Discuss the role of Backstop in the optimum allocation of renewable resource under perfect competition.
⇒We know that non-renewable resources are exhaustible resources and they deplete over time. However there may be a suitable resource or a backstop. For example solar energy may be the backstop of oil energy. Therefore the optimal depletion condition changes due to the presents of a back-stop. We know that the optimum depletion condition for a non-renewable resource is
P0=mc+P1-mc/1+r
Or P0=mc+uc
Where, P0 is the present price of a energy resource say oil & P1 is the future price of that oil. But this optimal condition will change when these exist a backstop. The change in the condition can be explain in the following manner.
Let us take the following equation for the two period time path of price when no backstop/substitute is consider, which is P1=mc+(P0-mc)(1+r)--------(i)
Now, extending this two any number of periods we have Pt=mc+(P0-mc)(1+r)---------(ii)
As time passes the prices grows away from marginal cost, rising as a rate that approaches the rate of interest (r). The uc component of price comes to dominate the fixed marginal extraction cost component.
Now the question is does price rise continuously and in deficiency when a substitute is available. The answer is obviously not when there exist a backstop. For example the solar energy sets on upper limit on the price of oil and it also determines the initial user to be added to the marginal cost of extraction in the following this two role of backstop functions are discussed.
Suppose the transaction (switch) from oil to the backstop takes place at time T. Then from the equation (ii) the price is given by
PT=mc+(P0-mc)(1+r)T---------------------(iii)
But with unlimited solar energy resource does not have any user cost. Therefore PT=mcb so after some the equation (iii) because
P0-mc=mcb-mc/(1+r)T---------------(iv)
This implies that the initial user cost at t=0, (P0-mc) is the difference between the cost of the backstop and the cost of the oil, discounted back from the debt of transaction (T) now substituting this expression for P0-mc into equation (iii) we obtain
Pt=mc+mcb-mc (1+r)T/(1+r)T-------------(v)
And this implies an expression for the price of oil in terms of the cost of backstop at any time t < T. We can rewrite the above equation as-
Pt=mc+mcb-mc/(1+r)t
So, if shows that user cost rises at the rate at r to (mcb-mc) and augmented marginal cost (amc) and the price rises to mcb at the time T, the switch point.
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