Question One. In $2012,$ according to the U$.$S$.$ Energy Information Administration $($EIA$)$ world known estimated reserves of oil $($i$.$e$.,$ the stock of oil $=$ S$)$ were roughly $1525$ billion barrels. In that same year, annual world consumption$/$extraction of oil $($R$0)$ was roughly $32$ billion barrels according to the International Energy Agency $($IEA$).$ Assume that all of the oil that is extracted $($R$)$ in a year is also consumed, so we can use the terms extraction and consumption interchangeably. Assume that world oil consumption $($i$.$e$.,$ extraction$)$ grows at a steady rate of $2\%$ through time in the future. Calculate the exponential exhaustion index for oil given current world oil market trends?

Question Two. For this question, use the same estimate of world reserves of oil, $1525$ billion barrels, that was used in the first question. Also, use the same estimate of annual world consumption of oil, $32$ billion barrels in $2012,$ as in Question One. Assume that the world oil price, Brent crude, averaged roughly $$122/$barrel during $2012($according to the U$.$S$.$ EIA$).$ Suppose that the long run price elasticity of demand for oil is $-0\mathrm{.}5.$ Also, assume that all of the oil that is extracted $($R$)$ in a year is also consumed, so we can use the terms extraction and consumption interchangeably. Assume an optimal extraction rate based upon a $3\%$ discount rate. Based you calculations on the following demand curve for oil, P$($R$),$ which has the following functional form:

P$($R$)=$ Ke$$aR

where the demand for oil is determined by:

$-$ the quantity of oil extracted $($i$.$e$.,$ consumed$),$ R

$-$ the parameter, K$,$ the choke price of oil

$-$ and a parameter, a

Given this information:

$1.$ What is the choke price for oil $($K$)?$

$2.$ What is the optimal year that the world should exhaust $($i$.$e$.,$run out$)$ of oil $($T$)?$

$3.$ What is the optimal level of consumption $($i$.$e$.,$ extraction$)$ of oil $($R$0)$ in $2012($i$.$e$.,2012$ is considered the $$initial year$)?$

$4.$ What is the optimal price of oil $($P$0)$ in the initial year of consumption$/$extraction$,2012?$

Question Three. Why do you think that the exponential exhaustion index and the optimal depletion methodology come up with such different answers about when the world $$runs out$$ of oil? Suppose that oil supply was added into the optimal depletion model, how do think that this would affect the answers you just derived?

Note: One omission in this analysis is that we have not factored in the environmental damage associated with oil use, so the term $$optimal$$ should be considered only from a private standpoint. We are not taking a social perspective in this example.