# The Index Number Problem: A Differential Geometric Approach

The Index Number Problem: A Differential Geometric Approach, a Ph.D. Thesis in Economics at Harvard by Pia Malaney, offers a groundbreaking analysis of index number theory using differential geometry, addressing long-standing inconsistencies in traditional economic indices. The thesis introduces the concept of an economic derivative, resolving discrepancies between indices like Paasche and Laspeyres by developing a unique differential geometric index equivalent to the Divisia index. This approach not only provides a consistent and accurate measure of economic changes but also offers practical applications in welfare analysis, challenging the traditional reliance on the Konus index.

In addition to resolving the index number problem, the thesis explores the welfare implications of using Divisia indices, particularly in relation to changing consumer preferences and psychological neutrality. Malaney further extends the analysis to household migration decisions under uncertainty, demonstrating how traditional models oversimplify the decision-making process by neglecting risk aversion and household-level dynamics. The work concludes by advocating for a more comprehensive understanding of economic indices and migration policies, highlighting the need for refined economic metrics and more nuanced policy approaches.

## Chapter Summaries

### Chapter 1: The Index Number Problem: A Differential Geometric Approach

Chapter 1 of The Index Number Problem: A Differential Geometric Approach begins by outlining the core issue in index number theory: the inconsistencies among various index numbers like the Paasche and Laspeyres indices. These inconsistencies lead to divergent qualitative assessments of economic phenomena, such as inflation and growth. The chapter emphasizes that these contradictions pose significant challenges for economists, undermining the reliability of these indices as economic diagnostic tools. The introduction sets the stage for the application of differential geometry to address these inconsistencies​​.

• Adapted Derivatives and Notions of Constancy: The chapter proceeds to introduce the concept of "notions of constancy," highlighting how traditional derivatives in calculus, such as the ordinary derivative, have been implicitly used in index number formulas. Malaney argues that these traditional notions are inadequate for economic applications. Instead, the chapter introduces the concept of "adapted derivatives," which are specifically designed to address the unique problems posed by index numbers. By using differential geometry, Malaney proposes an economic derivative that properly separates income and substitution effects, offering a more accurate measure of economic change​.
• The Differential Geometric Index: The central innovation presented in this chapter is the Differential Geometric Index, a unique index derived using adapted derivatives. This index aligns with the Divisia index, providing a consistent measure across different economic situations. The chapter includes a detailed discussion on the notation and mathematical framework required for constructing this index. The author uses a toy example featuring a simplified economy with two goods, coconuts and lumber, to illustrate the discrepancy between standard indices like Paasche and Laspeyres and how the differential geometric approach resolves this discrepancy​​.
• Resolution of the Index Number Problem: The chapter culminates in demonstrating that the Differential Geometric Index offers a solution to the traditional index number problem. By employing the adapted derivative, the chapter shows that discrepancies between all common bilateral index numbers disappear. This approach provides a unified framework that resolves the inconsistencies that have plagued index number theory. The Divisia index emerges as the unique, correct index when considering the proper economic derivative.

Chapter 1 concludes by emphasizing the significance of using differential geometry in economic analysis. It asserts that this new framework not only unifies previous results but also clarifies the questions economists should be asking. The Differential Geometric Index is posited as a superior measure for economic analysis, particularly for issues related to welfare and cost-of-living adjustments. The chapter sets the foundation for the subsequent chapters, which will further explore the implications and applications of this innovative approach​.

### Chapter 2: The Welfare Implications of Divisia Indices

Chapter 2 of The Index Number Problem: A Differential Geometric Approach explores the welfare implications of the Divisia index, particularly in relation to the Consumer Price Index (CPI) and the Konus index, often referred to as the "true" cost of living index. The chapter emphasizes the significance of accurately measuring the cost of living, given that 30% of federal outlays and 45% of federal revenues are indexed to the CPI, and a 1% overstatement in the CPI can cost the U.S. government \$280 billion over seven years​.

• The Divisia and Konus Indices: The chapter begins by comparing the Divisia index with the Konus index. While the Konus index theoretically offers a complete solution to measuring the true cost of living, it requires access to individuals' preference maps, which are generally unavailable. The Konus index also assumes static preferences, which is an unrealistic assumption in real-world scenarios where preferences change over time. The Divisia index, on the other hand, is data-based and does not rely on the static preference assumption, making it a more practical alternative​.
• Path Dependence and Welfare: One of the key characteristics of the Divisia index is its path dependence, which means that the index value depends on the specific sequence of prices and quantities over time. While path dependence is often viewed as a drawback, the chapter argues that it can be advantageous. For example, in scenarios where consumer preferences shift—such as during a fad—the Divisia index can more accurately reflect the changes in cost of living. This is illustrated through an example of two consumers who start and end with the same basket of goods but follow different consumption paths, resulting in different levels of inflation experienced by each. The Divisia index accounts for these differences, providing a fairer cost of living adjustment​.
• Psychological Neutrality and Welfare Implications: The chapter introduces the concept of "psychological neutrality," which refers to the assumption that changes in preferences do not affect welfare measurements. Under this assumption, the Divisia index provides an accurate measure of the cost of living that aligns with the Konus index when preferences are static. However, when preferences are dynamic, the Divisia index continues to provide a valid measure of welfare, whereas the Konus index may not​​.

Chapter 2 concludes that the Divisia index, with its path dependence and flexibility in accommodating changing preferences, offers a robust solution for measuring welfare and cost of living. Unlike traditional indices like the Paasche and Laspeyres indices, the Divisia index does not suffer from inconsistencies in welfare analysis, making it a superior choice for practical applications​.

### Chapter 3: Household Migration Decisions Under Uncertainty

Chapter 3 focuses on the complexities surrounding household migration decisions from rural to urban areas, a significant issue in development economics. The chapter highlights the inadequacies of traditional migration models, such as those proposed by Lewis (1954) and Todaro (1969), which primarily consider the decision-making process as an individual choice driven by expected income differences between rural and urban areas. However, these models often overlook key factors, including risk aversion and household-level decision-making, which are critical in understanding migration dynamics.

• Traditional Models and Their Limitations: The chapter critiques the classical Todaro model, which posits that individuals migrate based on a simple comparison of expected urban and rural incomes, assuming risk neutrality. This model suggests that policies aimed at reducing income disparities, such as raising rural wages or controlling urban wages, can effectively manage migration flows. However, Malaney argues that this approach is overly simplistic, as it fails to account for the risk aversion that typically characterizes low-income households. Moreover, the decision-making unit is not just the individual but often the household, which may have different risk preferences and constraints.
• Risk Sensitivity and Household-Level Decision Making: The chapter introduces a more nuanced model that incorporates risk sensitivity and household-level decision-making. It argues that households, rather than individuals, are the primary decision-makers in migration scenarios. This perspective is supported by evidence that migration decisions are often influenced by family resources and needs, as well as the potential for income diversification through remittances. The model presented in this chapter treats the migration decision as a portfolio allocation problem, where the household allocates its labor force between the "safe" rural market and the "risky" urban market.
• Perverse Migration Incentives: A critical insight of the chapter is the phenomenon of "perverse migration," where increasing rural wages may inadvertently lead to greater migration to urban areas. This counterintuitive result arises from the household's ability to "gamble" on the risky urban market when rural incomes provide a sufficient safety net. The model demonstrates that under certain conditions, particularly when households exhibit decreasing absolute risk aversion (DARA), increases in rural wages can actually enhance the attractiveness of urban migration, contrary to the predictions of traditional models.
• Policy Implications: The chapter concludes by emphasizing the need for a more sophisticated understanding of migration dynamics when designing policies to control rural-to-urban migration. It suggests that traditional policies, which focus solely on income adjustments, may not achieve the desired outcomes if they do not consider the underlying risk preferences and decision-making processes of households. Instead, a more holistic approach that includes measures to stabilize rural incomes and address household risk concerns may be necessary to manage migration effectively.

Overall, Chapter 3 provides a comprehensive analysis of the complexities involved in household migration decisions under uncertainty. It challenges the assumptions of traditional migration models and offers a more realistic framework that incorporates risk sensitivity and household-level dynamics. This chapter is pivotal in understanding the broader economic and social implications of migration and in designing effective policies to address these challenges.