### Payment flows in a single-region economy

The objective of this section is to develop the equation that transforms the vector of consumer payments to the associated vector of factor payments in a single region. We begin with the familiar primary input-output equation describing quantity relationships and its associated dual-price equation. We assume that all resources and products are measured in physical units and also possess a unit price. (Unpriced resources are easily accommodated, and the unit price is 1.0 in the base year if the resource or good is measured in nominal money values.) Given the *n* × 1 vector of consumer demand, **y**, and **A**, the *n* × *n* matrix whose columns describe the intermediate input requirements per unit of output, the familiar quantity model,

$$ \mathbf{x}=\mathbf{Ax}+\mathbf{y}\ \mathrm{or}\ \mathbf{x}={\left(\mathbf{I}\hbox{--} \mathbf{A}\right)}^{-1}\mathbf{y}, $$

(1)

solves for the outputs of goods and services, the *n* × 1 vector **x**. Given **F**, the *k* × *n* coefficient matrix whose columns quantify requirements for each of *k* resources per unit of sectoral output, the resource requirements are given by the *k* × 1 vector **φ**,

$$ \boldsymbol{\upvarphi} =\mathbf{F}\mathbf{x}. $$

(2)

The dual model determines unit prices, **p**, for the *n* sectoral outputs, based on exogenously specified resource prices, **π**,

$$ \mathbf{p}={\mathbf{A}}^T\mathbf{p}+{\mathbf{F}}^T\boldsymbol{\uppi}\ \mathrm{or}\ \mathbf{p}={\left(\mathbf{I}\hbox{--} {\mathbf{A}}^T\right)}^{-1}{\mathbf{F}}^T\boldsymbol{\uppi} . $$

(3)

Alternative scenarios may specify values for changes in demand, **y**; in technologies, represented by columns of **A** and **F**; or in factor prices, **π**. Note that scenarios specifying changes in technologies will in general impact both **x** and **p** even if **y** and **π** are unchanged.

We next define the *n* × 1 vector of consumer payments \( {\mathbf{y}}_{\mathbf{p}}=\widehat{\mathbf{p}}\mathbf{y} \) and the *k* × 1 vector of factor receipts \( {\boldsymbol{\upvarphi}}_{\boldsymbol{\uppi}}=\widehat{\boldsymbol{\uppi}}\boldsymbol{\upvarphi} \). The relationship between the two is established using Eqs. (1) and (2) and the definitions of the two variables:

$$ \begin{array}{l}{\boldsymbol{\upvarphi}}_{\boldsymbol{\uppi}}\kern0.36em =\widehat{\boldsymbol{\uppi}}\boldsymbol{\upvarphi} \\ {}\kern0.96em =\widehat{\boldsymbol{\uppi}}\mathbf{F}\mathbf{x}\\ {}\kern0.84em =\widehat{\boldsymbol{\uppi}}\mathbf{F}{\left(\mathbf{I}-\mathbf{A}\right)}^{-1}\mathbf{y}\end{array} $$

$$ {\boldsymbol{\upvarphi}}_{\boldsymbol{\uppi}}=\left[\widehat{\boldsymbol{\uppi}}\mathbf{F}{\left(\mathbf{I}-\mathbf{A}\right)}^{-\mathbf{1}}{\widehat{\mathbf{p}}}^{-\mathbf{1}}\right]{\mathbf{y}}_{\mathbf{p}}. $$

(4)

The CFM in this simple case is the familiar *k* × *n* matrix of total factor requirements per unit of final deliveries, **F(I − A)**
^{−1}, converted to money values. This conversion is achieved by pre-multiplying and post-multiplying the matrix of total factor requirements by the vectors of factor prices and the inverse of prices of goods, respectively. We denote the CFM as **Φ**:

$$ \boldsymbol{\Phi} =\widehat{\boldsymbol{\uppi}}\mathbf{F}{\left(\mathbf{I}-\mathbf{A}\right)}^{-1}{\widehat{\mathbf{p}}}^{-\mathbf{1}}, $$

(5)

which can be written in terms of its individual components as follows:

$$ \boldsymbol{\Phi} =\left[\begin{array}{ccc}\hfill {\phi}_{11}\hfill & \hfill {\phi}_{12}\hfill & \hfill {\phi}_{13}\hfill \\ {}\hfill {\phi}_{21}\hfill & \hfill {\phi}_{22}\hfill & \hfill {\phi}_{23}\hfill \end{array}\right] $$

This network is illustrated in Fig. 1. Setting *B* = (*I* − *A*)^{−1}, the elements of **Φ** are of the form

$$ {\phi}_{ij}=\frac{\pi_i}{p_j}\left({\displaystyle \sum_h{f}_{ih}{b}_{hj}}\right). $$

(6)

The components of the *j*th column of **Φ** describe how each money unit, say each dollar, of consumer payments for the product of sector *j* is ultimately distributed among the owners of the *k* factors. The matrix **Φ** is scenario-specific as the numerical values depend both on technological assumptions in **A** and **F** and on factor prices, **π**. In the case of the multiregional economy, as we will see in the numerical analysis below, the payment networks are in addition responsive to changes in final demand and to the size of factor endowments.

### Payment flows in a multiregional economy

We will be concerned with money flows in the global economy corresponding to payments for traded goods and services. The data structure for accommodating the information describing these transactions is the MRIO table. In a previous publication, Duchin and Levine (2015) define the MRIO table corresponding to a solution of the WTMBT and show how to construct it. Appendix 1 of this paper shows the equations for the WTMBT; for more detail, see Strømman and Duchin (2006).

The World Trade Model with Bilateral Trade (WTMBT) (Strømman and Duchin 2006) is a linear program that minimizes global factor use to satisfy consumption requirements while respecting regional factor constraints. Model results are the *mn* × 1 vectors of outputs, **x**, and prices, **p** (concatenations of *m* vectors of length *n*), the bilateral trade vectors, **e**
_{
ij
}, and two *mk* × 1 vectors of rents, **r**, received on fully utilized factors and of quantities of factor use, **φ**. The WTMBT database of inputs is combined with scenario results to derive **A**
_{
B
} and **Y**
_{
B
}, matrices forming the MRIO table, which are also required for the network analysis. See Appendix 3 for the definitions and derivation of **A**
_{
B
} and **Y**
_{
B
}.

The WTMBT is based on the logic of comparative advantage. It distinguishes bilateral trade flows by including the costs associated with international transportation of traded goods between any two regions as well as the world price for the goods. Along with the more familiar input-output objects, **A**
_{
i
}, **F**
_{
i
}, **y**
_{
i
}, and **π**
_{
i
} for each region *i*, the WTMBT database requires information on distances between pairs of regions and the mass of each product to be transported. This is accommodated in an *n* × *n* matrix, **T**
_{
ji
}, for transport between each pair of regions, *i* and *j*. We will assume that the *n*th sector, and therefore the *n*th row of **T**
_{
ji
}, quantifies the demand for international transport in ton-kilometers per unit of each good imported to region *i* from *j*. An empirical application may include several transport rows distinguished by mode of transport.

Also required for each region is **f**
_{
i
}, the *k* × 1 vector of resource endowments, which constrain a region’s production capacities. When a relatively low-cost producer runs into an endowment constraint, a higher-cost producer needs to enter the market, thus raising the world price of the good in question and allowing the lower-cost producers to earn rents on their scarce factors.

In a recent paper, Duchin and Levine (2015) construct an MRIO table for each WTMBT scenario outcome. The production portion of this table is the multiregional equivalent of the one-region matrix **A**; we call it the Big A matrix, denoted as the *mn* × *mn* matrix **A**
_{
B
}. In contrast to the one-region case, an element of **A**
_{
B
} specifies the quantity of the relevant input imported for a given sector from a particular region (with the associated transport services also accounted for if it is not domestically produced). Since bilateral trade flows are endogenous to the WTMBT, **A**
_{
B
} (unlike **A**) is scenario-specific even if there are no changes in technologies.

The matrix **A**
_{
B
} accounts for imports of intermediate goods as well as those produced domestically. Imported and domestically produced consumer goods are represented in the *mn* × *mn* matrix **Y**
_{
B
}. Each row of **Y**
_{
B
} corresponds to a specific sector in a specific region and contains the deliveries of the good produced by that sector in that region to consumers in all regions. We represent this flow as an import to the corresponding domestic sector, which in turn delivers it to the consumer. Therefore, the row has at most *n* non-zero elements, one for each region. The components of the *mn* × 1 vector **y**
_{
B
} are calculated as the row sums of **Y**
_{
B
}. Thus, each element of **y**
_{
B
} quantifies the amount of a consumer good produced in a region independent of where it is purchased and consumed. (By contrast, a component of the standard consumption vector, **y**, is the quantity of the good purchased by domestic consumers independent of where it was produced.) With **A**
_{
B
} and **y**
_{
B
} so defined, it follows that:

$$ \mathbf{x}={\left(\mathbf{I}\hbox{--} {\mathbf{A}}_{\mathbf{B}}\right)}^{-1}{\mathbf{y}}_{\mathbf{B}}, $$

(7)

where **x** is the *mn* × 1 vector of output. (This variable follows the standard definition of **x**, being the vector of output, and thus does not require a subscript B, which is reserved for vectors or matrices having a unique definition in the MRIO database. The price vector, **p**, is also defined in the standard way).

Finally, we define three more matrices. The first is **Y**
_{
OD
} (OD for off-diagonal), formed from **Y**
_{
B
} by replacing the diagonal elements (representing domestically produced consumer goods that are also sold domestically) with zeroes; consequently, **Y**
_{
OD
} contains only the exported consumer goods. The other matrices are **S**
_{
B
} and **T**
_{
S
}. The former is familiar as a requirement for building any MRIO database. Typically, it is an exogenous matrix of import shares coming from the different producing regions, but in our framework, it is an endogenous outcome of the WTMBT scenario analysis. The latter matrix, **T**
_{
S
}, is an input to the WTMBT and is used to incorporate the input requirements for international transport services into the **A**
_{
B
} matrix. The matrices **A**
_{
B
}, **Y**
_{
B
}, **S**
_{
B
}, **T**
_{
S
}, and **Y**
_{
OD
}, and the vector **y**
_{
B,} are described in Appendix 3 and, in more detail, in Duchin and Levine (2015).

We are now ready to derive the CFM that transforms consumer payments, **y**
_{
p
} (recall that \( {\mathbf{y}}_{\mathbf{p}}=\widehat{\mathbf{p}}\mathbf{y} \)), to the receipts of factor owners, **φ**
_{
π + r
} (where \( {\boldsymbol{\upvarphi}}_{\boldsymbol{\uppi} +\mathbf{r}}=\left(\widehat{\boldsymbol{\uppi}}+\widehat{\mathbf{r}}\right)\boldsymbol{\upvarphi} \)), for the multiregional case. We use a logic similar to the one-region derivation, substituting first for *x*, then for **y**
_{
B
}, and finally for **y**:

$$ \begin{array}{l}{\boldsymbol{\upvarphi}}_{\boldsymbol{\uppi} +\mathbf{r}}=\left(\widehat{\boldsymbol{\uppi}}+\widehat{\mathbf{r}}\right)\mathbf{F}\mathbf{x}\\ {}\kern2.16em =\left(\widehat{\boldsymbol{\uppi}}+\widehat{\mathbf{r}}\right)\mathbf{F}{\left(\mathbf{I}-{\mathbf{A}}_{\mathbf{B}}\right)}^{-\mathbf{1}}{\mathbf{y}}_{\mathbf{B}}\\ {}\kern2.04em =\left(\widehat{\boldsymbol{\uppi}}+\widehat{\mathbf{r}}\right)\mathbf{F}{\left(\mathbf{I}-{\mathbf{A}}_{\mathbf{B}}\right)}^{-\mathbf{1}}{\mathbf{S}}_{\mathbf{B}}\left(\mathbf{I}+{\mathbf{T}}_{\mathbf{S}}\right)\mathbf{y}\kern2.04em \end{array} $$

$$ {\boldsymbol{\upvarphi}}_{\boldsymbol{\uppi} +\mathbf{r}}=\left[\left(\widehat{\boldsymbol{\uppi}}+\widehat{\mathbf{r}}\right)\mathbf{F}{\left(\mathbf{I}-{\mathbf{A}}_{\mathbf{B}}\right)}^{-\mathbf{1}}{\mathbf{S}}_{\mathbf{B}}\left(\mathbf{I}+{\mathbf{T}}_{\mathbf{S}}\right){\widehat{\mathbf{p}}}^{-1}\right]{\mathbf{y}}_{\mathbf{p}} $$

(8)

where the *kn* × *mn* matrix **Φ**
_{
B
} is defined as follows:

$$ {\boldsymbol{\Phi}}_{\mathbf{B}}=\left(\widehat{\boldsymbol{\uppi}}+\widehat{\mathbf{r}}\right)\mathbf{F}{\left(\mathbf{I}-{\mathbf{A}}_{\mathbf{B}}\right)}^{-\mathbf{1}}{\mathbf{S}}_{\mathbf{B}}\left(\mathbf{I}+{\mathbf{T}}_{\mathbf{S}}\right){\widehat{\mathbf{p}}}^{-\mathbf{1}} $$

(9)

Equation (8) is used to determine the distribution of payments to factor owners using the CFM, defined as **Φ**
_{
B
} in Eq. (9), the multiregional counterpart of Eq. (5). As in the case of Eq. (5), the column of **Φ**
_{
B
} corresponding to a specific sector in a given region indicates the distribution one dollar's worth of consumer purchases of that sector’s output among the owners of all factors in all regions. Thus, the CFM generalizes the one-region total factor requirements matrix in that it is the result of a scenario analysis rather than the compilation of accounting data, it is multiregional in scope, it distinguishes resources from built capital and scarcity rents from other resource costs, and it incorporates the interregional transport of imports.

Since we are especially interested in the impact of scenario assumptions on scarcity rents, we supplement Eq. (8) by the following variable defined to contain only the rent payment: \( \boldsymbol{\upvarphi} =\widehat{\mathbf{r}}\mathbf{F}{\left(\mathbf{I}-{\mathbf{A}}_{\mathbf{B}}\right)}^{-\mathbf{1}}{\mathbf{S}}_{\mathbf{B}}\left(\mathbf{I}+{\mathbf{T}}_{\mathbf{S}}\right){\widehat{\mathbf{p}}}^{-\mathbf{1}}{\mathbf{y}}_{\mathbf{p}} \). When a factor is scarce, the size of the rent will reflect the differences in quality, ease of access, or desirable geographic location of one endowment relative to others.