Convert a SUT to an IOT

Understand the four conversion methods and choose how to allocate inputs or sales.

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A SUT records what each activity uses and produces, but an activity can make several products. Converting it to an IOT requires an assumption about how to assign those inputs or sales to a single productive set.

import mario

sut = mario.load_test("SUT")
iot = sut.to_iot(method="B", inplace=False)
iot.Z

Result · A product-by-product IOT. Its Sector labels and units come from the SUT’s commodities. inplace=False keeps the SUT intact.

Choose the assumption#

In this guide, product means MARIO’s Commodity and industry means Activity. A symmetric IOT uses the same set on both axes; its values need not be symmetric.

Method	IOT axes	What is held fixed?
A	Products	Input recipe of each product, across producers
B	Products	Input recipe of each industry, across its products
C	Industries	Customer proportions of each industry, across its products
D	Industries	Customer proportions of each product, across its producers

Check feasibility first. A/C require a square, invertible supply matrix and zero activity final demand (Ya). B/D also support rectangular supply and nonzero Ya. All methods retain the observed regional nodes; regional classifications need not be uniform.

Product-by-product IOT: A and B#

Both methods turn the activity columns of use into product columns. The resulting Z records products used to produce other products.

A · Product technology#

The recipe belongs to the product. A product requires the same input coefficients regardless of which industry makes it.

MARIO solves for product recipes that reproduce the observed industry inputs and product mixes. Inputs associated with secondary production are therefore assigned using that product’s recipe.

Example: cakes made by a bread bakery use the cake recipe, even though bread is that bakery’s main output. Their inputs need not follow the bakery’s average input mix.

Use A when a common recipe across producers is a defensible assumption. Negative entries can appear, even with nonnegative source flows. MARIO keeps them; it does not adjust the data or substitute a pseudoinverse for a singular supply matrix.

B · Industry technology#

The recipe belongs to the industry. All products of an industry receive its average input coefficients. MARIO allocates each industry’s inputs across its products in proportion to their shares of its output.

Example: if bread and cakes account for 80% and 20% of a bakery’s output value, B assigns 80% of each input to bread and 20% to cakes. Activity-level emissions are split in the same proportions.

Use B when proportional allocation within each industry suits the analysis. For a monetary SUT these are value shares, not physical quantities. B does not introduce negative flows when source flows are nonnegative; this property alone does not validate the allocation assumption.

Industry-by-industry IOT: C and D#

Both methods turn the product rows of use and final demand into industry rows. The resulting Z records deliveries from producing industries to using industries. Here, a sales structure means proportions sold to each industry or final-demand category.

C · Fixed industry sales structure#

The customer mix belongs to the industry. An industry sells every product in the same proportions to its customers. MARIO solves for industry sales patterns consistent with the observed product deliveries and supply mixes.

Example: if a producer’s inferred customer mix is 70% food manufacturers and 30% households, C applies that split to every product it supplies.

Use C when a common customer mix across an industry’s products is defensible. Like A, it requires a supply-matrix solve and can produce negative entries, including in final demand.

D · Fixed product sales structure#

The customer mix belongs to the product. Every producer of a product serves the same customer proportions. Equivalently, each buyer sources that product from industries in proportion to their shares of its total supply.

Example: if two industries supply 80% and 20% of all bread, D assigns each buyer’s bread purchases to those industries in an 80/20 split.

Use D when average supplier shares are appropriate for all buyers. The same shares apply to every buyer of a given regional product. As with B, nonnegative source flows give nonnegative transformed flows.

Matrix calculation#

The method operates on native SUT flows. The SUT’s current IT or PT setting does not choose the conversion: pass method explicitly. See SUT technology assumptions for the separate setting used when calculating directly on a SUT.

Equations for Z and Y

MARIO uses S with activity rows and commodity columns, and U with commodity rows and activity columns. Ya and Yc are the two final-demand blocks. See the SUT block diagram.

Let \(g=S\mathbf{1}+Y_a\mathbf{1}\) and \(q=U\mathbf{1}+Y_c\mathbf{1}\). Define diagonal output matrices \(G=\operatorname{diag}(g)\) and \(Q=\operatorname{diag}(q)\). On a balanced product account, \(q=S^{\mathsf T}\mathbf{1}\).

A — product recipes from a supply solve

\[Z=U S^{-\mathsf T}Q, \qquad Y=Y_c\]

B — industry inputs allocated over output

\[Z=U G^{-1}S, \qquad Y=Y_c+U G^{-1}Y_a\]

C — industry sales from a supply solve

\[Z=G S^{-\mathsf T}U, \qquad Y=G S^{-\mathsf T}Y_c\]

D — product purchases allocated over suppliers

\[Z=S Q^{-1}U, \qquad Y=Y_a+S Q^{-1}Y_c\]

Here \(S^{-\mathsf T}=(S^{\mathsf T})^{-1}\). A/C use linear solves; MARIO does not construct a dense inverse. For divisions by output, a fully inactive node is allowed, but zero output with nonzero flows or accounts raises an error. These operations do not balance the source or harmonize incompatible units.

Extensions, factors and direct final-demand accounts

The conversion rebuilds E, V, EY and VY alongside Z and Y. For A/B, activity accounts are allocated to products and existing commodity accounts are added. For C/D, activity accounts stay on activities; commodity accounts are allocated over intermediate and final uses.

With B, nonzero Ya also receives its share of activity inputs and accounts. With C/D, accounts attached to commodities consumed directly by final users contribute to EY or VY. Existing direct final-demand accounts are retained. Inspect productive and direct accounts together when comparing totals.

Inspect the result#

iot.get_index("Sector")
iot.units["Sector"]
iot.Y

Only the baseline is converted; other scenarios are absent from the IOT. Its public baseline name is retained. With the default inplace=True, the original object is replaced and the call returns None. Use inplace=False when comparing methods.

A/B inherit the commodity labels and units; C/D inherit the activity labels and units. Review negative flows and account totals before comparing footprints. For hybrid SUTs, check that the output sums and allocations are meaningful in the source units; conversion does not provide an allocation rule for combining incompatible physical units.

The method names and assumptions follow the Eurostat Manual of Supply, Use and Input-Output Tables (2008), Chapter 11. For the complete call contract, see db.to_iot().