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Roy's identity

Roy's identity is a fundamental theorem in microeconomic theory, named after the French economist René Roy, who derived it in his 1947 paper analyzing the distribution of income across goods, though a similar result was first obtained by Antonelli in 1886.^[1]^[2] It establishes a direct relationship between the indirect utility function v(\mathbf{p}, m)—which represents the maximum utility a consumer can achieve given prices \mathbf{p} and income m—and the Marshallian (uncompensated) demand functions x_i(\mathbf{p}, m) for each good i.^[3] Specifically, under standard assumptions of differentiability and a non-zero marginal utility of income, Roy's identity states that

x_i(\mathbf{p}, m) = -\frac{\partial v(\mathbf{p}, m)/\partial p_i}{\partial v(\mathbf{p}, m)/\partial m},

allowing economists to recover demand functions directly from the indirect utility function.^[3] This identity emerges from the envelope theorem applied to the consumer's utility maximization problem subject to the budget constraint, where the partial derivative of the indirect utility with respect to a price reflects the utility loss from that price change, scaled by the demand quantity and the shadow price of the budget (the marginal utility of income).^[3] Roy's identity is a cornerstone of duality theory in consumer choice, enabling the consistent derivation of structural relationships between direct and indirect representations of preferences, and it ensures that the indirect utility function satisfies key properties like homogeneity and monotonicity when demands do. Its derivation and properties are rigorously detailed in standard microeconomic texts, confirming its role in bridging expenditure minimization and utility maximization frameworks.^[3] Beyond theoretical foundations, Roy's identity has broad applications in empirical economics and policy analysis. It facilitates the estimation of demand systems from observable data on expenditures and prices, often simplifying econometric models by parameterizing the indirect utility function rather than demands directly. In welfare economics and public finance, it underpins derivations of optimal commodity taxation rules, such as the Ramsey rule, which prescribes tax rates inversely proportional to the elasticity of demand to minimize deadweight loss for a given revenue target.^[4] These applications extend to evaluating consumer surplus changes from price reforms and analyzing firm behavior under profit maximization, highlighting Roy's identity's enduring influence in modern economic research.

Introduction

Definition and Intuition

Roy's identity is a fundamental result in microeconomic theory that establishes a direct relationship between the indirect utility function and the Marshallian demand functions derived from utility maximization. Specifically, it states that the demand for good i, denoted x_i(p, I), can be obtained from the indirect utility function v(p, I) as

x_i(p, I) = -\frac{\partial v / \partial p_i}{\partial v / \partial I},

where p represents the vector of prices and I is the consumer's income.^[1] This identity allows economists to recover observable demand behavior from the underlying utility representation, facilitating analysis in both theoretical and empirical contexts. The economic intuition behind Roy's identity lies in the envelope theorem's implications for optimization problems. The partial derivative \partial v / \partial p_i captures the marginal disutility from an increase in the price of good i, reflecting substitution and income effects on the consumer's welfare, while \partial v / \partial I measures the marginal utility of income, which scales the overall responsiveness to changes in resources. By taking their ratio (with a negative sign to account for the cost increase), the identity reveals how demand balances price-induced trade-offs against income effects, providing a concise way to link maximal utility attainment to quantity choices without solving the full optimization explicitly. This perspective underscores the duality between direct and indirect approaches to consumer behavior, where the indirect utility encodes the solution to the expenditure-constrained maximization problem. Beyond consumer theory, Roy's identity has a dual application in producer theory, where analogous relationships derive factor demands and output supplies from the indirect profit function under cost minimization or profit maximization. On the producer side, it similarly connects marginal changes in profits with respect to input prices or output prices to optimal input demands, mirroring the consumer case but framed in terms of revenue and costs.

Historical Background

Roy's identity derives its name from the French economist René Roy (1894–1977), who independently derived and empirically applied the identity in his 1947 paper analyzing the allocation of personal income across various goods using empirical household budget data from Paris, though the core mathematical relationship was first derived by Antonello Antonelli in 1886.^[1]^[2] In this work, Roy examined statistical patterns of consumer expenditure to model demand behavior, providing an early empirical foundation for linking income distribution to consumption choices.^[5] Roy's analysis built upon earlier theoretical frameworks of consumer choice in the economic context of post-World War II France, where the country was grappling with reconstruction efforts, and his study contributed to understanding how limited resources influenced household spending patterns amid scarcity and policy reforms. Although Roy's contribution appeared before the widespread formalization of indirect utility functions in economic theory, it was retroactively identified as a crucial bridge in duality relationships during the 1970s revival of microeconomic duality theory.^[5] Economists like Daniel McFadden and W. Erwin Diewert highlighted its significance in their advancements, integrating it into rigorous frameworks for deriving demand systems from utility maximization problems.^[6] This recognition elevated Roy's identity from an empirical insight to a cornerstone of modern consumer and producer theory.

Theoretical Foundations

Prerequisite Concepts

In microeconomic theory, the foundation for analyzing consumer behavior begins with the utility maximization problem, where a consumer seeks to maximize their utility function u(x) subject to the budget constraint p \cdot x \leq I. Here, x \in \mathbb{R}^n_+ represents the vector of quantities of n goods consumed, p \in \mathbb{R}^n_+ is the vector of prices, and I > 0 denotes the consumer's income. Assuming the utility function u is continuous, strictly increasing, and quasi-concave, the solution to this problem yields the Marshallian demand functions x(p, I), which describe the optimal consumption bundle as a function of prices and income.^[7] The indirect utility function captures the maximum attainable utility from this optimization: v(p, I) = \max_{x} u(x) subject to p \cdot x \leq I. This function is increasing in income I, non-increasing in each price p_i, and homogeneous of degree zero in (p, I), meaning v(\lambda p, \lambda I) = v(p, I) for \lambda > 0. These properties arise from the underlying preferences and budget constraints, ensuring that scaling prices and income proportionally does not alter the achievable utility level.^[8]^[7] Dually, the expenditure minimization problem seeks the minimum cost to achieve a target utility level: e(p, u) = \min_{x} p \cdot x subject to u(x) \geq u, where u is the desired utility. The expenditure function e(p, u) is concave and homogeneous of degree one in prices p, non-decreasing in u, and non-decreasing in each p_i. Its partial derivative with respect to price p_i gives the Hicksian (compensated) demand h_i(p, u) = \frac{\partial e(p, u)}{\partial p_i}, a result known as Shephard's lemma, which holds under differentiability assumptions.^[9]^[7] The indirect utility and expenditure functions form conjugates through duality theory, satisfying the involution relations e(p, v(p, I)) = I and v(p, e(p, u)) = u. This duality links the primal utility maximization and dual expenditure minimization problems, allowing recovery of one function from the other and facilitating analysis of demand behavior across price and income changes.^[7]

Relation to Shephard's Lemma

Shephard's lemma provides a foundational result in duality theory, stating that the Hicksian demand for good i, denoted h_i(p, u), equals the partial derivative of the expenditure function e(p, u) with respect to the price p_i:

h_i(p, u) = \frac{\partial e(p, u)}{\partial p_i}.

This lemma links compensated demands directly to the gradients of the expenditure function, which minimizes the cost of achieving a fixed utility level u at prices p.^[10] Roy's identity serves as the dual counterpart to Shephard's lemma in consumer theory, deriving the Marshallian (uncompensated) demand x_i(p, I) from the indirect utility function v(p, I), which maximizes utility subject to a budget constraint with income I. While Shephard's lemma recovers Hicksian demands from the expenditure function, Roy's identity recovers Marshallian demands from the indirect utility function; together, these identities facilitate the complete recovery of demand systems through duality theory, allowing transitions between primal and dual representations of preferences.^[1]^[11] The mathematical connection between the two identities arises from the duality relation I = e(p, v(p, I)), which equates income to the expenditure required to achieve the maximized utility level. This equality bridges uncompensated and compensated demands, enabling the Slutsky equation to decompose price effects into substitution (Hicksian) and income components, with Shephard's lemma providing the substitution term.^[8] In producer theory, Shephard's lemma extends analogously to cost-minimizing input demands z(p, y) = \nabla_p c(p, y), where c(p, y) is the cost function for output y at input prices p. Roy's identity finds a parallel in the profit-maximizing output supplies derived from the indirect profit function, maintaining symmetry between cost minimization and profit maximization under duality.^[12] A key distinction lies in the role of income effects: Roy's identity incorporates them through the term \partial v / \partial I in its formulation, reflecting the budget constraint's influence on uncompensated demands, whereas Shephard's lemma, focused on compensated demands, excludes such effects entirely.^[3]

Mathematical Derivation

Standard Proof

Roy's identity establishes a relationship between the Marshallian demand function and the indirect utility function, derived directly from the first-order conditions of the consumer's utility maximization problem. Consider a consumer with a continuously differentiable, strictly quasi-concave utility function u(x) defined over the consumption bundle x \in \mathbb{R}^n_+, who maximizes utility subject to the budget constraint p \cdot x = I, where p > 0 is the price vector and I > 0 is income. The indirect utility function is defined as v(p, I) = u(x^*(p, I)), where x^*(p, I) is the optimal demand solving the maximization problem. To derive the identity, form the Lagrangian for the optimization: \mathcal{L}(x, \lambda) = u(x) + \lambda (I - p \cdot x), with first-order conditions (FOCs) given by \frac{\partial u}{\partial x_j}(x^*) = \lambda p_j for each good j = 1, \dots, n, and the budget constraint p \cdot x^* = I, where \lambda(p, I) > 0 is the Lagrange multiplier representing the marginal utility of income.^[13] Differentiate the indirect utility function with respect to income I:

\frac{\partial v}{\partial I} = \sum_{i=1}^n \frac{\partial u}{\partial x_i} \frac{\partial x_i^*}{\partial I}.

Using the FOCs \frac{\partial u}{\partial x_i} = \lambda p_i, this simplifies to \lambda \sum_{i=1}^n p_i \frac{\partial x_i^*}{\partial I}. Differentiating the budget constraint with respect to I (holding prices fixed) gives \sum_{i=1}^n p_i \frac{\partial x_i^*}{\partial I} = 1, so

\frac{\partial v}{\partial I} = \lambda(p, I).

^[13] Now differentiate v(p, I) with respect to the price of good i, p_i:

\frac{\partial v}{\partial p_i} = \sum_{j=1}^n \frac{\partial u}{\partial x_j} \frac{\partial x_j^*}{\partial p_i}.

Substituting the FOCs yields \sum_{j=1}^n \lambda p_j \frac{\partial x_j^*}{\partial p_i} = \lambda \sum_{j=1}^n p_j \frac{\partial x_j^*}{\partial p_i}. Differentiating the budget constraint with respect to p_i (holding income fixed) gives \sum_{j=1}^n p_j \frac{\partial x_j^*}{\partial p_i} + x_i^* = 0, so \sum_{j=1}^n p_j \frac{\partial x_j^*}{\partial p_i} = -x_i^*. Thus,

\frac{\partial v}{\partial p_i} = \lambda (-x_i^*) = -\lambda(p, I) x_i^*(p, I).

^[13] Combining these results, the Marshallian demand for good i is

x_i^*(p, I) = -\frac{\partial v / \partial p_i}{\partial v / \partial I},

since \lambda cancels in the ratio. This holds under the assumptions of continuous differentiability of u and v, strict quasi-concavity of u ensuring a unique interior solution, and positive prices and income.

Alternative Proof via Envelope Theorem

The envelope theorem provides a streamlined approach to deriving Roy's identity by focusing on the direct effects of parameter changes in optimization problems, without needing to compute indirect effects through endogenous variables. Consider the indirect utility function v(p, I) = \max_x u(x) subject to the budget constraint p \cdot x \leq I, where u(x) is the utility function, p is the price vector, I is income, and x is the consumption bundle. The Lagrangian for this problem is \mathcal{L}(x, \lambda; p, I) = u(x) + \lambda (I - p \cdot x). By the envelope theorem, the partial derivative of the value function with respect to a price p_i evaluated at the optimum x^* is \frac{\partial v}{\partial p_i} = \frac{\partial \mathcal{L}}{\partial p_i} \big|_{x^*, \lambda^*} = -x_i^* \lambda^*, where \lambda^* is the optimal Lagrange multiplier representing the marginal utility of income.^[14] Applying the envelope theorem further to the income parameter yields \frac{\partial v}{\partial I} = \lambda^*, the marginal utility of income at the optimum. Substituting this into the previous expression gives \frac{\partial v}{\partial p_i} = -x_i^* \frac{\partial v}{\partial I}, or rearranging, x_i^*(p, I) = -\frac{\partial v / \partial p_i}{\partial v / \partial I}, which is precisely Roy's identity linking Marshallian demands to gradients of the indirect utility function. This derivation bypasses the full chain rule differentiation of the first-order conditions required in the standard proof, offering a more efficient path. The primary advantage of this envelope theorem-based proof is that it avoids explicitly solving for the comparative statics dx/dp from the system of first-order conditions, making it particularly valuable for broader comparative statics analyses in parameterized optimization problems beyond consumer theory.^[14] This approach extends analogously to the producer side via Hotelling's lemma: for the indirect profit function \pi(p_w, p_o), input demands satisfy x_j^* = -\frac{\partial \pi}{\partial p_{w j}} and output supplies y_i^* = \frac{\partial \pi}{\partial p_{o i}}.^[14] The assumptions underlying this proof mirror those of the standard derivation—continuously differentiable utility or profit functions, strict quasi-concavity for unique optima, and linear budget or production constraints—but additionally require interior solutions to ensure the envelope conditions hold without boundary complications.^[14]

Applications and Extensions

In Consumer Choice

Roy's identity plays a central role in the estimation of consumer demand systems by enabling the derivation of Marshallian demand functions from an estimated indirect utility function, which simplifies empirical implementation and allows for tests of theoretical consistency such as homogeneity, symmetry, and adding-up restrictions derived from utility maximization.^[15] In flexible functional forms like the translog model, applying Roy's identity to a quadratic logarithmic specification of the indirect utility yields a system of budget share equations that capture nonlinear Engel curves and cross-price effects.^[15] Similarly, the Almost Ideal Demand System (AIDS), developed by Deaton and Muellbauer, employs Roy's identity to generate tractable share equations from a price-index-based indirect utility approximation, ensuring exact aggregation across consumers while maintaining flexibility. This methodology has been extensively used in industrial organization and public finance since the 1980s to analyze household expenditure patterns and evaluate market structures. Beyond demand estimation, Roy's identity facilitates precise welfare analysis by linking price changes to utility impacts through the Marshallian demands it generates. Specifically, the Marshallian consumer surplus (CS) for a change in the price of good i can be computed as

CS = \int_{p_i^0}^{p_i^1} -\frac{\partial v(p, I)/\partial p_i}{\partial v(p, I)/\partial I} \, dp_i,

where the integrand represents the demand for good i. This integral provides a money metric of welfare loss or gain along the Marshallian path, approximating exact measures like compensating variation in multi-good settings, though it is path-dependent unless preferences are quasilinear or homothetic.^[16]^[17] In policy applications, such as assessing the effects of excise taxes, Roy's identity supports simulations by differentiating an estimated indirect utility function to predict demand responses and aggregate welfare costs across affected households.^[18] For instance, in evaluating a value-added tax increase, researchers can trace shifts in consumption patterns and compute distributional impacts using AIDS-based estimates. Despite its utility, Roy's identity presupposes a continuously differentiable and invertible indirect utility function, with the marginal utility of income strictly positive; violations occur when goods are perfect substitutes, leading to non-differentiable kinks in the utility function and potential singularities in the derived demands.^[16]

In Firm Behavior

In producer theory, the indirect profit function \pi(p, w) serves as the analog to the indirect utility function in consumer theory, where p denotes the vector of output prices and w the vector of input prices. This function represents the maximized profit for a firm given prices, derived from duality principles that link the production technology to observable behavioral responses.^[19] The producer version of Roy's identity, often referred to as Hotelling's lemma, allows recovery of firm behavior from the profit function. Specifically, the input demands are given by x_j(p, w) = -\frac{\partial \pi}{\partial w_j}, while the output supply follows y_i(p, w) = \frac{\partial \pi}{\partial p_i}. These expressions stem from the envelope theorem applied to the profit maximization problem, enabling the derivation of factor demands and supplies directly from estimated profit functions without explicit specification of the underlying production function. For fixed output levels, conditional input demands can be recovered via Shephard's lemma from the cost function c(w, y), where x_j(w, y) = \frac{\partial c}{\partial w_j}.^[14] This framework finds prominent applications in estimating production frontiers, where input demands are recovered from parametric profit function estimates to infer technological parameters and efficiency. In agricultural economics, it has been widely used to analyze factor demands, such as labor and fertilizer inputs in crop production, by fitting flexible profit specifications to panel data on farm outputs and prices. For instance, studies on rice farming in developing economies estimate elasticities of input demands and output supplies, revealing how price changes affect resource allocation under varying market conditions.^[20] A representative example arises in duality models employing the generalized Leontief functional form for the profit function, where the identity connects observed input quantities to gradients of the cost or profit surfaces, facilitating tests of production hypotheses like returns to scale. This approach, pioneered in flexible functional forms, ensures global regularity properties while linking empirical input observations to theoretical cost minimization.^[19] Fundamentally, the identity bridges producer duality to Shephard's lemma, which governs conditional input demands under fixed output levels via derivatives of the cost function; in the restricted profit setting, the partial with respect to input prices yields the negative of these demands, reinforcing the duality between profit and cost representations of firm optimization.^[14]

Generalizations

Extensions of Roy's identity have been developed for dynamic settings, particularly in life-cycle consumption models from the 1980s onward, where time-separable utility functions lead to an intertemporal analog. In these frameworks, the identity relates period-specific demands to derivatives of the value function with respect to prices and assets. Specifically, for good i in period t, the demand is given by

x_{it} = -\frac{\partial V_t / \partial p_{it}}{\partial V_t / \partial A_t},

where V_t denotes the value function and A_t represents assets carried into period t. This form, derived using the envelope theorem, facilitates analysis of intertemporal allocation under borrowing constraints and uncertainty. In stochastic environments, such as random utility maximization (RUM) models, Roy's identity generalizes to recover choice probabilities as expected demands. The Williams-Daly-Zachary (WDZ) theorem provides the discrete-choice counterpart, stating that the probability of choosing alternative a is the partial derivative of the social surplus function S(u) with respect to the deterministic utility u_a: P(a|u) = \partial S(u) / \partial u_a. For the multinomial logit model, this yields P(a|u) = \exp(u_a / \sigma) / \sum_{j} \exp(u_j / \sigma), where \sigma is the scale parameter, linking observed choices directly to underlying utilities.^[21] Non-homothetic generalizations extend Roy's identity to preferences where income effects vary nonlinearly across goods, often using quasi-concave utility functions like constant differences of elasticity of substitution (CDES). These allow derivation of Marshallian demands that capture shifting expenditure shares, such as higher-income households allocating more to luxuries. For instance, applying Roy's identity to a non-homothetic indirect utility yields demand shares s_j = a_j + \sum_k \gamma_{jk} \log(p_k) + \beta_j \log(x_h / a(p)), where \beta_j > 0 for luxuries reflects income elasticity greater than one. Such forms are applied in heterogeneous agent models to analyze distributional effects in multisector growth dynamics.^[22] Despite these advances, Roy's identity has limitations in settings with quantity rationing or non-convex budget sets, where standard assumptions of unconstrained optimization fail. Under binding constraints, such as nonnegativity on consumption, the identity does not hold directly, necessitating alternatives like virtual prices that adjust the budget to mimic unconstrained equilibria. For example, virtual prices convert rationed demands into observable equivalents via dual approaches, enabling consistent estimation in microeconometric systems. Similarly, non-convexities from fixed hours or taxes require evaluating direct utility at kinked points rather than relying on differentiable indirect utility.^[23]^[24] Recent developments since the 2010s integrate Roy's identity with nonparametric methods for flexible estimation, particularly in heterogeneous agent contexts with additively separable unobservables. These approaches identify demands by applying the identity to expected indirect utilities, accommodating multiple maximizers without strong functional form assumptions. While machine learning techniques, such as deep neural networks for nonparametric regression, show promise in estimating underlying utilities for Roy's application, empirical implementations remain emerging.^[25]

References

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La distribution du revenu entre les divers biens - jstor
Par RENE RoY. PARMI les difficultes auxquelles se heurtent les chercheurs en ... Editeurs, 6 rue de la Sorbonne, Paris. 205. Page 2. RENE ROY. Statics and ...
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[PDF] Consumer Theory and the Envelope Theorem
The reader is invited to check Roy's Identity for themselves in the case of x. 2 Expenditure Minimization Problem. The consumer problem can be approached in ...<|control11|><|separator|>
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[PDF] I. Static Optimal Taxation
By Roy's Identity,. ∂Vi. ∂qk. = −αix i k where αi is the marginal utility ... This formula is known as the Ramsey Rule because it was first derived by ...
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[PDF] On Envelope Theorems in Economics and Mathematics - CORE
Rene Roy's identity was produced in Roy (1947) and the proof of the result is in line with. Auspitz and Lieben in that he uses the first order conditions of ...
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[PDF] The Paretian Heritage∗ - Department of Economics
The question naturally arose in Pareto's mind as to whether (a) the assumption that a total utility function exists which the consumer maximizes, or acts as if ...
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McFadden (1966) was the original author who appreciated the usefulness of duality theory in deriving the systems of input demand functions in the context of ...
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Free delivery 25-day returnsThe Expenditure Minimization Problem F. Duality: A Mathematical Introduction G. Relationships between Demand, Indirect Utility, and Expenditure Functions H ...
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The value function of (CP) is called the indirect utility function. Definition. The indirect utility function v : Rn × R → R is defined by. + v (p, w ) = max ...<|control11|><|separator|>
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Jul 18, 2016 · Bibliographic information ; Title, Cost and Production Functions ; Author, Ronald William Shephard ; Publisher, Princeton University Press, 1953.Missing: Shephard's lemma
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[PDF] HAL R. VARIAN
The first edition of Microeconomic Analysis was published in 1977. Af- ter 15 years, I thought it was time for a major revision. There are two types of ...
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Thus we find that Roy's Identity does hold. The dual and Shephard's Lemma. Now consider the dual problem of cost minimization given a fixed level of utility.
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How do we show that the indirect utility function is quasi-convex? We want to show that if v(p, m) ≥ v(p0,m0), then the indirect utility of.<|control11|><|separator|>
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Mar 22, 2004 · 2.1 Roy's Identity. One application of the envelope theorem is the derivation of Roy's identity. Roy's identity states that the individual ...
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May 9, 2009 · The AIDS model is derived from an expenditure function that can be interpreted as a second- order approximation to an arbitrary unknown function ...
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[PDF] Measurement of Welfare Change: A Review
Nov 15, 1995 · Compensating variation (CV) is defined as the amount of income that must be taken away from a consumer (positive or negative) after an economic ...
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Re-examination of uniform commodity taxes under a non-linear ...
From Roy's identity, ∂v(p+t, x) ∂p =−D(p+t, x) ∂v(p+t, x) ∂x , where D(p+t, x) is a conditional demand function of good 2 when the price is p+t and income x is ...
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An Application of the Shephard Duality Theorem: A Generalized Leontief Production Function ... Tong Li, Robert Rosenman Estimating hospital costs with a ...<|control11|><|separator|>
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Roy's identity applied to equation (5.1) yields the following Marshallian demand share for products in sector j: sj = aj +. X k γjk log(pk) + βj log xh a(p).Missing: generalizations | Show results with:generalizations
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IDENTIFICATION WITH ADDITIVELY SEPARABLE HETEROGENEITY
applying Roy's identity, and then taking expectations. Our proof does not require differ- entiability of the value function and allows multiple maximizers ...