

$$\beta_d a_d q_d + r\sigma^2 \beta_d^2 = |\gamma_d| a_{d'} q_{d'} + r\sigma^2 \gamma_d^2 \tag{21}$$

equating the total expected cost to HQ of a division manager’s exposure to each division, regardless of the correlation $\rho$. Cross-pay is always non-zero, $\gamma_d \neq 0$, even when divisions are uncorrelated.

Theorem 3 (uncertainty-averse HQ) yields pure equity at sufficiently high uncertainty:

$$\beta_d = \gamma_d = \frac{1}{1 + 3(1 - \hat{q}^d_d/\hat{q}^{HQ}_d)} < 1 \tag{25}$$

Relative-performance pay creates a short position for HQ in the other division, amplifying the beliefs disagreement between HQ (long position) and division managers (short position), raising the uncertainty discount and making equity strictly preferred.

The proofs use the envelope theorem applied to the minimax problem, with closed-form first-order conditions derived under the parametric beliefs approximation (eq. 11). The proofs of Theorems 1 and 2 appear in the appendix (pp. 2218-2221); Theorems 3, 4, 5, and 7 proofs are in the supplemental materials.

Equation coverage in the main text

The remaining numbered equations specify the uncertainty-neutral principal’s problem and the belief and effort subproblems. Equations 14-16, p. 2197, specialize HQ’s objective and the managers’ incentive and participation constraints:

$$\max_{\{w_d,a_d\}}\pi=\sum_{d\in\{A,B\}}E[Y_d(a_d)-w_d(Y)|q_d] \tag{14}$$ $$\max_{a_d}\min_{\hat{q}^d\in K^d}u_d=E[w_d|\hat{q}^d,a_d,a_{d'}]-\frac{r}{2}\operatorname{Var}(w_d)-c_d(a_d) \tag{15}$$ $$\min_{\hat{q}^d\in K^d}u_d=E[w_d|\hat{q}^d,a_d,a_{d'}]-\frac{r}{2}\operatorname{Var}(w_d)-c_d(a_d)\geq0 \tag{16}$$

For a given contract, managers’ worst-case beliefs solve eq. 17, p. 2198, subject to the relative-entropy constraint in eq. 18, p. 2198:

$$\min_{\hat{q}^d}u_d(\hat{q}^d)=E[w_d|\hat{q}^d,a]-\frac{r}{2}\operatorname{Var}(w_d)-c_d(a_d) \tag{17}$$ $$\ln\!\left(\frac{1}{1-(\hat{q}^d_A-q_A)/q_A}\right)+\ln\!\left(\frac{1}{1-(\hat{q}^d_B-q_B)/q_B}\right)\leq\eta_d \tag{18}$$

Effort is chosen after beliefs respond to the offered contracts (eq. 19, p. 2199):

$$\max_{a_d}u_d(a,\hat{q}^d(a,w))=E[w_d|\hat{q}^d,a_d,a_{d'}]-\frac{r}{2}\operatorname{Var}(w_d)-c_d(a_d) \tag{19}$$

When HQ is uncertainty averse, its belief minimization is given in eq. 22, p. 2204, with the core set in eq. 23, pp. 2204-2205:

$$\min_{\hat{q}^{HQ}\in K^{HQ}}\pi(\hat{q}^{HQ})=\sum_{d\in\{A,B\}}E[Y_d(a_d)-w_d(Y)|\hat{q}^{HQ}] \tag{22}$$ $$K^{HQ}=\left\{\hat{q}^{HQ}:D\!\left(\frac{\hat{q}^{HQ}_A-q_A}{q_A}\right)+D\!\left(\frac{\hat{q}^{HQ}_B-q_B}{q_B}\right)\leq\eta^{HQ}\right\},\quad D(x)=-\ln(1-x) \tag{23}$$

HQ’s relative exposure to the divisions is summarized by the hedging ratio in eq. 24, p. 2205:

$$H_d^{HQ}=\frac{(1-\beta_{d'}-\gamma_d)a_{d'}q_{d'}}{(1-\beta_d-\gamma_{d'})a_dq_d} \tag{24}$$

For the external-hedge extension, the three-variable core set and HQ and manager payoffs appear in eqs. 26-28, p. 2213:

$$K^i=\left\{\hat{q}^i:\sum_{j\in\{A,B,C\}}\ln\!\left(\frac{1}{1-(\hat{q}^i_j-q_j)/q_j}\right)\leq\eta_i\right\},\quad i\in\{A,HQ\} \tag{26}$$ $$\min_{\hat{q}^{HQ}\in K^{HQ}}\tilde{\pi}=(1-\beta)a_A\hat{q}^{HQ}_A+(1-\gamma)\mu\hat{q}^{HQ}_B-\psi\mu\hat{q}^{HQ}_C-s \tag{27}$$ $$\min_{\hat{q}^A\in K^A}\tilde{u}=s+\beta a_A\hat{q}^A_A+\gamma\mu\hat{q}^A_B+\psi\mu\hat{q}^A_C-\frac{r\sigma^2}{2}(\beta^2+\gamma^2+\psi^2)-\frac{a_A^2}{2\theta_A} \tag{28}$$

The rectangular-beliefs alternative in eq. 29, p. 2214, is a product of bounded intervals. Its high-HQ-uncertainty solution is eq. 30, p. 2215:

$$K^i(q^i)=\{\hat{q}^i:[q_A-\eta_i\leq\hat{q}^i_A\leq q_A+\eta_i]\times[q_B-\eta_i\leq\hat{q}^i_B\leq q_B+\eta_i]\} \tag{29}$$ $$\beta_d=\frac{1}{1+\left(1-\frac{\hat{q}^d_d}{\hat{q}^{HQ}_d}\right)+\frac{r\sigma^2}{\theta_d\hat{q}^{HQ}_d\hat{q}^d_d}} \tag{30}$$

Empirical specifications

The paper has no empirical sample or estimated regression. Its results are analytical propositions and comparative statics from the contracting model; Figures 1-7 illustrate parameterized model solutions rather than empirical estimates. Section 6 states testable implications about firm uncertainty and compensation choice but does not estimate them.

Datasets used

This paper is purely theoretical. It introduces no dataset.

Dataset	Role in paper	Wiki page
No empirical data used	Theory paper with numerical illustrations only	N/A

The empirical illustrations use baseline parameter values $q_A = q_B = 10$, $\theta_A = \theta_B = 2$, $\sigma = 10$, $r = 1$ (stated in Section 2 footnotes, p. 2195 and Figures 1-7).

When to read the full paper

Use the original if you are: designing incentive contracts under Knightian uncertainty or ambiguity aversion; trying to explain equity-based compensation for division managers or rank-and-file employees; studying why relative-performance pay is rare in practice despite its theoretical benefits; or extending the model to multitasking, labor-market equilibrium, or organization design with uncertainty. Theorem 1 (p. 2201) and Corollary 1 (p. 2203) are the key analytical results; Figure 3 and Figure 6 (pp. 2204, 2210) illustrate optimal contracts under uncorrelated and correlated cash flows respectively.

Attribution and rights

Source: peer-reviewed, The Review of Financial Studies 38(7), 2025. This distillation was initially extracted on 2026-06-06 and augmented and machine-verified against the PDF on 2026-10-04; it is not independently reproduced. The paper is published under Oxford University Press standard reuse rights (paywalled). Extract-only.

Dicks, David L., and Paolo Fulghieri. “Uncertainty, Contracting, and Beliefs in Organizations.” The Review of Financial Studies 38, no. 7 (2025): 2182-2225. DOI: 10.1093/rfs/hhaf005.