Input–output model

This article is about the economic model, for the computer interface see Input/output

An input-output model is widely used in economic forecasting to predict flows between sectors.

Wassily Leontief won a Nobel Prize in Economics for his development of these types of model for the national level. They are also used in local urban economics.

Irving Hock at the Chicago Area Transportation Study did detailed forecasting by industry sectors using input-output techniques. At the time, Hock’s work was quite an undertaking, the only other work that has been done at the urban level was for Stockholm and it was not widely known. Input-output was one of the few techniques developed at the CATS not adopted in later studies. Later studies used economic base-like techniques. Input-output concepts are simple. Consider the production of the ith sector. We may isolate (1) the quantity of that production that goes to final consumption (Ci) , (2) to total output (Xi) , and (3) flows (xij) from that industry to other industries. We may write a transactions tableau.

Table: Transactions in a Three Sector Economy

Economic Activities Inputs to Agriculture Inputs to Manufacturing Transport Final Demand Total Output Agriculture 5 15 2 68 90 Manufacturing 10 20 10 40 80 Transportation 10 15 5 0 30 Labor 25 30 5 0 60

or x11 + x12 + x13 + c1 = X1 x21 + x22 + x23 + c2 = X2 x31 + x32 + x33 + c3 = X3 x41 + x42 + x43 + c4 = X4

    What about production functions?  We know very little about these because all we have are numbers representing transactions in a particular instance (single points on the production functions).

x1 = F(x11, x12, x13, x14) x2 = g(x21, x22, x23, x24) . . . . . . . . .

Recall the neoclassical production function is an explicit function:

Q = f(K, L)

Where: Q = Quantity K = Capital L = Labor

and the partial derivatives (?Q/?K = fK > 0 ; ?Q/?L = fL > 0) are the demand schedules for input factors. Leontief, the innovator of input-output, uses a special production function. Using Leontief coefficients (aijs) we may manipulate our transactions information into what is known as an input-output table.

x11 = a11x1 x12=a12x2 x13=a13x3 x14=a14x4 . . . . Or xij=xjaij x41=a41x1 … … …

a11x1 + a12x2 + a13x3 + a14x4 + c1 = x1 . . . . . a41x1 + a42x2 + a43x3 + a44x4 + c4 = x4

gives

x1 - a11x1 - a12x2 - a13x3 - a14x4 = c1 . . . . . x4 - a41x1 - a42x2 - a43x3 - a44x4 = c4

rewriting

x1 (1- a11) - x2a12 - x3a13 - x4a14 = c1 . . . . . - x1a41 - x2a42 - x3a43 - x4(1-a44) = c4

Writing in matrix form, we may see how a solution may be obtained. Let:

Then: X = AX + C (I - A)X = C X = (I - A)-1C

There are many interesting aspects of the Leontief system, and there is an extensive literature. There is the Hawkins-Simon condition on producibility. There has been interest in disaggregation to clustered inter-industry flows, and the study of constellations of industries. A great deal of empirical work has been done to identify coefficients, and data have been published for the national economy as well as for regions. This has been a healthy, exciting area for work by economists because the Leontief system can be extended to a model of general equilibrium; it offers a method of decomposing work done at a macro level. 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 spatial economy and transportation implications of input-output, and began work in this area in the 1950’s developing a concept of interregional input-output. Take a one region versus the world case. We wish to know something about interregional commodity flows, so introduce a column into the table headed “exports” and we introduce an “input” row.

Figure: Adding Export And Import Transactions

Economic Activities 1 2 … … Z Exports Final Demand Total Outputs 1 2 … … Z Imports

A more satisfactory way to proceed would be to tie regions together at the industry level. That is, we identify both within region inter-industry transactions and among region inter-industry transactions. A not-so-small problem here is that the table (Figure X) gets very large very quickly. Input-output, as we have discussed it, is conceptually very simple. It is attractive. Although we have not discussed its extension to an overall model of equilibrium in the national economy, that extension is also relatively simple and attractive. But there is a downside. 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. Two additional difficulties are of interest in transportation work. There is the question of substituting one input for another, and there is the question about the stability of coefficients as production increases or decreases. These are intertwined questions. They have to do with the nature of regional production functions.

Forecasting and/or Analysis Using Input-Output:

    We limit this discussion to the use of input-output techniques in transportation; there is a vast literature on the technique as such.

Figure X: Interregional Transactions

Economic Activities Ag North Mfg … … Ag East Mfg … … Ag West Mfg … … Exports Total Outputs Ag North Mfg … … Ag East Mfg … … Ag West Mfg … …

As we see from the use of the economic base study, and as will be further elaborated later, UTP studies are demand-driven. The question we want to answer is, “What transportation need results from some economic development: what’s the feedback from development to transportation?” For that question, input-output is helpful. That’s the question Hock posed. There is an increase in the final demand vector, changed inter-industry relations result, and there is an impact on transportation requirements. Rappoport et al. (1979) started with consumption projections. These drove solutions of a national I-O model for projections of GNP and transportation requirements as per the transportation vector in the I-O matrix. Submodels were then used to investigate modal split and energy consumption in the transportation sector. Another question asked is: What is the impact of the transportation construction activity on an area? Though not asked in the UTP study, that’s a very popular question. One of the first studies made of the impact of the interstate used the national I/O model to forecast impacts measured in increased steel production, cement, employment, etc.

Table 3. Input-Output Model for Hypothetical Economy Total requirements from regional industries per dollar of output delivered to final demand

Purchasing Industry Agriculture Transport Manufacturer Services

Selling Industry Agriculture 1.14 0.22 0.13 0.12 Transportation 0.19 1.10 0.16 0.07 Manufacturing 0.16 0.16 1.16 0.06 Services 0.08 0.05 0.08 1.09 Total 1.57 1.53 1.53 1.34

The Maritime Administration (MARAD) has produced the Port Impact Kit for a number of years. This software illustrates the use of I/O models. Simply written, it makes the technique widely available. It shows how to calculate direct effects from the initial round of spending that’s worked out by the vessel/cargo combinations. The direct expenditures are entered into the I/O table, and indirect effects are calculated. These are the inter-industry-relations derived activities from the purchases of supplies, purchases, labor, etc. An I/O table is supplied to aid that calculation. Then, using the I/O table, induced effects are calculated. These are effects from household purchases of goods and services made possible from the wages generated from direct and indirect effects. The Corps of Engineers has a similar capability that has been used to examine the impacts of construction or base closing. The US Department of Commerce Bureau of Economic Analysis (BEA) (1997) model discusses how to use their state level I/O models (RIMS II). The ready availability of BEA and MARAD-like tables and calculation tools says that we will see more and more feedback impact analysis. The information is meaningful for many purposes. Feed forward calculations seem to be much more interesting for planning. The question is, “If an investment is made in transportation, what will be its development effects?” An investment in transportation might lower transport costs, increase quality of service, or a mixture of these. What would be the effect on trade flows, output, earnings, etc.? The first problem we know of worked on from this point of view was in Japan in the 1950’s. The situation was the building of a bridge to connect two islands, and the core question was of the mixing of the two island economies. A first consideration is the impact of changed transportation attributes, say, lower cost, on industry location, and/or agricultural or other resource based extra active activity, and/or on markets. A spatial price equilibrium model (linear programming, to be discussed later in notes on urban models) is the tool of choice for that. Input-output then permits tracing changed inter-industry relations, impacts on wages, etc. Harris (1974) uses that analysis strategy. He begins with industry location forecasting equations: treats equilibrium of locations, markets, and prices; and pays much attention to transport costs. An interesting thing about this and other models is that input-output considerations are no more than an accounting add-on; they hardly enter Harris’ study. The interesting problems are the location and flow problems.