In the first approach, which the authors label the transactions costs approach, the commodity space is the same as that used in models without transactions costs. In the second approach, which the authors label the valuation equilibrium approach, the commodity space is chosen so that the exchange problem can be formulated as an instance of the abstract exchange model described in Debreu (1954). The authors argue that the valuation equilibrium approach provides a tractable framework for quantitative studies of the effects of transactions costs on economy-wide resource allocation.