Rents in the era of resource scarcity: global payment flows under alternative scenarios

Abstract

Resource prices rise when more costly sources need to be exploited. When the world price increases, owners of low-cost sources receive scarcity rents. The magnitude of the rents depends on the range of resource qualities being simultaneously exploited and can represent a substantial transfer of wealth to those with property rights to large stocks of high-quality, easily accessible resources. These rents are bound to increase in the future along with the size of the human population, raised consumption expectations, and deployment of technologies that depend on a wider range of natural resources for their unique properties.

We report results for a set of scenarios for a three-region, four-sector, three-resource world economy, where progressively increased consumer demand requires a second and then a third region to extract ore. This numerical example illustrates how the amount of rents transferred to the low-cost producers depends on the size of the low-cost endowments relative to world demand and on the differential costs of extraction relative to the highest-cost producer.

The paper develops a framework for tracing global money flows from payments for specific consumer goods in one or more economies to receipts by owners of the embodied factors of production under alternative scenarios about the future. This objective requires a substantial generalization of methods utilized for ex post analysis of input-output databases for past years. For a world economy of m regions, n sectors, and k factors of production, we first compute scenario results using the World Trade Model with Bilateral Trade, then generate the Multiregional Input-Output database corresponding to each scenario, and finally compute a new Consumer-to-Factor Matrix from which the payment network is derived.

JEL Classification: F18, O13, C67, C61

“Growing populations, growing affluence, and the materials diversity of modern technologies are straining the resource capacities on which we draw. The situation need not inspire panic, but should instead stimulate more diligent and more comprehensive approaches to thebalance between supply and demand across the entire periodic table.”

Tom Graedel et al., On the Materials Basis of Modern Society

Appendices

Appendices

Appendix 1: The World Trade Model with Bilateral Trade (WTMBT)

The quantity and price models of the WTMBT are given below in Eqs. (10)–(13) and (14)–(17), respectively. The quantity equations are shown first—despite the fact that we focus here on the price model—by convention since in the past, most analyses have focused on the primal problem only.

$$ \mathrm{Minimize}\ \mathrm{C}={\displaystyle \sum_{\mathbf{i}}{\boldsymbol{\uppi}}_{\mathbf{i}}^T{\mathbf{F}}_{\mathbf{i}}}{\mathbf{x}}_{\mathbf{i}}, $$

(10)

$$ {\mathbf{x}}_{\mathbf{i}}+{\displaystyle \sum_{\mathbf{j}\ne \mathbf{i}}{\mathbf{e}}_{\mathbf{j}\mathbf{i}}}={\mathbf{A}}_{\mathbf{i}}{\mathbf{x}}_{\mathbf{i}}+{\mathbf{y}}_{\mathbf{i}}+{\displaystyle \sum_{\mathbf{j}\ne \mathbf{i}}{\mathbf{e}}_{\mathbf{i}\mathbf{j}}}+{\displaystyle \sum_{\mathbf{j}\ne \mathbf{i}}{\mathbf{T}}_{\mathbf{j}\mathbf{i}}}{\mathbf{e}}_{\mathbf{j}\mathbf{i}},\forall i $$

(11)

$$ {\mathbf{F}}_{\mathbf{i}}{\mathbf{x}}_{\mathbf{i}}\le {\mathbf{f}}_{\mathbf{i}},\forall i. $$

(12)

$$ {\mathbf{x}}_{\mathbf{i}}\ge \mathbf{0},\forall i. $$

(13)

The price dual takes the following form:

$$ \mathrm{Maximize}\ \mathrm{Z}={\displaystyle \sum_{\mathbf{i}}{\mathbf{y}}_{\mathbf{i}}^T{\mathbf{p}}_{\mathbf{i}}}-{\displaystyle \sum_{\mathbf{i}}{\mathbf{f}}_{\mathbf{i}}^T{\mathbf{r}}_{\mathbf{i}}} $$

(14)

such that

$$ \left(\mathbf{I}-{\mathbf{A}}_{\mathbf{i}}^T\right){\mathbf{p}}_{\mathbf{i}}\le {\mathbf{F}}_{\mathbf{i}}^T\left({\boldsymbol{\uppi}}_{\mathbf{i}}+{\mathbf{r}}_{\mathbf{i}}\right),\forall i, $$

(15)

$$ \left({\mathbf{p}}_{\mathbf{i}}-{\mathbf{p}}_{\mathbf{j}}\right)\le {\mathbf{T}}_{\mathbf{j}\mathbf{i}}^T{\mathbf{p}}_{\mathbf{i}},\forall i\ne j. $$

(16)

$$ {\mathbf{p}}_{\mathbf{i}},{\mathbf{r}}_{\mathbf{i}}\ge \mathbf{0},\forall i. $$

(17)

The price of each good in the producing region is determined by Eq. (15), with the highest-cost region, among those regions that actually produce and export the good, setting the price and receiving no rent, while all lower-cost producers charge the same price and receive scarcity rents as the difference between the price and their costs. The second price constraint, Eq. (16), describes price formation in importing regions, for which a strict equality holds. (See Strømman and Duchin, 2006 for proofs.) The WTMBT accounts for the role of interregional transport sectors in providing the transport service (Eq. 11) and in determining regional prices (Eq. 16). The T _ji matrices contain zeroes in all rows except the row (or rows) corresponding to the transport sectors. (The role of these matrices is illustrated in the numerical example of Section 4.) The nth element in each region’s n × 1 price vector, p _i , (the element corresponding to the transport sector) is the price per ton-kilometer, and the price for an imported good sold in region i includes the cost of purchasing the good and of transporting it. See Strømman and Duchin (2006) for more details.

Appendix 2. Database and results for the scenario analyses in Section 4

This appendix contains the WTMBT database for the scenario analysis in Section 4 for the economy of three regions, four sectors, and three factors of production. The model is shown in Appendix 1.

Regions: (1) industrialized, (2) agricultural, (3) mineral-rich.

Sectors: (1) agriculture, (2) manufacturing, (3) mining, (4) interregional transportation.

Factors: (1) labor, (2) ore, (3) land).

Exogenous data

The baseline A, F, and T matrices and the y, π, and f vectors for scenario S0 are as follows:

Consumer demand for all goods in the resource-rich region (the last four figures in the y vector) is increased for each subsequent scenario: by 10 % under S1, 20 % under S2, and 30 % under S3.

Model solutions

Scenario results are reported below in Table 2 for output, x, and prices, p, and in Table 3 for factor use, φ, and scarcity rents, r.

Table 2 ᅟ

Table 3 ᅟ

Appendix 3. Creating the MRIO database from the scenario results for the Consumer-to-Factor payment calculations of Section 3

The results of a scenario run with the WTMBT are organized as an MRIO database, and the Consumer-to-Factor Matrix is derived in Section 3 from the MRIO database. In this appendix, we show how the four MRIO matrices A _B , S _B , T _S , and Y _B (all of dimension mn × mn for m regions and n sectors) are derived from the WTMBT database and scenario results.

A _B is the MRIO equivalent of the standard input-output coefficient matrix for intermediate products for a single region. It differs in that the former distinguishes intermediate inputs according to where they are produced via S _B , and the transportation of imports to the purchasing region must be incorporated via T _S . These adjustments can be made only after the scenarios are run, as both S _B and T _S are determined endogenously. These objects are defined as follows:

$$ {\mathbf{A}}_{\mathbf{B}}={\mathbf{S}}_{\mathbf{B}}\left(\mathbf{I}+{\mathbf{T}}_{\mathbf{S}}\right)\mathbf{A}={\mathbf{S}}_{\mathbf{B}}\mathbf{A}+{\mathbf{S}}_{\mathbf{B}}{\mathbf{T}}_{\mathbf{S}}\mathbf{A} $$

(18)

where, for the three-region case,

The S _B matrix consists of a block diagonal sub-matrix Ŝ _ij for each pairing of regions, nine blocks in the case of three regions, containing the proportion for each good that is imported to region j from region i (or domestically produced when i = j) for the domestic intermediate or final use. The individual components can be defined as follows:

$$ {\widehat{\mathbf{S}}}_{\mathbf{ij}}=\left\{\begin{array}{l}{\left({\widehat{\mathbf{x}}}_{\mathbf{j}}+{\displaystyle \sum_{h\ne j}{\widehat{\mathbf{e}}}_{\mathbf{hj}}}\right)}^{-1}{\mathbf{x}}_{\mathbf{j}}\kern0.24em \mathrm{f}\mathrm{o}\mathrm{r}\kern0.24em i=j\\ {}\Big({\widehat{\mathbf{x}}}_{\mathbf{j}}+{\displaystyle \sum_{h\ne j}{\widehat{\mathbf{e}}}_{\mathbf{hj}}\Big){}^{-1}}{\widehat{\mathbf{e}}}_{\mathbf{ij}}\kern0.24em \mathrm{f}\mathrm{o}\mathrm{r}\kern0.24em i\ne j\end{array}\right. $$

(19)

where e _ij is the vector of exports from region i to region j. These proportions are also used to distribute transport services among the regional providers:

$$ {\mathrm{T}}_{\mathbf{Sj}}={\displaystyle \sum_{\mathbf{\mathsf{i}}\mathbf{\ne}\mathbf{\mathsf{j}}}{\mathrm{T}}_{\mathbf{ij}}{\widehat{\mathrm{S}}}_{\mathbf{ij}}} $$

(20)

These are the block diagonal sub-matrices that comprise T _S , and the matrix S _B T _S A adds the provision of interregional transport to A _B .

The S _B and T _S matrices are also used to derive the production in a region of consumer goods and their domestic consumption or destination as exports to other regions, Y _B (see the text for further details):

$$ {\mathbf{Y}}_{\mathbf{B}}={\mathbf{S}}_{\mathbf{B}}\left(\mathbf{I}+{\mathbf{T}}_{\mathbf{S}}\right)\widehat{\mathbf{y}}. $$

(21)

The mn × 1 vector y _B is comprised of the row-sums of Y _B . Its entries for a particular region quantify the final goods that are produced in that region regardless of where they are ultimately consumed.

$$ {\mathbf{y}}_{\mathbf{B}}={\mathbf{S}}_{\mathbf{B}}\left(\mathbf{I}+{\mathbf{T}}_{\mathbf{S}}\right)\mathbf{y}. $$

(22)

Next, we define Y _OD , the matrix containing the off-diagonal elements of Y _B and zeroes down the diagonal. Its non-zero elements represent only those final goods that are traded among regions.

Appendix 4. The Consumer-to-Factor Matrices for the scenarios in Section 4

The numerical values for the Consumer-to-Factor matrices, Φ _B , are given below for each of the scenarios: S0, S1, S2, and S3. The largest shifts are experienced in the second to last row, corresponding to receipts including scarcity rents by those with property rights to resources in region 3. See the text for a discussion of these results.

S0:

S1:

S2:

S3: