# Tracking Changes in Bilateral Trading Patterns in the Absence of PPP (Content)

Eric Weinstein's 2021 paper *Tracking Changes in Bilateral Trading Patterns in the Absence of PPP*, originally presented and hosted at the University of Chicago on November 10, 2021, addresses the complexities of measuring changes in trade between two nations when the law of one price and purchasing power parity (PPP) do not hold. Traditional trade indices struggle with these discrepancies because they assume a uniform relationship between prices and currencies. Weinstein proposes a novel approach by introducing a generalized matrix-valued trade index that accounts for shifts in the composition of traded goods and changes in currency valuations. Using tools from differential geometry and quantum theory, the paper constructs a framework that separates income and substitution effects in trade, allowing for a more nuanced understanding of how trade patterns evolve.

The trade index presented extends the traditional Divisia index into a multi-dimensional, matrix format to track bilateral changes accurately. It resolves the index number problem in economics, where different valuation methods often yield inconsistent results. Additionally, the paper introduces a corresponding price index for dual currencies and emphasizes the use of time-ordered products to handle non-commutative aspects of matrix operations. This approach provides a new way to assess intertemporal trade shifts, offering insights that go beyond classical economic analysis. Ultimately, Weinstein's work contributes a sophisticated mathematical framework that better captures the dynamics of global trade.

## Summary by Section[edit]

### Introduction[edit]

The paper *Tracking Changes in Bilateral Trading Patterns in the Absence of PPP* by Eric Weinstein addresses the complexities of measuring bilateral trade changes when the law of one price does not hold, complicating growth accounting. Traditional trade indexes, like Laspeyres and Paasche, assume Purchasing Power Parity (PPP) and proportional price movement, which often don’t apply in real-world situations. The author proposes a generalized trade index that employs a matrix-valued approach to handle these discrepancies, allowing for a more accurate representation of shifts in trade patterns over time. The framework uses techniques from differential geometry and quantum theory to address these problems effectively.

### The Geometric Interpretation of the Divisia Quantity Index[edit]

Weinstein builds on previous work with the Divisia quantity index, a conventional measure in economics that scales a basket of goods over time. He introduces a geometric interpretation, where different derivative operators are used to distinguish between income and substitution effects within economic data. By substituting the usual derivative with a specially adapted economic derivative, it becomes possible to more accurately measure changes in a trade basket, factoring in shifts in the composition of goods.

### Set-up[edit]

This section establishes the mathematical framework:

**Identification of the Fundamental Sub-spaces:**Defines two nations exchanging goods, each using separate pricing systems. These create subspaces used for projections in later constructions.**Construction of the Space of Economic Possibilities:**Extends from single-currency systems to dual-currency systems, defining the matrix-valued trade-value function for tracking bilateral trade changes.**Construction of the Economic Derivative:**Introduces a new derivative operator that distinguishes common substitution effects in both currencies to address discrepancies in valuation.**Construction of the Projection Maps:**Establishes projection maps onto the space of goods and mutual barters, separating income and substitution effects.**Time Ordered Products:**Uses time-ordered products from quantum physics to handle non-commutative properties of matrix-valued functions, necessary for constructing the generalized index.

### Construction of the Trade Index[edit]

The paper presents a matrix-valued trade index to address changes in trade patterns over time, capturing variations in export composition and discrepancies in currency values. The index helps countries determine whether shifts in trade result from real growth or changes in trading terms. It generalizes the single-basket Divisia index to a matrix-valued case, using time-ordered products to manage the non-commutative properties of matrix multiplication.

### Trade Index Number Problem[edit]

The traditional trade index number problem arises when bilateral indices, such as Laspeyres and Paasche, yield inconsistent results due to differences in valuation methods. The paper introduces adapted versions of these indices that utilize the matrix-valued trade index to resolve these inconsistencies, ensuring that both nations perceive the trade evolution similarly.

### Corresponding Bilateral Price Index for Bilateral Trade[edit]

A corresponding price index is introduced for dual currencies, allowing the tracking of changes in pricing patterns across two currencies. It parallels the construction of the quantity index but focuses on separating income and substitution effects from a pricing perspective rather than quantities traded.

### Conclusion[edit]

The paper concludes by highlighting the novelty of the generalized matrix-valued trade index, which tracks bilateral trading patterns without assuming PPP. Key insights include:

- The matrix nature of the trade index, which extends the Divisia index to multi-dimensional cases.
- The role of differential geometry in solving the index number problem.
- The necessity of time-ordered products due to non-commutative matrix properties.

Weinstein suggests that these techniques offer new opportunities for economic analysis, particularly in assessing complex trade relationships.

### References[edit]

The references provide background on index number theory and relevant mathematical techniques, including foundational works in economics and differential geometry, along with Weinstein's and Malaney's prior contributions.