2. Basic principles

1999-06-23
To Chap 1

  1. Monetary and real economy
  2. Payment balances
  3. Systems and flows between sectors
  4. Aggregates
  5. Resource flows
  6. Prices, wages & salaries
  7. Processes
  8. Self-supporting economic systems
  9. A simple market economy
  10. Matrix formulation
  11. Different economic systems with a public sector
  12. Dynamic models
  13. Exercises
  14. References

2.1 Monetary and real economy

Figure 2.1:1 Layers of monetary and real economy.

Payment flows, savings, debts and money issuing belong to the monetary economy. Labor, production of goods, services and investments belong to the real economy. These two layers of the economy are tied together by prices. Many transactions take place in both the monetary and the real economies, e.g. the purchase of a car comprises the payment for the car and the delivery of the car. The payment goes from the customer to the producer, the car passes from the producer to the customer. The amount of money paid is the price of the car. The price of labor is wages and salaries.

Some transactions take place only in the monetary economy, e.g. payment of taxes, savings and loans. Other economic activities take place only in the real economy, e.g. if you grow your own potato or the consumption of fixed capital in the production process. These produced or consumed commodities can be measured in monetary terms, but no payments take place during the process.

2.2 Payment balances

Common double bookkeeping uses T-accounts with a debit and a credit side. Those may be single accounts as cash, bank deposits, purchases, sales, debts or result and balance accounts. The principle is that the sum of all entries on the debit side equals the sum of all entries on the credit side.

Suppose that we want to keep accounts for the incomes and expenditures of an ordinary household. These are summarized in the table below which is valid for one year:

Credit = incomeDebit = expenditure
Cash, balance at the beginning of the year 1.000:-Income tax70.000:-
Salary during the year200.000:- Housing60.000:-
Child allowances10.000:- Other expenditures76.000:-
Interests earned4.000:- Savings in bank7.000:-
Cash, balance at the end of the year 2.000:-
Total income215.000:- Total expenditure215.000:-
Table 2.2:1 Income and expenditure for one household during one year.

For one normal household, the change in cash balance is much less than the other flows. In this case, the difference is 2000 - 1000 = 1000 currency units from the start of the year until the end of the year. For a large number of households, the collective change will be still less in relation to the other payments. Some households hold more cash, others hold less cash at the end of the year. The total wages and salaries, on the other side, is the sum from all households.

In order to make a simple model for a large number of households, we can ignore the cash balance. Thus, we make the approximation that, the total of all incomes equal the total of all expenditures. It is allowable to disregard the cash balance even in more sophisticated models, except for the financial sector and the central bank.

Note that bank savings is accounted for as expenditures. The total financial assets may well change during the year.

Let us call incomes and expenditures "payment flows". The situation can be depicted in the following figure.

Figure 2.2:1 Incomes and expenditures for a household.

The flows are denoted as a vector X(1), X(2), … X(7). Later on, we will compute the flows by solving an equation system. Then it is convenient to collect all flows in a long vector. The indices 1,2, … 7 identify the different quantities. The balance between debit and credit can now be written as the equation:

X(1) + X(2) + X(3) = X(4) + X(5) + X(6) + X(7) Eq. 2.2:1

which means that the sum of all payment flows to the household equals the sum of all payment flows from the households. This is the same equation that holds for electrical currents at a junction or branching point. In electrical theory, the equation is known as Kirchhoff's first law which says: "The sum of all currents to a junction is equal to the sum of all currents from that junction". The same law also holds for water or gas flows in piping. Then it is called the continuity equation. Because the same law holds for both economical and physical systems, similar methods can be used to compute the flows. The calculations will be somewhat different because the model equations for the economic sectors are different from the model equations (characteristics) for resistors and other electrical components.

Figure 2.2:2 Electrical currents to and from a junction or branching point.

We can put all terms to the left of the equal sign:

X(1) + X(2) + X(3) - X(4) - X(5) - X(6) - X(7) = 0 Eq. 2.2:2

Flows to the household have a plus sign and flows from the household has a minus sign. By following this convention, it is easy to set up a general balance equation.

This equation has seven variables. The values of six variables can be set to arbitrary values. The seventh variable is then determined by the equation above. The system is said to have six degrees of freedom. A general system has as many degrees of freedom as the number of variables minus the number of equations.

In order to determine the value of all seven variables, more equations are needed. These may be additional relations between the flows inside the sector, e.g. an assumption about the relative magnitude between different flows. These assumption are the model equations for the sector under consideration. Finally, a number of variables remain that cannot be determined from a model of the sector. The model of the sector will have as many degrees of freedom as the number of variables minus the number of equations. The remaining degrees of freedom are determined from circumstances outside the sector.

2.3 Systems and flows between sectors

A system is defined as the part of the real world that we want to study. The system is limited by the system border. A closed system has no flows across the border. An open system has flows across the border. The systems that we want to study consist of sectors of the society. Different sectors of the society are connected by flows.

As a consequence, the same flow will belong to two balance equations, one for the sector from which the flow comes and one equation for the sector to which the flow goes. The expenditure of someone is the income of someone else. The households pay for their private consumption to the seller. The taxes from the households become the income of the public sector. The wages and salaries paid by the companies become the income of the households. The picture below shows an example with two sectors A and B.

Figure 2.3:1 Flows between two sectors.

The balances for each of the two sectors can be described by the following equations:

Sector AX(1)+X(2)-X(3)=0
Sector BX(3)-X(4)-X(5)=0

Table 2.3:1 Balance equations for two sectors.

We have five variables and two equations which gives 5-2=3 degrees of freedom. Note that the flow X(3) is part of both equations. This flow ties the two sectors together.

Because the system has three degrees of freedom, another three equations are needed. If e.g. the flows X(1), X(2) and X(4) were known, then X(3) and X(5) could be calculated.

Alternatively, we can assume that X(1) and X(2) are given from the outside (exogenous) and that the ratio between X(4) and X(3) is given, e.g. X(4) = 0.4*X(3). All flows can be calculated also in this case.

2.4 Aggregates

The system in figure 2.3:1 with the sectors A and B can be described by one single sector AB:

Figure 2.4:1 Aggregated sectors A and B.

The flow X(3) is no longer visible. The balance equation can be written directly from the figure but can also be obtained by adding the equations in table 2.3:1. The sum is:

X(1) + X(2) - X(3) + X(3) - X(4) - X(5) = 0 Eq. 2.4:1

X(3) cancels and the result is:

X(1) + X(2) - X(4) - X(5) = 0Eq. 2.4:2

which is the balance equation for the sector AB. This system still has 3 degrees of freedom, 4 variables - 1 equation. The remaining variables has to be determined as before. Suppose that X(1) and X(2) are exogenous and that X(4) = 0.4 * X(3). X(3) can not be used so the last equation has to be written: X(4) = 0.4 * (X(1)+X(2)).

2.5 Resource flows

Flows of labor, finished goods, services, raw material and energy are different kinds of resource flows. Waste and discharges to the air can also be included. The recourses flow between different sectors of the society, e.g. the households supply labor to the disposal of the firms. Labor is measured as number of employees or (man-years/year).

Figure 2.5:1 Flow of labor.

Flow of goods can be shown in the same way:

Figure 2.5:2 Flow of goods.

It is a bit more difficult to measure the volume flow of goods and services. The units depend on the degree of detail in the model. The flow of cars can be measured as cars/year. If you measure iron ore, the unit may be tons/year. Services require a different unit. Man-years/year can be a general measure for the volume flow and corresponds to the amount of work that was needed for the production. We may call this unit "produced man-years/year", abbreviated to pmy/year. We will not use the value in money terms, e.g. billions of dollars per year for flows of goods and services. Money units are reserved for the payment for the commodities, that is payment flows.

One problem with the unit "produced man-years" is that the needed work depends on the type of technology and organization that is used in the production. It is necessary to define one type of technology during a certain year.

If the production of cars require different numbers of man-hours in Japan and in Sweden, then the technology in one of the two countries has to be used as a reference. In order to study the Swedish economy, it is appropriate to use the technology used in Sweden. Deviations from the selected norm gives a productivity factor. If the Japanese can produce 15 cars and Volvo 10 cars with the same amount of labor, then the Japanese productivity factor is 1.5 .

The productivity may also vary over time. If Volvo produces 10 cars with a certain amount of labor in the year 1990 and 20 cars in the year 1998, then the productivity factor is 2.0 in 1998 with 1990 as the base year. The considered need for labor is only the work needed for the production of the product itself. Previously work done in the form of machines (investments) are not counted. This work was already accounted for when the machines were manufactured.

All activities of the company are included in the work done: direct labor in the workshops, supervision, design, purchases, sales activities etc. The number of cars produced shall be compared to the total number of employees at the company the same year.

2.6 Prices, wages & salaries

The price tells how much to pay per unit of a commodity. This gives a relation between payment flows and resource flows. Flows of wages & salaries and labor are shown in the diagram below:

Figure 2.6:1 Flows of wages & salaries and labor.

The flow of wages & salaries is part of a circuit diagram for payment flows, the number of employees is part of a circuit diagram for resources. The relation between the total compensation and the number of employees is given by the equation:

X(1) = w * X(2)Eq. 2.6:1

The wage & salary relation is indicated by a dashed arrow in the diagram. The arrow means that the number of employees multiplied by the wage (salary) level w (dollars/man-year) gives the total compensation to all employees during one year.

The consumption of commodities can be described in the same manner:

Figure 2.6:2 Consumption expenditures and flow of goods and services.

The total expenditures of the households for their private consumption is calculated as the flow of goods and services (pmy/year) multiplied by the price p (dollars/pmy).

X(3) = p * X(4)Eq. 2.6:2


2.7 Processes

The activity in a sector of the society can be named a process. The manufacturing process in a company is shown in it's most simple form in the figure below (same indices as in figures 2.6:1 and 2.6:2):

Figure 2.7:1 The manufacturing process in a company.

Flows to the process are primary production factors and intermediate inputs. Flows from the process are called output or products. Labor, land, raw materials and energy are primary production factors. Intermediate inputs are outputs from other industries, e.g. finished goods or components such as wood for house construction and gear boxes for car manufacturing. Intermediate inputs may also be tools, office paper or services as transportation. Investments will be treated separately. Investments are not immediately consumed in the production process but are present in the form of real capital which is needed for the manufacturing process.

The need for labor (number of employees) is equal to the specific labor intensity, l (man-years/produced man-year) multiplied by the volume of products (produced man-years). The productivity factor f is equal to 1/l (pmy/man-year).

X(2) = l * X(4)Eq. 2.7:1

Including raw materials, energy and waste in the process gives the description below:

Figure 2.7:2 The production process in a company with a number of primary production factors.

The consumption of primary factors is given by the specific consumption coefficients, hr , he, hw , in relation to the volume of output. If the consumption of resources not shall rise as production rises, then the specific consumption coefficients have to decrease.

Labor (number of employees)X(2) = l * X(4) Eq. 2.7:1
Raw material (tons/year)X(5) = hr * X(4) Eq. 2.7:2
Energy (MWh/year)X(6) = he * X(4) Eq. 2.7:3
Waste, discharges (tons/year)X(7) = hw * X(4) Eq. 2.7:4

Note that the flows to the process and the flows from the process are measured by different units. Kirchhoff's 1:st law does not hold for different flows as labor, raw materials, energy and waste. Kirchhoff's 1:st law only holds for flows of the same kind.

The diagram above for one company may also be applicable for a whole sector. It may be the private sector as well as the public production. The diagram of the process belongs to the circuit diagram for recourses. The payment flows have a parallel and separate circuit diagram.

One part of the production output is nearly always used as an intermediate input to the production. This holds to a small extent for a single company, but to a great extent for the industrial sector as a whole.

Figure 2.7:3 Recalculation of intermediate products in the production process.

Here it is important to distinguish between flows of different kinds of resources. Both primary factors and intermediate products from previous production go into the process. The primary factors are of a different kind than the output. The junction symbol says that the flows to and from the junction are of the same kind. Kirchhoff's 1:st law holds for junctions but not for processes, as pointed out above.

If only primary factors F to the system and net output Y from the system are considered, then we get the same kind of process as in the figure 2.7:1 above. The details inside the process need not be shown as long as the relation between the input F and the output Y can be expressed by an equation of the type F = l * Y.

Chapter 3 describes input-output models which can be used to keep track of different kinds of production factors and different kinds of products.

2.8 Self-supporting economic systems.

The most simple self-supporting economic system can be described by only the flow of commodities.

Figure 2.8:1 A self-supporting household.

The figure shows one single household that produces for it's own needs.

If two households exchange commodities, then the diagram will be:

Figure 2.8:2 Two self-supporting and bartering households.

We can form an aggregate of the two households. They consume their combined production together.

Figure 2.8:3 Aggregate of two households, same numbering as above.

If one share of the production is offered to a market, we get the following system:

Figure 2.8:4 Self-supporting households and a market, new numbering of flows.

This type of economy requires some means of payment, money. The price, p, on the market determines the relation between purchase value and commercial exchange of commodities.

X(1) = p * X(3)Eq. 2.8:1

Commodities for self-support and barter, X(2), circulate independently from the market and from money.

2.9 A simple market economy

A pure market economy with all production in the companies may look like this:

Figure 2.9:1 A simple market economy.

This is an example of a closed feedback system. The payment flows form a closed loop. The households consume their whole income and the whole income of the companies is used for wages.

The resource flows do not form a closed feedback system. They have a starting point and an end point, both marked with small circles in the figure. The employees come from the households and work in the companies. Labor is converted into goods in the companies. The goods are sold to the households where they are consumed. The flow of goods terminates at the consumers. There is no direct relation between the consumption of goods and the labor offered by the households. A bigger consumption of goods does not imply that more labor is offered.

A more complete model could describe what happens to the goods when they are used up, how they continue into waste management etc. From an ecological viewpoint nothing comes from nowhere and nothing can disappear. The resources brought from the forest or dug from the mine, will terminate on a rubbish-heap or in the nature if not recirculated into the production. From a model viewpoint, the resource flow starts in on place and terminates in another place with transformations on the way. The degree of detail in the model determines to choice of start and end points.

The equations for the payment flows in figure 2.9.1 are:

HouseholdsX(1) - X(2) = 0
CompaniesX(2) - X(1) = 0

We have two variables, X(1) and X(2), and two equations which should give zero degrees of freedom for the system. It should be possible to calculate both X(1) and X(2) from these two equations. If the two equations are added, we get:

X(1) - X(2) + X(2) - X(1) = 0 that is 0 = 0

which means that the two equations are linearly dependent. Switching signs in one equation converts that equation into the other equation. Both equations mean the same thing and we have in practice 2 variables and only one equation which gives one degree of freedom. For a closed system, one balance equation can always be calculated from the other equations. One equation is superfluous and the balance equations give one condition less than the number of sectors in the model.

As previously pointed out, there is no relation of the Kirchhoff's first law type for the resource flows X(3) and X(4). Instead, the need for labor X(3) is determined from the specific labor intensity l for the production X(4).

X(3) = l * X(4)<=> X(3) - l* X(4) = 0

We have two price relations for the between the two sectors, w is the wage (salary) level (dollars/man-year) and p is the price (dollars/pmy), (pmy = produced man-year).

Wages and salariesX(1) = w * X(3)
ConsumptionX(2) = p * X(4)

The payment balances give that X(1) = X(2). Substitution into the price relations give:

w * X(3) = p * X(4)

Substitution of X(3) from the equations for the specific labor intensity l gives:

w * l * X(4) = p * X(4) or w * l = p. Because the specific labor intensity can be calculated from the productivity factor with the relation l = 1/f, we finally get:

Wage (salary) levelw = f * p Eq. 2.9:1

which means that the wage level w (dollars/man-year) equals the productivity factor f (pmy/man-year) times the price p (dollars/pmy). Note that the relation is valid only for a system with no profits in the enterprises.

Now we can summarize all equations of the equation system:

Balance of paymentsX(1) - X(2) = 0 Eq. 2.9:2
Labor needX(3) - l* X(4) = 0 Eq. 2.9:3
ConsumptionX(2) - p * X(4) = 0 Eq. 2.9:4

The equation system has 4 variables but only 3 equations, that is 4-3 = 1 degree of freedom. One equation is missing in order to calculate all flows. The equation system is a homogenous system with zeroes in the right member. The sizes of the flows do not matter as long as the ratios between flows are kept constant.

In order to completely know the flows, we assume that we know the production volume Y(1) and add the equation:

Production volumeX(4) = Y(1) Eq. 2.9:5

The production volume Y(1) can not be determined from the model but has to be known in some other way, Y(1) is an exogenous variable.

2.10 Matrix formulation

The equations Eq. 2.9:2 - Eq. 2.9:5 form an equation system that can be solved. With new numbering of the equations, it becomes:

Balance of paymentsX(1) - X(2) = 0 Eq. 2.10:1
ConsumptionX(2) - p * X(4) = 0 Eq. 2.10:2
Labor needX(3) - l* X(4) = 0 Eq. 2.10:3
Production volumeX(4) = Y(1) Eq. 2.10:4

Table 2.10:1

The coefficients l and p in the left member determine the ratios (relative magnitude) between the flows. The absolute size of the economy is given by the production volume Y(1).

The equation system can be reformulated using matrix formalism:

/ 1-1 00 \ / X(1) \ / 0 \
| 01 0-p| | X(2) |= | 0 |Y(1)
| 00 1-l| | X(3) | | 0 |
\ 00 01/ \ X(4) / \ 1 /

This is equivalent to the more compact form:

S(p,l) X = A Y Eq. 2.10:5

The system matrix S is a function of the price, p, and the specific labor intensity, l. The coefficients (parameters) in the system matrix are intensive quantities, that is they do not depend upon the magnitude of the flows. The vector X is the flow vector and the vector Y is the exogenous flows. X and Y are extensive quantities, that is,, they depend upon the magnitude of the flows (the size of the economy). A is a matrix that only contains 1 and 0 (ones and zeroes) and distributes the elements of the vector Y to give a column vector of the same length as the number of equations.

2.11 Different economic systems with a public sector

In order to emphasize to typical structure of each economic system, only payment flows are shown.

The Swedish economy has a public sector with many employees and with the main purpose to provide public services as education, medical care, justice etc.

Figure 2.11:1 Payment flows in the Swedish economy.
In this simple model, all households get paid from an employment in private or public sector. All taxes are used for wages and salaries to public employees. The private consumption of the households gives income to the companies. The companies do not pay profits. Transactions between companies are not shown.

The economy of the USA had at least during the 60:s and 70:s a public sector which main purpose was to finance defense and space programs.

Figure 2.11:2 Payment flows in the economy of the USA.

All production is done in the private sector (the companies). Education and health care is carried out by private enterprises. The households earn their income through employments in the companies and pay themselves for both consumption goods, education, health care etc. The public sector uses all the whole tax income for purchases from the companies. An alternative figure of the economy is given in the book "Economics" by Lipsey et.al. , see figure 6.2:1. The picture can be interpreted as that the companies make the payments of all the taxes, as payroll taxes, sales taxes, corporate income taxes and personal income taxes.

The economy in the former Soviet Union may be described as:

Figure 2.11:3 Payment flows in the former Soviet Union.

The public sector buys consumption goods and defense material from the state owned enterprises. The consumption goods are then sold to the households. The profits of the enterprises are delivered to the state. The households get their income from employments in public service and at the companies.

These circuit diagram are simplifications of the economic systems. No system is pure and there are mixed systems. In the USA, the public sector has been changed and delivers more services as education and health care. Subsidies to the households, in the case of poverty and unemployment (social security, welfare), have increased. The share of defense and space programs has decreased.

2.12 Dynamic models

The systems described above are static systems, they have no dynamic behavior (time dimension) built into the model. A general equation system for calculation the flows would be:

S(p) X = A Y + B U Eq. 2.12:1

In this case p denotes a vector of parameters. The right member has two parts, A Y and B U. The flows X are staticly determined if Y and U are given. In order to convert the system into a dynamic system, let Y represent flows that follow the inner dynamic laws of the system and let U be input signals (functions of time), exogenous variables that represent the environment of the system and not influenced by the behavior inside the system.

The vector Y is the state vector of the system. It's time derivative dY/dt or yearly change D(Y) depends upon the flows X. See also appendix A2.

Dynamic systems occur if accounts are accumulating in the system. That might be bank accounts (savings or loans), real capital in the form of investments or changes in stocks.

2.13 Exercises

Exc 2.1 The finances of the Swedish government
The incomes of the government from taxes and charges were 942 billion crowns (SEK), income from capital was 85 billion SEK and other incomes were 66 billion SEK during the year 1997. The costs were 1112 billion SEK.

Draw a circuit diagram for the payment flows to and for the governmental sector. Formulate the balance equation for the sector. Calculate the borrowing of the government = budget deficit.

Exc 2.2 The public sector
Make a circuit diagram for the public sector. Show two sectors: the central government and the municipal sector. The municipal sector includes all local communities. Show the following flows: tax incomes of central government, tax incomes of local governments, grants from central to local governments, wage and salary payments for central and local governments, pensions, investments, purchases and the borrowing of the central government.

Indicate in what other sector the flows start or terminate. Find statistics from public sources about the payment flows. Set up the balance equations for the system. Check if the flows to and from each sector balance. What differences do you find? Is the discrepancy big? Is there any flow that was omitted and has to be included?

Exc 2.3 Volvo in Gothenburg
Analyze the car manufacturer Volvo in Gothenburg. Consider two sectors: the Volvo company and its subcontractors. Include the following payment flows: sales of cars, purchases from subcontractors, wages and salaries, profits, the purchases of the subcontractors, investments, payroll taxes and other company taxes.

Draw a circuit diagram for the system. Set up the balance equations for both sectors. Show from where and to where each flow goes, i.e. what other sectors are in contact with the system and how.

What resource flows can you identify? Draw resource flows in a separate diagram. Which are the processes inside the Volvo and subcontractor sectors? Formulate the equations that describe the transformation of primary and intermediate resources to products for the market. Set up the equations for the price relations between payment flows and resource flows. How many degrees of freedom does the system have? What flows have to be exogenous?

Exc 2.4 A lighthouse watcher
A lighthouse watcher on a lonely island is employed by the maritime authorities. The watcher guides the see traffic that pays for the service to via fees to the harbors. The fees are forwarded via the government to the maritime authorities. The lighthouse watcher buys commodities at the merchant on the mainland.

Draw a circuit diagram for the system. Set up the balance equations. Show the resource flows in a separate circuit diagram. Show the processes. Show the price relations between the payment flows and the resource flows. Is there any flow that is not part of a price relation?

Exc 2.5 Extraction of raw materials
A land owner owns a virgin forest that has never been subject to any costs for planting or thinning. Now he cuts some wood and sells it to a sawing mill. He employs workers for the cutting and a transport service for the transport to the sawing mill. The timber is paid to the land owner by the saw mill at delivery. The land owner pays for cutting and transport. When the costs are paid, the land owner has a profit that he deposits on a bank account.

Draw a circuit diagram for the payment flows. Draw each actor as a sector. Set up the balance equations for the whole extraction operation. What payments are exogenous relative to the land owner. How is his profit calculated? Which are the different components of the price of the timber? Draw a circuit diagram for the resource flows (labor, timber). Show the forest as a separate unit at which the flow starts.

Exc 2.6 A third world country

A third world county has rural and urban population. The rural population is to a great extent self supporting but sells food to the urban population. The rural population buys industrial goods from the cities. The state receives aid from abroad. The urban population works in the factories and in the administration. There are companies that produce raw materials for export. The country imports oil and more advanced industrial goods.

List the different sectors of the economy. Draw a circuit diagram for the payment flows. Make the foreign countries as a separate sector so you get a closed system (because the foreign countries are included in the system). Draw a circuit diagram for the resource flows. Identify the processes in the different sectors.

Set up the equations for the payment balances and the conversion relations for the processes. Determine the number of degrees of freedom for the system.

Note that the description of the sectors and payments flows are not complete. Add what is missing. The ownership relations are important for the payment flows.

Exc 2.7 Swedish economy
Section 2.11 showed a simple model for the Swedish economy. Set up the balance equations for the payment flows. How many balance equations are there? How may degrees of freedom does the system have?

Add a circuit diagram for the resource flows (labor, goods, services). Identify and draw the processes in the diagram. Set up the equations for the processes. How many degrees of freedom are there now? Set up the equations for the price relations. Are those independent?

What exogenous variables have to be determined in order to determine all flows of the system?

Exc 2.8 The economy of the USA
Make the same exercise (as 2.7) for the model of the United States of America.

Exc 2.9 The economy of the former Soviet Union
Make the same exercise (as 2.7) for the model of the former Soviet Union.

2.14 References

Statistical references for the exercises:


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