Optimal Contracting with Altruistic Agents: Gaynor, Mehta & Richards-Shubik (2023)

What this is. The paper’s core results, the structural model (provider utility and government objective), the method (demand profile approach for supply contracting), and the empirical specifications with equations: enough to know what it found and how, without reading all 42 pages. To replicate or extend it, read the full source at the original.

TL;DR

The paper estimates a structural screening model of health-care provider behavior using 2008-2009 Medicare claims for epoetin alfa (EPO), an expensive drug used to treat anemia in dialysis patients with end-stage renal disease. Dialysis providers are heterogeneous in their degree of altruism toward patients and their marginal costs of administering EPO, both unobservable to Medicare. Using natural variation in patient hematocrit levels and quarterly variation in national Medicare payment rates, the paper recovers the joint distribution of provider types. It then derives optimal nonlinear payment contracts using the demand profile approach of Goldman, Leland, and Sibley (1984) and Wilson (1993), which handles multidimensional heterogeneity tractably. The optimal contracts completely eliminate medically excessive dosages (present for 75-86% of providers under the observed linear contract), reduce Medicare spending by 12-48%, and improve the government’s welfare objective by $87-$220 per patient per month. Aggregate gains are approximately $300 million per year. Like Clemens and Gottlieb (2014), who examine Medicare payment incentive effects broadly, this paper pushes further to derive and characterize the optimal contract for a specific treatment.

Core results

Magnitudes as reported; \*\*/\*\*\* = 5%/1%.

#	Result	Locator	Magnitude
R1	Optimal nonlinear contract eliminates medically excessive dosages for all provider types	Table 5, p. 1560	Share with medically excessive dosages: 82% / 75% / 86% under observed contract -> 0% / 0% / 0% under optimal nonlinear (low / medium / high hematocrit intervals); optimal linear contract reduces but does not eliminate this inefficiency (19% and 45% share remain in medium and high intervals)
R2	Optimal nonlinear contract reduces mean Medicare spending by 12-48%	Table 5, p. 1560	Mean monthly payments per patient: $744 -> $388 (-48%, low hematocrit); $541 -> $392 (-27%, medium); $437 -> $384 (-12%, high)
R3	Government welfare objective improves by $87-$220 per patient per month under optimal nonlinear contract	Table 5, p. 1560	Gains vs observed: $220 (low), $124 (medium), $87 (high) per patient per month; optimal linear achieves 70-85% of these gains but leaves medically excessive dosages in the medium and high intervals
R4	Aggregate gains from better contracting estimated at ~$300 million per year	Section VI, p. 1563	Rough approximation multiplying per-patient gains by patient-months; Medicare spent ~$2 billion per year on EPO during the study period
R5	Providers respond significantly to reimbursement rates: dosage rises by 6,390 units per $1 payment rate increase	Table 2, p. 1552	OLS reduced form: $\beta_2$ = 9.53 (SE 3.11), 6.39 (SE 2.12), 3.92 (SE 1.89) thousand units per $1/1,000u payment rate in low / medium / high hematocrit intervals; SEs clustered on dialysis center, 250 bootstrap replications
R6	Losses from asymmetric information about provider types are $1,739-$3,752 per patient per month	Section VB, p. 1561	Difference between full-information government objective and second-best achievable gains; approximately 8-43 times the gains achievable through optimal contracting ($220/$124/$87 in low/medium/high intervals), indicating costs of asymmetric information dwarf the gains from better contracting alone
R7	Optimal nonlinear contract reduces unjustified dosage variation by 26-52%	Table 5, p. 1560	Std dev of dosage: 9.7 -> 7.2 thousand units (-26%) for medium hematocrit; 5.2 -> 2.5 thousand units (-52%) for high; variation reduction reflects elimination of type-heterogeneity-driven overprovision

Overall (paper’s conclusion). The observed Medicare fee-for-service contract, which pays a constant marginal rate per EPO unit regardless of dosage, cannot be rationalized as optimal for any value of the government’s health weight given the estimated structural parameters. Moving to an optimal nonlinear contract with declining marginal payments would eliminate medically excessive dosages, reduce both mean and variance of treatment amounts, and improve the government’s welfare objective by hundreds of millions of dollars per year. The demand profile approach, applied here for the first time to supply contracting, is the tool used to handle the multidimensional provider heterogeneity (joint unobservability of altruism and marginal cost) that is central to this and many similar health-care payment settings.

Theory / model

The framework is a static screening model where the government (principal) pays a dialysis provider (agent) to treat a patient with end-stage renal disease (Section II, p. 1537). The patient arrives with baseline hematocrit $b$ and observed characteristics $x$; the provider chooses EPO dosage $a$ (total units administered per month). All three are observable by the government because providers report them on Medicare insurance claims.

Provider utility (equation 1, p. 1539): the provider values patient health, weighted by altruism $\alpha$, less the cost $za$ of administering the drug, plus the government’s payment $P(a; b, x)$:

$$u(a;\,\alpha,z,b,x,P) \;\equiv\; \alpha\, h(a;b,x) \;-\; za \;+\; P(a;b,x). \tag{1}$$

The provider utility specification follows the altruistic physician model of Ellis and McGuire (1986), extended here to allow heterogeneity in both altruism and costs. The health production function $h(a; b, x)$ is twice differentiable and strictly concave in $a$; it first increases then decreases in dosage. Dosages with $h'(a; b, x) < 0$ are “medically excessive.” The provider’s type $(\alpha, z)$ is unobserved by the government (asymmetric information): $(\alpha, z)$ has joint density $f(\alpha, z)$ on a compact support $[\underline{\alpha}, \bar{\alpha}] \times [\underline{z}, \bar{z}]$.

Government objective (equation 2, p. 1539): the government maximizes patient health (weighted by its own health preference $\alpha_g$) minus payments to the provider:

$$u_g(a;\,b,x,P) \;\equiv\; \alpha_g\, h(a;b,x) \;-\; P(a;b,x). \tag{2}$$

The government sets a potentially nonlinear payment policy $\{P(a; b, x)\}$ before provider types and patient health states are realized. Given $(b, x)$, the government maximizes the expectation of (2) over the distribution of types and their resulting treatment choices, subject to incentive compatibility (IC) and voluntary participation (VP) for each type $(\alpha, z)$:

$$\max_{P \in \mathcal{P}} \int_{\alpha,z} \Bigl[\alpha_g\,h\bigl(a^*(\alpha,z;b,x,P);\,b,x\bigr) - P\bigl(a^*(\alpha,z;b,x,P);\,b,x\bigr)\Bigr]\,f(\alpha,z)\,d\alpha\,dz,$$

subject to

$$\text{IC:}\quad a^*(\alpha,z;b,x,P) = \arg\max_{a \geq 0}\,u(a;\alpha,z,b,x,P),\quad \forall\,\alpha,z,$$ $$\text{VP:}\quad u\!\bigl(a^*(\alpha,z;b,x,P);\,\alpha,z,b,x,P\bigr) \;\geq\; \underline{u},\quad \forall\,\alpha,z.$$

Full-information first best (equation 3, p. 1540): under full information, the optimal treatment equates the government’s marginal benefit to the agent’s net marginal cost:

$$\alpha_g\,h'(a^{*FI}(\alpha,z)) \;=\; z - \alpha\,h'(a^{*FI}(\alpha,z)). \tag{3}$$

Altruism reduces the agent’s effective marginal cost, so first-best treatment amounts are higher with altruism than without. The full-information allocation never produces medically excessive dosages (where $h' < 0$), because both $\alpha$ and $\alpha_g$ are positive.

Method

The paper uses the demand profile approach of Goldman, Leland, and Sibley (1984) and Wilson (1993) to solve the optimal contracting problem with two-dimensional unobserved heterogeneity (Section IIC, p. 1541). Standard methods based on the revelation principle (Myerson (1981); Maskin and Riley (1984)) require a strict ordering of agent types so that the binding IC constraints reduce to adjacent-type comparisons; under multidimensional heterogeneity such a reduction is generally unavailable. The demand profile approach instead reformulates the government’s problem in terms of setting the marginal payment for each treatment amount, and separates it into independent subproblems.

Provider first-order condition under any differentiable contract $P$ (equation 4, p. 1541): the provider equates the net marginal cost to the marginal payment:

$$\underbrace{z - \alpha\,h'(a^*)}_{\text{nc}(a^*;\alpha,z)} \;=\; \underbrace{\dfrac{\partial P(a^*)}{\partial a}}_{p(a^*)}. \tag{4}$$

The net marginal cost $\text{nc}(a;\alpha,z) = z - \alpha h'(a)$ is upward sloping in $a$ (since $h'' < 0$). If the marginal payment curve is downward sloping, each net marginal cost curve intersects it at most once from below, which is the key regularity condition for the demand profile approach.

Demand profile (equation 6, p. 1542): $S(p, a)$ is the probability (over the type distribution) that the provider supplies at least amount $a$ when the marginal payment at $a$ equals $p$:

$$S(p, a) \;\equiv\; \Pr\!\bigl\{p(a) \;\geq\; z - \alpha\,h'(a)\bigr\}. \tag{6}$$

Decomposed government problem (equations 5 and 7, pp. 1540-1544): because of the regularity condition and quasilinearity of provider preferences, the government’s objective separates into independent maximizations, one for each treatment amount $a \in A$:

$$\max_{p(a)\in\mathbb{R}}\; S(p(a),\,a)\,\bigl[\alpha_g\,h'(a) - p(a)\bigr]. \tag{7}$$

Optimal contract first-order condition (equation 8, p. 1544):

$$\frac{\partial S(p^*(a),a)}{\partial p(a)}\,\bigl[\alpha_g\,h'(a) - p^*(a)\bigr] \;=\; S(p^*(a),\,a). \tag{8}$$

This equates the marginal benefit of raising the marginal payment (the change in the probability of provision times the government’s marginal health valuation) to the marginal cost (the probability that $a$ is already being provided). The optimal total payment $P^*$ is recovered by integrating $p^*(a)$ over dosage. The optimal marginal payment declines toward and past the health-maximizing dosage level, ensuring that no medically excessive dosages arise in the second-best allocation (a standard no-distortion-at-the-top result holds at the highest treatment amount; all others are distorted downward).

Empirical specifications

Estimation proceeds in three steps (Section IV, p. 1547).

Step 1: Reduced-form OLS. The health function is quadratic (equation 9, p. 1547):

$$h(a;b,x) \;=\; H \;-\; \tfrac{1}{2}(\delta\,a + b - \tau'x)^2, \tag{9}$$

where $\delta$ converts EPO units into hematocrit points and $\tau'x$ is a patient-characteristics index. Under a linear contract with constant marginal payment $p_1$, the provider’s first-order condition (4) yields (equation 10, p. 1547):

$$a^*(\alpha,z;b,x,P^L) \;=\; \frac{\tau'x - b}{\delta} + \frac{p_1 - z}{\alpha\,\delta^2}. \tag{10}$$

Decomposing the marginal cost as $z_{ik} = \mu_z + \zeta_{ik}$ and adding an idiosyncratic shock $\eta_{ijt}$, the estimating equation within hematocrit interval $k$ is (equation 11, p. 1549):

$$a_{ijt} \;=\; \underbrace{\!\left[\frac{-1}{\delta_k}\right]\!}_{\beta_1^k}\! b_{jt} \;+\; \underbrace{\!\left[\frac{1}{\alpha_{ik}\delta_k^2}\right]\!}_{\beta_2^k}\![p_{1t} - \mu_z] \;+\; \underbrace{\!\frac{\tau_k'}{\delta_k}\!}_{\beta_3^k}\! x_{jt} \;+\; \underbrace{\!\left[\frac{-\zeta_{ik}}{\alpha_{ik}\delta_k^2}\right]\!}_{\nu_i^k} \;+\; \eta_{ijt}. \tag{11}$$

This is estimated by OLS separately within each of three hematocrit intervals ($b \in (30, 33]$, $(33, 36]$, $(36, 39]$). The regression includes age, sex, CCI indicators, and month and year dummies. Standard errors are clustered on dialysis center (250 bootstrap replications). Identification rests on: (i) natural month-to-month variation in patient hematocrit $b_{jt}$ (not manipulated by providers), which identifies $\beta_1^k$; and (ii) quarterly variation in the national Medicare payment rate $p_{1t}$, set by an administrative formula (106% of average sales price lagged six months) that no individual facility influences, which identifies $\beta_2^k$. The mean marginal cost $\mu_z = \$8.58$ per 1,000 units is set externally from facility cost reports (acquisition cost $7.53 + administration cost $1.05).

Step 2: Structural parameter recovery. Structural parameters $\delta_k$, $\tau_k$, and the joint distribution $F_k(\alpha, z)$ are recovered analytically from the reduced-form moments within each hematocrit interval. The joint distribution of $(\ln\alpha, z)$ is bivariate normal with four unknown parameters per interval. Using Stein’s lemma and properties of the log-normal distribution, these are identified from the first and second moments of the random coefficient $\beta_2^k$ and random effect $\nu^k$ in equation (11), estimated via a semiparametric auxiliary regression of the residuals (Section IVB, p. 1549-1550; Online Appendix F for full details).

Step 3: Optimal contract construction. The government’s health weight $\alpha_g = 52.6$ is calibrated from a statistical life year value and EPO dose-response estimates from clinical trials (Online Appendix G.2). Type distributions are truncated at the 0.5th and 99.5th percentiles to ensure compact support. The demand profile $S(p, a)$ is computed from the estimated distributions; optimal marginal payments $p^*(a)$ are solved numerically from equation (8) for each treatment amount and hematocrit interval and integrated to obtain $P^*$. The paper verifies that the regularity condition (no provider type has a net marginal cost curve with multiple intersections with the optimal marginal payment curve) holds in the estimated model (Online Appendix I).

Datasets used

Dataset	Role in paper	Wiki page
Medicare outpatient claims (CMS 2008-2009b, 20% sample)	Primary estimation data: monthly EPO dosages, baseline hematocrit, patient demographics (age, sex), Charlson Comorbidity Index, payment rates; 919,745 claims after exclusions	No page yet
Renal Dialysis Facilities Cost Report Data (CMS 2008-2009a)	Annual per-facility EPO acquisition costs used to set mean marginal cost $\mu_z$; publicly available from CMS	No page yet
Medicare Part B ASP Drug Pricing Files (CMS 2008, 2009)	National quarterly payment limits for EPO (the source of payment rate variation); public administrative data	No page yet
Medicare Beneficiary Summary File (MedPAR; CMS 2007-2009a)	Patient age and sex, linked to claims	No page yet

Sample: January 2008 to December 2009 (monthly, US). Final sample restricts to patients with hematocrit in 30-39 percent range: 919,745 claims, 74,260 unique patients, 5,148 dialysis providers.

When to read the full paper

Read the original if you are: designing optimal nonlinear payment contracts for provider-administered drugs or any supply-contracting problem with multidimensional unobserved agent heterogeneity; applying the demand profile approach beyond its original monopoly-pricing context; estimating structural models of provider behavior with altruism, including the identification argument and semiparametric moment-recovery details (Section IVB and Online Appendices E, F); or studying the welfare costs of asymmetric information in health-care reimbursement. The locators above point to the exact tables and figures.

Attribution and rights

Source: peer-reviewed, American Economic Review 113(6), June 2023. This distillation was extracted by an LLM on 2026-06-25 and is not human-verified or independently reproduced. The paper is paywalled; only text extracts are reproduced here under extract-only terms.

Gaynor, Martin, Nirav Mehta, and Seth Richards-Shubik. “Optimal Contracting with Altruistic Agents: Medicare Payments for Dialysis Drugs.” American Economic Review 113, no. 6 (June 2023): 1530-1571. DOI: 10.1257/aer.20210208.