12. Dynamic Model D1

2002-06-02
To Chap 11
Variable names in formulas will be revised later, so they conform with the English language.  
  1. Payment flows of model D1
  2. Public finances
  3. Public production
  4. Transfers
  5. Households
  6. Financial sector
  7. Private sector
  8. Profits
  9. Sales tax and other indirect taxes
  10. Abroad
  11. References

12.1 Payment flows of model D1

Dynamic model D1 is an extension of previous models. The additions include:

The model is not fully described by the picture 12.1:1 above. Savings, borrowings and account balances are described more in detail by picture 11.1:1. Flows of labor are described by picture 10.1:3. Volumes of commodity flows are described by picture 10.1:1.

This is the first model that does not have the private consumption as an exogenous variable. As the number of employees is dynamically calculated, the total earnings and net incomes will vary with time. The net incomes will in turn determine the consumption volume and the need for labor.

The equations for payment balances will not be explained from now on. The complete equations are found in the Excel calculus for the model D1. The equations follow the rules described earlier (chap 5) and can be set up by the use of pictures 12.1:1, 11.1:1, 10.1:3 and 10.1:1. Only the model equations of the individual sectors will be described below, that is the assumptions made about the relations between the different flows of a sector and how they depend upon exogenous quantities.

Picture 12.1:1 Payment flows of Dynamic Model D1

12.2 Public finances

The public finances has no model equations. Balance of payments is accomplished by adjusting the public borrowing X(1).

12.3 Public production

Earnings in public sector X(3)-ow*X(6)/1000=0 eq. (3
Purchases from private sector oi*X(2)-X(4)+oi*X(13)=0 eq. (4
Public employers contribution to social security oa*X(3)-X(5)=0 eq. (5
Number of employees X(6)=Y(1) eq. (6

Table 12.3:1 Model equations for public production.

The total earnings of the public employees X(3) depend upon the yearly earning per employee ow (dollars/employee*year) and the number of publicly employees X(6). The purchases of the public sector X(4) are assumed to be a constant share oi of the total costs X(2)+X(13) of the public production. Employers contribution to social security X(5) is the share oa of the total earnings X(3). The number of employees X(6) is the number of employees Y(1) that have been budgeted for that year.

The number of employees in the public production Y(1) is assumed to be adjusted so that the next year will give a predetermined budget deficit oBu in the public finances. The number of employees will change all the time, at equilibrium, the public sector will have as many employees as it can afford provided the predetermined budget deficit.

Number of new employees dY(1) = -(1000/ow) * (X(1) - oBu) eq. (D1a
Number of employees next year Y(1)(t+1) = Y(1)(t) + dY(1) eq. (D1b

Table 12.3:2 Adjustment of the number of employees in the public sector.

Y(1) is a state variable. The values of the state variables vary with time. Different sets of state variables can be chosen, but the number is always given and equals the smallest number needed to determine the state of the system. The concept "state" is explained in references (1) and(2) and in appendix A2. In simple words, it can be described as a position (in a n-dimensional space) for the system that continuously is changing with time. When the state is known, then all other variables X(.) for the system can be calculated. The state at time t+1 depends upon the sate at the previous time t and the dynamic properties of the system.

The difference between public borrowing X(1) and predetermined budget deficit oBu is the need for cuts a certain year (at time t). If the borrowing is bigger than the planned budget deficit, a number dY(1) of employees have to be fired. If a surplus occurs, more employees can be hired. The following year (at time t+1), the public sector will have Y(1)(t+1) employees. The model assumes a planning period for one year at a time.

12.4 Transfers

Benefits to the households X(7)=tBidr eq. (7
Benefits to unemployed X(8)-twa*X(16)/1000=0 eq. (8
Pensioners X(9)-twp*X(17)/1000=0 eq. (9

Table12.4:1 Transfers from the public finances to the households.

Note: Transfers X(7)+X(8)+X(9) are not shown as a unit in picture 12.1:1. (There is shortage of variables, due to that subsidies X(14) has been given it's own variable).

The benefits to the households tBidr consist of children's allowance, housing allowance, sickness insurance etc. Their extent are determined by factors outside of the model (exogenously). The benefits to the unemployed X(8), depends upon the compensation level twa (dollars/person*year) and the number of unemployed X(16). The unemployed in the model include all at working age that do not have a regular employment.

12.5 Households

Income taxes hs*X(3)+hs*X(8)+hs*X(9)-X(11)+hs*X(19)+hs*X(30)=0 eq. (11
Savings of the households X(12)=hSpar eq. (12
Fees of the households X(13)=hAvg eq. (13
Capital supply from households to companies X(15)=hKt eq. (14
Number of unemployed X(6)+X(16)+X(24)=hNw eq. (15
Number of pensioners X(17) = hNp eq. (16

Table 12.5:1 Model equations for the households.

Income taxes X(11) are calculated with the same rate hs for all kinds of income. The number of unemployed X(16), that is not working, is calculated as the balance hNw of the population between the age of 16 and 64 years. The other quantities are exogenous. The capital supply to companies includes purchases during the issue of new shares.

12.6 Financial sector

The model of the financial sector is simplified and uses only net savings/borrowings during the year and net balances of financial assets and liabilities. The sector has been described in chapter 11. The net balances Y(3) - Y(6) are four of the state variables of the system. The change from one year to the next by the amount that have been saved or borrowed during the year, see table 11.1:1.

12.7 Private sector

Subsidies X(14) = pSubv eq. (18
Earnings in private sector X(19)-pw*X(24)/1000=0 eq. (19
Investments X(20)=pInv eq. (20
Imports piinv*X(20)-X(21)+pik*X(32)+piexp*X(34)=0 eq. (21
Private employers contribution to social security pa*X(19)-X(22)=0 eq. (22

Table 12.7:1 Model equations for payments to/from the private sector.

The amount of subsidies X(14) is determined by rules that are no part of the model. The total earnings X(19) depends upon the level of wages and salaries pw (dollars/person*year) in the private sector and the number of employees X(24). The investments X(20) are exogenous in this model. A future model will estimate the investments from the amount of real capital, volume sold, a strategy for maximizing the profits etc. The imports X(21) are estimated as shares piinv, pik and piexp of the private investments X(20), the goods and services sold at suppliers price X(32) and the exports X(34). The values of the parameters have previously been estimated in chapter 5. The employers contributions to social security X(22) amount to the share pa of the total earnings X(19).

The number of employees in the private sector X(24) changes according to the need for labor. The need for labor is assumed to depend upon the changes in stocks X(28). The companies fire or employ labor in order to adjust the capacity. The goal is that the next year, there will be no increase in stocks, nor will the companies take from the stocks of goods.

Need for new labor dY(2) = -(1/pf) * X(28) eq. (D2a
Number of employees next year Y(2)(t+1) = Y(2)(t) + dY(2) eq. (D2b
Number of employees X(24)=Y(2) eq. (24

Table 12.7:2 The need for labor in the private sector.

The number of employees Y(2) is the last of the six state variables in this model. The increase in stocks is X(28) (pmy/year) is the balance between production, imports and the consumtion of goods and services, see chapter 10. When the increase in stocks is positive, the the number of employees is decreased by dY(2). The productivity factor pf (pmy/wmy) determines how many people will be fired or employed.

Note: pmy = produced man-years, wmy = worked man-years. A produced man year (pmy) is the volume of goods that are produced during a certain base year with the technology used during that year. The productivity factor is a measure of the standard of technology, the productivity of the organization and the daily working hours. It measures the production volume during a certain year as compared to the production volume during the base year, using the same amount of labor both years.

Both the wage level pw and the productivity factor pf are weighted averages for three categories of employees, A, B and C. See table 10.1:3.

12.8 Profits

Dividends X(30)=pUtdv eq. (30

Table 12:7 Dividends.
The share of the profits that is distributed to the capital owners is decided by the management of the private enterprises, thus the prefix p. The dividends pUtdv are treated as an exogenous variable to the system.

12.9 Sales tax and other indirect taxes

Sales tax and other indirect taxes X(31)-moms*X(32)=0 eq. (32
Volume of goods and services at producers price X(32)-prDo*X(33)/1000=0 eq. (33

Tabell 12.9:1 Indirect taxes and volume of products sold.
The indirect taxes are for simplicity called sales tax (Sw. moms). The indirect taxes also include tax on alcohol, tobacco, gasoline etc. The tax rate moms is calculated from statistical data for indirect taxes X(31) and the value of commodities sold at producers price X(32). The tax rate is calculated to approx. 0.31 (31%) as compared to the nominal 0.25 (25%). The higher tax rate on special goods as alcohol etc. increases the average tax rate. The value of the sold commodities X(32) is the price per produced man-year prDo times the sold volume (33). Prefix pr for price, Do for domestic.

12.10 Abroad

Export, value X(34)=uExp eq. (34
Export, volume X(34)-prExp*X(35)/1000=0 eq. (35

Tabell 12.10:1 Exports of goods and services.
The size (value) of the export uExp is in principle decided abroad. The value X(34) is determined by the price per produced man-year prExp and the exported volume X(35) measured as produced man-years.

12.11 References

  1. Torkel Glad och Lennart Ljung: Reglerteknik: Grundläggande teori, Studentlitteratur, Lund 1989. (Control systems: Basic theory).
  2. Lennart Ljung och Torkel Glad: Modellbygge och simulering, Studentlitteratur, Lund 1991. Kapitel 3.4. (Model design and simulation).

The Excel calculus for the dynamic model D1 can be downloaded here (462 kB, corrected and translated into English 2 June 2002). 

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