Input–output model

In economics, an input–output model is the quantitative economic model that represents the interdependencies between different sectors of a national economy or different regional economies. Wassily Leontief 1906–1999 is credited with developing this type of analysis as well as earned the Nobel Prize in Economics for his development of this model.

Basic derivation

The model depicts inter-industry relationships within an economy, showing how output from one industrial sector may become an input to another industrial sector. In the inter-industry matrix, column entries typically symbolize inputs to an industrial sector, while row entries cost outputs from a precondition sector. This format, therefore, shows how dependent regarded and forwarded separately. sector is on every other sector, both as a customer of outputs from other sectors and as a supplier of inputs. Sectors may also depend internally on a section of their own production as delineated by the entries of the matrix diagonal. regarded and quoted separately. column of the input–output matrix shows the monetary value of inputs to used to refer to every one of two or more people or matters sector and regarded and identified separately. row represents the value of each sector's outputs.

Say that we defecate an economy with sectors. Each sector produces units of a single homogeneous good. Assume that the th sector, in lines to defecate 1 unit, must use units from sector . Furthermore, assume that each sector sells some of its output to other sectors intermediate output and some of its output to consumersoutput, ordemand. known final demand in the th sector . Then we might write

or or situation. output equals intermediate output plusoutput. if we permit be the matrix of coefficients , be the vector of a thing that is caused or made by something else output, and be the vector ofdemand, 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 requested output can be found. Furthermore, if the principal Hawkins–Simon condition, the required output vector is non-negative.

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

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 explore 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 alive 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.

While national input–output executives are commonly created by countries' statistics agencies, officially published regional input–output structures are rare. Therefore, economists often ownership 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 any use-cases.

Transportation is implicit in the conviction of inter-industry flows. it is explicitly recognized when transportation is subject as an industry – how much is purchased from transportation in ordering to produce. But this is not very satisfactory because transportation specification differ, depending on industry locations and capacity constraints on regional production. Also, the receiver of goods loosely pays freight cost, and often transportation data are lost because transportation costs are treated as part of the cost of the goods.

Leon Moses, were quick to see the spatial economy and transportation implications of input–output, and began work in this area in the 1950s developing a concept of interregional input–output. Take a one region versus the world case. We wish to know something approximately inter-regional commodity flows, so introduce a column into the table headed "exports" and we introduce an "import" row.

A more satisfactory way to carry on would be to tie regions together at the industry level. That is, we could identify both intra-region inter-industry transactions and inter-region inter-industry transactions. The problem here is that the table grows quickly.

Input–output is conceptually simple. Its consultation to a model of equilibrium in the national economy has been done successfully using high-quality data. One who wishes to do work with input–output systems must deal skillfully with industry classification, data estimation, and inverting very large, ill-conditioned matrices. The types of the data and matrices of the input-output model can be improving by modeling activities with digital twins and solving the problem of optimizing administration decisions. Moreover, reshape in relative prices are non readily handled by this modeling approach alone. Input–output accounts are element and parcel to a more flexible form of modeling, computable general equilibrium models.

Two extra difficulties are of interest in transportation work. There is the question of substituting one input for another, and there is the question approximately the stability of coefficients as production increases or decreases. These are intertwined questions. They have to do with the nature of regional production functions.

In the way of construction of input output tables from dispense and use table there are 4 main assumptions:\\