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The household is assumed to produce and consume a vector of commodities.
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Also, let B denote the budget for the representative consumer and p = ( p 1, …, p N ) the price vector for commodities, where p j designates the price of the j commodity.
where p ′ is a row vector of prices of produced commodities by each sector and v ′ denotes a row vector of primary inputs per one monetary unit of price; I is the identity matrix; ( I − A ) − 1 is the Leontief inverse L or total requirements matrix.
The quantity of j commodity is represented by x j, while the whole set of commodities is represented by a vector x = ( x 1, …, x N ).
Vector q, whose elements represent the material footprint intensities of commodities supplied to Japanese households, is calculated as shown in Eq. 1: {mathbf{q}} = {mathbf{d}}left( {{mathbf{I}} - {mathbf{A}}} right)^{ - 1} (1}.
The inner product of the commodity portfolio x and the price vector p denoted as p ⋅ x.j is the type of commodities; δ is the distance for accessing the commodities (or it can also be understood as the necessary cost per unit of time for purchasing the commodities).
Then, for all the buildings, using bottom-up models, it is possible to describe and to project a certain number of reference processes supplied by a series of commodities (coal, natural and city gases, LPG, dh vectors, electricity, etc).
An index of commodities slumped.
Agudelo is that most precious of commodities.
The dance of commodities resumes.
Given a real wage rate and a consumption vector, the model indicates that the paths of prices, royalties, rents, intensities of commodity production, and resource-searching processes can be determined once a sequence of profit rates as well as the initial amounts of commodities and resources are given.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com