2.1 A SAM-based model for project evaluation
Consider the social accounting matrix equation for a generic scenario:
where X is an n,1 vector of activity levels for productive sectors, and incomes for factors and institutions and \(Q = I - A\), the SAM coefficient matrix.
We can consider an investment project as an additional activity and augment the size of the SAM by adding a column and a row of transactions corresponding, respectively, to the outlays and the receipts of the project cash flow. In order for the inflows and outflows to balance, this entails, in particular, accounting, among the receipts, for any financing flow and, among the expenditures, any profits distributed to factors of production and other stakeholders. We can then write two new equilibrium conditions for the situation “without” and “with project” SAM as:
$$X_{s} = A_{s} X_{s} ,$$
(2a)
$$X_{c} = A_{c} X_{c} .$$
(2b)
In (2a) and (2b), \(A_{s}\) and \(A_{c}\) are n + 1, n + 1, SAM matrices augmented of one column and one row to represent, respectively, the situation without and with the project. The matrix without the project \(A_{s}\), in particular, can either contain an additional column and row of zeros, for the case of full project additionality, or the data of the cash flow of an alternative project as a counterfactual.
Subtracting Eq. (
1) from Eqs. (
2a) and (
2b), we obtain, after some manipulation:
$$X_{c} - X_{s} = A_{c} (X_{c} - X_{s} ) + (A_{c} - A_{s} )X_{s} ,$$
(3a)
$$~X_{c} - X_{s} = A_{s} (X_{c} - X_{s} ) + (A_{c} - A_{s} )X_{c} .$$
(3b)
As noted in the literature on structural decomposition (e.g., Rose and Casler
1996), the two expressions (
3a) and (
3b) signal an index number problem. In the remainder of this paper, we will assume that the differences between (3a) and (3b) are small enough that they can be ignored or otherwise solved by appropriately averaging the results obtained by separately applying the two equations (Koppany
2017, p. 619).
Both the
\(A_{s}\) and the
\(A_{c}\) matrix are singular, but we can decompose them into a non-singular square submatrix of coefficients of endogenous variables and rectangular submatrices of coefficients of exogenous variables:
$$A_{i} X_{i} = \left[ {\begin{array}{*{20}c} {A_{{ee,i}} } & {A_{{ex,i}} } \\ {A_{{xe,i}} } & {A_{{xx,i}} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {X_{{ei}} } \\ {X_{{xi}} } \\ \end{array} } \right]{\text{ }}\quad {\text{for}}\,\;~i = s,c.$$
(4)
In (4)
\(X_{{ei}}\) and
\(X_{{xi}}\) are vectors, respectively, of endogenous and exogenous activity levels and
\(A_{{ee,i}} ,~A_{{ex,i,~~~}} A_{{xe,i,~~~}} A_{{xx,i}}\) corresponding submatrices from partitioning of
\(A_{i} .~~\) Developing the expression, we can re-write (2) and (3) as follows:
$$X_{{ei}} = A_{{ee,i}} X_{{ei}} + A_{{ex,i}} X_{{xi}} ;\;\quad i = s,c$$
(5)
This expression identifies one part of the system (\(A_{{ex,i}} X_{{xi}} )~\) as a vector of exogenous demand levels and one part ((\(I - A_{{ee,i}} )X_{{ei}} )\) as a corresponding vector of endogenous supply levels necessary to satisfy the direct and indirect demand generated by the exogenous demand levels.
Subtracting the endogenous vector without the project from the one with the project, we obtain:
$$X_{{ec}} - X_{{es}} = A_{{ee,c}} (X_{{ec}} - X_{{es}} ) + (A_{{ee,c}} - A_{{ee,s}} )X_{{es}} + A_{{ex,c}} (X_{{xc}} - X_{{xs}} ) + (A_{{ex,c}} - A_{{ex,s}} )X_{{xs}} .$$
(6)
More synthetically, expression (
6) can be re-written in difference notations as:
$$\Delta X_{e} = A_{{ee,c}} \Delta X_{e} + \left( {\Delta A_{{ee}} } \right)X_{{es}} + A_{{ex,c}} \Delta X_{x} + \Delta A_{{ex}} X_{{xs}} .$$
(7)
Solving for the endogenous variables, we obtain:
$$\Delta X_{e} = (I - A_{{ee,c}} )^{{ - 1}} \left[ {A_{{ex,c}} \Delta X_{x} + \left( {\Delta A_{{ee}} } \right)X_{{es}} + \left( {\Delta A_{{ex}} } \right)X_{{xs}} } \right].$$
(8)
Expression (
8) indicates that the variation of the endogenous variables of the model may be the consequence of three different shocks, all filtered through the matrix of multipliers of the endogenous sectors: (i) the autonomous variation of the exogenous variables (capital formation, exports or a specific vector of project expenditures); (ii) the variation of the SAM coefficient submatrix of the transactions within the endogenous accounts, and (iii) the variation of the SAM submatrix of the transactions between the endogenous and the exogenous accounts. Intuitively, the exogenous activities increases aggregate demand through the value chains quantified in the SAM, but may also introduce technological change, and induce a new pattern of transactions, boost or reduce existing connections and create new ones.
If one of the exogenous accounts is a specific investment project, in particular, consider the exogenous variation measured by the project cash flow over a time horizon \(t = 0,1, \ldots ..T\) and the changes in the SAM coefficients due to the changes of the project cash flow every year.
Indicating with the t subscript the time, the term \(A_{{ex,c,t}} \Delta X_{{x,t}} = A_{{ex,c,t}} \left( {X_{{xc,t}} - X_{{xs,t}} } \right)\) is the increase in project expenditure in the tth year, while (\(\Delta A_{{ex}}^{t} )X_{{xs}}^{t}\), is the change induced by the project into the counterfactual SAM matrix of the same period without the project. With no competing alternative (\(X_{{xs,t}} = 0)\), we have:\(~A_{{ex,c,t}} \Delta X_{{x,t}} = A_{{ex,c,t}} X_{{xc,t}}\), i.e., the project cash flow. This includes, as all columns of the SAM, the demand increases (with respect to the situation without the project) generated by the expenditures of the project with respect to all sectors. These expenditures include both costs and net benefits of the project such as the payments made to project stakeholders as for example the net margins paid to capital and the other net benefits, accounted in gross terms in a corresponding row of the SAM. The term \(\left( {\Delta A_{{ext}} } \right)X_{{xt}} = (A_{{ext + 1}} - A_{{ext}} )X_{{xs}}^{t}\) represents the structural impact of the technology embodied in the project. This impact may be due to different project requirements in terms of use of intermediate inputs and value added in comparison to existing technologies. Project impact is thus evaluated as the sum of two components, one depending on the change in the scale of the cash flow, and one depending on the change of the weights of the different items of the project transaction vector in a new SAM updated to account for the transactions introduced by each phase of the project.
The present value at rate of discount
r of project impact can be directly derived from Eq. (
8):
$$\mathop \sum \limits_{{t = 0}}^{T} \frac{{\Delta X_{{et}} }}{{(1 + r)^{t} }} = \mathop \sum \limits_{{t = 0}}^{T} \frac{1}{{(1 + r)^{t} }}(I - A_{{ee,t + 1}} )^{{ - 1}} \left[ {A_{{ex,t + 1}} \Delta X_{{xt}} + \left( {\Delta A_{{eet}} } \right)X_{{et}} + \left( {\Delta A_{{ext}} } \right)X_{{xt}} } \right].$$
(9)
However,
\(A_{{ee,t + 1}}\) will approximately remain constant if the project is small enough, and
\(\Delta A_{{eet}} \cong 0\), so that expression (
9) can be approximated on the basis of the initial SAM for the endogenous accounts:
$$\mathop \sum \limits_{{t = 0}}^{T} \frac{{\Delta X_{{et}} }}{{(1 + r)^{t} }} = (I - A_{{ee}} )^{{ - 1}} \mathop \sum \limits_{{t = 0}}^{T} \frac{1}{{(1 + r)^{t} }}\left[ {A_{{ex,t + 1}} \Delta X_{{xt}} + \left( {\Delta A_{{ext}} } \right)X_{{xt}} } \right].$$
(10)
Expression (
10) allows to estimate the present value of project impact using a single SAM and its variations as the direct and indirect effects of the present values of the project cash flows. In turn these are defined as the sum of two components: (i) the yearly project outlays for a given structure of the interdependencies between the project and the rest of the economy, and (ii) the yearly increases in the same outlays due to the variation of these interdependencies brought about by the changes of project outlays over time.