4.2.2.1 The Expenditure Approach

The most useful method which economists use for measuring the GDP is the expenditure approach. The GDP is equal to the total expenditures for all final goods and services produced within the country in a stipulated period [9].

(4.1)

Y is GDP; C is personal consumption expenditure (PCE); I is Investment, gross private domestic investment; G is Government consumption expenditure and gross investment; NX is Net export, Export minus Import. Table 4.1 shows this method for USA GDP in 2019.

Personal Consumption expenditures (C) consist of expenditure people pay for goods and services. Goods includes durable goods and nondurable goods. Durable goods are something that lasts for a long time (usually for more than one year); motor vehicles and parts, furnishings and durable household equipment, and recreational. Nondurable goods are those that last for less than one year; basically a citizen's daily life consumptions. Food and beverages purchased for off‐premise consumption, clothing and footwear, gasoline or other energy goods, and other items. Services consist of household consumption expenditures (for services) and final consumption expenditures of nonprofit institutions serving households. Household consumption expenditures contain housing and utilities, health care, transportation services, recreation services, food services, insurance, and other services. Final consumption expenditures of nonprofit institutions serving households contain the gross output of nonprofit institutions minus receipts from sales of goods and services by nonprofit institutions.

PCEs are the highest part of the GDP for 2019; reaching 67.9% of the GDP of the United States.

Gross private investment (I) consists of fixed investment and change in private inventories. Fixed investment has two parts; nonresidential and residential investment. Nonresidential investment contains structures, equipment, and intellectual property products. Change in private inventories includes farm and nonfarm investment. Gross private investment contains 17.5% of the GDP. Almost all of these investments are about fixed and nonresidential investments.

Net exports and imports (NX) include exports and imports of goods and services. A trade deficit occurs when the number of goods and services a country imports is more than goods and services export. The United States mostly has a trade deficit, and it was −2.8% of GDP in 2019.

Government consumption expenditures and gross investment (G) have two parts; federal, and state and local for the US. Federal consists of national defense and nondefense, and state and local consists of government consumption expenditures and gross investment. The government consumption expenditure and gross investment was 17.5% of the GDP in 2019, and most of it was from consumption expenditures of states and local governments.

4.2.2.2 The Income Approach

Another approach for measuring the nation's economy is the income approach. This approach counts the income earned and costs spent during production. The Gross domestic income (GDI) should equal the GDP because of the different sources used to measure it. You can see this method in Table 4.2 for USA income in 2019. There are some differences between these two kinds of measurements and is known as the “statistical discrepancy.”²

The name of the accounts in this table can show which activity counts; but an account that is less speaks about is the consumption of fixed capital (CFC).

CFC is an account considered to degrade and corrode the economy’s capital. This item was $3474.4 billion in 2019, and most of this includes the private sector. In the Bureau of Economic Analysis, this account is defined as, “The charge for the using up of private and government fixed capital located in the United States. It is the decline in the value of the stock of fixed assets due to wear and tear, obsolescence, accidental damage, and aging. For general government and for nonprofit institutions that primarily serve individuals, CFC serves as a measure of the value of the current services of the fixed assets owned and used by these entities” [10]. The System of National Accounts (SNA) defined this as, “Consumption of fixed capital (CFC); the decline in the value, in current prices, of the stock of capital goods held by producers resulting from physical deterioration, normal obsolescence, or normal accidental damage” [11].

In the later sections we speak about this account in more depth.

4.2.2.3 The Value‐Added Approach

The value‐added approach is another source for measuring economic production. It concerns the values that each institutional unit adds to the components which produces a good or service. This approach estimates the sum of employee compensation, taxes on production, and imports fewer subsidies and gross operating surplus. Table 4.3 explains this approach. Each item in the following table includes one of these three components. In the following table, you can see the share of industries in domestic production.

As you can see, these three types of measuring the GDP are almost equal, but the points of view are different.

4.2.2.4 Income, Consumption, Saving, and Investment

Saving and investment are two factors related to the main topic of this section. To explore these two factors, we must familiarize ourselves with the relation between income, investment, consumption, and saving. All of the consumption, investment, or saving in the private and governmental sectors include these items.

In an economy, income is distributed between savings or consumption. Thus, this relation is between these three items:

(4.2)

where Y is the income of the economy, C is consumption, and S is savings. Savings is the part of income that society prefers to put away without spending, for example, in a bank account.

On the other side, this relation is between consumption and income:

(4.3)

C₀ is the undebatable consumption in an economy and is called autonomous expenditure.

There is another definition for C and Y:

(4.4)

Where MPC means marginal propensity consumption.

Furthermore, this relation is between saving and income.

(4.5)

Also, marginal saving is:

(4.6)

The relation below is between marginal consumption and marginal savings:

(4.7)

In sum,

(4.8)

(4.9)

This equation shows the relation between savings and consumption in an economy. As you can see, if autonomous consumption increased, the level of savings would decrease.

4.2.2.5 Gross National Product

Production of final goods and services by a country's production factors are the GNP, regardless of where they are produced. The three primary production factors are land, labor, and capital [8]. Everywhere these factors generate an output, it measures as that country's GNP.

The differences between GDP and GNP are about the border of a country; all of the goods and services produce within a country's borders, regardless of its citizenship, are the GDP.

An example of these two types of estimations is the Honda plant in Marysville, Ohio. This is a Japanese firm owned by the Honda Corporation. The workers of this company are US workers, so the output of this plant is included in the US GDP. The wages paid to US workers are US GNP; Honda's profits count as Japanese GNP but not as Japanese GDP because the products are created within the other country's borders [8].

4.2.3.1 Production Account, the Intermediate Consumption, and the Consumption of Fixed Capital

The most crucial point in this standard relating to this book's purpose is the production account. This account displays the account sequence for institutional units, and sectors show how income is generated, distributed, and used in an economy. The economy‐wide production account includes the account aggregation for an institute or a sector and as a whole for the entire economy. The production account allows the study of industrial activity and permits the compilation of supply and use tables, and inputs‐output tables, extending this table in the other sections of this chapter [11].

The production account shows the value‐added, and it is one of the main balancing items in the SNA. Table 4.5 shows the production account in uses view. However, this account does not cover all transactions within production processes or show the final production (output); it is sample information for corrosion engineers [11].

The intermediate consumption consists of inputs of goods and services in the production process. It neither covers the cost of the fixed assets use nor any expenditures on the acquisition of fixed assets. Measuring the intermediate consumption is troublesome because the borderline between consumption and gross fixed capital formation is not always easy to determine in practice. One example of this dichotomy appears when repairs and maintenance occur; ordinary maintenance and repair for fixed assets constitute intermediate consumption, these actions improve their performance and increase their capacities. Drawing a line between common repairs and necessary improvements is problematic, although SNA provided enough recommendations for this issue [11].

CFC is a flow of physical deterioration, normal obsolescence, or accidental damage such as storms, fires, or human errors that decline the current value of fixed assets. This must measure accidental damage, normal obsolescence, and physical deterioration, so we calculate it because of changing value‐added estimation. The CFC is one of the most important elements of the SNA. This account calculated fixed assets owned by producers. It declines over time and use for output in the production process. In addition, capital consumption measures all types of structures, such as railway tracks or roads that are scrapped or demolished. These types of assets are owned or maintained by government units. This account does not consist of precious metals or stones, or natural asset degradations such as oil, coal, etc. Some of the losses infrequently occur as major earthquakes, volcanic eruptions, tidal waves, or hurricanes. These items do not include accidental damage, and they consist of cost of production. CFC is one of the most troublsome accounts to define conceptually and estimate in practice [11].

There is an important reason for calculating the CFC account. The assets may have been purchased in the past, the consumption occurs in the future years, and its prices are different from the original supply. The CFC account reflects underlying resource costs and relative demands at times of production, but, the cost value with a set of current prices is used for output and intermediate consumption. The historic costs⁴ of fixed assets become irrelevant for this account calculation, as a result, a reduction in the PV derives from the CFC [11].

The final item is the gross or net value added in this section. The gross term is used before you deduct the amount of the CFC. When it is deducted, the value‐added becomes the net. This term applies to all the major balancing items in the accounts from value‐added to savings [11].

4.2.5.1 Technical Coefficients

In the nation‐wide subdivided sectors to n + 1 sectors they produce n + 1st final demand. Imagine we have two sectors, with the names i and j; the output of sector i is x_i, and sector j absorbs this output as its input with x_ij. Also, there are many other sectors to absorb x_i, thus the final demand of sector i is x_in + 1, and its quantity is identified by y [12].

The output quantity of sector i absorbed by sector j per unit of its total output j es described by the symbol a_ij and called the input coefficient of the product of the sector i into the sector j [12].

(4.12)

The matrix arranged by all of the a_ij items is called structural matrix of that economy.

There is a balancing between the total output and the combined inputs of the product of each sector. We can show these in a set of n equations:

(4.13)

In this set of equations, x₁, x₂, ⋯x_n are a set of outputs, and y₁, y₂, ⋯y _n are the set of the final bill of goods absorbed by the economy such as households, government, industries, and so on. Combinations of the (4.12) and (4.13) make the following equations:

(4.14)

If we solve these equations for unknown x's, we can find the number of goods absorbed by parts of an economy are y₁, y₂, ⋯y_n . Thus, Eq. (4.14) becomes:

(4.15)

For our example these equations mean total demands for x₁ are combinations of all demands of these inputs in all parts of the economy, a sector y₁ with its coefficient A₁₁, plus sector y₂ with its coefficient A₁₂, and so on.

The constant A_ij shows, one unit increase in x_i (of the ith sector) if the y_i increase in their demands. The output x_i for i = j is directly affected (and also indirectly) and for i ≠ j affected indirectly. Sector j increases in their outputs, while sector i provides additional inputs for other sectors including sector j as y_j. The computational view tells us each magnitude of coefficient A in the solution equations, (4.15) in general, depends on all the input coefficient a appearing on the left‐hand side of the system of equilibrium Eq. (4.14) [12].

The matrix A in the mathematical language appears as

(4.16)

This matrix shows the constants appearing on the right‐hand of the solution (4.15), and these constants are the inverse of this matrix:

(4.17)

Because there are many outputs, x₁, x₂, ⋯x_n, which are inputs for other outputs, y₁, y₂, ⋯y_n, the inverted matrix, A_ij, are nonnegative. There is a sufficient condition for this,

(4.18)

and of all its principal submatrices,

(4.19)

It means this matrix should be positive; this is the so‐called Hawkins–Simon condition. If this condition satisfies for one sector, then it satisfies for all other sectors, too. The material interpretation of this condition is if each sector function of an economic system absorbs the output of other sectors, directly and indirectly, it can not only sustain itself, but can also make positive delivery to final demand for smaller and smaller subsystems and so on, and this can make a sustainable production system . However, if one of them cannot pass the test, it makes abound and cause a leak that will destroy the sustainability of all systems [12].

In other words, the condition of the sustainability of an economy is the sum of the coefficients of each column and its structural matrix should be less than or equal to 1, if at least one of the column sums is less than 1 [12].

In an open economy, exports can be positive and imports can be negative for components of final demand.

4.2.5.2 Price and the Input–output Table

Another item of an economy determining an open input‐output system is price, and there are a set of equations to show the price of each productive sector of the economy. These equations consist of prices received per unit of its output and the total outlay incurred in the course of the production; these two items must be equal. The equations below show the total outlay of an economy output, these outlays include both purchases of inputs and also value‐added for other exogenous sectors [12].

(4.20)

In these equations, p_i is the price of each product of endogenous sectors, and each of these equations describes the balance of a receiving price and payments for output. This price usually contains wages, interest on capital, and entrepreneurial revenues of households, taxes, and other final demand sectors [12].

Similar to solving Eqs. (4.14) for x_i's, in these equation we solve for p_i's to determine the prices of all products from the value added of each sector [12].

(4.21)

In Eq. (4.21) the price p_i is a dependent variable of sector j with value added, v_i, earned per unit of its output in sector i. In these equations too, the constant A_ij measure these dependent variables, p_i's [12].

There is a fantastic relationship between these two set of equations; the a_ij's in the rows of q. (4.14) and corresponding column of the q. (4.20) are related to each other. This means the A_ij coefficients in the row of output matrix build the corresponding coefficients in the columns of the price solutions matrix, q. (4.21).

The following identity is derived from qs. (4.15) and (4.21), because of the internal consistency of the price and the quantity of relationships within an open input–output system.

(4.22)

This equation’s sides show two main principles of an economy. First, there are some endogenous parts of this system and exogenous parts. The left‐hand side shows paying out of endogenous parts of the system to the exogenous sectors of this system; on the right‐hand side it combines values (quantities × prices) of their respective products delivered by all endogenous sectors to the final demand. It shows the accounting identity between the receiving and spending national income.

The price analysis has another viewpoint relating to the stocks used in the production processes. Stocks are a particular kind of goods such as machine tools, industrial buildings, and “working inventories” of primary or intermediate materials. These stocks are produced by industry i and employed by industry j, and measure its value based on per‐unit of the sector j output. To show this, capital requirements represent a matrix B. Each column of the matrix B describes the physical capital requirements of a particular industry. In the same way, the corresponding column of matrix A has its “current inputs” requirements [12]. The capital coefficient b_ij is:

(4.23)

The price analysis can advance by splitting the right‐hand side of the equation into two parts. The value‐added is divided into two parts: capital, invested in building, machinery, and other stocks of goods required for the production of output; and wages. In addition, there is a return on capital representing the value of all productive stocks multiplied by the given rate of return [12].

Finally, the price of goods and services has a relationship with wage rates, the rate of return on capital:

(4.24)

where W is a column vector of wage costs paid by different industries per unit of their respective outputs [12].

4.2.5.3 Dynamic Input–output Analysis

To describe and analyze the process of economic growth, we employ a set of linear differential equations representing dynamic input–output relationships.

(4.25)

In this equation, the column vector Y(t) represents the amounts of various goods and services in year t produced by sectors to households and other final users. X(t) and X(t + 1) are the column vectors representing the output levels of different industries in times t and (t+1). A is the matrix of input coefficients in Eq. (4.16), and B is the matrix of capital coefficients in Eq. (4.23). The capital stock produced in year t is used in year (t+1).

These sections are to familiarize you with the main principles of the input–output model introduced by Professor Leontief. In the following section we discuss Corrosion Economics and the input–output model by Battelle Institute, which extended these primary principles for estimating corrosion cost. To describe and analyze the process of economic growth, we employ a set of linear differential equations representing dynamic input–output relationships.

4.3.1.1 Matrix of Technical Coefficients

Matrix A is the matrix of direct technical coefficients. These matrix cells show the dollar worth of inputs that is required to produce one dollar of output. The worth of inputs are shown in the row, and the worth of outputs are shown in the column of this matrix. The sum of column coefficients in this matrix is inputs per dollar of its output.

To design matrix A we add three items together to obtain the matrix cells; these items are value‐added, social savings, and foreign trade. These three rows are outside the matrix A.

Value‐added is the intermediate matrix speaking about this item's nature in economics.

Social savings is a new row added to the transaction table and matrix A, and it is a hypothetical item. In World I, which is the real world, this row is empty. This item measures the labor and resources not used in the absence of corrosion; in other words, the reduction in the GNP that would take place if our world were corrosion‐free. It has an indirect influence on the sizes of all the intermediate direct coefficients [16].

Foreign trade consists of the imports of intermediate and final demand for goods and services. These imports subtend competitive and uncompetitive. The items included in the Battelle model are “noncompetitive” imports and show in the matrix as a row across the table. Because the competitive imports constitute a large proportion of the intermediate demand, these imports are not included. Demands such as crude oil contained the competitive items in this matter, for they became heavily negative total outputs [16]. To modify this problem, the Battelle carries all imports as a row outside the intermediate and inverted matrices. Thus, the total input becomes:

(4.26)

where TI is total input; TDI is total domestic intermediate input; DVA is domestic value added; IM is imports.

Regardless of domestic or imported, every direct coefficient in the A‐matrix is technologically correct. An import column calculates by multiplying each sector's output by the sector's import coefficient. These calculated values can add as a column to the table so that:

(4.27)

and

where TDO is total domestic output; TIO is total intermediate output; TFD is total final demand; TI is total inputs [16].

Finally, the column sum of the A‐matrix plus the sum of the three above items (value‐added, imports, and social savings) is equal to one since the coefficients are in terms of proportions. The sum of the column coefficients of matrix A is equal to per dollar of the output of the column sector's use of its domestic intermediate inputs.

In World I, the industry's total output is equal to the column industry's total inputs, also equal to the sum of cells of matrix and value added and import rows filling with its dollars.

4.3.1.2 Matrix of Capital Coefficients

Matrix B is the matrix of stocks used in the production processes and called the capital matrix, as you learned in the Leontief input–output model. The previous matrix, A, measures dollars worth of input per dollar of output, while B matrix measures dollars worth of capital per dollar of output. However, because there are many other definitions for the capital stock matrix, the capital matrix used in this study is for estimating this purpose.

In this matrix, the sectors listed across the top of the matrix show capital users, sectors listed across the side of the matrix indicate capital producers. There are a great many empty rows in this matrix because the capital producers are low.

Although the capital stocks excess is for more than one year and include durable goods, they need to grow and replace. The growth in the demand for an industry's output is the reference of the growing need for capital stocks, regardless of whether these growths are demand for final or intermediate goods, or foreign or domestic. In addition, capital replacement may relate to technological change or the age of existing capital. For these scenarios, matrix B changes in this manner:

(4.28)

where B is the standard capital coefficient matrix (expressed in the form of ratios of capital stock to output); G is matrix of annual growth rates of total capacity (capital); and R is matrix of annual replacement rates of capital.

The growth matrix – Growth in demands calls for growth in the capital matrix to determine this capacity. We need the ratio g from the first approximation growth rates of total output (X). If using K as the capacity, we use g in this manner:

(4.29)

We can substitute a comparable ratio based on X (which approximates capital formation change based on this total output of the sector of the economy) so that it changes approximately equal to the direct or indirect growth of general demand.

(4.30)

The replacement matrix – All the stock capital has a life period; this age‐structure relates to the steady growth at the rate g. In other words, the replacement rate is a function of the growth rate and the replacement life of that stock. The replacement rate (r) of each sector is a combination of both replacement life expectancy and sector growth.

Matrix of replacement lives (U) is a full matrix, like B‐matrix. Every U_ij corresponding with a nonzero cell in B show a replacement life expectancy value. These values may take from a given number of years or a range of years. Each column has an average annual growth rate, g_j, derived with Eq. (4.30) and used to describe the entire cycle of replacement lives.

To show the current deterioration of stock capital which has formed during a period u years, and assuming smooth growth, we must consider the reasons for growth and replacement during this period. In this sequence of years, K₀ is the first year and K_u − 1 is the end of this period. The total value of the current capital stock can show as ∑K:

(4.31)

where K₀, K₁, … is the growth and replacement capital purchase in year 0,1,2, … respectively, and U is the life of first replacement of the capital stock.

Now, to obtain the annual replacement rate of capital for these sectors we have:

(4.32)

In a given annual growth rate of the sector g, each K becomes:

(4.33)

As you can see, K₀ is common to all terms, thus:

(4.34)

Then this can transfer into:

(4.35)

To build the replacement matrix, R, we establish a value of r_ij for each corresponding value of u_ij.

Now is the time to set up the basic I/O model equation with recent principles together.

4.3.1.3 Input–output Model

The Leontief input–output equation can be written with this standard matrix notation:

(4.36)

where X is total output; A is the direct coefficient matrix; and FD is total final demand.

The rephrase of this equation says to produce the total output, so we must cover both intermediate input requirements and final demand.

Taking account of the investment demand rewrite Eq. (4.36) in this manner:

(4.37)

where is total capital coefficient matrix [which we saw in Eq. (4.28)]; and is total stipulate (noncapital) final demand.

Thus, Eq. (4.37) can be written:

(4.38)

where XBG is growth capital and XBR is replacement capital, then the sum of these two terms is the total capital.

Then we extract the total output from Eq. (4.38):

(4.39)

4.3.1.4 Final Demand

The third major component of the I/O model is final demand. This final demand is the consumption we learned of in the first section (in estimating GDP), personal consumption, government purchases, exports, and net inventory change. In theory, the final demand summation is equal to GNP; but in reality, it needs to be adjusted. The adjustment of GNP done by adding gross private domestic investment to and subtracting imports from the consumptions.

In the final demand, the assumption is that we have full employment in the economy [16]. One of the four main concepts of economics is the employment rate. In economic principles, full employment occurs when 5% of working‐age people, aged18 to 65, are unemployed, and 95% of people are employed.

The consumption of households, government expenditure, exports, and inventory change are the four types we have discussed. In the following paragraphs, I bring these types of consumption components in the Battelle final demand.

In the Battelle approach, we have a three‐dimensional matrix for household consumption. These dimensions are income behavior class, age of head of the family, and the family size. This concept can estimate the level of consumption by a family and the entire economy as a whole [16].

For this estimation, the Battelle report designated a separate analysis to measure how families with different income and different size characteristics spend their income in certain categories.⁶ A regression equation expresses how many average families in a given income class spend their consumptions in one year. The total amount gathered with this method uses a PCE column in the I/O table's final demand sector. The governmental expenditure is this expenditure in the normal situation, not about the emergency programs, for example when a country is engaged in a war. Foreign trade in the final demand matrix is about the gross exports. In the A‐matrix we have a row for imports.

The last item contained in the final demand is the inventory change. It is about the change in the stock goods over many years. This issue determinate is crucial in practice; thus, economists use econometrics methods to calculate this part. In other words, the inventory change means how much goods every industry has at the end of the year in their warehouse, and this amount changes year by year. You can see this account in the table of GDP expenditure approach.

4.3.1.5 World I, World II, World III

As mentioned in the beginning of this section, there are three worlds for an economy; World I is the existing economy, World II introduce a hypothetical world without any corrosion, and World III is a hypothetical world with efficient corrosion control.

To introduce these worlds we reduce the information of World I to coefficient and range change. This means the World II and World III flow is the adjusted coefficient of World I. Adjustment of capital/output coefficient is expressed as a percentage, excess capacity for an industry, or for an entire economy; World I. Adjustment of replacement life is expressed as a change in the range of years; for example, if for World I we use 10–30 years, for World II use 20–30 years. Changes in final demands are expressed as a percentage change of World I values [16].

The differences between World I and World II is the national cost of corrosion and shows resources wasted cause of corrosion. The differences between World III and World I shows the avoidable costs of corrosion. These avoidable costs could not occur if economically preventive practices were used throughout an economy. Moreover, the differences between World III and World II measures presently unavoidable costs [17]. In all, there are differences between industries in this concept, and it relates to the users of goods and producers of goods.

4.3.1.6 Estimating Corrosion Cost by Battelle

We introduced the Battelle input/output model and described its components, in the following, we bring the Battelle models to estimate the corrosion cost in the industries.

Disaggregation – You saw in the direct technical coefficient matrix that certain cells within the final demand vectors disaggregate. This disaggregation is about the sectors producing durable goods and the corrosion prevention materials, and maintenance and repair of construction sectors. These sectors are disaggregated into corrosion‐related sectors and non‐corrosion‐related sectors. Then, we introduce some annual purchasing methods for these kinds of goods and their replacement and growth.

(4.40)

Social capital/infrastructure is defined as consisting of final demand of durable goods that last for more than one year.

The growth in the purchase of capital is:

(4.41)

where SK is social capital (in abbreviations we usually use K for capital).

Thus:

(4.42)

and

(4.43)

This equation can be rewritten in this way:

(4.44)

g is the growth purchases in one year, for u years this growth becomes a rate of change:

(4.45)

The annual replacement of social capital is:

(4.46)

Substituting qs. (4.44) and (4.46) into (4.40) makes:

(4.47)

Now solve Eq. (4.47) for (SK_t) because we need to know about the social capital in time t:

(4.48)

Cost parameters – There are four different parameters for estimating the cost (i) the direct cost of corrosion per unit of output, (ii) the total direct costs of corrosion per sector, (iii) the direct and indirect cost of corrosion per unit of output, and (iv) the total direct and indirect cost of corrosion for each sector.

4.3.1.6.1 The Direct Cost of Corrosion per Unit of Output

Comparing production requirements in Worlds I and II and III show the direct per‐unit cost of corrosion for each sector [16]. This production requirement is the intermediate requirement (direct technical coefficients including value‐added) per unit of the sector's output, capital requirements per unit of output of the producing sector multiplied by the sector's growth rate, and the annual replacement of capital per unit of production. These component summations show a total requirement necessary to produce a unit of a given sector's output. Below are the steps to represent the differences between these three world requirements.

Step 1. Calculate the direct requirements per unit of production:

(4.49)

where g_j is the annual rate of growth of sector j's output; SS_w,j is the social savings entry for sector j in World w (SS_j is zero for World I); C_wj is the total capital/output ratio for sector j in World w; and ρ_wj is the total annual replacement of capital required to produce one unit of section j's output in World w.

In Eq. (4.49), 1 − SS_w,j is equal to the sum of direct technical coefficients plus value‐added, and is a more direct approach to obtaining that sum.

Step 2. Calculate total direct per unit costs:

(4.50)

Step 3. Calculate reducible direct per unit costs:

(4.51)

A devised account for keeping the cost of various corrosion‐related expenses is the value SS_w,j. This account contains any reductions in operating costs that those reductions relate to; both intermediate inputs and value‐added. The input changes are determined through industry surveys, and the value‐added adjustments estimated in the following ways.

Value‐added adjustments – There are several adjustments in each industry's value‐added coefficients, and it shows the differences between World I, World II, and World III. These adjustments group in the common term; Z_wj. This value factors the sector j in World I (w shows the number of the world). This value Z_wj is the value‐added that subtracts from sector j in World I and adds this value to row sector, SS, world w, and sector j. SS is the cumulation of net corrosion savings in the column of direct technical coefficients. Thus Z_wj is defined as:

(4.52)

A_wj shows the reduction in the cost of holding capital that the total capital requirement reduced. This occurs when the prime interest rate reflects the fact that some firms cannot obtain capital at the prime interest rate.

(4.53)

∆_wj shows the reduction in a net accumulation of the depreciation account. Gross using of the capital and the gross accumulations in the depreciation account are measured in D. The value of replacement accounted for final demand is measured in ρ. The difference between them represents the net accumulation of in the depreciation⁷ account. ∆ shows the reduction in the net accumulation in World I to Worlds II and III.

(4.54)

ρ_wj represents the annual replacement of capital that is used by sector j in World w:

(4.55)

where r_wij is the World w annual replacement rate of capital produced by sector I and used by sector j; c_wij is the World w capital produced by sector I and required to produce a unit of sector j's output.

Then the average replacement rate, RR_wj, can be applied to sector j's total capital:

(4.56)

where C_wj is the total capital/output ratio of sector j.

Referring to the replacement matrix we have:

(4.57)

where g is the growth rate of sector j; and U is the World w average replacement life of capital used by sector j.

Now combine qs. (4.56) and (4.57) and solve them for the average replacement life of all capital, U_wj, used by sector j:

(4.58)

It is not possible to calculate the straight life depreciation of capital, D_wj, owned by sector j:

(4.59)

4.3.1.6.3 Direct and Indirect per Unit Costs of Corrosion

We can obtain the total direct and indirect costs by calculating the total direct and indirect requirements needed to produce the vectors of direct requirements, as shown in the previous sections.

This calculation involves multiplying the vectors of direct requirements associated with each sector by the appropriate World I, II, or III inverse (the inverse in this position refers to (I − A)⁻¹). The resulting vectors sum, and for each sector World II summation subtracts from the World I sum., then the World III summations subtract from the World I sum. Mathematically the procedure may be represented by the following steps:

Step 1. Form a V_wj of which the ith row is defined as:

(4.60)

where V_wij is the ith row of vector V_wj; a_wij is the direct technical coefficient, World w, row i, column j; g_j is the growth rate for sector j; c_wij is the capital supplied by sector I to sector j per unit of j's output in World w; and r_wij is the annual replacement by j of capital supplied by i (capital per unit if j's output) in World w.

Step 2. Form the matrix product:

(4.61)

where is the inverse matrix for World w; and X_wj is the resulting output vector corresponding to sector j.

Step 3. Sum columns X_wj:

(4.62)

where X_wij is the ith row of X_wj.

Step 4. Calculate total direct and indirect per unit cost:

(4.63)

Step 5. Calculate reducible direct and indirect per unit cost:

(4.64)

4.3.2.1 Life‐Cycle Cost Model

In the LCC model, estimation of cost sum up all the costs of a component in the lifespan and makes its life cost. This section brings two kinds of equations to use to estimate the LCC.

4.3.2.1.1 The Dr. Reddy Formulation

This equation was developed by V. Ratna Reddy and his co‐author in the “Life‐cycle Cost Approach for Management of Environmental Resources” book. This equation is quite comprehensive to show the cost of a system.

The basic functional form of the equation is:

(4.65)

where LCC_xt is life‐cycle costs of specified products/services; CapExhw_xt is Capital expenditure on hardware (initial construction cost); CapExsw_xt is Capital expenditure on software; CapManEx_xt is Capital management expenditure (rehabilitation cost); CoCap_xt is Cost of capital; DsCost_xt is Direct support costs; IDsCost_xt is Indirect support costs; OpEx_xt is Annual operation and maintenance cost; CoEExt_xt is Cost of environmental externalities; x represents product or service, and t represents year.

This equation brings the LCC of an asset by either including the expenditure on the capital process of life or the environmental externalities⁸ and social sustainability. However, quantifying some of these costs is difficult to determine, as these costs must annualize for standardizing cost. Some of these costs are for the past or future; thus, we must make these costs to the PV to compare them with other investments [18]. Accordingly, Eq. (4.65) can be written as:

(4.66)

where pvf is present value factor (1 + r) ^t ; r is rate of interest or inflator and; t is time period.

Using the inflation rate or the prevailing interest rate may be appropriate for estimating the PV or worth [18].

The other view that Dr. Reddy brings is the risk based in analyzing the cost of a lifecycle. Risk and uncertainty are features of some of the components. Systems may not follow a schedule and fail randomly. Natural factors and unexpected climate events create various risk and uncertainties impressing the estimation of the component's cost.

(4.67)

where Psf_xt is probability of risk.

This formulation is appropriate for a wet environment if the groundwater is substantial. This view of the LCC helps to estimate the real‐life cost in every climate and location [18].

4.3.2.1.2 The Professor Adeli Formulation

The other view of estimating the life‐cycle cost is from Professor Hojjant Adeli and his co‐author in the book “Cost Optimization of Structures.” They introduce a formulation for measuring the LCC of steel structure, but we can use it for any other components.

(4.68)

where C_Lifecycle, C_Initial, C_Maintenance, C_Inspection, C_Repair, C_Operating, C_Failure, and C_Dismantle are the total life‐cycle, initial, maintenance, inspection, repair, operating, probable failure, and dismantling costs of a steel structure, respectively; i is the discount rate of money; y_n1, , , , , are the years when the associated costs incur; and ∑ denotes the summation of all the costs of the same category during the life of the structure [20].

You read in the PV section that is a factor to convert the cost value to the PV discounted. These two models are two various models for estimating the cost of a life cycle in a system. In the first model, the LCC of an asset, you see system externalities, such as its effect on the climate, being profound for these estimations. In the second model, the main point is bringing all costs to the present, whether they are future or past.

If you need to know about more models or estimation of the cost read the book “Life Cycle Costing for Engineers” by Professor B. S. Dhillon; in chapter 4 he introduces some LCC models and cost estimation methods.