Indirect utility function
The indirect utility function, often denoted as v(\mathbf{p}, w), represents the maximum level of utility a consumer can achieve given a vector of prices \mathbf{p} and income or wealth w, formally defined as v(\mathbf{p}, w) = \max_{\mathbf{x}} u(\mathbf{x}) subject to the budget constraint \mathbf{p} \cdot \mathbf{x} \leq w, where u(\mathbf{x}) is the direct utility function and \mathbf{x} is the consumption bundle.[1][2] This function arises in consumer theory by substituting the Marshallian demand functions—the optimal consumption choices—back into the direct utility function, transforming utility from a function of quantities into one of observable market variables like prices and income.[3] Key properties of the indirect utility function include homogeneity of degree zero in prices and income, meaning v(\lambda \mathbf{p}, \lambda w) = v(\mathbf{p}, w) for any \lambda > 0, which reflects that scaling all prices and income proportionally does not change the maximum attainable utility.[1][2] It is also non-decreasing in income (strictly increasing under local non-satiation of preferences) and non-increasing in prices, ensuring that higher income allows for at least as much utility while price increases reduce it.[1][2] Additionally, the function is continuous in \mathbf{p} and w (assuming continuity of the direct utility) and quasi-convex, implying that the set of price-income pairs yielding utility no better than a given level is convex.[1][2] Roy's identity further connects it to demand theory by relating the partial derivative with respect to a price to the corresponding Marshallian demand: x_i(\mathbf{p}, w) = -\frac{\partial v / \partial p_i}{\partial v / \partial w}.[1] In economic analysis, the indirect utility function plays a central role in welfare economics, enabling the measurement of consumer welfare changes from price or income shifts, such as through compensating or equivalent variations.[2] It establishes duality with the expenditure function, where the minimum expenditure to achieve a utility level \bar{u} at prices \mathbf{p} is the inverse in a sense, facilitating derivations of Hicksian demands and cost-of-living indices.[2] This framework is foundational for empirical applications in demand estimation and policy evaluation, linking theoretical preferences to market data without directly observing utility.[3][2]Definition and Basics
Formal Definition
The indirect utility function, denoted as v(\mathbf{p}, m), represents the maximum level of utility a consumer can achieve given a vector of prices \mathbf{p} for goods and services and a budget m. Formally, it is defined as the solution to the consumer's utility maximization problem: v(\mathbf{p}, m) = \max_{\mathbf{x} \geq \mathbf{0}} \, u(\mathbf{x}) \quad \text{subject to} \quad \mathbf{p} \cdot \mathbf{x} \leq m, where \mathbf{x} is the consumption bundle, u(\cdot) is the direct utility function representing preferences over \mathbf{x}, and the dot denotes the inner product. This definition assumes the direct utility function u(\cdot) is continuous, strictly increasing, and quasi-concave, ensuring the existence of a unique interior solution to the maximization problem under standard conditions in consumer theory. The concept originated in the late 19th century with Giovanni Battista Antonelli's work on integrability and demand functions, where he first introduced the indirect utility function as a means to express utility in terms of prices and income.[4] It was further developed in the early 20th century by Eugen Slutsky in his foundational analysis of consumer budget constraints and substitution effects, laying groundwork for modern demand theory.[5] The framework was formalized in contemporary terms through Hendrik S. Houthakker's extension of revealed preference theory, which connected observable demand data to the existence and properties of utility representations, including indirect forms.[6] The term "indirect" arises because v(\mathbf{p}, m) evaluates utility based on observable economic variables—prices and income—rather than directly on the unobservable quantities consumed, providing a bridge from market data to preference revelation via the optimal choice.Interpretation in Consumer Theory
The indirect utility function, denoted v(p, m), represents the maximum level of welfare or satisfaction that a consumer can achieve given a vector of prices p and income m, reflecting the outcome of optimal consumption choices under the budget constraint.[1] This measure serves as a key welfare index in economic analysis, enabling comparisons of consumer well-being across different price environments or income levels, and it underpins policy evaluations such as cost-of-living adjustments in inflation indexing.[7] For instance, changes in v(p, m) can quantify the welfare effects of price shifts, informing adjustments to social benefits or tax policies to maintain equivalent living standards.[7] In revealed preference theory, the indirect utility function encodes consumer choices by linking observed demand behavior directly to underlying preferences, without requiring direct observation of the utility derived from specific consumption bundles.[8] It provides an axiomatic foundation for inferring preference orderings from price and income data, where axioms of revealed favorability—analogous to those in standard revealed preference—ensure consistency with rational utility maximization.[8] This approach allows economists to test whether observed demands are consistent with a well-behaved indirect utility function, facilitating empirical validation of consumer rationality solely through market data.[8] The interpretation of the indirect utility function relies on several core assumptions in consumer theory, including local non-satiation, which ensures that more consumption is always preferred, preventing indifference at any finite bundle.[9] Convexity of preferences guarantees that the consumer's demand is well-defined and that indifference curves bow toward the origin, supporting the existence of an interior solution to the maximization problem.[9] Additionally, the framework assumes Walrasian demand, where consumers exhaust their budget and respond continuously to price and income changes, ensuring the indirect utility captures genuine optimization behavior.[1] Regarding utility measurement, the indirect utility function operates within an ordinal framework, preserving the ranking of preference orderings across situations but not assigning cardinal values that imply interpersonal or absolute comparisons of satisfaction levels.[1] Any monotonic transformation of the underlying direct utility function yields an equivalent indirect utility that maintains these ordinal properties, emphasizing relative rather than absolute welfare assessments in consumer theory.[1]Mathematical Derivation
From Direct Utility Maximization
The indirect utility function arises from the consumer's primal problem of utility maximization subject to a budget constraint. Consider a consumer with a continuous, strictly increasing, and strictly quasi-concave utility function u(\mathbf{x}), where \mathbf{x} = (x_1, \dots, x_n) is the vector of goods consumed, facing prices \mathbf{p} = (p_1, \dots, p_n) > \mathbf{0} and income m \geq 0. The problem is to maximize u(\mathbf{x}) subject to \mathbf{p} \cdot \mathbf{x} \leq m and \mathbf{x} \geq \mathbf{0}. Assuming an interior solution exists, form the Lagrangian \mathcal{L}(\mathbf{x}, \lambda) = u(\mathbf{x}) + \lambda (m - \mathbf{p} \cdot \mathbf{x}). The first-order conditions are \frac{\partial u}{\partial x_i} = \lambda p_i for all i = 1, \dots, n, along with the budget exhaustion condition \mathbf{p} \cdot \mathbf{x} = m. These conditions, together with the second-order conditions ensured by strict quasi-concavity of u, yield the unique Marshallian demand functions \mathbf{x}(\mathbf{p}, m) that solve the system. Substituting the optimal demands back into the utility function gives the indirect utility function v(\mathbf{p}, m) = u(\mathbf{x}(\mathbf{p}, m)), which represents the maximum attainable utility given prices and income. The strict quasi-concavity of u and positivity of prices guarantee the existence and uniqueness of this solution under standard continuity assumptions. By the envelope theorem applied to the Lagrangian, the partial derivative of the indirect utility with respect to income is \frac{\partial v}{\partial m} = \lambda, where \lambda > 0 is the Lagrange multiplier interpreting the marginal utility of income. This follows directly from differentiating v(\mathbf{p}, m) while holding \mathbf{p} fixed, as the indirect effect through demands cancels out, leaving only the direct effect on the constraint.Properties of the Indirect Utility Function
The indirect utility function v(\mathbf{p}, m), arising from the solution to the consumer's utility maximization problem, exhibits several essential mathematical properties that align with the axioms of consumer preferences, such as continuity, monotonicity, and convexity. A key property is homogeneity of degree zero in prices and income: v(t\mathbf{p}, tm) = v(\mathbf{p}, m) for any scalar t > 0. This means that proportional increases in all prices and income leave the maximum attainable utility unchanged, reflecting the absence of money illusion in rational consumer behavior. The property follows directly from the homogeneity of degree zero in the Marshallian demand functions, as the optimal consumption bundle \mathbf{x}(\mathbf{p}, m) satisfies \mathbf{x}(t\mathbf{p}, tm) = \mathbf{x}(\mathbf{p}, m), yielding the same utility evaluation. By Euler's theorem applied to this homogeneous function, \sum_i p_i \frac{\partial v}{\partial p_i} + m \frac{\partial v}{\partial m} = 0.[10] The indirect utility function is non-increasing in each price p_i: \frac{\partial v}{\partial p_i} \leq 0. Under the standard assumption of local non-satiation, this effect is strict (\frac{\partial v}{\partial p_i} < 0), as higher prices reduce purchasing power and force the consumer to a lower indifference curve. This can be shown using Roy's identity, \frac{\partial v / \partial p_i}{\partial v / \partial m} = -x_i(\mathbf{p}, m), where x_i > 0 due to local non-satiation and \frac{\partial v}{\partial m} > 0, implying the negative price derivative. Shephard's lemma provides an indirect link through duality, confirming the consumption response to prices.[11] It is also strictly increasing in income: \frac{\partial v}{\partial m} > 0. Greater income expands the budget set, enabling access to higher utility levels consistent with monotonic preferences. This marginal utility of income is positive because any additional resources allow for a preferred consumption bundle under local non-satiation.[10] Finally, the indirect utility function is quasiconvex in prices and income, meaning the sublevel sets \{(\mathbf{p}, m) \mid v(\mathbf{p}, m) \leq \bar{v}\} are convex for any \bar{v}. This property ensures smooth and consistent welfare adjustments to changes in economic conditions, preserving the convexity of preferences in the dual space.[10]Connections to Other Economic Functions
Relationship with Expenditure Function
The expenditure function, denoted e(\mathbf{p}, u), is defined as the minimum expenditure required to achieve a given utility level u at prices \mathbf{p}, formally expressed ase(\mathbf{p}, u) = \min_{\mathbf{x}} \mathbf{p} \cdot \mathbf{x} \quad \text{subject to} \quad u(\mathbf{x}) \geq u.
This function serves as the dual problem to the utility maximization problem underlying the indirect utility function v(\mathbf{p}, m).[2] A fundamental duality theorem establishes the inverse relationship between these functions: for any prices \mathbf{p} and income m, the income equals the expenditure needed to achieve the indirect utility level, m = e(\mathbf{p}, v(\mathbf{p}, m)); conversely, for any utility u, the indirect utility from the expenditure to achieve u recovers the original utility, u = v(\mathbf{p}, e(\mathbf{p}, u)).[2]
The proof relies on the equivalence of optimization solutions: the bundle \mathbf{x}^* that maximizes utility subject to the budget constraint also minimizes expenditure subject to the utility constraint, assuming continuity of preferences and local nonsatiation.[2] This duality implies that the expenditure function can be obtained by inverting the indirect utility function: solving v(\mathbf{p}, m) = u for m yields e(\mathbf{p}, u), allowing recovery of the minimum cost to attain a specific utility from observable prices and income-derived utility levels.[2] In welfare economics, the expenditure function facilitates the construction of true cost-of-living indices, such as the Konüs index, which measure the ratio of expenditures needed to maintain a reference utility level across price changes, often derived from estimates of the indirect utility function to assess real income adjustments.[7]