Market allocation factor

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Market allocation factor is a method to balance out market situations when either the demand or supply increase or decrease. To be able to predict this, elasticity, market allocation factor, or other methods need to be used.

Question

How to predict market balances when either the demand or supply increases or decreases?

Answer

How many per cent does the supply or demand change if the price of the product increases 1 €/product unit?

Market allocation factor(relative change per (€/unit))
Obs	Market	Activity	Use	Fuel	Unit	Amount
1	Fuel wood	Petroleum refineries	Input	Renewables and waste	MWh	1
2	Fuel wood	CHP plants	Input	Renewables and waste	ton	1
3	Fuel wood	Heating	Output	Renewables and waste	MWh	1
4	Fuel wood	Other energy consumption	Output	Renewables and waste	MWh	1
5	Traffic fuel	Petroleum refineries	Output	Petrochemical products	MWh	1
6	Traffic fuel	Final energy use	Output	Petrochemical products	MWh	1
7	Traffic fuel	Petroleum refineries	Output	Petrochemical products	MWh	1
8	Traffic fuel	Petroleum refineries	Output	Petrochemical products	MWh	1
9	Global oil	Petroleum refineries	Input	Crude oil	MWh	1
10	Global oil	CHP plants	Input	Crude oil	ton	1
11	Peat	CHP plants	Input	Coal and peat	ton	1
12	District heat	CHP plants	Output	Heat	MWh	1
13	National electricity	CHP plants	Output	Electricity	MWh	1
14	Emission trade	CHP plants	Output	CO2e	ton	1
15	Emission trade	Final energy use	Output	CO2e	ton	1
16	Cardboard	Other energy consumption	Output	Cardboard products	ton	1

⇤--#: . The input/output column does not make sense. The meaning depends on the part of the energy balance table (supply, transformation, use) where the activity is. Can it be dropped? --Jouni 12:09, 24 April 2012 (EEST) (type: truth; paradigms: science: attack)

+ Show code - Hide code

# NOW how do we tell the energy need in the balance sheet?
# THIS CODE DOES NOT REALLY WORK.

############# Price is a function that finds the price for the energy in a given situation and calculates the new supplies and demands.
#     energy = data.frame containing the energy supplies and demands.
#     cost   = data.frame containing the cost curves.
#     conditions = vector of column names that should be looked at separately, e.g. different policy options. The vector always contains Fuel and Use.
price <- function(energy, cost, conditions = NULL) {
	conditions <- c("Fuel", "Use", conditions)
	c <- as.data.frame(as.table(tapply(energy$Amount, energy[conditions], sum)))
	c$Freq <- ifelse(c$Use == "Demand", -c$Freq, c$Freq)
	c <- as.data.frame(as.table(tapply(c$Freq, c$Fuel, sum)))
	b <- as.data.frame(as.table(tapply(cost$b, cost$Fuel)))
	a <- as.data.frame(as.table(tapply(cost$a, cost$Fuel)))
	a$Freq <- a$Freq + 0.000000001
	p <- c((-b + sqrt(b^2 -4*a*c))/(2*a), (-b-sqrt(b^2-4*a*c))/(2*a))
	p <- p[which.min(abs(p))] # Assumes that the root that is closer to the current price is the correct one.
	return(p)
}

Rationale

Explanations of the calculations

The logic of the code is the following:

Energy contains information about the total amounts of energy in a given Activity * Fuel pair per time unit (typically a year). (For example, 54 ktoe/a petrochemical products are used in road transport.)
Trans contains descriptions of all activities that result in the production or use of energy. (For each ktoe used in road transport, 1 ktoe is imported.
When these two are merged, we get a table that explains where the energy came from and where the energy went, assuming that all energy flow goes through that activity.
The ratio of the actual energy stock to the energy stock in the activity table is used as the factor to scale the whole activity (not only the energy). This is done by merging the data.frame with factors again with the activities (tran) and multiplying the factor with the activities. This gives the BAU situation.
We assume that current energy production and use is distributed in such a way that the cost curves are in balance. If the price changes, both the supply and demand will change according to their cost curves. The new balance can be found from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Sigma f_i(p) = 0} where f_i are the cost curves (supply functions producing positive values and demand functions negative values. When p is solved from the equation, values of all supply and demand functions can be solved. If all functions

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_i(p) = a_i p^2 + b_i p + c_i,}

then individual functions and also the sum of all functions can be expanded and then easily be solved by using the equation for polynomial roots:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Sigma f_i(p) = \Sigma (a_i p^2 + b_i p + c_i) = 0}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p = \frac{-\Sigma b_i \pm \sqrt{(\Sigma b_i)^2 - 4 \Sigma a_i \Sigma c_i}} {2 \Sigma a_i}}

Cost curves are parameterised in the way that the current price p = 0 and therefore the current supply or demand is c_i.

When a demand or supply changes, a new price p must be calculated based on f_i, and then the supplies and demands of other actors can be calculated, resulting in a new balance.

Taxes are complicating this picture. They do not actually change the shape of a cost curve (if it is proportional to energy content) but they shift a supply curve to right (or a demand curve to left if the tax is payed by the buyer and not the seller) by the amount of the tax. In other words, instead of p we solve Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Sigma f_i(p - t) = 0} , i.e.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p = \frac{-\Sigma (b_i - 2 a_i t) \pm \sqrt{(\Sigma b_i - 2 t \Sigma a_i)^2 - 4 \Sigma a_i (\Sigma c_i + t^2 - t \Sigma b_i)}} {2 \Sigma a_i},}

where t is the amount of tax to the suppliers. With the same formula, it is also easy to calculate particular taxes, where the tax per unit energy is different to different suppliers. Then we simply use t_i instead of t.

How to estimate parameters a, b, and c?

Let's start from a simple approach. If the cost curve is linear, then a = 0. If all suppliers or all buyers change their behaviour in the same way (in relative terms) to price changes, then b_i are proportional to c_i. If we don't care about the absolute values yet, we can simply start from these assumptions and say: a = 0, b = c, c = the current amount of demand or supply, and p = 0.

The curves can be generalised to that they are polynomials with higher degree than just 2. The computations are more laborious, but the computer solves that problems with no additional burden to the user.

----#: . We could use a tool that estimates the parameters based on some points that a user gives on a graph about his/her personal cost curve. --Jouni 06:53, 2 February 2012 (EET) (type: truth; paradigms: science: comment)

References

Related files