Machines and infrastructure contribute to product footprints through the industries that use them. Capital endogenization incorporates those contributions into footprint analysis. Two common approaches to capital endogenization are the flow matrix method and the augmented matrix method.
The flow matrix method adds capital input coefficients K to the ordinary input coefficients A:
\[ A_{\mathrm{cap}} = A + K \]
Capital goods enter the same product rows as intermediate goods. The matrix keeps its original dimensions after capital endogenization. This method is compact, but the combined matrix merges intermediate goods and capital goods. No distinction between intermediate inputs and capital inputs is therefore preserved.
The other way is the augmented matrix method. It adds rows for capital-good products and columns for capital formation activities:
\[ A_{\mathrm{cap}} = \begin{pmatrix} A & U_C \\ U_K & 0 \end{pmatrix} \]
Here, \(U_C\) represents intermediate goods converting into capital goods. \(U_K\) contains capital use of industry activities. Capital formation activities consume intermediate goods and supply capital goods. Industries then use those capital goods. Each capital product keeps its own identity. This structure increases the matrix dimensions and can increase computational cost. It also makes capital flows explicit.
With the same capital requirements and capital formation, the two methods can produce the same footprints. In the simplified block model, \(K = U_C U_K\). The benefit of augmentation comes from explicit accounts, and the consistency checks they permit. For better transparency, BONSAI is moving to the augmented matrix approach to make capital flows easier to trace.
Coming soon in BONSAI v2.5.0.
