Methodology 5: Input-output analysis.
We can Leontief function to estimate how much input each is required to get one unit of output of any good or service.
AX : what the economy needs to satisfy demand D or Y (amount of input per unit of output)
X -AX = D (demand) or net production
X(I-A) = D to satisfy all demand
If we want to know how much input I need to satisfy my demand I can do it as follows:
XI = D*(I-A)^-1
or
(I-A)^-1*(I-A)*X=(I-A)^-1*D
X=(I-A)^-1*D
We know I (identify) A, and D (demand), we can get X by isolating it.
IEooc Methods5 Lecture2: Multiregional input-output analysis
A is equal to the normalized values the inter-input matrix. A is the product requirement per unit of output.
The market balanced says that everything produced is consumed or use by other industries.
The Leontief inverse allow us to get the factor of conversion from demand Y to requirements X.
The Leontief inverse can be approximated or expanded by a power series to describe the requirements of every step of the supply chain.
Note that instead of A, the technical recipe, we need E, the emissions for each output sector.
Market balance is possible, but not the industry balance due to different units of measurement of different inputs. Similar products and industries can be added together.
How to build a multiregional A matrix:
- single region A matrices from national statistical offices
- share of imports of each product used for each region, from national statistical offices
- bilateral trade data of product between regions (UN ComTrade database)
- This will allow us to create trade coefficient matrices
- Crs (i) denote the share of product i from region r in total use of product i in regions
We can do the same input requirement calculation as for the single region case
X = (I- A)^-1 *y
The difference here is that y is a matrix and not a vector as it includes different region final consumption and its origin.
We can also add environmental extensions:
b = S*x = S *L*y
This can be used for labor, water, co2 and other footprint.
IEooc_Methods5_Exercise1: Basics of Input-Output Modelling
- Make sure that the four system descriptions of the IO-model as shown in figure 1 are equivalent i.e. every description can be transformed into another without loss of information. How?
The data in the table of figure 1a account for both columns and rows. Consequently, you can extract the balance for rows and columns from this table:
x_i = sum_j Z_ij+y_i
x_j=sum_ij Z_ij + v_j
They can then be written as matrix equation using e as a summation vector consisting of nothing but 1s.
x = Z*e+y
x= Z^T*e+v
Every entry of Z, v, y and x is a flow between two nodes. Visualizing the non-explicit nodes in Fig. 1a you receive following image: Every row in table a) is the output of one node, a distribution node for product i to be exact. Every column of table a) is the input of one node namely a transformation node for making product j. All flows vi originate from a node outside the system. All flows yi end in a node outside the system. Within the system you can find node groups: j for transformation/production and i for goods distribution. A flows Zij go from one distribution node i to a production node j.
The upper formula is the market balance for the different goods produced. It implies that total production x either leaves the system to supply final consumers (y) or is used up by other industries (Z·e). The lower formula is the balance of the industries that produce the different goods. Every single good has exactly one producer. The formula implies that the quantity of goods (x) produced is the sum of the consumed precursor products (ZT ·e) and added value v
How many processes are there in an IO-system with 18 industries and products
36. 18 industries and 18 markets.
How do you identify the matrix for production functions A using the given system variables? What exactly is the meaning of a matrix entry for A? What does an A column, A row stand for?
A = Z x_hat^-1
A_ij shows the quantity of good i needed to produce one unit of good j, e.g., 12 kWh of electricity per kg aluminum. Column A_j describes the ‘recipe for production’ for good j, hence the quantity of all industrial goods required to produce one unit of j, e.g., electricity per ton of Al, aluminum oxide per ton of Al, fluorspar per ton of Al, etc. Row A_i describes the relative quantities of good i required for the production of the different industrial goods produced, e.g., electricity per ton of Al, electricity per ton of paint, electricity per service unit, electricity per produced motor vehicle, etc
How do you obtain the matrix of market shares B using the given system variables? What exactly is the meaning of a matrix entry of B? What does a B column, B row stand for?
Other than for A, for B the rows and not the columns are divided by the output vector x. Analogous to A you can then find.
B = x^-1*Z
B_ij denotes the inter-industrial flow of good i to industry j per output xi. B_ij hence describes the share of consumption of industry j in the total production of good i. Column B_j describes the consumption share of industry j in relation to total production of all goods. For example, Al industry consumes 3 % of total electricity, 98 % of total aluminum oxide, 65 % of total fluorspar, etc. Row B_i defines the share of all industries in the consumption of good i, for example 3 % of electricity goes towards Al industry, 2 % towards steel industry, 5 % to agriculture, etc.
Determine the formula to calculate Leontief Inverse L using given system variables! What does a single entry for L stand for? What’s the difference between A and L?
L = (I-A)^-1 = (I-Z*x^-1)^-1
We want to find the relationship between L and the system variables x, y, v, and Z. This we can do by replacing A with the definition of A. L is the matrix for the demand-driven IO model. The element L_ij determines how much of good i needs to be produced in the upstream chain to deliver one unit of good j to the final consumer. A contains data describing individual industrial processes; it also defines exactly one step of the supply chain of all products. L on the other hand includes all (infinite number of) steps of the entire supply chain of all products. It does not describe individual industrial processes only but a complex industrial network.
Determine the formula to calculate Gosh inverse G using given system variables! What does a single entry for G stand for? What’s the difference between B and G?
G= (I-B^T)^-1 = (I-Z^T*x^-1)^-1
We want to find the relationship between G and the system variable x, y, v, and Z. This we can do by replacing B with the definition of B. G is the matrix for the supply-driven IO model. An element G_ij defines how many of good i can be produced in the whole economy given one unit of added value in industrial sector j. B defines data for individual, simultaneously existing markets and also defines exactly one step of the distribution of goods. G on the other hand includes all (infinite number of) steps of the distribution of goods and it does not describe individual markets only but a complex industrial network
Describe two specific examples for when to use Leontief and Gosh matrices. During the cold war era the hypothesis was that the Leontief model represents the capitalistic economic system while the Gosh model represents the socialistic planned economic system. How did people come up with this idea? Do you agree/disagree?
Example Leontief: I buy a bottle of apple juice and calculate the output of the different industrial sectors (electricity supply, transport, refinery, …) of the bottle’s upstream production chain using L. Or calculating the total global economy output which was produced directly and indirectly for the German final demand of goods using the total demand of Germany, 2015, separated into different goods.
Example Gosh: I assume certain added value in a specific industrial sector and then calculate the distribution of added value among different goods in the economy using the Gosh model. Or: There are several growth-limiting nutrients in an eco-system, e.g., nitrate, and I determine their impact on different species and subsequent changes in the whole food chain using the Gosh model.
The different species in the eco-system represent the processes and the number of organisms is the output of the processes, which are either consumed by other organisms (Z) or increase the inventory of the own species (y). People used to believe that the symmetry of the two models represent the at that time rival economic systems.
The Leontief IO model is demand-driven and was applied in the USA during the war for example to improve planning of the industrial system dealing with the changes caused by the new war economy. On the other hand, planned economies stipulate the distribution of goods, or the consumption share, of every single industrial sector, hence people believed that the economic system of Gosh should be applied.
The system of stipulated market shares is unrealistic. Its application would mean that for each increase of production of x_i of 10 %, for example, all other sectors that obtain some fraction of x_i would receive 10 % more, no matter if they need it to produce output or not. That means the model assumes that with increasing tire production the ceteris paribus clause applies and hence automatically an increase of motor vehicle production can be observed.
Consequently, based on the Gosh model’s unrealistic production function assumptions it usually does not get applied to investigate changes in the system. Actually, even countries that applied planned economy used only the Leontief model. The Ghosh model is useful, however, to trace portions of value added through the economy. It can be applied as attributional model to trace the fate of capital services or subsidies, or of natural resources for a model in physical units
Exercise: "Multiregional input-output analysis (Excel-based)." This exercise contains a simple application of the MRIO analysis: construction of supply chains, carbon footprint calculations of final consumers in the EU, investigation of fine particulate matter and mercury emissions along the supply chain. Prerequisites: Matrix algebra on paper and Excel
Due to the international trade, the production of high emission and work intense goods is shifted from rich to poor countries (outsourcing). One side effect of this trend is that national emission values are no longer necessarily a representation of the actual emission intensity of the goods consumed by the final consumers within the individual countries. The MRIO analysis can reveal emissions along the supply chain. Calculation methods of the MRIO analysis are further the basis of calculations for life cycle assessments (LCA).
In this exercise we analyze a simple MRIO model. In the provided excel workbook ‘IEooc_Methods5_Exercise2_MRIO_Data.xlsx’ you can find an MRIO- X, matrix A, a matrix for total demand Y, along a stressor matrix S. The data was derived through aggregation of EXIOBASE2, a MRIO data base with 200 products and 48 regions. The dimensions of the matrices (3 regions, 11 product groups, 170 emissions) were chosen so any calculations with excel is possible as well as a measure to keep the model clean and clear.
1) What are the dimensions of the MRIO-A-matrix and what do the dimensions mean? What are the units of A and what do they mean? Cell H10 value is 0.25787…, what does this number mean? Cell AE10 value is 0.012854…, what does this number mean?
The dimensions of A are 33 x 33. A contains information regarding the use of 11 products from 3 regions within 11 industrial sectors in 3 regions. Since all inter-industrial-flows in this system are in MEUR (million euros) all coefficients in A are in MEUR/MEUR.
Hence A defines how many MEUR of product x from China are needed to produce product y within the EU. For example: a_7_5 = 0.258, (cell H10) means that the food sector in the EU obtains 0.258 MEUR services from the EU per MEUR output. A_7_5 = 0.0129, (cell AE10) means that energy providers in ROW (average of all countries except China and EU28) obtain 0.0129 MEUR of services from the EU per MEUR output.
2) What are the dimensions of the MRIO-Y-matrix and what do the dimensions mean? What are the units of Y and what do they mean? Cell AN7 value is 1,348,186. …, what does this number mean? Cell AO32 value is 14718…, what does this number mean?
The dimensions of Y are 33 x 3. Y contains information regarding the final consumption of 11 products from 3 regions within the 3 regions. All final consumption flows in this system are in MEUR (million euros).
Hence Y defines how many MEUR of product x from China are consumed by final consumers within the EU. For example: Y_4_1 = 1248186 MEUR, (cell AN7) means that the total domestic final consumer demand for consumer products within the EU was around 1350 billion Euros. Y_29_2 = 0.0129, (cell AO32) means that the final consumer demand in China for services from ROW was 14 billion EUR.
3) What are the dimensions of the MRIO-S-matrix and what do the dimensions mean? What are the units of S and what do they mean? Cell I39 value is 2,544,378. …, what does this number mean? Cell T62 value is 0.1392…, what does this number mean?
The dimensions of S are 170 x 33. S contains information regarding the emission intensities of 170 types of emissions of 11 industrial sectors in 3 regions. All inter-industrial-flows in this system are in MEUR (million euros) and the emissions have varying units and are defined on the sheet.
Hence the units of the coefficients in S are [given unit]/MEUR. Hence S defines how many emissions of type x are emitted during the production of product y required in the EU. For example: S_1_6 = 2544378 kg/MEUR, (cell I39) means that the energy supply sector in the EU emits 2540 t CO2 into the air from combustion processes per MEUR output. S_24_18 = 0.1392, (cell T62) means that the energy supply sector in China emits 0.14 kg mercury into the air from combustion processes per MEUR output.
4) Tom buys a desk, an air journey, and two Snickers. Susi buys 10 kg of potatoes and a car. How do you calculate the combined carbon footprints of the bundles of goods for Ralf and Susi?
The Leontief IO-model is a linear model. This means that the carbon footprint of the combined bundles of goods of Ralf and Susi is the sum of all bundles of goods footprints from Ralf and Susi which are further the sum of all single goods within the bundles.
5) Determine the total global industrial production x_EU28 needed to meet the total final demand of all consumers in EU28! What are the dimensions of vector x_EU28 and what does it mean? Compute a series expansion for the Leontief inverse and specify the first six summands x_0 to x_5 along with the final result (limit value x_infinity). Examine the results for the sectors ‘landfill’ and ‘mining’! How do the total outputs behave within the three regions compared to final demands of goods ‘agricultural products’ and ‘mining products’? How do the single steps of the value chain develop with increasing order of x_i and how can the development be explained?
Below we then create the fields for the first five powers starting with A, A^2, A^3, A^4 and A^5. The powers we then obtain gradually from A by simply applying successive multiplication processes (MMULT, matrix function!). By clicking on the single cells in A^x you can inspect the applied formulas. Eventually we determine L using the known function. Using the values for A through A^5 as well as L we can evaluate the results
6) Determine the total emissions of ‘CO2-combustion, air’, CH4-combustion, air’, CO2-non combustion, air’, ‘CH4-non-combustion, air’, and ‘Hg-combustion, air’ for the first six summands for x_EU28 and the total output x_EU28! How big is the share of a) emission of zeroth order and b) steps 1-5 of the supply chain compared to total CO2 emissions? What’s the explanation for these results?
These results are shown in table ‘Total emissions for EU final demand, iteration’. Next, we multiply the rows of the stress matrix S with the values of x accordingly. ‘CO2-Combustion, Air’ applies to the first row of S, ‘Hg-combustion-air’ applies to the 24th row. The results of this calculation can be found in table ‘Total emissions for EU final demand’. Discussion: The emissions of the zeroth order CO2 combustion emissions account for only a quarter of total emissions, the other 70 % are also not emitted during production of the consumed goods but further up the supply chain.
The zeroth order emissions for methane (non-combustion) are especially low with only 5 %. Most of the emissions (95 %) are emitted during higher stages of the supply chain. This makes sense since non-combustion methane are produced especially in agriculture and natural gas production which are both sectors that supply little directly to final demand. It’s the opposite with Hg emissions. Here the contribution of the order 0 is almost 55 % which can be traced back to electricity production emissions.
7) Determine the emissions of ‘CO2-combustion, air’ for each sector and region of x_EU28. Do your calculations in appropriate units (ex. Mt). What are the sector-region pairs that contribute the most emissions to total CO2 emissions? How do CO2 emissions split up among the three regions?
Discussion: The total consumption-based CO2 emissions in EU28 in 2007 were 4.78 Gt. Of this 57 % can be allotted to the EU, 13 % to China and 30 % to ROW. The biggest share of emissions come from energy supply in the EU (1300 Mt), service sector of the EU (920 Mt), energy supply in ROW (700 Mt) as well as material production sector in ROW (370 Mt)
8) Determine the global CO2 and fine particulate matter (PM2.5 combustion air) emissions of one EU citizen generated through combustion processes using the average annual consumption of goods of 30,000 Euros (basic price w/o sales tax or other levy and taxes). How do the emissions split up among a) the three regions and b) the 11 sectors?
This calculation we perform on a different sheet ‚Unit Demand‘. First we determine the average bundle of good for 30,000 Euros by taking the associated raw data from total consumption in the EU and then downscale with the help of the rule of three to 30,000 EURO = 0.03 MEUR (column D top). Then we determine total output x for this demand (column D, bottom). As in task 7 we copy and transpose the relevant rows from S (row 1 and 18), column H and I. Through simple multiplication of x with S we receive the final result (column E and F).
In total sum we incur 12 tons of CO2 and 4.2 kg PM2.5 emissions. The share of emissions for the EU is 55-60 %, China 13-14 % and 25-30 % for ROW. The biggest share for CO2, just like in exercise 7 (linear model!), stems from the energy supply sector in the EU (3.3 t), the service sector of the EU (2.4 Mt), the energy supply in ROW (1.8 t) as well as the material production sector in ROW (1 t). The biggest emitter of particulate matter in the personal supply chain is the service sector (transport!) with 1.63 kg, followed by material production in ROW (0.43 kg) and the energy supply in the EU (0.35 kg)
Appendix
The series <I + A + A^2 + A^3 + ... + A^N> converges on the Leontief inverse (I-A)^(-1) as N approaches infinity. In this format, I can be thought of as initial demand for a given product, A represents the first tier inputs in the supply chain required to produce I, A^2 represents the second tier inputs in the supply chain needed to produce the first tier inputs, A^3 is the the third tier inputs needs to produce the second tier inputs, etc. Viewed in this way, we can see that one intuitive explanation of the Leontief inverse is as a summary of the supply chain relationships that make up the economy.
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