|Figure 1: A Photo Probably From The Same Week I visited A Joint Production Process|
I thought I would continue thinking about the joint production example in the previous post. I want to consider the price equations for three processes that might be operated with this apparatus in the course of a full day. In the first process, labor operates the main generator and pump for 12 off-peak hours. The second and third processes execute in parallel during 12 peak hours. Labor operates the main generator alone in the second process. And, in the third process, labor operates the secondary generator alone.
Assume this electric company takes the wage, the costs of operating the main and secondary generators, and the cost of operating the pump as given. What rate of profits and relative prices of peak and off-peak hours of electricity justifies the utility in operating these processes (when this apparatus is new and no quasi-rent is being charged)? This post gives an incomplete outline answering this accounting question.2.0 Assumptions And Price Equations
Some definitions follow:
- p1 = cost of operating main generator for 12 hours.
- p2 = cost of operating pump for 12 hours.
- p3 = cost of operating secondary generator for 12 hours.
- p4 = 1 = price of a unit of non-peak hours of electricity.
- p5 = price of a unit of peak hours of electricity.
- p6 = price of a unit of pumped and stored water.
- w = the wage.
- r = the rate of profits (for a 12 hour period).
- b41 = Units of off-peak hours of electricity produced in 12 hours when the pump is operating
- b53 = Units of peak-hours of electricity produced in 12 hours by the secondary generator.
- a01 = Person-hours of labor needed to operate the main generator and pump for 12 hours
- a02 = Person-hours of labor needed to operate the main generator alone.
- a03 = Person-hours of labor needed to operate the secondary generator alone.
I have taken a unit of non-peak hours of electricity as the numeraire. Assume that electricity is measured in units such that the output of the main generator operating alone is one unit of electricity. Since the pump is operating during the production of off-peak hours of electricity, the electricity generated during this period is less than one-unit:
0 < b41 < 1.
Measure pumped and stored water in units such that the amount pumped in 12 hours is a unit. The second law of thermodynamics implies the following additional constraint:
0 < b41 + b53 < 1.
Finally, I assume that less labor is required to operate the main generator alone than is required to operate it with the pump:
0 < a02 < a01.
These assumptions allow one to specify the following price equations:
(p1 + p2)(1 + r) + a01w = b41 + p6
(p1)(1 + r) + a02w = p5
(p3 + p6)(1 + r) + a03w = b53p5
The price equations show that wages are paid out of the surplus, not advanced. The price equation for the first process shows that it produces a joint product.3.0 The Solution Prices
The solution prices are:
w = [(b53p1 - p3 + b41)(1 + r) - (p1 + p2)(1 + r)2]/[a01(1 + r) - a02b53 + a03]
p5 = (p1)(1 + r) + a02w
p6 = [(b53p5 - a03w)/(1 + r)] - p3
In a more thorough analysis, one would consider when the wage-rate of profits curve is downward sloping, when the price of peak-hours electricity is positive, and when the price of pumped and stored water is positive. As is typical in price theory, prices depend on the distribution of income. The analysis uncovers the accounting price for pumped and stored water. Since this is a long-period model, consumer demand enters only in determining the scale at which this facility is constructed. Prices can be found without ever considering consumer demand schedules.4.0 Discussion and Conclusions
A fuller development would look at the depreciation of the pump and generators. If one were to look at the economy as a whole, instead of just this electric company, one would want to include processes for producing pumps and generators, perhaps with inputs that include electricity. And one could add further complications. Anyways, I think I have justified, in this post, the (unoriginal) claim that Sraffa's book has empirical implications.