Time-path of Fossil Fuel’s Price- A Math Eco article

Time-path of Fossil Fuel’s Price- A Math Eco article

The article is trying to explain the price time-path of fossil fuels. It is a known fact that day by day foil fuels. Some believe that they are renewable; while, others think that since it takes 200 million years to renew, it is a non-renewable resource. However, complete exhaustion is near.

The Dynamics of Monopoly (1924), written by G.C. Evan and cited by Alpha W. Chiang in Elements of Dynamics of Optimization, deserves the credit. The mathematical model works around Euler’s equation with vertical transversality conditions. The consideration for the time horizon was successful after verifying a lot of environmental journals and articles. According to most environmentalists, by 2030 the fossil fuels shall be exhausted.

This research article formulates a model with simple classroom-inspired equations of demand and cost. Euler’s equation is the base to build the model to find the optimal time-path of price. Vertical transversality conditions are the key highlight.

Key Word: Time-path, Fossil Fuel, Price path

Introduction: Time-path of Fossil Fuel’s Price

Organic materials are decomposed and left under the earth for millions of years to form fossil fuels. It is generally an anaerobic decomposition process. They contain a high percentage of carbon, and hence they are highly combustible. Examples including are coal, petroleum, natural gases so on.

Ages after ages, we are told that fossil fuels will exhaust. According to Colin Campbell, an Oxford Ph.D. and chief geologist at Amoco, fossil fuels won’t get depleted for years. However, other research stories contradict saying that we have already consumed 90% of the reserves. In the opinion of most environmentalists, before the depletion of the fuel reserves, science will invent or discover a replacement for these reserves. However, our research concerns the economic impact and the price time-path of the market scenarios.

Dynamic optimization methods try to achieve an optimal path to address an economic problem over a technical planning period. Dynamic optimization allows a multistage decision-making solution.

Time-path price


The adjustment costs are associated with input factors. It affects how the firms react to relative price change. Since the inputs are depleting at an increasing rate, the product price is also increasing. Many research works are looking into the static working of the perfect competition. However, very few take up the dynamic trajectory path for natural resources. The work in this sector is minimal and requires the attention of economists along with environmentalists.

At the beginning of 2004, a demand-driven shift led to an abrupt rise in the price of fossil fuels. According to Peter Christoff, Australia is the highest contributor to the fossil fuel supply problem. It also led to the price shock. Few dynamic models are researching the extraction of renewable resources. This research article tries to elucidate the same.

Price Time-path Background

To build the model pillars, I use a straight-line demand equation, Q = a+bP. (a>0 and b<0).


  • Q is the output demanded, and 
  • P is the price of fossil fuel over a constant time. 

Fossil fuels are consumer goods as well as raw materials or intermediate goods. To keep the model relatable, let us consider and treat them as final goods.

According to the law of demand, the quantity demanded & the price of the commodity are inversely related. However, over time the stock of fossil fuel (x = Q) depletes. Therefore, dx/dt < 0. Since overtime, the quantity available falls, the price of the scarce product should rise. Hence, we had to incorporate a variable term (x) that would depend on time.

Therefore, Q= a + bP + h.P’. (h.P’ > 0).
Since we are looking at the problem with a dynamic approach, we need to incorporate the time factor in the model. It means we need to include a change in price over time as a new variable [dp/dt].

Now, let us take a look at the cost function. Again, to keep things easy, we have used the quadratic cost function.

Let E denote the Fossil Fuel extraction cost, which is constant. As the units extracted increase, the total cost of extraction falls. It reaches the minimum point and then starts to rise again. Thus, forming the U-shaped curve.

C(Q) = AQ2+BQ+E; A>0, B>0, E>0, E is the fixed cost of production, which is never zero.

Price Time-path Theoretical Model

The profit function:

Π = TR – TC,                                      TR = total revenue, TC = total cost.

TR       = Q.P

            = (a + bP + h.P’) P

TC= C(Q) = AQ2+BQ+E


Π         =          {(a + bP + h. P’) P} – { AQ2+BQ+E}

Π         =          {aP +bP2 + hP.P’} – { A(a + bP + h. dP/dt )2+B(a + bP + h.P’)+E}

Profit is a function of P and P’.

Π (P, P’) =       {aP +bP2 + hP.P’} – { A(a + bP + h. dP/dt )2+B(a + bP + h.P’)+E}

Our motive here is to maximise Π (P, P’) subject to P(t=0) =P0 and P(t=T) = PT

Euler’s Equation:

Fy= Fy’t + Fy’ydy/dt + Fy’y’d2y/dt2

Therefore, the Euler equation in our case shall be

Πp = Πp’t + Πp’pdp/dt + Πp’p’ d2P/dt2

Further, using the ‘Fundamental Methods of Mathematical Economics’ we try to frame a general solution.

P’’+ jP’ + kP = l

The general solution is P(t) = Pc + Pp,= D1er1t + D2er2t + P᷃ ,

D1er1t + D2er2t is the complementary solution (Pc)

P᷃ is the particular solution (Pp)

Optimal Price Time-path

Now. Let us move to the Optimal price path using proper equations instead of absolute algebraic equations. Until now, what we did and solved is easily available in dynamic books. We, try to implement the same working equations but for fossil fuels with actual price and expected time of exhaustion.

We choose demand and cost functions of fossil fuels based on literature review.

C= 1/20 Q2 + 500……………………………………………………(1)

Q = 80 – 4P + 50P’……………………………………………………(2)

Π = TR -TC = P.Q – C(Q) = P(80 – 4P + 50P’) – 1/20 Q2 – 500……………………………..(3)

Π = P(80 – 4P + 50P’) – 1/20 (80 – 4P + 50P’)2 – 500

Π = 80P – 4P2 + 50P’P – 1/20 {6400 + 16P2 + 2500P’2 + 2(4000)P’ – 2(320P) – 2(200PP’)} – 500………………………………………………………….(a+b+c)2 formula.

Π = 112P – 24/5P2 + 70PP’ – 125P’2 – 400P’ – 500……………………………. (4)


Π is a function of P and P’. Where P’ = dP/dt.

Figuring out the Euler equation.

As mentioned earlier, Πp = Πp’t + Πp’pdp/dt + Πp’p’ d2P/dt2

112- 48/5P + 70P’ = Πp

70P – 250P’ – 400 = Πp’

70 = Πp’p

-250 = Πp’p’

0 = Πp’t

Substituting all the values back into the Euler’s equation.

112- 48/5P + 70P’ = 0 + 70dP/dt -250 d2P/dt2

250 d2P/dt2 = 48/5P – 112 ………………………(5) Euler’s equation after normalization.

Transversality condition

As per literature, fossil fuels shall reach their peak by 2030. And the papers which I read were published in the year 2010 -2011 mostly. Therefore, I begin with the approximate prices of fossil fuels around that time. The prices were $120/ barrel. According to research, the expected rise in the fossil fuel price during the depletion period shall be 275%. So approximating the price expectation, we assume that P(T) = $400/ barrel by 2030.

Since we are using vertical transversality condition. Therefore, we state that,

p’]t=T = 0

70P – 250P’ – 400 = 0…………………………………. (7) Vertical Transversality condition

General Solution

Now, we try to seek a general solution.

As the formulae suggest, P(t) = Pc + Pp,= D1er1t + D2er2t + P᷃

Where, P(t) is the general solution

D1er1t + D2er2t is the complementary solution (Pc)

P᷃ is the particular solution (Pp)

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Hence, we finally reach the definite solution of the dynamic optimization solution.


Our assumption had extreme high price change and therefore it led us to some difficult situations. Mostly under such conditions we are supposed to assume and limit the variation in price to some region P0 ± M, where M is some fixed amount. The explosive time-path of price is very evident. Since as the stock of fossil fuels deplete, the scarcity increases and hence the dearth will obviously lead to an explosive time-path. 

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Ms. Shreya Roy
Ms. Shreya Roy

Ms. Roy is a PhD research scholar at Indian Institute of Foreign Trade. She has expertise in Trade, Money-Finance, Labor- Eco and Math-Eco. She is a lover of art and philanthropist at heart. Click on the social media icons below to follow her.

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3 thoughts on “Time-path of Fossil Fuel’s Price- A Math Eco article

  1. Very informative & knowledgeable article… Today i get to know why the prices of fossil fuel are getting higher
    So that due to high rates of petrol & diesel.. people will shift from fossil fuel vehicle to electric vehicle
    It can’t be done with a sudden movement that why they are increasing the price & slowly people will shift to electric vehicle and it will also help the environment too😌😌

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