demand). Call final demand in the th sector . Then we might write or total output equals intermediate output plus final output. If we let be the matrix of coefficients , be the vector of total output, and be the vector of final demand, then our expression for the economy becomes which after re-writing becomes . If the matrix is invertible then this is a linear system of equations with a unique solution, and so given some final demand vector the required output can be found. Furthermore, if the principal minors of the matrix are all positive (known as the Hawkins–Simon condition), the required output vector is non-negative. Example Consider an economy with two goods, A and B. The matrix of coefficients and the final demand is given by Intuitively, this corresponds to finding the amount of output each sector should produce given that we want 7 units of good A and 4 units of good B. Then solving the system of linear equations derived above gives us Further research There is extensive literature on these models. There is the Hawkins–Simon condition on producibility. There has been research on disaggregation to clustered inter-industry flows, and on the study of constellations of industries. A great deal of empirical work has been done to identify coefficients, and data has been published for the national economy as well as for regions. The Leontief system can be extended to a model of general equilibrium; it offers a method of decomposing work done at a macro level. Regional multipliers While national input–output tables are commonly created by countries' statistics agencies, officially published regional input–output tables are rare. Therefore, economists often use location quotients to create regional multipliers starting from national data. This technique has been criticized because there are several location quotient regionalization techniques, and none are universally superior across all use-cases. Introducing transportation Transportation is implicit in the notion of inter-industry flows. It is explicitly recognized when transportation is identified as an industry – how much is purchased from transportation in order to produce. But this is not very satisfactory because transportation requirements differ, depending on industry locations and capacity constraints on regional production. Also, the receiver of goods generally pays freight cost, and often transportation data are lost because transportation costs are treated as part of the cost of the goods. Walter Isard and his student, Leon Moses, were quick to see the