Each annual release of IMPLAN's regional data has a unique national structural matrix file.

### National I-O Structural Model

The structural matrices are the basis of the Inter-Industry flows (the flow of dollars between Industries). There are two structural matrices: the Use Matrix and the Make Matrix. The Make Matrix shows the production of Commodities made by each Industry. The Make matrix has two coefficient forms: Byproducts Matrix and Market Shares Matrix. The Use Matrix shows the use of Commodities by each Industry. The Use Matrix, in coefficient form, is the Absorption Matrix. The commodities an industry buys per dollar of output is also known as the Production Function.

#### Make/Byproducts Matrices

**Make/Byproducts Matrices: **These matrices show the value of all the commodities each industry produces. These matrices are thus IxC (rows are industries, columns are commodities).

- Make Matrix = production values
- Byproducts Matrix = coefficients, calculated by dividing each value by the row total (rows sum to 1). The Byproducts Matrix shows the commodity make-up of each industry's production.
- Market Share Matrix = coefficients, calculated by dividing each value by the column total (columns sum to 1). The Market Shares Matrix tells us the proportional share each industry has of the region's production of a commodity.

The Make Matrix represents the make, or production, of commodities by a given industry. The Make Table from the latest BEA Benchmark I-O Study of the U.S. is price-updated to the current year and forms the basis for the IMPLAN model. Rearranging the U.S. Make Matrix into IMPLAN format allows us to divide each row element by the row total to create a Byproducts Matrix.

Since the Absorption Matrix are coefficients and we do not have Total Commodity Output (TCO) controls, it is not necessary to RAS the Make Table. Accepting the Byproducts Matrix now makes it possible to calculate TCO as the sum of each column of Total Industry Output (TIO), distributed across the matrix.

#### Use/Absorption Matrices

**Use/Absorption Matrices: **These matrices show the commodity purchases each industry makes in order to produce its output. They are CxI matrices (rows are Commodities, columns are Industries).

- Use Matrix = industry outlays for intermediate goods and services.
- Absorption Matrix = coefficients, calculated by dividing each value in the Use matrix by the column total (columns sum to 1). The absorption matrix is also known as the production function.

The creation of the Use Matrix is more complex than the Make Matrix. The TIO and TCO data are first estimated as just described. Value-Added is estimated as described here and Final Demand is estimated as described here. We then bridge the BEA Use Tables to the 546 IMPLAN sectors. One step in this process is splitting out the aggregated retail industries, which is done based on income - i.e., Employee Compensation, Proprietor Income, and Other Property Income. In general, there is very little relative difference in production functions amongst the IMPLAN retail trade industries that conform to a BEA retail aggregate. There are, however, differences in Value-Added - specifically Employee Compensation, Proprietor Income, and Other Property Income.

The current 546 scheme uses the production functions from the 2012 BEA benchmarks. Each succeeding BEA benchmark has become more aggregated. We have updated the more detailed sectors from previous benchmarks and controlled all production functions to the 2012 BEA benchmark.

Industry | Commodity | Factors | Institution | Trade | Total | |

Industry | Make | Exports | TCO | |||

Commodity | Use | Final Demands | Exports | TIO | ||

Factors | Value Added | Exports | ||||

Institutions | Sales | Transfers | Transfers | Exports | ||

Trade | Imports | Trade | Imports | |||

Total | TIO | TCO |

Once we have a preliminary set of National structural matrices, the absorption table can be adjusted. Value Added, Final Demands, TIO and TCO are placed in a table as shown above. At this point, all National information except for the final Use is complete. To complete the national Use Matrix, the intermediate industry and commodity output values are calculated. The intermediate outputs are used as new row and column control totals for the Use Matrix. The Use Matrix is then RAS'd (Ratio Allocation System) {bi-proportionally adjusted} to match the new row and column totals.

After the adjustments are made, the national model balances with total Value Added equaling total Final Demand. TIO equals TCO, making intermediate industry and commodity output equivalent.

The domestic trade flows for states and counties are created based on IMPLAN's gravity model.

### Matrix RAS

The RAS procedure, sometimes called the Ratio Allocation System or Richard A. Stone system, actually refers to the appearance of the variables describing the coefficient matrix in the original paper: (rAs). RAS procedures are used to re-balance matrices. This process is used numerous times throughout the IMPLAN database development process. The procedure requires a matrix of size M x N and a vector of row size N and column size M totals. Pre- RAS matrix row and column totals will not equal the vector of row and column totals.

The RAS procedure forces the matrix to sum to the vector of row and column totals. This is accomplished by calculating the difference between the new and old row and column totals and distributing the differences iteratively until the differences drop to zero. This allocates the vector row and column elements to the matrix based on the matrices distribution pattern. The result is a new matrix consistent with the vector of row and column totals.

### Regional Absorption

The national gross absorption coefficients (percentages) are adjusted to each region's value added per output ratio for each sector. For example, if an industry in a particular region has higher value-added per dollar of output, then that industry must have correspondingly lower Intermediate expenditures per dollar of output, so the national gross absorption coefficients get adjusted downward proportionately so that the sum of absorption coefficients + VA/Output = 1. To get the regional absorption coefficients (net of imports), each gross absorption coefficient is multiplied by the RPC for that commodity, which varies across regions.

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