A New Theory to Forecast the Price of Non Renewable Energy Resources with Mass and Energy-Capital Conservation Equations.

The mass and energy-capital conservation equations are employed to study the time evolution of mass and price of non-renewable energy resources, extracted and sold to the market, in case of no-accumulation and no-depletion; i.e. when the resources are extracted and sold to the market at the same mass flow rate. The Hotelling rule for non-renewable resources, i.e. an exponential increase of the price at the rate of the current interest multiplied the time, is shown to be a special case of the general energy-capital conservation equation when the mass flow rate of extracted resources is unity. The mass and energy-capital conservation equations are solved jointly to investigated the time evolution of the extracted resources. The parameter PIFE, “Price Increase Factor of Extracted resources”, is the difference between the interest rate of capital, typically the inflation rate, and the mass flow rate of extraction of non-renewable resources. The price of the extracted resources increases if PIFE is greater than zero, i.e. the mass flow rate of extraction is smaller than the inflation rate. The price is constant if PIFE is zero, i.e. the mass flow rate of extraction is equal to the inflation rate. The price is decreasing with time if PIFE is smaller than zero, i.e. the mass flow rate of extraction is greater than the inflation rate. The price of selling resources varies with time according to the relation between the parameters PIFE and PIFS, “Price Increase Factor of Selling resources”, which is the difference between the extraction rate and the interest rates of selling resources, prime or discount rate. The price of selling resources increases with time if the initial price is greater than CIPS, “Critical Initial Price of Sold resources”, which depends on the initial price of extracted resources, the interest rate of non-extracted resources, and the difference between PIFS and PIFE or is greater than CIPES, “Critical Initial Price Extreme of Selling resources”, which depends on the initial price of extracted resources, the interest rate of non-extracted resources, and PIFS. The price of selling resources increases temporarily with time if the interest rates of non-extracted and extracted resources are equal, i.e. PIFE is equal to PIFS, and the initial price is greater than CIPES, “Critical Initial Price Extreme of Selling resources”. The price evolutions of the difference between selling and extracted resources are investigated according to the relation between extraction rate and interest rate of extracted and selling resources. The price difference increases with time if PIFS is greater than PIFE of the extracted resources and the initial price is greater than the critical price of selling resources, which depends on the initial price of extracted resources and the interest rate of non-extracted and extracted resources. The price difference decreases with time if PIFS is greater than PIFE and the initial price is smaller than the critical price of selling resources. The other cases are discussed extensively in the paper. The price evolution of non-renewable resources versus the consumption rate is investigated with the aim of constructing the energy supply curve. The case studied is without accumulation nor depletion of the resources and the mass and energy-capital conservation equations are solved under the condition of the same mass flow rate of extraction and sale. The energy supply curve of extracted resource is dependent on the new parameter, RINE, “Rate of Interest of Non-extracted resources on the Extraction rate”. The energy supply curve of selling resource is dependent on the new parameter, RISE, “Rate of Interest of Sold resources on the Extraction rate”, in case the rate of interest of non-extracted resources, rN, is nil. The energy supply curve of selling resources is dependent also on two dimensionless parameters, “Dimensionless Critical Initial Price of Sold resources”, i.e. DCIPS, and “Dimensionless Critical Initial Price Extreme of Sold resources”, i.e. DCIPES. The energy supply curve of selling resources is investigated under different relations between three parameters, i.e. extraction rate and interest rates of extracted and selling resources. New trends are observed in the economic market of non-renewable energy resources. The energy supply curve of the difference between selling and extracted resource is dependent on two dimensionless parameters, “Critical Initial Price Difference”, i.e. CIPD, and “Critical Extreme of the Initial Price Difference”, i.e. CEIPD. The price difference between selling and extracted resources is investigated versus the dimensionless mass flow rate of extraction. The evolution is dependent on four parameters: RINE, RISE, DCIPS, and DCIPES.