1.0 Introduction

This is now merely fragments of an essay. The old version of this page is now Section 2 below.

...the principal usefulness of the balance sheet lies in its indicating how fit the firm is in an an environment which is essentially unpredictable. In such an environment, agents move forward by taking stock of their position from time to time and then making the appropriate adjustments. Specifically, agents look at their balance sheets at each instant and adjust their production levels up or down according to whether the reproduction rate of profit is above or below average. In analogous fashion, they check their transactions accounts at each moment and raise or lower prices according to whether excess demands are positive or negative. That is, agents seek to reduce their states of stock and flow disequilibrium by employing cross-dual adjustment processes. Of course, in the exceptional cases where reasonable forecasts can be made, agents modify their plans to deal with the anticipated changes (e.g., by raising prices in the face of anticipated resource or quantity constraints so as to secure the proper rents), but the basic approach to disequilibrium dynamics is evolutionary...

...It is assumed that agents - unsure of the underlying economic structure - tend to rely on their balance sheets to obtain an idea of how well they are doing. Indeed, given the environment is at its most unpredictable when stationarity is dropped, the use of optimisational methods is most dubious and the argument for the use of evolutionary ones (with their attendant concepts of fitness, balance-sheet comparisons, and so on) is extremely strong. (Sharpe 1999)

I have always tried to read Sraffa's magnum opus as if it was an accountant's manual, supplemented by ingenious constructive devices to prove the solvability of systems of equations. (Velupillai 2003)

2.0 An Algorithm For Technology Represented In Terms Of Dated Labor Inputs

Consider a firm choosing within a set of techniques for producing a given commodity. Each technique exhibits Constant Returns to Scale and can be represented as a series of dated labor inputs. The first element in this series is the amount of labor that must be applied in the technique during the current production cycle to produce one unit of the commodity. The second element is the amount of labor that must be applied one cycle ago to produce the capital goods required for use by the first labor input in the series. And so on.

The cost of producing a single unit of the produced commodity with a given technique is:

l0 w + l1 w (1 + r) + l2 w (1 + r)^2 + ... + ln w (1 + r)^n + ...

where w is the wage for an unit of labor, r is the rate of profits, and (l0, l1, l2, ...) is the series of dated labor inputs representing the technique. The wage is paid at the end of each cycle for the labor expended during that cycle.

The following algorithm yields a cost-minimizing technique and the rate of profits for a given wage:

  1. Pick a technique.
  2. For the given wage, find the rate of profits, r, that, when used to cost up the labor inputs, equates that cost with the value of a single unit of the produced commodity.
  3. Cost up the labor inputs for all techniques with this wage and rate of profits.
  4. Choose the technique with the minimum of the costs found in Step 3.
  5. If the technique for which the rate of profits was obtained in Step 2 can be selected in Step 4, stop. You have found a cost-minimizing technique and the rate of profits.
  6. Else go to Step 2 and repeat with the newly selected technique.

An Example

Consider a firm whose managers know two techniques for producing widgets. Table 1 shows the amount of labor inputs required to produce a widget for each technique. The interesting qualitative properties of this example depend merely on the difference in labor inputs between the techniques each year. Thus, if every input of labor in both techniques was increased by the same amount (e.g., 25 person-years), the results of this example would look much the same. One instance discussed in the literature is a choice between using a plot of land for either grazing or mining; mining requires a high initial expenditure and a costly cleanup phase at the end of the use of the land. Other empirical examples have been discussed in the literature, usually in the context of environmental economics or geography.

TABLE 1: A Choice Between Techniques With Dated Labor Inputs
YEAR
BEFORE
OUTPUT
LABOR HIRED
FOR ALPHA
TECHNIQUE
LABOR HIRED
FOR BETA
TECHNIQUE
0 33 Person-Years 0 Person-Years
1 0 Person-Years 52 Person-Years
2 20 Person Years 0 Person-Years

Suppose the managers follow the given algorithm for determining the cost-minimizing technique for a given wage. In Step 1, they can select either techique. Figure 1 shows the rate of profits they will calculate in Step 2. Two wage-rate of profits curves are shown, one for each technique. These curves intersect at two points, once at a wage of (1/78) Widgets per Person-Year and a rate of profits of 50% and again at a wage of (5/286) Widgets per Person-Year and a rate of profits of 10%. These intersections are called "switch points" in the jargon. Rate Of Profits by Technique versus Wage
Figure 1: Rate of Profits by Technique, First Example

Suppose the Alpha technique was selected at Step 1. Figure 2 shows the costs that would be calculated at each wage by Step 3. The Alpha technique is cheaper for wages between the switch points, while the Beta technique is cheaper at low and high wages. If the firm faced an intermediate wage, the algorithm would terminate with the choice of the Alpha technique. Otherwise, one following the algorithm would continue by calculating the rate of profits for the Beta technique.

Cost Of Techniques At Alpha Rate Of Profits Versus Wage
Figure 2: Cost Of Techniques At Alpha Rate Of Profits, First Example

On the other hand, consider the case where the Beta technique is selected at Step 1. Figure 3 shows the costs of the two techniques at each wage when the rate of profits for Beta is found at Step 2. Here, too, the Beta technique is cost-minimizing at low and high wages, while Alpha is cost-minimizing at intermediate wages. Figures 2 and 3 demonstrate that it does not matter which technique is selected at Step 1 of the algorithm; in either case the algorithm terminates with the same choice for the cost-minimizing technique.

Cost Of Techniques At Beta Rate Of Profits Versus Wage
Figure 3: Cost Of Techniques At Beta Rate Of Profits, Second Example

Consider a case where the Beta technique is cost minimizing, that is, at an extreme wage. Suppose the firm was producing a constant level of output, say one widget per year. In any given year, the firm would be employing 52 workers. On the other hand, if the firm were to adopt the Alpha technique to produce a steady-state output of one widget per year, the firm would employ 53 workers per year. Thirty three of these workers would be working to produce the widget available in the given year from a two thirds-completed widget in last year's inventory. The firm would have a one third-completed widget in the current-year inventory. And the firm would be employing 20 workers to produce a partially-completed widget to age for a year in next year's inventory. Figure 4 summarizes this discussion. The figure shows the labor intensity of the cost-minimizing technique against the wage.

Wage Versus Labor Intensity Of Cheapest Technique
Figure 4: Labor Intensity Of Cheapest Technique, First Example

By following the above algorithm, I have examined the different choices available to this firm. The algorithm shows the maximum rate of profits the firm can make. Using this rate of profits to cost up all the techniques at the given wage, one finds that the costs for no technique are less than the costs for the technique from which the rate of profits is found. And one finds that for some range of wages, the firm will find a more labor-intensive technique cheaper at a higher wage.

So much for the theory that wages and employment are determined by the intersection of well-behaved supply and demand curves in the labor market.

3.0 An Algorithm For Technology Represented In Terms Of Dated Outputs

Now suppose each technique known by the firm consists of one period's labor directly producing a machine and the outputs produced from the machine over successive periods. The machine is a produced capital good. Continue to assume that Constant Returns to Scale prevail.

The net present value of the revenue obtained from the commodities produced with the produced capital good is:

y1/(1 + r) + y2/(1 + r)^2 + ... + yn/(1 + r)^n + ...

where r is the rate of profits, and (y1, y2, y3, ...) is the series of the value of outputs produced at the end of each cycle.

The following algorithm yields a cost-minimizing technique and the rate of profits for a given wage:

  1. Pick a technique.
  2. For the given wage, find the rate of profits, r, that, when used to discount the outputs, equates the revenue with the cost of the labor used to produce the machine from which the outputs are generated.
  3. Discount the output for all techniques using this rate of profits. Using the given wage, subtract the cost of the labor input from the net present value of the output, thereby obtaining net revenue.
  4. Choose the technique with the maximum of the net revenue found in Step 3.
  5. If the technique for which the rate of profits was obtained in Step 2 can be selected in Step 4, stop. You have found a cost-minimizing technique and the rate of profits.
  6. Else go to Step 2 and repeat with the newly selected technique.

An Example

Consider a firm whose managers know of three different types of machines for producing widgets. Each type of machine is produced with an expenditure of one person-year labor in one period. Each type of machine lasts for three periods. Table 2 shows the outputs of widgets that flow out of the machines in each period in which they survive. Assume free disposal of each machine at the end of the three periods.

TABLE 2: A Choice Between Techniques With Dated Outputs
YEAR
AFTER
MACHINE
PRODUCED
OUTPUT FOR
ALPHA-TYPE
MACHINES
OUTPUT FOR
BETA-TYPE
MACHINES
OUTPUT FOR
GAMMA-TYPE
MACHINES
1 140 Widgets 40 Widgets 0 Widgets
2 10 Widgets 250 Widgets 296 Widgets
3 150 Widgets 7 Widgets 7 Widgets

Rate Of Profits by Technique versus Wage
Figure 5: Rate of Profits by Technique, Second Example

Cost Of Techniques At Alpha Rate Of Profits Versus Wage
Figure 6: Net Revenue By Technique At Alpha Rate Of Profits, Second Example

Cost Of Techniques At Beta Rate Of Profits Versus Wage
Figure 7: Net Revenue By Technique At Beta Rate Of Profits, Second Example

Cost Of Techniques At Gamma Rate Of Profits Versus Wage
Figure 8: Net Revenue By Technique At Gamma Rate Of Profits, Second Example

Wage Versus Labor Intensity Of Cheapest Technique
Figure 9: Labor Intensity Of Cheapest Technique, Second Example

4.0 An Algorithm For An Input-Output Model Without Joint Production

5.0 An Algorithm For An Input-Output Model With Joint Production

This example is due to Woods (1990)

TABLE x: Processes Exhibiting Joint Production
INPUTS Process I Process IIa Process IIb
Labor 5 Person-Yrs 10 Person-Yrs 10 Person-Yrs
Iron 18 Tons 12 Tons
Coal 10 Cwt
OUTPUTS
Iron 48 Tons 12 Tons 12 Tons
Coal 10 Cwt 30 Cwt 30 Cwt

References

(Sharpe 1999) Kieran Sharpe, "On Sraffa's Price System", Cambridge Journal of Economics, V. 23, pp. 93-101.

(Velupillai 2003) K. Vela Velupillai, "The Unreasonable Ineffectiveness of Mathematics in Economics", Economics for the Future, Cambridge (UK), 17-19 September.

(Woods 1990) J. E. Woods, The Production of Commodities:An Introduction to Sraffa, Humanities Press.