Industry Leontief Production Functions in IMPLAN


Backward linkages are the interconnection of an industry to other industries from which it purchases inputs to produce its output. IMPLAN's most current data contains 546 defined industries. Learn more about how these industries are defined and how select the most appropriate industry for your analysis by reviewing this Picking an Industry article.

Each industry purchases Commodities from Industries to operate their business, which generates the initial round of backward linkages for each industry. Each cell in the matrix below represents a payment from the Industry column to the Commodity row. In other words, the dollars spent by a given Industry to purchase operational Inputs is represented by the columns in the matrix below. These dollars are referred to as Intermediate Inputs. IMPLAN's data source (see production function section) on Input purchases by Industry is only available at the National level. IMPLAN is an annual model so the input-output data in the matrices will be based on a given year. Intermediate Inputs for each Industry can be converted to a Spending Pattern by dividing each expenditure by the total value of all Intermediate Inputs for the given Industry, producing a percentage for each Intermediate Expenditure that sum to 100% for the given Industry. 



Industries of course spend money on things other than Intermediate Inputs. The Production Function of an Industry in IMPLAN determines how an Industry will allocate Output to continue to operate. To complete each Industry’s Production Function, we’ll need to incorporate the other production values. Industries must pay taxes or Taxes on Production and Imports. They also require Labor, who are paid Labor Income, to create the final good or service. The Production Function also leaves room for profit to be earned and to cover things like the purchase of equipment, technically referred to as Other Property Income. These additional factors of a Industry's Production Function are called Value Added.  The combination of a Industry’s spending on Value Added and Intermediate Inputs equals the Industry’s Total Output.  IMPLAN collects sub-national data on Value Added and Total Output by Industry. 

Using regional dollar values for Total Output and Value Added for each Industry, a Value Added Coefficient can be calculated (Value Added / Total Output). Knowing the Production Function represents 100% of the distribution of total Output for a Industry, this regional Value Added Coefficient is used to adjust the National Spending Patterns so that the sum of the coefficients allocated to each Commodity purchase, Total Gross Absorption, plus the Value Added Coefficient, equals 100%, while maintaining the ratio of the Commodity purchases to one another.

These regionalized Spending Patterns are further regionalized by converting Gross Absorption to Regional Absorption. The Commodities purchased by each Industry cannot all be sourced from within a closed economy; some Commodity purchases are imported from other regions. Therefore, each Commodity has a model-specific Regional Purchasing Coefficient.

Multiplying each Gross Absorption for a Industry by the Commodity’s RPC produces the Regional Absorption for the Industry and Region.  The sum of Regional Absorption and the Value Added Coefficient for each Industry will be less than 100%. Because an RPC tells us how much demand is sourced locally, 1 minus the RPC tells us how much demand will be sourced non-locally, according to the model for each Commodity. You can think of this 1-RPC value as the import rate of the Commodity in the Region.  The difference between 100% and the sum of Regional Absorption and Value Added for each Industry is the portion of dollars IMPLAN will model as leaving the Region when the effect of production in each Industry is analyzed. Dollars that model does not continue to circulate through the Region’s economy are considered leakages. The import of Commodities is one form of leakage.  





Recalling the definition of backward linkages, these linkages are from Industries to other Industries, but our matrices currently maps Industry’s purchases of Commodities, so IMPLAN converts Commodities to Industries. This is another way in which regionality is factored. Commodities in IMPLAN are numbered 3001 through 3536 to distinguish them from Industries. The Industry number for the primary producer of a Commodity can be translated by removing 3000 from the Commodity number, but Commodities can be sold/purchased from multiple sources. For some Commodities there are multiple producers and some Commodities can be stored in inventory and sold in a later year. Therefore, Commodity Output can be tied to production by an Industry or Industries, and/or sales by an Institution (which includes inventory). The sum of the sources of a Commodity is called Commodity Supply. The portion of Commodity Supply that comes from each of these sources is called a Market Share. These Market Shares are based on the Commodity value from each source divided by the total value of the Commodity Supply. The Commodity value from each source and the Commodity Supply are specific to the Data Year and Region. Therefore, to convert the Regional Absorption for each Commodity to be Industry based, the Regional Absorption is distributed to the suppliers of the Commodity based on the Market Shares (found in Region Details > Social Accounts > Balance Sheets > Commodity Balance Sheet > Industry-Institutional Production). Through this conversion, the y-axis will become Industries 1 through 536 and a new Industry based Spending Pattern is produced for each Industry. An Industry could be producing more than one Intermediate Expenditure Commodity, meaning they may make up some of the Market Share for more than one Commodity in a given Industry’s Spending Pattern. The portion of production dedicated to producing each commodity in a given Industry is called a Byproduct Coefficient (found in Region Details > Social Accounts > Balance Sheets > Industry Balance Sheet > Commodity Production). 

Market Share refers to the portion of the total locally-produced supply of a Commodity that is produced by an individual Industry or Institution in the Study Area. Learn more at Industry Leontief Production Functions in IMPLAN.

For example, if $100 of Commodity X is produced by three Industries (Industry A, $50; Industry B, $30; Industry C, $20) their respective Market Shares would be 50%, 30% and 20%. Market Shares are often used during impact analysis to distribute the total demand for a Commodity among the Industries and Institutions that produce it. When a Commodity has some market shares from Institutional sales (such as government production or inventory), these portions of the local Commodity output are leakages. 

Market Shares allocated to Institutions (including Inventory) will not be modeled as continuing to circulate through the Region’s economy, treating these portions as a leakage.

The leakage due to institutional sales further decreases the sum of the Production Function for each Industry. This conversion makes the top matrix a true business to business Input-Output Matrix.




The Income earned by individuals working to produce an Industry's output also affects the economy when that Income Is spent. Incorporating these effects classifies the IMPLAN model as a Social Accounting Matrix (SAM) Model.

To incorporate Households and Institutions into the model, an additional matrix for Final Demand is added. Final Demanders are, by definition, Institutions.  Final Demand is the value of goods and services produced and sold to final users to meet demand, whether it is local demand or an export.  Final Demand for the Region can be found in the Region Details > Overview > Final Demand table.

Particularly in the case of government purchases, some of these purchases can be incorporated in creating some other good or service, such as materials purchased to build a new highway.

To determine how payroll transitions from compensation to disposable income, a final fourth matrix is added, completing the larger matrix. Considering the spending of Income allows for further regionalization of the model. The full cost of Labor is called Labor Income in IMPLAN.

Not all dollars earned by an individual or household are spent by the household. There are several ways in which some dollars earned leak out of the model. All Labor Income, regardless of which Industry it was earned in, cycles through the model in the same way. The Employee Compensation and Proprietor Income columns, summing together to be Labor Income, make payments to the rows.  Converting these column values from dollars to percentages of the column total gives us a pattern for the region and year that reflects the transfer of dollars earned as Labor Income to Household Income dollars. 

Labor Income is distributed to the Household Income groups based on the percentage each of type Labor Income paid to each Household Income Group. Labor Income also makes payments to the Federal Government and may make payments to the State and Local Government depending on the Region. These are the payment of Payroll Taxes from Labor Income. Again, since we don’t know how, where, or when the government will use fiscal revenue, taxes paid are treated as a leakage and do not continue to circulate in the Region’s economy in the model. 

The region specific in-commuting rate is also applied to Labor Income paid to wage and salary workers to leak income earned by non-residents out of the model. The in-commuting rate is the portion of Employee Compensation that will be estimated to leak out of the Region due to employees earning income within the Region, but commuting home, outside of the Region, where they spend their money (Region Details > Social Accounts > I x C Social Accounting Matrix > I x C Aggregate SAM > EC column, Foreign and Domestic trade rows divided by column total). These columns will always be blank for Proprietor Income as it is assumed to be earned by residents of the Region. 

Income earned by Households locally is distributed to IMPLAN’s nine Household Income groups (categorizing ranges of income). 

The journey of Household Income, the payment to Household rows from Labor Income, continues in the Household columns. Household Income primarily makes a payment to Commodities. Households also transfer money to other households. Personal taxes are reflected as the payment to government rows from the Household Income Group columns. Savings of household income is reflected as the payment to the Capital row from the Household Income Group columns. Converting these column values of payments to government and capital to percentages of the column totals gives us the effective Personal Tax and Savings rates, respectively. 

We can break down the purchase of Commodities for each Household Income Group in Region Details > Study Area Data > Household Commodity Demand. Converting each Household Income group column from dollars to percentages of the column totals give us the Spending Pattern for each Household Income group. Like Industries’ purchases of Commodities, the amount of each Commodity purchased by households that will be modeled as a local purchase is also determined by the Commodity's RPC. Applying the average RPC for each Commodity would reflect the local spending per dollar of Household Spending on each Commodity. This information is reflected in dollars in Region Details > Social Accounts > Reports > Household Local Commodity Demand. The Commodities purchased by Households are converted to purchases to Industries also using the Market Shares.



Written September 26, 2019