The consumption-real exchange rate anomaly with extensive margins

• The Backus-Smith can be attenuated with changes in the number of product varieties.
• The key is a wealth effect due to changes in extensive margins.
• Another key is the statistical inefficiency in measuring them.
• Market incompleteness is also required to achieve the result.

This paper investigates a consumption-real exchange rate anomaly from the open macroeconomics literature known as the Backus-Smith puzzle. We both analytically and quantitatively examine how an expansion of trade along extensive margins can contribute to the puzzle's resolution. Our argument is based on 1) a wealth effect due to changes in the number of product varieties, 2) statistical inefficiency in measuring the number of product varieties, and 3) market incompleteness. Contrary to complete asset markets which, in general, feature overly strong risk sharing properties, changes in the number of product varieties under incomplete markets may produce a wealth effect under high trade elasticity. Since statistical agencies systematically fail to capture the welfare impact arising from that changes, data-consistent terms of trade and real exchange rates tend to appreciate due to this positive wealth effect. This provides a realistic correlation between data-consistent real exchange rates and consumption.