Added math and explination

In_Situ_Policy.tex

@@ -139,6 +139,7 @@ PhD candidate of Economics, University of Wyoming
 \input{Supporting/Acro.tex}
 \input{Sections/Introduction.tex}
 \input{Sections/Data.tex}
+\input{Sections/Profit.tex}
 \input{Sections/Extended_Production.tex}
 
 

R-Code/WELL_PROFIT.r

@@ -0,0 +1,24 @@
+#Test of Change in Well Profit relative to operating costs COP
+COP <- 1
+D <- 0.3
+r <- log(1.1)
+CRES <- COP*2
+RUN <- function(){print((COP/D)*(1-COP/(COP-r*CRES)-log(COP+r*CRES)+log(COP)))}
+RUN()
+CRES <- COP*3
+RUN()
+CRES <- COP*4
+RUN()
+#PRINT SHIFT
+print("SHIFT by 100")
+COP <- 100
+CRES <- COP*2
+RUN()
+CRES <- COP*3
+RUN()
+CRES <- COP*4
+RUN()
+for(i in 1:20
+
+
+

Sections/Appendix/Math.tex

@@ -1,4 +1,5 @@
 \section{Derivation of \cref{EQINFWELL}}
+	\label{AINFWELL}
 \begin{equation*}
 	\pi_{w}=\int_{t=0}^{T}\left[ \left(P_{ur}\cdot q_{i}\cdot e^{-Dt}-C_{op}\right)e^{-rt}\right] \,dt-C_{Drill}-C_{Res}\cdot e^{-rT} \
 \end{equation*}
@@ -34,4 +35,52 @@
 \end{equation*}
 
 
+\section{Derivation of \cref{EQPROD}}
+	\label{APROD}
+\begin{equation}
+	\Delta Q =\int_{T_{1}^{\star}}^{T_{1}^{\star}+\Delta T} q_{i}\cdot e^{-rt}\,dt
+\end{equation}
+\begin{equation*}
+	\Delta Q = -\frac{q_{i}\cdot e^{-D \left(T_{1}^{\star}+\Delta T\right)}}{D}+\frac{q_{i}\cdot e^{-D T_{1}^{\star}}}{D}
+\end{equation*}
+
+\begin{equation*}
+	\Delta Q = -\frac{q_{i}\cdot e^{-D T_{1}^{\star}}\cdot e^{-D \Delta T}}{D}+\frac{q_{i}\cdot e^{-D T_{1}^{\star}}}{D}
+\end{equation*}
+
+\begin{equation*}
+	\Delta Q = q_{i}\cdot e^{-D T_{1}^{\star}}\frac{-e^{-D \Delta T}+1}{D}
+\end{equation*}
+
+\begin{equation*}
+	\Delta Q = \frac{q_{i}\cdot e^{-D T_{1}^{\star}}}{D}\left(1-e^{-D \Delta T}\right)
+\end{equation*}
+
+\begin{equation*}
+	\Delta Q = \frac{q_{i}\cdot e^{-D T_{1}^{\star}}}{D}\left(1-e^{-D \frac{\ln(C_{op}-r C_{Res})-\ln(C_{op})}{D}
+}\right)
+\end{equation*}
+\begin{equation*}
+	\Delta Q = \frac{q_{i}\cdot e^{-D T_{1}^{\star}}}{D}\left(1-e^{-\ln(C_{op}-r C_{Res})+\ln(C_{op})}\right)
+\end{equation*}
+
+\begin{equation*}
+	\Delta Q = \frac{q_{i}\cdot e^{-D T_{1}^{\star}}}{D}\left(1-\frac{C_{op}}{C_{op}-r C_{Res}}\right)
+\end{equation*}
+
+\begin{equation*}
+	\Delta Q = \frac{q_{i}\cdot e^{-D \frac{\ln(P_{ur})+\ln(q_{i})-\ln(C_{op})}{D} }}{D}\left(1-\frac{C_{op}}{C_{op}-r C_{Res}}\right)
+\end{equation*}
+
+\begin{equation*}
+	\Delta Q = \frac{q_{i}}{D}\cdot e^{-\ln(P_{ur})-\ln(q_{i})+\ln(C_{op})}\left(1-\frac{C_{op}}{C_{op}-r C_{Res}}\right)
+\end{equation*}
+
+\begin{equation*}
+	\Delta Q = q_{i}\frac{C_{op}}{D\cdot P_{ur}\cdot q_{i}}\left(1-\frac{C_{op}}{C_{op}-r C_{Res}}\right)
+\end{equation*}
+
+\begin{equation*}
+	\Delta Q = \frac{C_{op}}{D\cdot P_{ur}}\left(1-\frac{C_{op}}{C_{op}-r C_{Res}}\right)
+\end{equation*}
 

Sections/Extended_Production.tex

@@ -1,15 +1,72 @@
+\section{Profit Function}
+\begin{equation}
+	\label{EQPROFITALL}
+	\pi=\sum_{t=0}^{T}\left[\left( P_{ur} \cdot \left(W_{t}^{\alpha}-W_{t-1}^{\alpha}\right)-\left(C_{Drill}+C_{Res}\right)\cdot\left(W_{t}-W_{t-1}\right)\right)\frac{1}{(1+r)^t}\right]-C_{Facility}
+	\end{equation}
+Subject to: \(\beta\cdot C_{Facility}\ge W_{t}^{\alpha}-W_{t-1}^{\alpha}\), and \(\gamma_{Drill}\ge W_{t}^{\alpha}-W_{t-1}^{\alpha}\)
+\newline
+Where \(P_{ur}\) is the price of uranium, \(W_{t}\) is the total number of wells drilled in a aquifer at time \emph{t}, \(\alpha\) is a constant between zero and one representing the decline in ore grade across the reservoir, \(C_{Drill}\) is the cost to drill a well, \(C_{Res}\) is the cost to restore the water affected by a well, \emph{r} is the yearly discount rate of the firm, \(C_{Facility}\) is the investment cost in the uranium processing facility, \(\beta\) is a factor that converts the dollars spent to construct a uranium processing facility to output capacity, and \(\gamma_{Drill}\) is the maximum available drilling capacity in the region. 
+
+Assuming neither constraint binds the optimal number of wells to drill is \(W_{T}^{\star}=\left[\frac{\alpha P_{ur}}{C_{drill}+C_{res}}\right]^{1-\alpha}\). Adding the restoration costs to wells lowers the number of wells, and total uranium output based on the coefficient \(\alpha\) this is applied to a time series regression to predict elasticity of uranium producer on using the natural log of output.
+
+
 \section{Extended Production}
+We next evaluate how restoration requirements effect the total profits of specific wells in a in situ operation. In this analysis the choice to drill a production well in the uranium roll front has been made, making capital investment sunk. The operating cost of the well is assumed to be constant. The extraction rate of a single will group, must stay in a certain range to keep a constant pressure gradient in produced aquifer. This limits the choice of pumping rates and makes this assumption within the realm of possibility. 
+
+The rate of uranium production of a well is assumed to follow an exponential decline curve, a method frequently used in oil and gas production \citep{mccain2017}. This model predicts the uranium extraction rate of the well as a function of the original extraction rate, and the total time of production. In each production cycle where lixivant is injected into the reservoir a portion of the remaining uranium is dislocated in to the water, which is then brought to surface. As more uranium is extracted there is less remaining uranium to be dissociated slowing extraction rate. This makes a exponential decline curve model a plausible method of matching uranium production declines in wells.
+
+Under the assumption no operating costs, and a exponential decline curve the marginal choice of producers is whether or not to drill a well, and not inter well production changes are made based on prices \citep{anderson2018}. However, in the current setting operating costs cannot be assumed to be low, but they are assumed to be constant. This means that the optimal time to operate a set of uranium recovery wells is a choice variable of the producer. The total profits of a uranium well are modeled in \cref{EQPROFIT}.
+
 \begin{equation}
 	\label{EQPROFIT}
 	\pi_{w}=\int_{t=0}^{T}\left[ \left(P_{ur}\cdot q_{i}\cdot e^{-Dt}-C_{op}\right)e^{-rt}\right] \,dt-C_{Drill}-C_{Res}\cdot e^{-rT} \
 \end{equation}
-
+Where \emph{D} is the decline rate of the well, and r is the instantaneous private discount rate. Since the terminal time \emph{T} is a choice variable the optimal time to operate the well can be found with:
 \begin{equation*}
 	\frac{d \pi_{w}}{dT}= \left(P_{ur}\cdot q_{i}-C_{op}e^{-rT}+r\cdot C_{Res}\right)e^{-rT}=0 \
 \end{equation*}
-
+This yields a optimal operating time in \cref{EQINFWELL}, with the increase in operting time induced by the resotration requirement \(\Delta T^{\star}\) identified in \cref{TIMEDIFF}. The full derivation is provided in \cref{AINFWELL}
 \begin{equation}
 	\label{EQINFWELL}
 	T^{\star}=\frac{\ln(P_{ur})+\ln(q_{i})-\ln(C_{op}-r C_{Res})}{D}
 \end{equation}
-This result is calibrated Wyoming mine operation plans. The added well operation time induced by the restoration requirements is \(\frac{ln(C_{op})-ln(C_{op}-r C_{Res})}{D}\). We use the Strata Ross project as a bassline, in situ operation, due to the high data granularity. 
+\begin{equation}
+	\label{TIMEDIFF}
+	\Delta T^{\star}=\frac{\ln(C_{op}-r C_{Res})-\ln(C_{op})}{D}
+\end{equation}
+A few points can eb taken from \cref{EQINDWELL}. First the operating time of well increases as the uranium price increase, or the ore grade (\(q_{i}\)) of a well increase. This is intuitive since both values increase returns. The faster the decline rate, the shorter the operation life of a well as the resource is depleted sooner. 
+
+The most interesting term is the subtraction of \(\ln(C_{op}-r C_{Res})\). As either the cost of restoration of the private discount of the firm increases, the well operating life increase. This  is because the restoration cost introduces a new incentive. The restoration costs must be paid after the well stops producing uranium. By producing the uranium for slighly longer, this large cost can be avoided.
+
+The magnitude of discounted value of the restoration costs compared to the current operating costs decide if a well should remain in operation. In fact if the discounted restoration costs is larger than operating cots, then the well will never cease operation. Even if no uranium is produced, the avoided restoration costs are worth more than net present cost of operating the well in perpetuity.
+
+This effect may explain one perplexing data point. In 2018, uranium prices were \$28 dollars per pound which is well bellow the expected profit thersholds of the observed mines. However, a single Wyoming operation remined open, which declining produciton. This would be consistent with the mine running existing wells as a means to avoid restoration costs. It is not clear that this operation would never  shut down, but the added incetive to avoide these costs can explain the temporry conounce udner low uranium prices. 
+
+This result is calibrated Wyoming mine operation plans. We use the Strata Ross project as a baseline in situ operation, due to the high data granularity. The decline rate of the average well is calculated to be 0.316, based on a five year operation time with 80\% total recovery. These wells have a discounted total operating cost of 1.7 million dollars, and a restoration costs of 1.8 million dollars. Applying \cref{TIMEDIFF} yields a increase in operating time of 4 months. This operation has the lowest restoration costs of any of the five identified projects, since there is lower calcite in the surrounding aquifer. A counter factual example is created where the average restoration cost of other projects is applied to the Strata Ross Project. In this counterfactual the average increase in well operation time is one year and three months. 
+
+\emph{Change in Production}
+\begin{equation}
+	\Delta Q =\int_{T_{1}^{\star}}^{T_{1}^{\star}+\Delta T} q_{i}\cdot e^{-rt}\,dt
+\end{equation}
+\begin{equation*}
+	\Delta Q = -\frac{q_{i}\cdot e^{-D \left(T_{1}^{\star}+\Delta T\right)}}{D}+\frac{q_{i}\cdot e^{-D T_{1}^{\star}}}{D}
+\end{equation*}
+
+\begin{equation}
+\label{EQPROD}
+	\Delta Q = \frac{C_{op}}{D\cdot P_{ur}}\left(1-\frac{C_{op}}{C_{op}-r C_{Res}}\right)
+\end{equation}
+
+\emph{Total Profit Change}
+\begin{equation}
+	\Delta \pi_{w}=\Delta Q\cdot P_{ur}-\Delta T\cdot C_{op}
+\end{equation}
+
+\begin{equation*}
+	\Delta \pi_{w}=\frac{C_{op}}{D}\left(1-\frac{C_{op}}{C_{op}-r C_{Res}}\right)-\frac{C_{op}}{D}\left(\ln(C_{op}-r C_{Res})-\ln(C_{op})\right)
+\end{equation*}
+
+\begin{equation}
+	\Delta \pi_{w}=\frac{C_{op}}{D}\left[1-\frac{C_{op}}{C_{op}-r C_{Res}}-\ln(C_{op}+r C_{Res})+\ln(C_{op})\right]
+\end{equation}
+

Sections/Profit.tex

@@ -0,0 +1 @@
+\section{Mine Profit}