Indirect Utility Function

Background§

The indirect utility function is a fundamental tool in microeconomics, particularly in consumer theory. It provides an indirect relationship between prices and utility by indicating the maximum utility a consumer can achieve given their budget constraint and the current market prices of goods and services.

Historical Context§

The development of the indirect utility function dates back to early 20th-century advancements in consumer theory. Economists such as John Hicks and Paul Samuelson made significant contributions in formalizing the concepts, including the relationships between utility, expenditure, and demand functions.

Definitions and Concepts§

The indirect utility function, $V(p, M)$, expresses the maximum utility a consumer can attain with a given income $M$ and a vector of prices $p$ for goods and services. This contrasts with the direct utility function, which depends directly on quantities of goods consumed.

The expenditure function, $E(p, U)$, measures the minimum expenditure needed to achieve a certain utility level $U$ given prices $p$. An essential link between these concepts is expressed by Shephard’s Lemma, which states that the partial derivative of the expenditure function with respect to prices yields the Hicksian (compensated) demand function: $\frac{∂E}{∂p_i} = h_i(p, U)$.

Major Analytical Frameworks§

Classical Economics§

Classical economics focuses on the production and costs without specific attention to the utility functions seen in later theories.

Neoclassical Economics§

Neoclassical economists heavily utilize utility functions, including the indirect utility function, to understand consumer choices, market demand, and welfare implications of price changes.

Keynesian Economics§

While the primary focus is on aggregate demand and government policy, understanding consumer behavior at the micro-level via utility functions supports Keynesian analysis.

Marxian Economics§

Marxian analysis primarily addresses production relations and labor exploitation, hence uses different conceptual tools rarely reliant on utility functions.

Institutional Economics§

Institutional economics may consider utility functions to understand consumer behavior in the context of economic institutions and social structure.

Behavioral Economics§

Indirect utility functions can be compared to observed behavior patterns in testing standard economic assumptions about rational behavior.

Post-Keynesian Economics§

This school may employ utility functions to analyze individual consumption patterns but typically places stronger emphasis on macroeconomic relationships.

Austrian Economics§

Austrians focus on preference orderings and marginal utility but may still reference indirect utility in broader critiques of equilibrium analysis.

Development Economics§

Indirect utility functions can be essential for analyzing household behavior in developing countries and assessing the impact of policy changes on welfare.

Monetarism§

Through its focus on money supply and price levels, monetarism indirectly considers utility functions to understand consumer reactions to monetary policy changes.

Comparative Analysis§

The indirect utility function, compared to the direct utility function, is more operationally useful in welfare analysis and policy design as it directly links income, prices, and utility levels. It allows economists to infer changes in consumer well-being from variations in income or price without converting back to quantities of goods consumed.

Case Studies§

Price Subsidy Program§

In this study, policymakers use the indirect utility function to assess consumer welfare changes when implementing subsidies on basic goods.

Tax Reforms§

Analyzed through the $V(p, M)$ framework, tax policies are evaluated in terms of their impact on different income groups.

Suggested Books for Further Studies§

• “Microeconomic Theory” by Andreu Mas-Colell, Michael D. Whinston, and Jerry R. Green
• “Advanced Microeconomic Theory” by Geoffrey A. Jehle and Philip J. Reny
• “A Course in Microeconomic Theory” by David M. Kreps

Related Terms with Definitions§

Direct Utility Function§

The direct utility function represents utility as a function of quantities of goods consumed.

Expenditure Function§

Represents the minimum cost needed to achieve a given level of utility considering the prices of goods.

Hicksian Demand Function§

Derived from the expenditure function, demonstrating the quantity of goods demanded to maintain a fixed utility level as prices change.

Shephard’s Lemma§

Relates the derivative of the expenditure function to the Hicksian demand; $\frac{∂E}{∂p_i} = h_i(p, U)$.