# understanding input-output modeling for assessing potential economic impact

Maybe this dry, geeky essay won’t shock your soul, but it describes input-output models, their assumptions, & their uses for estimating economic impact of grand public investments.

David L. Passmore https://davidpassmore.net (Distinguished Professor, Emeritus, Penn State; Academic Visitor, University of Pittsburgh)
2022-04-12

Input-output modeling is a data-based, computational, and often prospective approach to unwinding the impact of projects, events, and changes in local economies. An input-output model accounts for goods and services that are required as inputs by industries to produce outputs of goods and services necessary to satisfy supply requirements for inputs to other industries and delivery directly to consumers .

Input-output modeling is a way to measure the effects of a project, program, activity, or event on the structure or conditions of economic life in a country, area, or period. Use of input-output modeling is pervasive in the assessment of economic impact. However, this modeling technique often is misapplied and debauched by various rakehells of academia and consulting to fulfill patently politically-motivated spending of public funds. For these reasons, understanding the ways and means of input-output modeling is essential for weighing critically any assertions about economic impact.1

In this paper, I outline the details of the input-output model and its assumptions, outline common uses (and some concerns about abuses) of the model to estimate potential economic impact, and list selected software and systems available for input-output modeling. In doing so, I do not get into any depth in the vast literature on input-output modeling. Nor do I establish a critical or skeptical stance about this modeling or its uses. For the most part, I offer in this blog posting a technical description, unadorned by advocacy or patronage, of input-output modeling. I will refer to the technical base offered in this blog posting to support my reviews and commentaries about a variety of economic impact reports that I will make in other blog postings.

### Specification of the Input-Output Model

#### Production and Consumption

Consider an economy with $$I$$ producing industries and $$J$$ purchasing industries, where $$I = J$$. Some of the output of industry $$i$$ is sold to industry $$j$$ for use in its own production.

Let $$\textbf{X}$$ stand for a square matrix of transactions among industries, with elements $$x_{ij}$$ containing the dollar value of goods and services sold by producing industry $$i$$ to purchasing industry $$j$$. Sales of the product of industry $$i$$ to itself — that is, $$x_{i=j}$$ — are possible. For instance, the coal industry also might purchase coal for use in its own production processes.

Output of industry $$i$$ not purchased by industries for their own production instead is sold directly to consumers to meet demands for personal consumption in households, government purchases of defense and non-defense goods and services, fixed investments such as housing and other structures, and exports net of imports. Let $$\textbf{y}$$ represent an $$I$$–length vector whose elements display the dollar value of goods and services delivered by producing industry $$i$$ to fulfill final demands of the economy for personal consumption, government purchases, fixed investments, and net exports. Vector $$\textbf{y}$$ represents the “final demand” for goods and services in an economy and is known commonly in the parlance of national income and product accounting practice as gross domestic product .

#### Direct Requirements

Quantities in an input–output model most commonly are expressed as monetary values (e.g., as dollars, rubles, yuan, or some other form of currency used as a medium of exchange) rather than as quantities of goods and services, although some energy and resource input–output models are scaled in physical quantities such as megawatts of electricity, kilograms of waste, hectares of land, or liters of water .

The production sector of the economy, then is defined by $$\textbf{X}$$, and $$\textbf{y}$$ is the consumption sector. The total output of the economy by producing industry, contained in $$I$$–length vector $$\textbf{x}$$,2 is equal to the sum of production and consumption, or

$\tag{1} \textbf{x}=\textbf{X}+\textbf{y}. \end {equation}$

Define $$\textbf{A}$$, a direct requirements matrix with elements $$a_{ij}=x_{ij}/x_{j}$$ showing the proportion of industry $$i$$ output required for production by industry $$j$$. Equation (1) now becomes

$$$\tag{2} \textbf{x}= \textbf{Ax} + \textbf{y}.$$$

Assumptions undergird all models, which are not the entity itself but are an abstraction of particular features of interest. Accordingly, input-output models rest on important assumptions that simplify and complicate…both at the same time.

Elements in $$\textbf{A}$$ are assumed to be fixed, homogeneous, and linear within purchasing industries. I dwell on these assumptions because they are at the very core of the operation as well as the credibility of input-output modeling.

Fixed

Inputs required by purchasing industry $$j$$ from producing industry $$i$$ do not change over time. More technically, zero elasticity of price or quantity substitution exists among inputs to purchasing industry $$j$$. Increasing or decreasing industry j output requires increasing or decreasing industry i inputs in the same proportion. “In other words,” according to the Bureau of Economic Research for the U.S. Department of Commerce, “…businesses can use as many inputs as needed without facing higher prices.” An implication is that no bottlenecks in supply are recognized in static input-output models, even though economic difficulties during the COVID-19 pandemic show that supply constraints are a disruptive fact of economic life. In these ways, coefficients in $$\textbf{A}$$ are assumed to be fixed.

Homogeneous

All firms and establishments in purchasing industry $$j$$ use the same, fixed proportion of industry $$i$$ inputs. That is, no variation exists in resource and product markets within an industry. Same suppliers; same customers; same technology of production. Every producer in an industry does the same thing and is affected by market forces in the same way. In this sense, industries are rendered homogeneous in input-output modeling.

An aside about terminology and classification applicable to the the assumption of homogeneity of industries in input-output modelings:3

1. A fundamental rule followed in most input-output modeling is that the least common denominator of analysis occurs the level of an industry, which is an aggregation of activity by highly similar goods or services and business activity, and not by firm or within establishments run by firms. Industry output is produced by firms, which are for-profit business organizations, such as a corporations, limited liability companies, or partnership. A number of firms can contribute to an industry. As I shall explain in the next item in this side discussion, industries are not mutually exclusive collections of firms. Separate industries might include some of the same firms. Also, firms might own or manage multiple establishments at which industry production occurs. An establishment is a single location where business is conducted or where services/operations are performed.4 In general, the production flow moves from establishments within firms and, then, to industries.

2. To add complexity to the input-output accounting scheme, some firms produce multiple goods and services that are classifiable in multiple industries. For instance, the production of food-safe, non-denatured alcohol distilled from grain5 is accounted as output of the Denatured Alcohol Manufacturing Industry (NAICS6 325193, ). Grain remaining after distillation is a co-product7 product of denatured alcohol manufacturing and could be classified as a product of the Other Animal Food Manufacturing Industry (NAICS 311119, ). One single process, two separate products.

3. An economy is understood as an open system that can gain or lose income or products from outside its boundaries through trade. Moreover, any aggregations such as firms, cities, geopolitical regions, countries, or the entire world could form the boundaries of an economy examined. As described the Encyclopedia of Social Measurement , large, diversified economies producing many goods and services can demonstrate large economic impacts because (a) a large region can fulfill its demand for goods and services from within the region, (b) transportation of goods and services with the region is simpler than hauling in these resources, and (c) economic activity in large region tends to agglomerate sales to surrounding regions. For these reasons, specification of the kind and level of aggregation of industry data can have a profound effect on an assessment of economic impact.

Linear

Neither economies nor diseconomies of scale occur as producing industry $$j$$ output increases or decreases (i.e., linear). Also, this assumption implies that scale – and the institutional knowledge aggregated from collective work – has no bearing on the dollar value of resources required for production or for the functioning of the product market.

##### Are These Assumptions Realistic?

How correct are these assumptions? How important even is verisimilitude of the assumptions with reality?

These three assumptions – fixed, homogeneous, linear – about the nature of direct input-output requirements are, of course, highly restrictive and probably are unrealistic attributes to assign to most economies. In spite of that, many analysts choose to believe that these assumptions have proved tenable within limits and useful for economic analysis in static contexts, even without adding additional complexity to expand the simple model represented by equation (2). Rather, the use of the model often rests on the verisimilitude of its assumptions with reality. Many analysts have relied on the pragmatic view that these assumptions are close enough to short-term reality to allow answers to economic problems that do not lend themselves to other analytic techniques .

The John Locke Foundation is one critic of common input-output modeling applications. The Foundation asserts that many modeling efforts fail to consider opportunity costs – that is, forgone benefits that could accrue from an option not chosen. Most studies using input-output analysis only consider benefits of a project, but fail to consider or model the costs of a project.

Of course, any new economic activity viewed without costs always will demonstrate unalloyed benefits. Something is larger than nothing, right? The “big-benefits-with-few-costs” approach is quite compelling to legislators, policy-makers, and funding agencies seeking to convince the public that projects, events, or changes are indispensable for economic growth and community revitalization . The John Locke Foundation asserts that input-output modeling is so often conducted “by professional consulting firms who are hired by interest groups to do the studies. Furthermore, seldom do those who actually perform the studies have formal training in economics. Instead their expertise is in using…proprietary [input-output] models ”.

Ignoring opportunity costs and other modes of failure to think ahead to consider full economic consequences prompted Frédéric Bastiat, French economist best known for his journalistic writing in favor of free trade and the economics of Adam Smith, wrote in his last published pamphlet in 1850 that:

In the economic sphere an act, a habit, an institution, a law produces not only one effect, but a series of effects. Of these effects, the first alone is immediate; it appears simultaneously with its cause; it is seen. The other effects emerge only subsequently; they are not seen; we are fortunate if we foresee them.

There is only one difference between a bad economist and a good one: the bad economist confines himself to the visible effect; the good economist takes into account both the effect that can be seen and those effects that must be foreseen .

I do not mean to ignore these importance of these assumptions and criticisms of them. I will take up these matters directly as I review reports of economic impact in other postings on this blog.

#### Indirect and, then, Total Requirements (The Leontief Inverse)

Introducing for mathematical convenience an $$I x J$$ identity matrix, $$\textbf{I}$$, and algebraically rearranging terms in equation (2), the relationships among production, consumption and economic output is expressed as

$$$\tag{3} \textbf{x}=(\textbf{I} - \textbf{A})^{-1}\textbf{y},$$$

where $$(\textbf{I} - \textbf{A})^{-1}$$ is called a total requirements matrix that shows the additional dollar value of output from industry $$i$$ needed directly or indirectly to deliver a dollar’s worth of output from industry $$j$$ to final demand in the economy. The matrix inverse, $$(\textbf{I} - \textbf{A})^{-1}$$, often is described as the Leontief Inverse,8 in honor of Wassily Leontief , an economist who won the Nobel Memorial Prize in Economic Sciences in 1973 for developing economic input–output analysis theory and methods . Actual economies can have large numbers of industries, so $$\textbf{A}$$ often is highly dimensional. As a consequence, $$(\textbf{I} - \textbf{A})^{-1}$$ usually is solved numerically by computational techniques involving Laplace transforms and Lagrange multipliers.

Taking the difference between the total requirements matrix and the direct requirements matrix produces an indirect requirements matrix. Indirect requirements are generated from the long string of transactions among industries representing intermediate production that must occur to generate a final product for consumption. The addition of households as both a row and column in $$\textbf{X}$$ allows inclusion of the effects of household production and purchasing into the model. This addition augments the total requirements matrix to include not only direct and indirect requirements but also to show induced effects of household spending on the economy as it stimulates purchases of goods and services. The difference between the total requirements matrices, net of the indirect requirements matrix, including household spending produces an induced requirements matrix.

#### Multipliers

Indirect and induced requirements matrices display how each dollar spent on goods and services delivered to consumers multiplies itself throughout the economy. Ratios between [(direct requirements $$+$$ indirect requirements) $$/$$ direct requirements] as well as ratios between [(direct requirements $$+$$ indirect requirements $$+$$ induced requirements) $$/$$ direct requirements] are called Type I and Type II multipliers, respectively. Pre–multiplying the direct, indirect, and induced requirements matrices by, say, total employment parses the impact of delivery of final demand to consumers into direct, indirect, and induced employment components. In a similar manner, multipliers for other economic indicators – such as economic output, value added, income, taxes – can be computed.

People who commission input-output modeling of potential economic impact often are most interested in multipliers to demonstrate that an investment of public funds in a project can benefit an entire community. For example, an analysis of the potential economic impact of a petrochemical manufacturing facility in Beaver County (north of Pittsburgh, PA) concludes

…operations will involve approximately 240 to 450 new jobs for Beaver County residents at the petrochemical facility. With multipliers, total new jobs held by Beaver County residents are anticipated to be between 777 and 1,444. Annual labor income increases in Beaver County will be between approximately $73 and$120 million. Over the 40-year operational life of the Project, labor income increases within Beaver County are estimated to total between $1.5 and$2.4 billion. Annual value added in Beaver County would be between $260 and$846 million. Over the assumed 40-year operation period, this equates to a present value of $10.3 to$16.7 billion.

This analysis and other assertions about economic impact of operation of the petrochemical facility supported state and local tax breaks for the developer:

Pennsylvania’s fiscal incentive package — which includes a large tax credit along with job-training incentives — was also crucial to the company’s decision to both proceed with the project and locate it in Pennsylvania, Visser said.

In 2012, state lawmakers authorized a tax credit for companies that use ethane that invest more than $1 billion and employ more than 2,500 workers during construction. Shell was given a 25-year tax break worth about$66 million a year starting in 2017, or about \$1.65 billion in total. The state also extended a tax-free Keystone Opportunity Zone in Beaver County and provided other grants.

“I can tell you with the hand on my heart that without these fiscal incentives, we would not have taken this investment decision,” Visser [Royal Dutch Shell representative] said.

### Use of Counterfactuals to Identify Potential Economic Impact

#### What is a Counterfactual?

Input-output modeling is useful for describing of the anatomy and physiology of an economy. However, these models also offer a basis for simulating economic scenarios that are “counter to the facts” to inform risk assessment.

A counterfactual is a hypothetical scenario that runs “counter to the facts” of an existing reality. A counterfactual condition entails “What if…?” questions that represent thought experiments about propositions that run counter to the facts. For instance, how would an economy have grown and changed if an activity or event did not occur? How do supply chain constraints or disruptions affect an innovation? The potential impact of an activity, event, or constraint is the difference between what would have happened with and, then, without the activity, event, or constraint.

Researchers long have used the counterfactual approach to consider strategic decisions, decisive acts, and paths of activity that would have altered the course of history. For instance, in Ab Urbe Condita Libri ,9 Titus Livius Patavinus, a Roman historian, contemplated an alternative in 4th century BC by questioning, “What would have been the results for Rome if she had been engaged in war with Alexander?” Lewis indicated that the modern counterfactual approach goes back at least to the 18th century Scottish Enlightenment philosopher David Hume.

Historians, essayists, and novelists have applied counterfactual thinking productively in a variety of situations. For instance, in 1931 Winston Churchill wrote an essay, “If Lee Had Not Won the Battle of Gettysburg,” that examined the counterfactual viewpoint about what would have happened if the Confederacy had won the American Civil War10 (see counterfactual examples regarding the American Civil War ). In 1962, Philip K. Dick, a novelist in the science fiction genre, published The Man in the High Castle , a narrative of a thought experiment that considered the consequences if Nazi Germany and Imperial Japan had won the Second World War.

#### Constructing Counterfactuals Within Input-Output Modeling

Input–output modeling can help examine counterfactuals. The typical way to conduct such a counterfactual approach is to construct and compare two economic scenarios. The first scenario contains a possible alteration in the economy – the counterfactual – that would be identified in input-output terms an economic changes resulting from, say, an innovation (e.g., a new production process), event (e.g., a disaster), policy (e.g. immigration regulation), or financial rule (e.g., taxing authority). A second scenario merely maintains the “status quo” of the economy under the assumption that the economy simply grows and changes consistently according to historical trends. The difference between the first and the second scenarios is the potential economic impact of the economic alteration specified in the first scenario.

The results of such counterfactual economic analyses are not economic forecasts. Rather, such analyses isolate the importance only of one narrow group of changes in an economy by showing the gross effect of these changes rather that the effect of changes net of all other potential changes taken together that could affect an economy. A counterfactual used with input–output methods typically is specified under the assumption of ceteris paribus – that is, with the assumption of “all other things remaining the same” or, as is much more restrictively used in physics and other sciences , “if, and only if, all other things are the same and correct” More restrictively, ceteris absentibus, a term meaning literally “others absent,” probably best describes the approach for economic counterfactuals .

#### An Example of a Counterfactual Specification

A potential impact of a counterfactual is computed by calculating the difference between (a) an economic scenario created by a counterfactual condition exemplifying a production or consumption shock and (b) a status quo scenario representing an economy anticipated under historical trends in growth and change. Shown in Table 1 are three classes of counterfactuals that are possible in a hypothetical case for the production and consumption of transparent aluminum, a material I introduce only to offer an example of the structure of potential counterfactuals in input-output analysis.

Table 1. Three hypothetical macroeconomic counterfactuals specified to assess risk to production and consumption of transparent aluminum

Counterfactual description Change specified in macroeconomic model
Change in consumption of transparent aluminum $$\Delta y_{i'}$$, where element $$y_{i'}$$, is total final demand for industry $$i'$$, the transparent aluminum industry (where $$i'\subseteq I$$).
Change in production function for the transparent aluminum industry $$\Delta a_{j'}$$, where vector $$a_{j'}$$ is a column vector representing the proportion of inputs from $$I$$ industries purchased as inputs to the transparent aluminum industry (where $$j'\subseteq J$$).
Change in production function of a purchaser of transparent aluminum $$\Delta a_{j}$$, where vector $$a_{j}$$ is a column vector representing the proportion of inputs from $$I$$ industries purchased as inputs to any industry other than the transparent aluminum industry (where $$j' ≠ j$$).

Transparent aluminum, a ceramic composite technically known as aluminum magnesium oxynitride, is an extremely durable, sintered crystalline material composed of magnesium, aluminum, oxygen, and nitrogen (AlN-MgO-MgAl2O4) that is optically transparent (≥80%) in the near-ultraviolet, visible, and midwave-infrared regions of the electromagnetic spectrum. The name of this material comes from its similarity to a material referenced in the science fiction movie, Star Trek IV: The Voyage Home.11

Transparent aluminum is a promising material that is four times harder than fused silica glass, 85% as hard as sapphire, and resists corrosion, radiation, and oxidation . Products requiring both durability and transparency – such as armored glass, windshields on airplanes and sea vessels, windows for laser and infrared communications, personal protective gear, smartphone and watch screens, and sensor components – are among the potential uses for transparent aluminum.

##### Change in consumption of transparent aluminum

Using equation (3), implementing this counterfactual involves substitution of $$y_{i'}$$ for $$y_{i}$$ and solving for $$(\textbf{x}=(\textbf{I} - \textbf{A})^{-1}\textbf{y'}$$, where $$y'$$ is a new column vector of total final demand for industry $$i'$$, the transparent aluminum industry ($$i'\subseteq I$$). The result is a direct change in transparent aluminum output, along with indirect and induced changes of the total output of all industries to the extent that they have direct or indirect interindustry linkages to the transparent aluminum industry.

##### Change in production function for the transparent aluminum industry

Changing a production function involves altering the proportion of total products from other industries purchased by the transparent aluminium industry. This change could result from efficiencies affected in the production of transparent aluminium. Or, supply chain constraints or shortages might have required changes in the manner that transparent aluminium is produced. Such a change is manifest in alterations in $$\textbf{A}$$. As a result, $$(\textbf{x}=(\textbf{I} - \textbf{A})^{-1}$$ also is altered, reestablishing the input-output identity to $$(\textbf{x}=(\textbf{I} - \textbf{A'})^{-1}\textbf{y}$$ where $$\textbf{A'}$$ is $$\textbf{A}$$ altered by $$a_{j'}$$, a column vector representing the proportion of inputs from $$I$$ industries purchased as inputs to the transparent aluminum industry ($$j'\subseteq J$$). The result observed, holding transparent aluminum output constant, is indirect and induced changes of the total output of all industries to the extent that they have direct or indirect interindustry linkages to the transparent aluminum industry.

##### Change in production function of a purchaser of transparent aluminium

The consequence of this scenario is a similar change similar to changes in the production function for the transparent aluminum industry. However, the result observed, holding output constant in the industry making the production function change, is indirect and induced changes of the total output of all industries, including the transparent aluminum industry, to the extent that they have direct or indirect interindustry linkages to the industry making the production function change.

### Selected Software and Systems for Input-Output Modeling

Many software and systems are available to analyze input-output data from the national income and product accounts of the United States (viz description of accounts by the Bureau of Economic Analysis of the U.S. Department of Commerce . Three prominent sets of software and systems are IMPLAN, RIMS II, and IO-Snap.

#### IMPLAN

IMPLAN (IMpact for PLanning and ANalysis) is a prominent example of software used estimate the direct, indirect, and induced impacts in regional economies. The IMPLAN economic impact modeling system consists of software tools and region–specific data to enable in–depth examinations of state, multi–county, county, sub–county, and metropolitan regional economies. IMPLAN seems to have a large share of the market for input-output modeling. Some writers (e.g., ) describe IMPLAN as the “gold standard” for input-output analysis.

#### RIMS II

RIMS II is a product of the Bureau of Economic Analysis of the U.S. Department of Commerce that is based on a set of national input-output accounts showing the goods and services produced by each industry and the use of these goods and services by industries and final users.

#### IO-Snap

IO-Snap is a software application that includes all aspects of fundamental input-output analyses. Data can be edited and displayed in a variety of formats and compared and contrasted across geographical definitions at different times.

## Last updated on

[1] "2022-04-22 07:43:38 EDT"

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1. I am not pretending to advance an original or new conception of input-output modeling. Many competent descriptions of this model are evident on the Internet and extant scholarly literature. Frankly, I wrote this posting for more personal reasons: I abide by the apothegm that a person knows nothing about a topic until that person had to write about it.↩︎

2. Although the notation seems awkward to have an element of matrix $$\textbf{X}$$ indicated by what would appear to most to be the scalar $$x$$ and the column vector of total output indicated by $$\textbf{x}$$, I am following notation conventions that are common in specifications of the mathematical properties of input-output modeling.↩︎

3. And to most economic analysis that uses public data to focuses on industry attributes.↩︎

4. If employees normally do not work at a single physical location, the establishment is identified by the main office that supervises employees’ activities or is the base at which employees carry out these activities. ↩︎

5. Wheat, barley, corn, sorghum, or cereals containing 55%-70% starch.↩︎

6. The North American Industry Classification System (NAICS) is the standard used by Federal statistical agencies in classifying business establishments for the purpose of collecting, analyzing, and publishing statistical data related to the U.S. business economy .↩︎

7. A co-product, a desirable secondary good that is generated during the manufacturing process and can be sold or reused profitably, is different than a by-product, a material of value that is produced as a residual of, or incidental to, the production process. Generally, co-products generate revenue; by-products are associated with costs.↩︎

8. -1 refers to an inverse of a matrix, not a reciprocal.↩︎

9. Book digitized by Google from the library of the University of Michigan and uploaded to the Internet Archive by user tpb.; 5 volumes 20 cm; Pars 1. liber I-VI. – pars 2. liber VII-XXIII. – pars 3. liber XXIV-XXX. – pars 4. liber XXXI-XXXVIII. – pars 5. liber XXXIX-XLV. Epitom. lib. XLVI-CXL. – pars 6. Fragmenta et Index↩︎

10. If Churchill’s title seems like a non sequitur, refer to the explanation of his rhetoric at http://www.troynovant.com/Franson/Churchill/If-Lee-Not-Won-Gettysburg.html.↩︎

11. Kirk and the Enterprise crew must save the whales to, in turn, save planet Earth. They must contain the whales in water on the Enterprise. The crew travels back in time to 20th Century Earth, when transparent aluminum had not been invented yet. Scott, the Enterprise’s chief engineer, shares the formula for transparent aluminum with a 20th Century scientist so that the Enterprise could be outfit with the material crucial to building tanks for the whales.I dunno…it’s supposed to be one of the light-hearted Star Trek movies.↩︎