The Uncompensated Law of Demand in an exchange economy

Recently in these Letters, Franks and Bryant (2017) provided new necessary and sufficient conditions for the Uncompensated Law of Demand (ULD) when consumer income is exogenous and not price dependent. We now consider price dependent incomes, and provide new necessary and sufficient conditions for the ULD to hold for a consumer in an exchange economy. Like Franks and Bryant (2017), these conditions avoid unobservables such as preferences or utility and hence are 'Revealed Preference' in character. Our Theorem 1 establishes that consumers in an exchange economy satisfy the ULD if and only if they consume exactly and only their initial endowment. Interestingly, unless a consumer lives in a state of personal autarky, there will always be a price variation for which their behaviour violates the ULD.