## Saturday, August 25, 2012

### Peak And Off-Peak Electricity As A Joint Product (Continued)

 Figure 1: A Photo Probably From The Same Week I visited A Joint Production Process
1.0 Intrduction

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.

Nick Rowe said...

If the PPF between peak and off-peak consumption of electricity is a straight line, with a constant slope, then yes, the relative price of the two goods, which is equal to that slope, will be determined by technology without reference to preferences. But if the PPF is curved, you need to look at indifference curves as well, to find the slope at the tangency point.

Take a dam which has limited storage capacity, or limited generating capacity, and the PPF won't be a straight line. It's going to have a kink in it. And you need to look at preferences to see if you are at the kink, and if you are at the kink, what relative price would clear both markets.

Robert Vienneau said...

The post itself demonstrates that Rowe's comment is not well-taken. Note that the comment makes no distinction between when the non-substitution theorem holds and when it does not. Or between the role of land and of capital.

On the other hand, if I decide to draw the Production Possibilities Frontier in my next post, I will have been prodded to do that by this comment.

Greg Hill said...

Robert and Nick,

I’m trying to follow this dialogue, but it’s difficult (for me at least). Let me pose a different, but hopefully related, question.

Suppose you’re running a state-owned electric utility like B.C. Hydro and you want to set prices to ensure the efficient use of B.C.’s hydropower. Further, suppose B.C. Hydro produces peak and off-peak power, that the MC of peak power > MC of off-peak power, but that B.C. Hydro faces a revenue constraint such that it can’t price both peak and off-peak power at MC. Can you set efficient prices for peak and off-peak power, given the revenue constraint, without knowing the demand curves for peak and off-peak power?

Robert Vienneau said...

Sraffa's equations are consistent with Marginal Cost pricing. Talk about tangencies between utility isoquants and PPFs misses all the interesting issues here.

Nick Rowe said...

Greg: my point here is a much simpler one. Assume Marginal Cost pricing (i.e. there is no revenue constraint). The MC of producing an additional KWH at a particular time will depend on the quantity produced at that time. And the quantity produced will depend on demand at that time, which depends on consumers' preferences.

For example, the MC of producing an extra KWH of electricity at (say) noon will probably be very low if quantity produced is low and there is lots of spare capacity, and MC at noon will get higher (maybe in steps or maybe smoothly) as more and more is produced, so you either need to build more/bigger dams, pump water uphill during off-peak, switch to oil, or whatever. So there's an upward-sloping MC curve for producing electricity at noon.

And, quantity demanded at noon will generally be a decreasing function of the price at noon. There's a downward-sloping demand curve, which depends on consumer preferences.

So you need *both* the upward-sloping MC curve *and* the downward-sloping demand curve to figure out what the MC and price at noon will actually be. Knowledge of the technology and the wage rate and price of oil won't be enough to tell you what the price will be (unless the MC curve is flat because MC doesn't depend on quantity produced).

Neo-Ricardian models ignore consumer preferences and demand. That only works if MC curves are flat. But what is interesting about the peak/off-peak electicity problem is precisely the fact that MC curves aren't flat. The bigger the difference between peak and off-peak production of electricity, the bigger the difference between peak and off-peak MC.

Robert: strictly speaking, what you have in your model here is a (set of) straight line isocost curves between the two goods peak and off-peak electricity, for given technology and given input prices. So preferences will affect the quantities produced in peak and off-peak, but won't affect the MC of peak and off-peak (taking input prices as given). A non-linear technology would give you curved isocost curves (or isocost curves with lots of kinks). Then you would need preferences and the tangency point to tell you the slope of the isocost curve, and the ratios of the peak/off-peak Marginal Costs.

BTW, in the classic wool-mutton example of joint production, the isocost curves are reverse-L-shaped. Your example of joint production here, with its straight line isocost curves, is at the exact opposite extreme to the classic wool-mutton example.

Greg Hill said...

Robert and Nick,

Thanks for taking the time to respond. It's a lot to think about, but well worth it.

BruceMcF said...

"If the PPF between peak and off-peak consumption of electricity is a straight line, with a constant slope, then yes, the relative price of the two goods, which is equal to that slope, will be determined by technology without reference to preferences. But if the PPF is curved, you need to look at indifference curves as well, to find the slope at the tangency point.

Take a dam which has limited storage capacity, or limited generating capacity, and the PPF won't be a straight line. It's going to have a kink in it. And you need to look at preferences to see if you are at the kink, and if you are at the kink, what relative price would clear both markets.
"

So in other words, given that there is a level of reverse pumped hydro capacity results in a straight line PPF between peak and off-peak electricity, whether you have to invoke indifference curves from fictitious utility functions in the short term is an investment decision in the long term.