SM5005 Chap.4 Building the Technical Coefficient Matrix
Building the Technical Coefficient Matrix
Normalize flows into production recipes
The transaction matrix reports absolute purchases, so large sectors naturally have large entries. The technical coefficient matrix removes that scale by dividing each transaction in column j by the total output X_j of the using sector. In matrix notation, A = T X-hat inverse, where X-hat is the diagonal matrix formed from sector outputs.
Entry A_ij = T_ij / X_j is the input from sector i required per unit of output from sector j.
The denominator carries the interpretation
Because a column is the recipe of a using sector, all entries in that column share the same denominator. Dividing by the supplier output would answer a different question about sales distribution and would not create the matrix required by the Leontief model.
A dimension and label check is faster than debugging a plausible-looking answer later.
Coefficients are conditional, not timeless constants
A coefficient reflects the technology, price structure, sector aggregation, geography and year of its table. It is often treated as fixed within the model scenario, but that is an assumption used to calculate the production response.
Structural change, substitution and price movements can make coefficients inappropriate for another period or decision.
Check the matrix before inversion
Verify that no output denominator is missing or zero, that monetary units are consistent and that columns align with the output vector. Inspect large values and column patterns against the sector descriptions.
A transpose error can survive visual inspection, so test one coefficient manually from the raw transaction and the named using-sector output.
What this chapter covers
- 01
From absolute transactions to normalized inputs
- 02
A equals T times inverse diagonal output
- 03
Entry Aij equals Tij divided by Xj
- 04
Column recipes for using sectors
- 05
Fixed-coefficient assumption
- 06
Dimension and manual-cell checks
Worked example · free
Calculating and interpreting a coefficient
- 1Identify the correct denominator.
- 2Calculate the coefficient.
- 2Interpret the result as a production requirement.
- 2Reject the supplier-output denominator.
Key terms
- Coefficient Matrix
- The matrix of input requirements per unit of using-sector output.
- Diagonal Matrix
- A matrix with values only on its main diagonal, used here to scale columns by sector output.
- Production Recipe
- The set of technical coefficients in a column for one producing sector.
- Fixed Technology
- The modelling assumption that input coefficients remain unchanged for the scenario.
- Dimension Check
- Verification that matrix sizes and sector order permit the intended economic operation.
Building the Technical Coefficient Matrix FAQ
Why is the inverse output matrix placed on the right of T?
Right multiplication by a diagonal matrix scales columns. Each transaction column must be divided by the output of the sector that uses those inputs. Placing the scaling matrix on the other side would scale rows and change the economic meaning.
Can a technical coefficient exceed one?
It can in monetary tables under some price, classification or accounting circumstances, so a large entry is not automatically impossible. It is still a prompt to check units, valuation, aggregation, signs and the denominator before accepting it.
What assumption makes A usable for scenario calculations?
The standard calculation holds the production recipe fixed while final demand changes. This permits a linear response but ignores capacity constraints, substitution, economies of scale and structural change. The scenario should not be interpreted as a forecast beyond that assumption.
Assessment move
Rebuild one column from raw transactions and its output denominator. Explain every entry in words, write the matrix dimensions and use a tiny numerical example to verify that right-side diagonal scaling divides columns rather than rows. Calculate the same cell once as T_ij divided by X_j and once through matrix multiplication so the cell and matrix forms agree.
Add the compatible value-added share and diagnose any failure to account for the full output using the table's valuation conventions. In a spreadsheet, colour is not enough: use stable sector keys and inspect the formula references in a high-value cell, a cross-region cell and a small service cell.
Deliberately sort the output vector separately, observe that the dimensions still match, and then recover the error with named-sector checks. Make a unit note explaining why monetary input per monetary output is dimensionless while still tied to a price basis and year.
Finally, write the fixed-recipe assumption beside A and list one scenario where substitution or structural change would make that assumption a material limitation. As a final retrieval drill, close the table and reconstruct the cell formula, matrix formula, denominator rule and interpretation from memory.
Reopen it only to correct the first failed link, then repeat with a cross-region coefficient whose origin and destination labels differ. Say aloud which sector supplies and which sector uses the input before recording the coefficient.
