When the trading process is characterized by search frictions, traders may be rationed so markets need not clear. We build a general equilibrium model with transferable utility where the uncertainty arising from rationing is incorporated in the definition of a commodity, in the spirit of the Arrow-Debreu theory. Prices of commodities then depend not only on their physical characteristics, but also on the probability that their trade is rationed. The standard definition of competitive equilibrium is extended by replacing market clearing with a matching condition which describes a trading technology that is not frictionless. This condition relates the rationing probabilities of buyers and sellers to ratio of buyers to sellers in the market via an exogenous matching function with constant returns, as in standard search-theoretic models. When search frictions vanish, our model is equivalent to the competitive assignment model of Gretsky, Ostroy and Zame (1992). We adopt their approach, which uses linear programming techniques and duality theory, to derive the welfare and existence theorems in our search environment. Our competitive equilibrium notion is equivalent to that of directed (or competitive) search. The strength of our formulation and the linear programming approach is that they allow us to generalize the constrained efficiency and existence results in the directed search literature to a much broader class of economies. Our framework also opens the door to the use of linear programming algorithms for computing equilibria