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			\newcommand{\R}{\mathbb{R}}
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            \newcommand{\KLev}{\pmb{K}}
            \newcommand{\NFALev}{\NLev}
            \newcommand{\NFARat}{\NRat}
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            \newcommand{\PopU}{\mathcal{U}}
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\begin{document}
\textbf{Population of the employed in period t:} A population of size $\EmpGro^t$ was born in the current period $t$ and is employed.  A population of size $\EmpGro^{t-n}$ was born $n$ periods ago, and a $\erate^n$ portion of this population is still employed today.  Summing across all previous generations, the total current employed population is:
\begin{equation*}
\PopE_t =  \sum_{n=0}^{\infty} \EmpGro^{t-n} \erate^n = \EmpGro^t \cdot \sum_{n=0}^{\infty} (\frac{\erate}{\EmpGro})^n = \EmpGro^t \frac{1}{1-\frac{\erate}{\EmpGro}} = \frac{\EmpGro^{t+1}}{\EmpGro - \erate}
\end{equation*}

\vspace{0.5cm}

\textbf{Population of the unemployed in period t:} For any period $t-n$, a population of size $\urate \PopE_{t-n}$ entered unemployment at this time.  Individuals in this population are still alive in proportion $\PLives^{n-1}$.  Summing across all previous generations of newly unemployed, the total population of unemployed is:
\begin{equation*}
\PopU_t = \sum_{n=1}^{\infty} \PLives^{t-n} \urate \PopE_{t-n} = \sum_{n=1}^{\infty} \urate \PLives^{t-n} \frac{\EmpGro^{t-n+1}}{\EmpGro - \erate} = \frac{\urate}{\EmpGro} \PopE_t \sum_{n=1}^{\infty} (\frac{\PLives}{\EmpGro})^n = \frac{\urate}{\EmpGro} \PopE_t \frac{1}{1 - \frac{\PLives}{\EmpGro}} = \frac{\urate \PopE_t}{\EmpGro - \PLives} = \frac{\urate \EmpGro^{t+1}}{(\EmpGro - \urate)(\EmpGro - \PLives)}
\end{equation*}

\vspace{0.5cm}

\textbf{Wage equation:} GDP in period $t$ is $\GDPLev_t = \KLev_t^\kapshare (\ptyLev_t \LLev_t)^{1-\kapshare}$.  The wage rate is the marginal product of labor:
\begin{equation*}
\Wage_t = \frac{\partial \GDPLev_t}{\partial \LLev_t} = \KLev_t^\kapshare (1-\kapshare)(\ptyLev_t \LLev)^{1-\kapshare}/\LLev_t = (1-\kapshare)\GDPLev_t
\end{equation*}

\vspace{0.5cm}

\textbf{Euler equation for the employed consumer:} The consumer's Euler equation is, as usual: $u'(\pmb{c}_t) = \Discount \Rfree E[u'(\pmb{c}_{t+1})]$.  For an employed consumer with a CRRA utility function, this can be written as:
\begin{equation*}
u'(\cLevE_t) = \Discount \Rfree (\erate u'(\cLevE_{t+1}) + \urate u'(\cLevU_{t+1})) = \Discount \Rfree (\erate (\cLevE_{t+1})^{-\CRRA} + \urate (\cLevU_{t+1})^{-\CRRA})
\end{equation*}

\newpage

\textbf{Proof of eq:Explicit:} First, note that for an employed consumer, total pay grows by: $\frac{\Wage_{t+1}\labor_{t+1}}{\Wage_t \labor_t} = \PGro$, as wages grow at rate $\WGro$ while labor supply grows by $\XperGro$ each period.  Thus we can see that $\frac{\cLevE_{t+1}}{\cLevE_t} = \frac{\cRatE_{t+1}\Wage_{t+1}\labor_{t+1}}{\cRatE_t \Wage_t \labor_t} = \frac{\cRatE_{t+1}}{\cRatE_t} \PGro$.  Similarly, $\frac{(\cLevE_{t+1})^{-\CRRA}}{(\cLevE_t)^{-\CRRA}} = (\frac{\cRatE_{t+1}}{\cRatE_t})^{-\CRRA} \PGro^{-\CRRA}$.  Thus we can divide both sides of the employed consumer's Euler equation by $(\cLevE_t)^{-\CRRA}$ to arrive immediately at eq:Explicit.

\vspace{0.5cm}

\textbf{Proof of eq:cedel:} Substituting in the definition of the growth patience factor, eq:Explicit now says: $1 = \PatG^\CRRA (\erate (\frac{\cRatE_{t+1}}{\cRatE_t})^{-\CRRA} + \urate (\frac{\cRatU_{t+1}}{\cRatE_t})^{-\CRRA})$.  Divide both sides by $(\frac{\cRatE_{t+1}}{\cRatE_t})^{-\CRRA}$ to get:
\begin{equation*}
(\frac{\cRatE_{t+1}}{\cRatE_t})^{\CRRA} = \PatG^\CRRA (\erate + \urate (\frac{\cRatU_{t+1}}{\cRatE}_{t+1})^{-\CRRA}) \Rightarrow (\frac{\cRatE_{t+1}}{\cRatE_t}) = \PatG (\erate + \urate (\frac{\cRatE_{t+1}}{\cRatU_{t+1}})^{\CRRA})^{1/\CRRA} \Rightarrow (\frac{\cRatE_{t+1}}{\cRatE_t}) = \PatG (1 + \urate((\frac{\cRatE_{t+1}}{\cRatU_{t+1}})^{\CRRA} - 1))^{1/\CRRA}
\end{equation*}

\vspace{0.5cm}

\textbf{Proof of eq:ce t+1:} First, combine eq:normibc with the solution to the unemployed consumer's problem to find that: $\cRatU_{t+1} = \MPCU \frac{\Rfree}{\PGro} (\bRatE_{t+1} - \cRatE_{t+1} + 1)$.  Next, take eq:cedel, multiply both sides by $\cRatE_t$ and raise both sides to the $\CRRA$ power to get:
\begin{equation*}
(\cRatE_{t+1})^{\CRRA} = \PatG^\CRRA (\cRatE_t)^{\CRRA}(1 + \urate((\frac{\cRatE_{t+1}}{\cRatU_{t+1}}))^{\CRRA} - 1))
\end{equation*}

Next, isolate the factors of $(\cRatE_{t+1})^\CRRA$ to find that:
\begin{equation*}
\erate \PatG^\CRRA (\cRatE_{t})^\CRRA = (\cRatE_{t+1})^\CRRA (1 - \PatG (\cRatE_t)^\CRRA \urate (\cRatU_{t+1})^{-\CRRA})
\end{equation*}

Divide the equation by the factor on $(\cRatE_{t+1})^\CRRA$ and raise to the power $1/\CRRA$:
\begin{equation*}
\cRatE_{t+1} = \PatG \erate^{1/\CRRA}\cRatE_t (1 - \urate \PatG^\CRRA (\frac{\cRatE_t}{\cRatU_{t+1}})^\CRRA)^{-1/\CRRA}
\end{equation*}

Now we can substitute in the equation for $\cRatU_{t+1}$ and rearrange a bit:
\begin{equation*}
\cRatE_{t+1} = \PatG \erate^{1/\CRRA}\cRatE_t (1 - \urate \PatG^\CRRA (\frac{\cRatE_t}{\MPCU \frac{\Rfree}{\PGro} (\bRatE_{t+1} - \cRatE_{t+1} + 1)})^\CRRA)^{-1/\CRRA} = \PatG \erate^{1/\CRRA}\cRatE_t (1 - \urate (\frac{\PatG \PGro}{\Rfree \MPCU})^\CRRA (\frac{\cRatE_t}{\bRatE_{t+1} - \cRatE_{t+1} + 1})^\CRRA)^{-1/\CRRA}
\end{equation*}

Now we only have to show that $\frac{\PatG \PGro}{\Rfree \MPCU} = \frac{1 - \MPCU}{\PLives \MPCU}$ and we are done.  $\PatG \PGro = (\Discount \Rfree)^{1/\CRRA}$  Further, $(\Discount \Rfree)^{1/\CRRA}/R = (1 - \MPCU)/\PLives$, and so we arrive at the result.

\vspace{0.5cm}

\textbf{Proof of eq:bTarg:} To be completed.

\vspace{0.5cm}

\textbf{Proof of eq:SuOpLev:} Because $\frac{\BLevE_t}{\GDPLev_t}$ is constant and $\GDPLev_t$ grows by $\EmpGro \WGro$ each period, $\BLevE$ must also grow by the factor $\EmpGro \WGro$.  Individuals who became unemployed $n$ periods ago have $(\Rfree(1 - \MPCU))^n = (\PLives(\Discount \Rfree)^{1/\CRRA})^n$ portion of their original wealth.  This starting wealth is $(\EmpGro \WGro)^{-n} \frac{\BLevE_t}{\GDPLev_t} \urate$ (a $\urate$ portion of total wealth belongs to newly unemployed individuals).  Summing across all past cohorts to become unemployed, this is:
\begin{equation*}
\frac{\BLevU_t}{\GDPLev_t} = \urate \frac{\BLevE}{\GDPLev} \cdot \sum_{n=0}^\infty (\frac{\PLives (\Discount \Rfree)^{1/\CRRA}}{\EmpGro \WGro})^n = \urate \frac{\BLevE}{\GDPLev} \frac{1}{1 - \frac{\PLives(\Discount\Rfree)^{1/\CRRA}}{\EmpGro \WGro}} = \frac{\BLevE}{\GDPLev} \frac{\EmpGro \WGro \urate}{\EmpGro \WGro - \PLives(\Discount\Rfree)^{1/\CRRA}}
\end{equation*}

\vspace{0.5cm}

\textbf{Proof of eq:NOpLev:} First, note that $\pmb{S} = \BLevE + \BLevU \Rightarrow \frac{\pmb{S}}{\GDPLev} = \frac{\BLevE}{\GDPLev} + \frac{\BLevU}{\GDPLev}$.  Using the result from eq:SuOyLev, we can also write that: $\frac{\pmb{S}}{\GDPLev} = \frac{\BLevE}{\GDPLev}(1 + \frac{\EmpGro \WGro \urate}{\EmpGro \WGro - \PLives(\Discount\Rfree)^{1/\CRRA}})$.  $\BLev$ and $\KLev$ both grow by factor $\EmpGro \WGro$ in each period, thus $\KLev_{t+1} = \EmpGro \WGro \KLev_t$ and $\BLev_{t+1} = \EmpGro \WGro \BLev_t$.  Using this, eq:NFALev can be written as:
\begin{equation*}
\NFALev_t = \EmpGro \WGro (\BLev_t \Rfree^{-1} - \KLev_t) \Rightarrow \frac{\NFALev_t}{\GDPLev_t} = \EmpGro \WGro (\frac{\BLev_t}{\GDPLev_t} \Rfree^{-1} - \frac{\KLev_t}{\GDPLev_t}) = \EmpGro \WGro(\Rfree^{-1}(1 + \frac{\EmpGro \WGro \urate}{\EmpGro \WGro - \PLives(\Discount\Rfree)^{1/\CRRA}})\frac{\BLevE}{\GDPLev} - \frac{\kapshare}{\Rfree - \DeprFac}
\end{equation*}

\vspace{0.5cm}
\textbf{Proof of eq:bTargWithSocIns:} To be completed.

\vspace{0.5cm}

\end{document}