4. A simple production system

06 August 1999
To Chapter 3

  1. Real flows.
  2. Solution with constant investment level.
  3. Solution with constant investment share.
  4. Numerical simulation.
  5. Payment flows.
  6. Exercises.
  7. References.

4.1 Real flows

A very simple production system can be modeled with only two sectors:

  1. Households which supply the work force and consume produced commodities.
  2. Producers which produce commodities and invest in fixed capital and stocks.

This simple example will have no payment flows. The example may apply to a single farmer or fisherman that produces for his own needs. Payment flows will be added later. Then the system can be applied to producers that are different from the households. By first studying a system with no payment flows and later a system with payment flows, the different properties of the systems can be compared.

Figure 4.1:1 Real flows of a simple production system

There are seven flows X(1) - X(7), three state variables Y(1) - Y(3) and two exogenous variables U(1) -U(2). They are:

Real flowsVariable Units
Work force X(1)number of people = man-years/year = my/year
Work doneX(2)worked man-years / year = wmy/year
Total product volumeX(3)produced man-years / year = pmy/year
InvestmentsX(4) pmy/year
Consumption of fixed capital = Wear X(5) pmy/year
Change in stocksX(6)pmy/year
Private consumption = Goods & services X(7) pmy/year
State variables   
Available work forceY(1)number of people
Fixed capitalY(2)pmy
Exogenous variables = Input signals   
Decided investmentsU(1)pmy/year
Decided change in stocksU(2)pmy/year

Table 4.1:1 Real flow, state variables and exogenous variables

There are two processes: The daily work of a number of people and the conversion of work into products. It is assumed that the fixed capital (machinery etc.) has to be attended during the whole workday and that only a maximum number of people can be employed by a certain amount of fixed capital. The work done depends upon the number of workers and the daily working hours. The productivity factor tells how much products are produced from a certain amount of work. The productivity factor is a property of the fixed capital but depends also upon the skill of the workers and the organization of the work.

The volume of investments/year is decided by the production management (exogenous to the model). The consumption (wear) of fixed capital is assumed to be share of the total fixed capital. The amount of fixed capital next year is the amount of fixed capital at the beginning of current year plus the investments minus the wear during the current year.

The processes are described by the following equations:

Process inputs/outputsEquation Eq. no
Work forceX(1) <= Y(1)
X(1) <= fWfCp * Y(2)(t);
Eq. 4.1:1
Eq. 4.1:2
Work done X(2) <= whDay/whNom * X(1) Eq. 4.1:3
Products X(3) = fPdWk * X(2)Eq. 4.1:4
Flows & flow balance   
InvestmentsX(4) = U(1); 0 <= X(4) <= X(3) Eq. 4.1:5
WearX(5) = fWrCp * Y(2)(t)Eq. 4.1:6
Change in stocksX(6) = U(2);
X(6) >= -Y(3)(t) / Dt
Eq. 4.1:7
Eq. 4.1:8
Private consumptionX(7) = X(3) - X(4) - X(6) Eq. 4.1:9
State variables   
Available work forceY(1)(t+Dt) = Y(1)(t) Eq. 4.1:10
Fixed capitalY(2)(t+Dt) = Y(2)(t) + (X(4) -X(5))*Dt Eq. 4.1:11
StocksY(3)(t+Dt) = Y(3)(t) + X(6)*Dt >= 0 Eq. 4.1:12

Table 4.1:2 Process equations

These equations have a number of process parameters:

Description Parameter Units
Employment factorfWfCp(my/year)/pmy = employees / pmy
Daily working hourswhDayhours/day
Nominal working hourswhNomhours/day
Productivity factorfPdWkpmy/wmy
Wear of fixed capitalfWrCppmy/year/pmy = 1/year
Time stepDt year

Table 4.1:3 Process parameters

There is a limited number of workers available, Y(1). The number of workers are the same all years. (Other growth models can be used).

The employment factor tells how many people, X(1), can be employed by the use of a certain amount of fixed capital Y(2)(t). Less people may be employed than the maximum number. The amount of fixed capital in use is assumed to be constant during the year. The investments and the consumption of fixed capital (wear) are accumulated at the end of the year to give the amount of fixed capital during the next year, Y(2)(t+1).

The amount of work done, X(2) is measured as worked man-years = the work done by one worker working 8 hours a day, whNom, during one year. The amount of work done is proportional to the daily working hours, whDay, and the number of workers, X(1). Less work may be done if the whole work day is not utilized. Over-time is accounted for as a higher value of whDay and part-time as a smaller value of whDay.

The volume of products produced during one year, X(3), is proportional to the work done, X(2), and the productivity which is a property of the fixed capital.

The volume of investments during each year, X(4), is decided for each year. The investments can not exceed the total production. The consumption of fixed capital, X(5), is a factor fWrCp of the total fixed capital, which corresponds to a life time = 1 / fWrCp of the fixed capital.

The total production, X(3), is shared between consumption, X(6), investments, X(4), and change in stocks, X(6). Investments minus wear is added to the fixed capital each year. The change in stocks is positive when stocks increase, negative when commodities are withdrawn from the stocks. The stock volume has to be positive. Time t denotes the current year and time t+1 the next year. The flows X(1) - X(7) are all valid for the current year t.

All consumption goods and services, X(7), are consumed by the households. The surplus may be saved in stocks for a later year.

4.2 Solution with constant investment level.

Let us assume that the available workforce always is sufficient, that the fixed capital is fully utilized (equal signs in Eq. 4.1:2) and (3)), and that no stocks are accumulated (X(6) = 0). The investment level, U(1), is constant. Then the equations of table 4.1:1 can be summarized as follows:

Flow / StateEquation Eq. No
Private consumption X(7) = fPdWk * whDay/whNom * fWfCp * Y(2)(t) - U(1) Eq. 4.2:1
Fixed capitalY(2)(t+Dt) = (1 -Dt*fWrCp) * Y(2)(t) + U(1)*Dt Eq. 4.2:2

Table 4.2:1. Simplified equations with constant investment rate.

The solution of equation (4.2:2) can be found by trying the equation:

Y(2)(t) = a*kt + b

and inserting into equation (14) and using dt=Dt in the exponent (D can not be typed in the exponent).

a*kt+dt + b = (1 -Dt*fWrCp) * (a*kt + b) + U(1)*Dt which gives the solution:

k = (1 -Dt*fWrCp)1/dt and
b = U(1) / fWrCp
a = Y(2)(0) - b

The final expression for the fixed capital is:

Y(2)(t) = (Y(2)(0) - U(1)/fWrCp) * (1-Dt*fWrCp)t/dt + U(1)/fWrCp

Y(2)(0) is the fixed capital at time t = 0. Y(2)(~) = U(1)/fWrCp is the equilibrium level that will be reached after a long time. By that time the investments are equal to the consumption of fixed capital: U(1) = fWrCp * Y(2)(~). The annual adjustment rate is k = (1-Dt*fWrCp)1/dt. Dt has to be chosen small enough to make Dt*fWrCp << 1 or if Dt = 1 year, the annual wear fWrCp should be a small fraction of a year. The magnitude of the factor k will be less than one, which means that the amount of fixed capital will be closer to the equilibrium level for each year.


Figure 4.2:1. Growth at constant investment rate.

From equation (4.2:1), X(7) = fPdWk * whDay/whNom * fWfCp * Y(2)(t) - U(1), it can be seen that a higher level of private consumption can be achieved if the productivity, fPdWk, is high, if the work days, whDay, are long and if many workers can be employed by the fixed capital, fWfCp. The private consumption will be lower in the short run if the investment level, U(1), is higher.

The consumption level in the long run can be calculated if the equilibrium level of fixed capital, Y(2)(~) = U(1)/fWrCp, is inserted into equation (4.2:1): X(7) = ( fPdWk * whDay/whNom * fWfCp / fWrCp - 1 ) * U(1). We see that a higher investment rate (pmy/year) gives a higher private consumption level.

Figure 4.2:2. Growth at different constant investment rates.

4.3 Solution with constant investment share.

The previous example assumed that the investment rate was constant. Let us investigate what happens if the investment share of the total production is constant. Set U(1) = fInvPd * X(3), fInvPd = investments pmy/year per production volume pmy/year.

As before, the available workforce always is sufficient, the fixed capital is fully utilized, and that no stocks are accumulated (X(6) = 0). Then the equations of table 4.2:1 can be rewritten as follows:

Flow / StateEquation Eq. No
Private consumption X(7) = (1 - fInvPd) * fPdWk * whDay/whNom * fWfCp * Y(2)(t) Eq. 4.3:1
Fixed capitalY(2)(t+Dt) = (1 + ( fInvPd * fPdWk * whDay/whNom * fWfCp - fWrCp) *Dt ) * Y(2)(t) Eq. 4.3:2

Table 4.3:1. Simplified equations with constant investment share.

Define a new factor fPdCp = fPdWk * whDay/whNom * fWfCp = the annual production volume per amount of fixed capital. The new expression for the fixed capital is:

Y(2)(t) = Y(2)(0) * (1 + ( fInvPd * fPdCp - fWrCp )*Dt )t/dt

and the private consumption level:

X(7) = (1 - fInvPd) * fPdCp * Y(2)(t)

Both fixed capital and consumption will exhibit exponential growth. Now there is no limit for the amount of fixed capital and level of consumption.


Figure 4.3:1. Growth at constant investment share.

As before, a higher investment share gives a lower private consumption in the short run but a higher consumption in the long run.

Figure 4.3:2. Growth at different constant investment shares.

4.4 Numerical simulation.

The analytical solutions to the equations required that the fixed capital was fully utilized. A numerical simulation can simulate the general case with the limitation of the available work force and varying the parameters as time passes. The simulation uses the time step, Dt, is 0.5 year.

Imagine a farmers family with three people who can be employed in the production. They start with a small fixed capital (tools, machinery and live stock). Let us assume a constant rate of investments in more fixed capital until the limitation of available workforce is reached. Let the following investments be in higher productivity but using the same number of workforce.

Figure 4.4:1. Flows of a growing economy

From year 1995 until year 1998 are directed so that more people can be employed. The total production rises until all three people of the family are fully busy. Is discovered that further investments of the same kind only results in equipment that can no be used due to shortage of workforce. From 1999 investments are made in more efficient equipment so more can be produced with the same workforce. See productivity factor in the diagram below.

Figure 4.4:2. Parameters of a growing economy

Because the investments during 1998 were made in the same kind of equipment as before and the average employment factor decreased from 6 workers / pmy of fixed capital to 4.9 workers / pmy.

Figure 4.4:3. Workforce and employment

From 1998 and onwards, it would have been possible to employ 3.4 people with the available equipment.

4.5 Payment flows.

Let us introduce hypothetical payment flows and assume that no profits are generated. Money is not necessary in this simple economy. It can be used for studying the relation between wages and prices. At this simple circumstances, the relation between wages and prices is well defined. The payment flows are shown in the figure below.

Figure 4.5:1. Simple payment flows with no profits.

Both payment balance equations, for the households and for the producers, will give the same result:

X(8) = X(9) , Eq. 4.5:1.

Let the wage level be wa (CU/wmy) and the price level pr (CU/pmy). CU = currency units, wmy = worked man-years (8 hours a day), pmy = produced man-years. The we get the following relations between payment flows and real flows:

Wages, WorkX(8) = wa * X(2) Eq. 4.5:2
Expenditures, ProductsX(9) = pr * X(7) Eq. 4.5:3

Table 4.5:1. Payment and real flows.

From the payment balance X(8) = X(9) we get

wa * X(2) = pr * X(7) , Eq. 4.5:4

which is the relation between wages and prices.

The wage level is shown below, assuming a given price of the products and all other circumstances being the same as in paragraph 4.4 above:

Figure 4.5:2. Wage and price levels.

At the base year 1995, 1 wmy = 1 pmy. We can draw the following conclusions if we use the same base year all the time:

At the beginning, the wage level is lower than the price level because some products are used for investments, and that share of the production is not paid for. The rate of investments was constant in this example so the investment share decreases as the production increases. Maximum production with productivity factor = 1 is reached in year 1999. Then it is decided to invest in increased productivity with the same workforce. The production rises still more and is shared between the same number of workers. The wages pass the prices in year 2000.

This leads to two well known conclusions:

  1. One part of the work result is accumulated by the owners and is not paid for. (Surplus value according to Marx).
  2. Higher productivity gives higher production and higher standard of living. (Classical economic theory).

Profits are introduced in chapter 5. Then the rigid relation between prices and wages will not hold any more.

4.6 Exercises.

  1. Investigate the influence of the other parameters of table 4.1:3. How does shorter working days change the employment level and the production volume?
  2. Change the simulation of paragraph 4.4 so the investment share is constant all the time. Use the Excel work sheets as above and change equation (5). Reproduce diagrams 4.1:1-3 and 4.5:2 for this case. Give your comments.
  3. Imagine a society with the simple economy above. I uses no money. What rules would be necessary? How would commodities be distributed? How would work be organized? Can you imagine any circumstances when this is a viable option?

4.7 References.

  1. No references for the moment.

The Excel calculus for the simple production system can be downloaded here.

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