## INTRODUCTION

Input-Output (I-O) Analysis is designed to show the ripple effects of a given economic activity in other Industries. Production in a given Industry supports demand for production in other Industries throughout the economy, both due to supply chain spending and spending by workers. This spending is derived from the I-O and Social Accounting Matrix (SAM) model and often referred to as the multiplier effect.

Multipliers are calculated via matrix algebra, but most commonly and easily understood as unitless ratios. Multipliers measure an Industry's connection to the wider local economy by way of input purchases, payments of wages and taxes, and other transactions. Multipliers are the total production impact within a Region for every unit of Direct production. Total production will vary depending on the method of inclusion (whether the Induced Effects are included or not). Applying multipliers to a Direct economic activity will result in Indirect and Induced economic effects as described in this article.

## ECONOMIC EFFECTS

The economic impact of an Industry consists of the following three types of effects which sum to the total effect.

### DIRECT EFFECT

A Direct effect is the initial exogenous change in final demand or set of expenditures applied to the I-O multipliers for impact analysis. These initial changes are determined by an analyst to be a result of the activity or policy being analyzed. Direct effects can be positive or negative. Applying these initial changes to the multipliers in IMPLAN will then display how the Region will respond economically to these initial changes.

For Industry, Commodity, and Institutional Spending Pattern Events, the Event Value entered is assumed to be final demand. Thus, the Event Value(s) entered in these Event Types will be used to determine the Direct Effect of the Event. Labor Income, Household Income, and Industry Spending Pattern Events do not generate Direct Effects because there is no initial Industry spending.

### INDIRECT EFFECT

Indirect effects are the business-to-business purchases in the supply chain taking place in the region that stem from the initial Industry input purchases. As the Industry specified in an Event spends its money in the region with their suppliers, this spending is shown through the Indirect Effect.

Labor Income and Household Income Events do not generate Indirect Effects because no initial Industry is specified in these Event Types.

### INDUCED EFFECT

The Induced Effects stem from Labor Income being spent, after removal of taxes, savings, and commuter income. This includes spending throughout the selected region(s) associated with the Industries specified in the Event (Direct Effect) and those impacted through the supply chain (Indirect Effects).

IMPLAN does not assume that 100% of this Labor Income is spent, nor that it is spent locally. IMPLAN removes payroll taxes, personal income taxes, savings, in-commuter income, and non-local purchases before spending the rest locally. These leakages and expenditures are based on information in the SAM.

## MULTIPLIERS

Multipliers are a measure of an Industry's connection to the wider local economy by way of input purchases, payments of wages and taxes, and other transactions. Multipliers are the total production impact within the Region for every unit of direct production. Total production will vary depending on the method of inclusion.

Type I Multiplier measures an industry's connection to the wider local economy by way of input purchases only (no induced effects, no institutions internalized). Type SAM Multiplier (where SAM stands for Social Accounting Matrix) measures an industry's connection to the wider local economy by way of input purchases, payments of wages and taxes, and other transactions.

The calculations for each type of Multiplier is as follows:

**Indirect Multiplier = **(Indirect Effect) / (Direct Effect)

**Type I Multiplier = **(Direct Effect + Indirect Effect) / (Direct Effect)

**Induced Multiplier = **(Induced Effect) / (Direct Effect)

**Type SAM Multiplier = **(Direct Effect + Indirect Effect + Induced Effect) / (Direct Effect)

Both Type I and Type SAM multipliers represent the Total Effect to All Industries in a Region per Direct Effect, or a Summary Multiplier. The Detail Multiplier breaks the Summary Multiplier into partial Multipliers associated with purchases from other Industries.

### HOW MULTIPLIERS ARE CALCULATED

Multipliers are calculated via matrix algebra, but most commonly and easily understood as unitless ratios. While the complex process of creating the Social Accounting Matrix is not described here, the results of those calculations are a complete transactions table showing what every Industry needs to purchase in order to make its products and the value of every Industry's labor payments (Labor Income), taxes (Taxes on Production & Imports), and profits (Other Property Income) and what each Household income group buys.

We also know how much of each Commodity is produced locally, which Industries or Institutions produce it in the local economy, and how much of the production is attributed to each producer. The combination of these factors allows the application to determine, based on the entered or estimated value of Industry Output, how much of each Commodity will be required to meet the change in production of the target Industry (Gross Absorption) and how much can be obtained from local vendors (Regional Absorption). After multiple rounds of purchases are accomplished and all the spending not attributed to local vendors is lost to leakage, the resulting values spent locally on each Commodity can be summed to show the total purchasing requirements for that Commodity from the local economy in dollars and cents.

The Output of any given Industry (Xi) goes to meet intermediate demands for the Industry (XiAi), where Ai represents the production function for Industry i, plus Final Demand for the Industry (Yi). That is, the Output of Industry i (Xi) depends on all other Industries' Output (X) times their requirements for Industry i's Output as one of their inputs (as determined by their production functions, (A)), plus the Final Demand for Industry i's Output (Yi). Thus, if X = the matrix of all Industries' Output and Y = the matrix of Final Demand for all Industries and A = the matrix of production functions for all industries, then X = AX + Y. Solving for Y we get Y = X (I-A)-1. (I-A)-1 is known as the Leontief Inverse. For any one particular Industry this equation becomes Yi = Xi (I-A)-1. This equation tells us the Output of each and every Industry that is required to meet Final Demand of Industry i. If we wanted to know how much each Industry's Output would change in response to a change in the Final Demand of Industry i, we would modify the equation to ∆Yi = ∆Xi (I-A)-1. In other words, to meet a change in the Final Demand for Industry i (∆Yi) we need to increase that Industry's Output (∆Xi) plus the Output of that Industry's input suppliers (∆Xi*A).

### AVERAGE OUTPUT MULTIPLIER RANGE

In IMPLAN, one can find Summary or Detail Multipliers for *Output*, *Employment*, *Labor Income*, and *Total Value Added,* as well as for each of the components of Value Added: *Employee Compensation*, *Proprietor Income*,* Other Property Income*, and *Tax on Production and Imports*.

Typically we expect Output Type SAM Multiplier ranges to follow this general pattern:

- at a county level an Output Multiplier is between 1-2
- at a state level an Output Multiplier is 2-3
- at a national level the expected range is 2-7.

However, individual regions may vary greatly depending on their concentrations of activity.

### AGGREGATE MULTIPLIERS

Multiplier specificity is a key to accuracy within an analysis. The more disaggregate an Industry specification is the more accurate the results of the analysis will be, as the Multiplier for each Industry reflects:

- the target Industry's specific purchasing pattern
- its specific relationships for Labor Income / Worker
- its specific relationships for Output / Worker
- its specific relationships for Other Property Income / Output
- its specific relationships for Taxes on Production & Imports / Output

However, there are times when it may be necessary to aggregate Industries together in order to perform an analysis. Please read more about Aggregation Bias and Aggregating Industries if you are unable to attach your dollar value to a specific Industry in IMPLAN.

### MRIO MULTIPLIERS

The MRIO methodology followed by IMPLAN works iteratively. That is, we do not construct a single, multi-regional SAM and thus do not have pre-calculated multi-regional multipliers. However, MRIO multipliers can be calculated after an analysis is run.

IMPLAN uses each region's local multipliers and loops through the analyses between regions following the commodity trade rates based on the previous regions' demands. As a result, the IMPLAN MRIO methodology only needs the local multipliers for each region in the analysis along with the trade between those regions to loop through the individual analysis until the trade between regions is sufficiently small.

## OTHER TYPES OF MULTIPLIERS

There are actually five total types of multipliers! While IMPLAN utilizes the Type SAM, here is an outline of each type just for fun.

- The Type I Multiplier only includes direct and indirect effects.
- The Type II Multiplier adds induced effects but only utilizes a single Household row and column.
- The Type SAM Multiplier, used by IMPLAN, builds on the Type II multiplier. The Type SAM multiplier uses more information than the Type II so it is more consistent with reality. The Type SAM incorporates both the Household and Labor Income as both rows and columns, and thus allows for direct tax payments and commuting, which the Type II does not. The Type SAM with the households internalized is considered the best conservative estimate and maximizes the use of the social accounts detail available in the model.
- The Type III Multiplier has not been the multiplier of choice since the DOS version of IMPLAN written by the Forest Service. The Type III multiplier used population change as a driver for the induced effect, so all jobs were treated equally (e.g., a fast food worker had the same induced impact as a medical doctor). With the Type II and Type SAM multipliers, income is the driver, such that the higher the worker's income, the larger the induced effect.
- The Type IV Multiplier takes this a step further and separates out spending patterns between employed and unemployed residents (Miller & Blair, 2009, P. 256).

## VIDEOS

The Multiplier Effect in Economics

## REFERENCES

Miller, R.E. and P.D. Blair. 2009. *Input-Output Analysis: Foundations and Extensions*, *Second Edition*. New York: Cambridge University Press.

Written August 30, 2023