First, only two of the versions mention anything about international trade. This underscores that the essence of all SS versions is the link between product prices and factor prices imposed by the “zero-profit conditions” equating price with average cost that must hold in all perfectly competitive industries with actual production.1 These zero-profit conditions imply a systematic relationship between the entire set of product prices facing domestic producers and the entire set of factor prices paid by these producers. Analytically, the economy’s entire set of zero-profit conditions can be written as follows:
where P is an (N x 1) vector of N domestic product prices, W is an (M x 1) vector of M domestic factor prices, and A is an (N x M) technology matrix (which might also depend on W) whose aij element tells the number of units of factor i required to produce one unit of product j. Each row of equation (1) corresponds to one of the N products, and in words the equation says for each product price equals average costs. Importantly, notice that factor prices are not indexed by
industry: with the assumption of perfect interindustry factor mobility each factor has only one national price. This is a key aspect of all versions of the SS theorem.
Holding constant technology, equation (1) can be rewritten in terms of percentage changes for (sufficiently small) changes. This yields the following equation:
where P* is an (N x 1) vector of N domestic product-price changes, W* is an (M x 1) vector of M domestic factor-price changes, and 9 is an (N x M) initial cost-share matrix (which depends on technology and, perhaps, W) whose 9 ij element tells the share of factor i in the average costs incurred producing one unit of product j.2 Changes in product prices faced by domestic firms generate changes in domestic factor prices paid by firms as described by equation (2). This is true whether product-price changes are caused by international trade or any other force. Indeed, many SS versions apply even to countries in autarky with all products nontraded. The empirical implication of this is the need for a way to determine what portion of observed product-price changes are attributable to international trade. Equation (2) will be an important reference point. credit