5. Payment flows and sector strategies
|08 September 1999 |
To Chapter 4
The two payment flows expenditures and wages, shown in figure 4.5:1, gave a too simple model. It gave a strict relationship between prices and wages levels. A more realistic model has to allow for independent prices and wage levels which also implies that the producers may get a profit. We also add a possibility for savings. Let the savings flow also include other kinds of capital supply to the producers, such as investments in shares. Negative savings are the same as loans.
Figure 5.1:1. Payment flows of a simple production system.
The numbering of the flows is continued from chapter 4. This set
of payment flows gives a lot of flexibility to the model. It allows
for the households to spend more or less than their incomes. It
can also show the effect of price increase on the consumption
The payment balances will be:
|Households||X(8) - X(9) - X(10) + X(13) = 0||Eq. 5.1:1|
|Gross profits||X(10) + X(11) - X(12) - X(13) = 0||Eq. 5.1:2|
|Producers||- X(8) + X(9) - X(11) + X(12) = 0||(superfluous)|
Table 5.1:1. Payment balances of a simple production system.
One of the equations is superfluous as usual. The last equation is arbitrarily excluded.
The process equations from chapter 4 can be rewritten as follows. The equal sign is for simplicity used in all places. The equations with the decision variables U(1) and U(2) are removed. They will be replaced with the strategy equations described in the next paragraph 5.2 .
|Process inputs/outputs||Equation||Eq. no|
|Work force||X(1) = fWfCp * Y(2)(t)||Eq. 5.1:3 (was Eq. 4.1:1)|
|Work done||X(2) = whDay/whNom * X(1)||Eq. 5.1:4 (Eq. 4.1:3)|
|Products||X(3) = fPdWk * X(2)||Eq. 5.1:5 (Eq. 4.1:4)|
|Flows & flow balance|
|Wear||X(5) = fWrCp * Y(2)(t)||Eq. 5.1:6 (Eq. 4.1:6)|
|Private consumption||X(7) = X(3) - X(4) - X(6)||Eq. 5.1:7 (Eq. 4.1:9)|
Table 5.1:2 Simplified process equations.
We also have to include the price equations from chapter 4. A new equation is included for the investments which are assumed to have the same price as the sold products:
|Wages, Work||X(8) = wa * X(2)||Eq. 5.1:8 (was Eq. 4.5:2)|
|Expenditures, Products||X(9) = pr * X(7)||Eq. 5.1:9 (Eq. 4.5:3)|
|Investments, Products||X(12) = pr * X(4)||Eq. 5.1:10 (new)|
Table 5.1:3. Price equations.
Now we can see that have 13 flows to determine. We have 10 equations above. Another three equations are needed. They are described next.
The strategies describe the behavior of the actors in the different sectors. The strategies can be either free or formal.
Free strategies are close to the real world behavior where decisions are taken by humans, maybe with the use of decision rules and a variety of background information. Free strategies can also be used in the design of computer games where the user (player) is asked for his/hers decision at each step in time.
Formal strategies are more suited for simulations. They constitute a mathematical model of the behavior of a sector. The models can be classified as linear or nonlinear, static or dynamic. We get four possible classes of models:
Table 5.2:1. Classes of formal strategies.
A static strategy is based upon information for one sample in time only. It has no memory of historical data. A dynamic strategy considers historical data and can also use average or trend values or other combinations of past data.
A linear model expresses the decision variables as linear expressions of the background data (e.g. U=a*X(1)+b*X(2)+ ) . They are simple to use because the final equation system will be an ordinary linear equation system. Nonlinear models uses inequalities (X(1)<=Y(1)) and other combinations of variables such as X(1)*X(2) or any other nonlinear functions. The nonlinear models require special mathematical methods for each case.
The model above requires that the strategies can be expressed as tree equations in addition to the 10 equations of paragraph 5.1.
The can decide how much goods and services to purchase or how much money to save or spend. The households can only make one independent decision, the other variables depend upon the first decision. The background variables are the wages, dividends, number of employees and total work force. The general strategy can be expressed as:
( X(7), X(9), X(10) ) = f ( X(8), X(13), X(1), Y(1) ) Eq. 5.3:1
|Decision variables||Background variables|
|Consumption of goods and services, X(7), pmy/year||Wages, X(8), CU/year|
|Expenditures, X(9), CU/year||Dividends, X(13), CU/year|
|Savings, X(10), CU/year||Number of employees, X(1), people|
|Total work force, Y(1), people|
Table 5.3:1. Household strategy variables.
The producers can decide about flows and product prices. As stated above, three additional equations for the flows are needed, one equation has just been defined for the households. This means that the producers can make two independent decisions about the flows from their sector.
The producers can decide about change in stocks, investments and dividends. The can also decide that less work is done than the workforce can do and to employ less people than the fixed capital allows. Both cases mean that the full production capacity is not used. The background variables are the wear of fixed capital, incomes = household expenditures, gross profits, total available work force, volume of fixed capital and stocks. The general strategy can be expressed as:
( X(1), X(2), X(6), X(12), X(13) ) = f ( X(5), X(9), X(11), Y(1), Y(2), Y(3) ) Eq. 5.4:1
|Decision variables||Background variables|
|Number of employees, X(1), people||Wear of fixed capital, X(5), pmy/year|
|Work done, X(2), wmy/year|
Wages, X(8), CU/year
|Incomes, X(9), CU/year|
Goods and services, X(7), pmy/year
|Change in stock, X(6), pmy/year||Gross profits, X(11), CU/year|
|Investments, X(12), CU/year|
Investments, X(4), pmy/year
|Total work force, Y(1), people
Fixed capital, Y(2), pmy
|Dividends, X(13), CU/year||Stocks, Y(3), pmy|
Table 5.4:1. Producers strategy variables.
The equivalence between pairs of variables come from the wage level and price relations, eq. 5.1:8, 5.1:9 and 5.1:10.
The Excel calculus for the simple production system can be downloaded here.
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