\providecommand{\aboveMin}{\blacktriangle}
\providecommand{\aAgg}{\ensuremath{\mathsf{A}}}
\providecommand{\adj}{\ensuremath{\mathrm{j}}}
\providecommand{\adjPar}{\ensuremath{\omega}}
\providecommand{\aE}{\aRat^{e}}
\providecommand{\aFunc}{\ensuremath{\mathrm{a}}}
\providecommand{\Age}{\ensuremath{Z}}
\providecommand{\age}{\ensuremath{z}}
\providecommand{\ALevBF}{\ensuremath{\mathbf{A}}}
\providecommand{\aLevBF}{\ensuremath{\mathbf{a}}}
\providecommand{\ALev}{\ensuremath{\pmb{A}}}
\providecommand{\aLev}{\ensuremath{\pmb{a}}}
\providecommand{\Alive}{\ensuremath{\cancel{\mathfrak{D}}}}
\providecommand{\Alt}{\grave}
\providecommand{\ARat}{\ensuremath{A}}
\providecommand{\aRat}{\ensuremath{a}}
\providecommand{\ASS}{\ensuremath{\ensuremath{A}}}
\providecommand{\aSS}{\ensuremath{\ensuremath{a}}}
\providecommand{\ATarg}{\ensuremath{\check{A}}}
\providecommand{\aTarg}{\ensuremath{\check{a}}}
\providecommand{\BE}{\BRat^{e}}
\providecommand{\bE}{\bRat^{e}}
\providecommand{\BLevBF}{\ensuremath{\mathbf{B}}}
\providecommand{\bLevBF}{\ensuremath{\mathbf{b}}}
\providecommand{\BLevE}{\ensuremath{\BLev^{e}}}
\providecommand{\BLevU}{\ensuremath{\BLev^{u}}}
\providecommand{\BLev}{\ensuremath{\pmb{B}}}
\providecommand{\bLev}{\ensuremath{\pmb{b}}}
\providecommand{\BRat}{\ensuremath{B}}
\providecommand{\bRat}{\ensuremath{b}}
\providecommand{\bRatE}{\ensuremath{b}^{e}}
\providecommand{\BRatE}{\ensuremath{B}^{e}}
\providecommand{\BTarg}{\ensuremath{\check{B}}}
\providecommand{\bTarg}{\ensuremath{\check{b}}}
\providecommand{\BU}{{B}^{u}}
\providecommand{\bU}{{b}^{u}}
\providecommand{\cAgg}{\ensuremath{\pmb{C}}}
\providecommand{\ccRat}{\ensuremath{\mathsf{c}}}
\providecommand{\CDF}{\ensuremath{\mathcal{F}}}
\providecommand{\CEndFunc}{\ensuremath{\mathfrak{C}}}
\providecommand{\cEndFunc}{\ensuremath{\mathfrak{c}}}
\providecommand{\cEss}{{c}^{e}}
\providecommand{\CE}{\CRat^{e}}
\providecommand{\cE}{\cRat^{e}}
\providecommand{\CFunc}{\ensuremath{\mathrm{C}}}
\providecommand{\cFunc}{\ensuremath{\mathrm{c}}}
\providecommand{\cFunc}{\ensuremath{\mathrm{c}}}
\providecommand{\cFuncBelow}{\ensuremath{\uline{\mathrm{c}}}}
\providecommand{\cFuncAbove}{\ensuremath{\bar{\mathrm{c}}}}
\providecommand{\cFuncMax}{\ensuremath{\bar{\bar{\mathrm{c}}}}}
\providecommand{\CGroOverG}{\ensuremath{\Upsilon}}
\providecommand{\CGroOverR}{\ensuremath{\Phi}}
\providecommand{\CGroPF}{\ensuremath{\Lambda}}
\providecommand{\cGroPF}{\ensuremath{\lambda}}
\providecommand{\CGroPS}{\ensuremath{\chi}}  % Precautionary Saving boost to consumption growth
\providecommand{\Chi}{\ensuremath{\mathrm{X}}} % capital chi is sometimes useful, and not native to LaTeX
\providecommand{\chiFunc}{\pmb{\chi}}
\providecommand{\CLevBF}{\ensuremath{\mathbf{C}}}
\providecommand{\cLevBF}{\ensuremath{\mathbf{c}}}
\providecommand{\CLevE}{\CLev^{e}}
\providecommand{\cLevE}{\cLev^{e}}
\providecommand{\cLevFunc}{\ensuremath{\pmb{\cFunc}}}
\providecommand{\CLevU}{\CLev^{u}}
\providecommand{\cLevU}{\cLev^{u}}
\providecommand{\CLev}{\ensuremath{\pmb{C}}}
\providecommand{\cLev}{\ensuremath{\pmb{c}}}
\providecommand{\Cons}{\ensuremath{C}}
\providecommand{\cons}{\ensuremath{c}}
\providecommand{\cPDVFunc}{\ensuremath{\mathbb{C}}}
\providecommand{\CPDV}{\ensuremath{\text{PDV($C$)}}}
\providecommand{\cPPP}{\cons^{\prime\prime\prime}}
\providecommand{\cPP}{\cons^{\prime\prime}}
\providecommand{\cP}{\cons^{\prime}}
\providecommand{\CRatE}{\CRat^{e}}
\providecommand{\cRatE}{\cRat^{e}}
\providecommand{\CRatU}{\CRat^{u}}
\providecommand{\cRatU}{\cRat^{u}}
\providecommand{\CRat}{\ensuremath{C}}
\providecommand{\cRat}{\ensuremath{c}}
\providecommand{\CARA}{\ensuremath{\alpha}}
\providecommand{\CRRA}{\ensuremath{\rho}}
\providecommand{\CTargE}{\CTarg^{\null}}
\providecommand{\CTarg}{\ensuremath{\Target{C}}}
\providecommand{\cTarg}{\ensuremath{\Target{c}}}
\providecommand{\cTargE}{\ensuremath{\Target{c}^{e}}}
\providecommand{\cTargTarg}{\Target{\Target{\cRat}}}
\providecommand{\corr}{\varrho} 
\providecommand{\curr}{1}  % Permits a change of notation to T-1 and T, t and t+1, or whatever
\providecommand{\Curr}{t}   % Permits a change of notation to T-1 and T, t and t+1, or whatever
\providecommand{\CU}{\CRat^{u}}
\providecommand{\cU}{\cRat^{u}}
\providecommand{\debtLim}{\mathsf{d}}
\providecommand{\Debt}{\ensuremath{D}}
\providecommand{\debt}{\ensuremath{d}}
\providecommand{\depr}{\ensuremath{\delta}}
\providecommand{\Discount}{\ensuremath{\beta}}
\providecommand{\DiscRate}{\ensuremath{\vartheta}}
\providecommand{\DiscAlt}{\ensuremath{\beth}}
\providecommand{\DiscAltuAdj}{\ensuremath{\underline{\underline{\beth}}}}
\providecommand{\Dvdnd}{\ensuremath{\mathbf{D}}}
\providecommand{\dvdnd}{\ensuremath{\mathbf{d}}}
\providecommand{\DivGro}{\ensuremath{\mathsf{G}}}
\providecommand{\divGro}{\ensuremath{\mathsf{g}}}
\providecommand{\Div}{\ensuremath{D}}
\providecommand{\div}{\ensuremath{d}}
\providecommand{\EEndMap}{\ensuremath{\mathsf{E}}}
\providecommand{\EpShkInv}{\Ex[\pShk^{-1}]}
\providecommand{\InvEpShkInv}{\acute{\psi}}
\providecommand{\effUnits}{\ensuremath{X}}
\providecommand{\eFunc}{\ensuremath{\mathrm{e}}}
\providecommand{\ek}{\ensuremath{\lambda}}
\providecommand{\empState}{\xi} % employment state indicator variable 
\providecommand{\EPrem}{\Phi} % equity premium
\providecommand{\eprem}{\phi} % log equity premium
\providecommand{\erate}{\ensuremath{\cancel{\mho}}}
\providecommand{\Err}{\ensuremath{Z}}
\providecommand{\err}{\ensuremath{z}}
\providecommand{\err}{\ensuremath{z}}
\providecommand{\error}{\ensuremath{\epsilon}}
\providecommand{\Estdr}{\ensuremath{\sigma_{\risky}}} % Standard deviation of log return on risky asset
\providecommand{\Evarr}{\ensuremath{\sigma_{\risky}^{2}}} % Variance of log return on risky asset
\providecommand{\EVarr}{\sigma_{\Risky}^{2}} % Variance of level return on risky asset (when returns norm dist)
\providecommand{\expend}{\ensuremath{\xi}}
\providecommand{\xpend}{\ensuremath{\xi}}
\providecommand{\Ex}{\ensuremath{\mathbb{E}}} % Expectations
\providecommand{\FFunc}{\ensuremath{\mathrm{F}}}
\providecommand{\fFunc}{\ensuremath{\mathrm{f}}}
\providecommand{\FDist}{\ensuremath{\mathcal{F}}}
\providecommand{\fDist}{\ensuremath{\mathcal{f}}}
\providecommand{\FLev}{\ensuremath{\pmb{F}}}
\providecommand{\fLev}{\ensuremath{\pmb{f}}}
\providecommand{\fP}{\ensuremath{\mathrm{f}^{\prime}}}
\providecommand{\fPP}{\ensuremath{\mathrm{f}^{\prime\prime}}}
\providecommand{\FP}{\ensuremath{\mathrm{F}^{\prime}}}
\providecommand{\FPP}{\ensuremath{\mathrm{F}^{\prime\prime}}}
\providecommand{\GovNW}{\ensuremath{N}}
\providecommand{\govNW}{\ensuremath{n}}
\providecommand{\GovSpend}{\ensuremath{X}}
\providecommand{\govSpend}{\ensuremath{x}}
\providecommand{\h}{\ensuremath{h}}
\providecommand{\hFunc}{\ensuremath{\mathrm{h}}}
\providecommand{\H}{\ensuremath{H}}
\providecommand{\Hi}{\hat}
\providecommand{\Habit}{\ensuremath{H}}
\providecommand{\Habit}{\ensuremath{H}}
\providecommand{\habit}{\ensuremath{h}}
\providecommand{\Ham}{\ensuremath{\mathcal{H}}} % Hamiltonian
\providecommand{\HLev}{\ensuremath{\pmb{H}}}
\providecommand{\hLev}{\ensuremath{\pmb{h}}}
\providecommand{\HRat}{\ensuremath{H}}
\providecommand{\hRat}{\ensuremath{h}}
\providecommand{\hEnd}{\ensuremath{\mathfrak{h}}}
\providecommand{\hEndMin}{\ensuremath{\ushort{\mathfrak{h}}}}
\providecommand{\hMin}{\ensuremath{\ushort{\h}}}
\providecommand{\ILev}{\ensuremath{\pmb{I}}}
\providecommand{\iLev}{\ensuremath{\pmb{i}}}
\providecommand{\impg}{{\imath}_{\pGro}}
\providecommand{\ImpG}{{\Im}_{\PGro}}
\providecommand{\impr}{{\imath}_{\rfree}}
\providecommand{\ImpR}{{\Im}_{\Rfree}}
\providecommand{\Inc}{{Y}}
\providecommand{\inc}{{y}}
\providecommand{\Inv}{{I}}
\providecommand{\inv}{{i}}
\providecommand{\IRat}{\ensuremath{I}}
\providecommand{\iRat}{\ensuremath{i}}
\providecommand{\itc}{\ensuremath{\zeta}} 
\providecommand{\PostITC}{\ensuremath{\cancel{\zeta}}} 
\providecommand{\jFunc}{\ensuremath{\mathrm{j}}}
\providecommand{\kapRent}{\ensuremath{\varkappa}}
\providecommand{\kapShare}{\ensuremath{\alpha}}
\providecommand{\Kap}{{K}}
\providecommand{\kap}{{k}}
\providecommand{\KLevBF}{\ensuremath{\mathbf{K}}}
\providecommand{\kLevBF}{\ensuremath{\mathbf{k}}}
\providecommand{\KLev}{\ensuremath{\pmb{K}}}
\providecommand{\kLev}{\ensuremath{\pmb{k}}}
\providecommand{\KFunc}{\ensuremath{\mathrm{K}}}
\providecommand{\kFunc}{\ensuremath{\mathrm{k}}}
\providecommand{\KRat}{\ensuremath{K}}
\providecommand{\kRat}{\ensuremath{k}}
\providecommand{\kTarg}{\ensuremath{\Target{k}}}
\providecommand{\kTargE}{\ensuremath{\Target{k}^{e}}}
\providecommand{\Labor}{\ensuremath{L}}
\providecommand{\labor}{\ensuremath{\ell}} % 
\providecommand{\labShare}{\ensuremath{\nu}}
\providecommand{\Leisure}{Z} % 
\providecommand{\leisure}{z} % 
\providecommand{\LGro}{\ensuremath{\Lambda}}
\providecommand{\lGro}{\ensuremath{\lambda}}
\providecommand{\LLevBF}{\ensuremath{\mathbf{L}}}
\providecommand{\lLevBF}{\ensuremath{\pmb{\ell}}}
\providecommand{\lLev}{\ensuremath{\ell}}
\providecommand{\LLev}{\ensuremath{\pmb{L}}}
\providecommand{\LLev}{\pmb{\ensuremath{L}}}
\providecommand{\LRat}{\ensuremath{L}}
\providecommand{\Lo}{\check}
\providecommand{\MaxMaxMPC}{\ensuremath{\bar{\bar{\kappa}}}}
\providecommand{\MaxMinMPC}{\ensuremath{\hat{\underline{\kappa}}}}
\providecommand{\MinMinMPC}{\ensuremath{\ushort{\kappa}}}
\providecommand{\MaxMPC}{\ensuremath{\bar{\kappa}}}
\providecommand{\MaxMPS}{\ensuremath{\PatR}}
\providecommand{\Mean}{\ensuremath{\mathbb{M}}} % Mean
\providecommand{\m}{\ensuremath{m}}
\providecommand{\mEss}{\check{m}^{e}}
\providecommand{\ME}{\MRat^{e}}
\providecommand{\mE}{\mRat^{e}}
\providecommand{\MRatE}{\MRat^{e}}
\providecommand{\mRatE}{\mRat^{e}}
\providecommand{\mFunc}{\ensuremath{\mathrm{m}}}
\providecommand{\MinMPC}{\ensuremath{\uline{\kappa}}}
\providecommand{\MinMPS}{\ensuremath{\pZero^{1/\CRRA} \PatR}}
\providecommand{\MLevBF}{\ensuremath{\mathbf{M}}}
\providecommand{\mLevBF}{\ensuremath{\mathbf{m}}}
\providecommand{\mLevBF}{\ensuremath{\mathbf{m}}}
\providecommand{\mLevE}{\mLev^{e}}
\providecommand{\MLev}{\ensuremath{\pmb{M}}}
\providecommand{\mLev}{\ensuremath{\pmb{m}}}
\providecommand{\MPCE}{\MPC^{e}}
\providecommand{\MPCFunc}{\ensuremath{\pmb{\kappa}}}
\providecommand{\MPCPPF}{\ensuremath{\Pi}}
\providecommand{\MPCP}{\ensuremath{\pi}}
\providecommand{\MPCU}{\MPC^{u}}
\providecommand{\MPC}{\ensuremath{\kappa}}
\providecommand{\MPSFunc}{\ensuremath{\pmb{lambda}}}
\providecommand{\MPS}{\ensuremath{\lambda}}
\providecommand{\MRat}{\ensuremath{M}}
\providecommand{\mRat}{\ensuremath{m}}
\providecommand{\MSS}{\ensuremath{\breve{M}}}
\providecommand{\mSS}{\ensuremath{\breve{m}}}
\providecommand{\mTarg}{\check{m}}
\providecommand{\MU}{{M}^{u}}
\providecommand{\mU}{{m}^{u}}
\providecommand{\next}{2}  % via search-and-replace
\providecommand{\Next}{t+1} % 
\providecommand{\nFunc}{\ensuremath{\mathrm{n}}}
\providecommand{\NLev}{\ensuremath{\pmb{N}}}
\providecommand{\nLev}{\ensuremath{\pmb{n}}}
\providecommand{\NRat}{\ensuremath{N}}
\providecommand{\nRat}{\ensuremath{n}}
\providecommand{\n}{\ensuremath{n}}
\providecommand{\OLevBF}{\ensuremath{\mathbf{O}}}
\providecommand{\oLevBF}{\ensuremath{\mathbf{o}}}
\providecommand{\OLev}{\ensuremath{\pmb{O}}}
\providecommand{\oLev}{\ensuremath{\pmb{o}}}
\providecommand{\ORat}{\ensuremath{O}}
\providecommand{\oRat}{\ensuremath{o}}
\providecommand{\PatGAdj}{\text{\pmb{\Thorn}}_{\acute{\PGro}}}
\providecommand{\patgAdj}{\text{\thorn}_{\acute{\pGro}}}
\providecommand{\PatGG}{\text{\pmb{\Thorn}}_{\WGro}}
\providecommand{\patgg}{\text{\thorn}_{\wGro}}
\providecommand{\patghat}{\hat{\text{\thorn}}_{\pGro}}
\providecommand{\PatG}{\text{\pmb{\Thorn}}_{\PGro}}
\providecommand{\patg}{\text{\thorn}_{\pGro}}
\providecommand{\PatR}{\text{\pmb{\Thorn}}_{\Rfree}}
\providecommand{\patr}{\text{\thorn}_{\rfree}}
\providecommand{\Pat}{\text{\pmb{\Thorn}}}
\providecommand{\pat}{\text{\thorn}}
\providecommand{\pDeadRate}{\ensuremath{\grave{\cancel{\mathsf{d}}}}}
\providecommand{\pDead}{\ensuremath{\mathfrak{D}}}
\providecommand{\pDieRate}{\ensuremath{\grave{\mathsf{d}}}}
\providecommand{\PDies}{\ensuremath{\mathsf{D}}}
\providecommand{\pDies}{\ensuremath{\mathsf{d}}}
\providecommand{\PDV}{\ensuremath{\mathbb{P}}} % PDV
\providecommand{\PGroAdj}{\ensuremath{\underline{\PGro}}}
\providecommand{\pGroAdj}{\ensuremath{\underline{\pGro}}}
\providecommand{\PGrouAdj}{\ensuremath{\underline{\underline{\PGro}}}}
\providecommand{\pGrouAdj}{\ensuremath{\underline{\underline{{\pGro}}}}}
\providecommand{\PGro}{\ensuremath{\Gamma}}
\providecommand{\pGro}{\ensuremath{\gamma}}
\providecommand{\phiFunc}{\ensuremath{\digamma}}
\providecommand{\PIHMPC}{\ensuremath{\varkappa}}
\providecommand{\PInc}{\ensuremath{P}}
\providecommand{\Plabor}{P} % Permanent labor income in levels
\providecommand{\PLabor}{\ensuremath{P}} % Permanent labor income in levels
\providecommand{\PLevBF}{\ensuremath{\pmb{P}}}
\providecommand{\pLevBF}{\ensuremath{\pmb{p}}}
\providecommand{\PLev}{\ensuremath{\mathbf{P}}}
\providecommand{\pLev}{\ensuremath{\pmb{p}}}
\providecommand{\pRat}{\ensuremath{p}}
\providecommand{\PLives}{\ensuremath{\cancel{\PDies}}}
\providecommand{\pLives}{\ensuremath{\cancel{\pDies}}}
\providecommand{\pNotZero}{\ensuremath{\cancel{\wp}}}
\providecommand{\pSav}{\ensuremath{\phi}}
\providecommand{\PopE}{\mathcal{E}}
\providecommand{\popE}{e}
\providecommand{\EmpGro}{\ensuremath{\Xi}}
\providecommand{\empGro}{\ensuremath{\xi}}
\providecommand{\PopGro}{\ensuremath{\Xi}}
\providecommand{\popGro}{\ensuremath{\xi}}
\providecommand{\PopLev}{\ensuremath{\pmb{N}}}
\providecommand{\PopU}{\mathcal{U}}
\providecommand{\Pop}{\ensuremath{L}}
\providecommand{\power}{\ensuremath{\eta}}
\providecommand{\pPDVFunc}{\ensuremath{\mathbb{P}}}
\providecommand{\PPDV}{\ensuremath{\text{PDV($P$)}}}
\providecommand{\Price}{\ensuremath{\mathsf{P}}}
\providecommand{\kPriceAfterITC}{\ensuremath{\mathscr{P}}}
\providecommand{\ProdFunc}{\ensuremath{\mathrm{F}}}
\providecommand{\ProdFunc}{\ensuremath{\mathrm{F}}}
\providecommand{\prodFunc}{\ensuremath{\mathrm{f}}}
\providecommand{\prudEx}{\ensuremath{\omega}}
\providecommand{\prud}{\ensuremath{\eta}}
\providecommand{\PShk}{\Psi} % 
\providecommand{\pShk}{\psi} % 
\providecommand{\pShkMin}{\ushort{\psi}} % 
\providecommand{\pshk}{\psi} % 
\providecommand{\PtyGro}{\ensuremath{\Phi}}
\providecommand{\ptyGro}{\ensuremath{\phi}}
\providecommand{\PtyLev}{\ensuremath{\mathrm{A}}}
\providecommand{\ptyLev}{\ensuremath{a}}
\providecommand{\PtyLab}{\ensuremath{\mathrm{Z}}}
\providecommand{\ptyLab}{\ensuremath{z}}
\providecommand{\pZero}{\ensuremath{\wp}}
\providecommand{\q}{\ensuremath{\koppa}}
\providecommand{\RevFunc}{\ensuremath{\pmb{\Pi}}}
\providecommand{\revFunc}{\ensuremath{\pmb{\pi}}}
\providecommand{\Rev}{\ensuremath{\Pi}}
\providecommand{\rev}{\ensuremath{\pi}}
\providecommand{\rfree}{\ensuremath{\mathsf{r}}}   % The net return on the safe asset at an annual rate
\providecommand{\Rfree}{\ensuremath{\mathsf{R}}}   % The return factor on the safe asset - unfortunately mathfrak fonts don't come through for tth 
\providecommand{\R}{\Rfree}
\providecommand{\RFunc}{\ensuremath{\mathrm{R}}}
\providecommand{\RGross}{\ensuremath{\breve{\mathsf{R}}}}
\providecommand{\rGross}{\ensuremath{\breve{\mathsf{r}}}}
\providecommand{\riskyshare}{\ensuremath{\varsigma}}
\providecommand{\Risky}{\ensuremath{\mathbf{R}}}    % The return on the risky asset
\providecommand{\risky}{\ensuremath{\mathbf{r}}}  % The net return on the risky asset annual rate
\providecommand{\Rnorm}{\ensuremath{\mathcal{R}}}    % Normalized version of riskless return factor
\providecommand{\rnorm}{\ensuremath{\mathit{r}}}    % Normalized version of riskless rate of return
\providecommand{\Rport}{\ensuremath{\pmb{\mathfrak{R}}}}    % Portfolio -weighted return
\providecommand{\rport}{\ensuremath{\pmb{\mathfrak{r}}}}    % Portfolio -weighted return
\providecommand{\Rprod}{\ensuremath{\mathscr{R}}}
\providecommand{\rprod}{\ensuremath{\mathscr{r}}}
\providecommand{\RProd}{\ensuremath{\mathsf{R}}}
\providecommand{\rProd}{\ensuremath{\mathit{r}}}
\providecommand{\RCpnd}{\ensuremath{\mathbf{R}}}
\providecommand{\Save}{S} % Saving (income minus consumption) 
\providecommand{\save}{s} % saving (income minus consumption) 
\providecommand{\Seniority}{\ensuremath{\mathsf{X}}}
\providecommand{\seniority}{\ensuremath{\mathsf{x}}}
\providecommand{\SE}{\SRat^{e}}
\providecommand{\sE}{\sRat^{e}}
\providecommand{\sFunc}{\ensuremath{\mathrm{s}}}
\providecommand{\SLevE}{\SLev^{e}}
\providecommand{\sLevE}{\sLev^{e}}
\providecommand{\SLevU}{\SLev^{u}}
\providecommand{\sLevU}{\sLev^{u}}
\providecommand{\SLev}{\ensuremath{\pmb{S}}}
\providecommand{\sLev}{\ensuremath{\pmb{s}}}
\providecommand{\SRatE}{\SRat^{e}}
\providecommand{\sRatE}{\sRat^{e}}
\providecommand{\SRatU}{\SRat^{u}}
\providecommand{\sRatU}{\sRat^{u}}
\providecommand{\SRat}{\ensuremath{S}}
\providecommand{\sRat}{\ensuremath{s}}
\providecommand{\sRat}{\ensuremath{s}}
\providecommand{\STargE}{\Target{\SRat}^{\null}}
\providecommand{\sTargE}{\Target{\sRat}^{\null}}
\providecommand{\STarg}{\Target{\SRat}}
\providecommand{\sTarg}{\Target{\sRat}}
\providecommand{\STargTarg}{\Target{\Target{\SRat}}}
\providecommand{\sTargTarg}{\Target{\Target{\sRat}}}
\providecommand{\Stocks}{{S}}
\providecommand{\stocks}{{s}}
\providecommand{\straight}{\Pi}
\providecommand{\Surplus}{\ensuremath{Z}}
\providecommand{\surplus}{\ensuremath{z}}
\providecommand{\SU}{\SRat^{u}}
\providecommand{\sU}{\sRat^{u}}
\providecommand{\Target}{\check}
\providecommand{\TaxCorp}{\ensuremath{\tau}}
\providecommand{\TaxFree}{\ensuremath{\cancel{\Tax}}}
\providecommand{\TaxLev}{T}
\providecommand{\TaxPaid}{\ensuremath{T}}
\providecommand{\TaxRate}{\ensuremath{t}}
\providecommand{\TaxNetTrans}{\ensuremath{Z}}
\providecommand{\taxNetTrans}{\ensuremath{z}}
\providecommand{\tax}{t}
\providecommand{\taxDep}{\ensuremath{\partial}}
\providecommand{\Tax}{\ensuremath{\tau}}
\providecommand{\TaxComb}{\ensuremath{\mathcal{T}}}
\providecommand{\TaxCombInv}{\ensuremath{\mathcal{T}^{-1}}}
\providecommand{\tBak}{\ensuremath{\pmb{n}}}
\providecommand{\TEatBak}{\ensuremath{\mathtt{q}}}
\providecommand{\TEat}{\ensuremath{\TEnd}}
\providecommand{\TEndBak}{\ensuremath{\mathsf{p}}}
\providecommand{\TEnd}{\ensuremath{T}}
\providecommand{\TermTime}{\ensuremath{T}}
\providecommand{\tFwd}{\ensuremath{n}}
\providecommand{\tHorOfm}{\ensuremath{\pmb{n}}}
\providecommand{\tHor}{\ensuremath{\mathsf{n}}}
\providecommand{\timeRate}{\ensuremath{\vartheta}}
\providecommand{\tinyAmount}{\ensuremath{\epsilon}}
\providecommand{\hours}{\ensuremath{\mathfrak{h}}}
\providecommand{\Hours}{\ensuremath{\mathfrak{H}}}
\providecommand{\TMap}{\mathscr{T}}
\providecommand{\tNow}{\ensuremath{t}}
\providecommand{\tShkAll}{\xi} % 
\providecommand{\TShkEmp}{\Theta} % 
\providecommand{\tShkEmp}{\theta} % 
\providecommand{\tShkEmpMin}{\ushort{\theta}} % 
\providecommand{\tshkEmp}{\theta} % 
\providecommand{\TShk}{\Xi} % 
\providecommand{\tShk}{\xi} % 
\providecommand{\tshk}{\xi} % 
\providecommand{\tSS}{\ensuremath{t}}
\providecommand{\tThen}{\ensuremath{\tau}}
\providecommand{\DeprFac}{\ensuremath{\daleth}}
\providecommand{\Depr}{\ensuremath{\daleth}}
\providecommand{\unins}{\ensuremath{\zeta}}
\providecommand{\Severance}{\ensuremath{\varsigma}}
\providecommand{\uPDVFunc}{\ensuremath{\mathbb{U}}}
\providecommand{\uPPP}{{\ensuremath{\mathrm{u}^{\prime\prime\prime}}}}
\providecommand{\uPP}{{\ensuremath{\mathrm{u}^{\prime\prime}}}}
\providecommand{\uP}{{\ensuremath{\mathrm{u}^{\prime}}}}
\providecommand{\urate}{\ensuremath{\mho}}
\providecommand{\util}{{\ensuremath{\mathrm{u}}}}
\providecommand{\InvEpShkInv}{\underline{\psi}}
\providecommand{\uInvEpShkuInv}{\underline{\underline{\psi}}}
\providecommand{\ValAlt}{\ensuremath{\mathcal{V}}} % middle-of-period Value function
\providecommand{\valfn}{\ensuremath{\mathrm{v}}} % middle-of-period value function
\providecommand{\vk}{\ensuremath{\lambda}}
\providecommand{\Value}{\ensuremath{\mathrm{V}}} % middle-of-period Value function
\providecommand{\VEndFunc}{\ensuremath{\mathfrak{V}}}
\providecommand{\vEndFunc}{\ensuremath{\mathfrak{v}}}
\providecommand{\VEnd}{\ensuremath{\mathfrak{V}}} % end-of-period Value function
\providecommand{\vEnd}{\ensuremath{\mathfrak{v}}} % end-of-period value function
\providecommand{\vEss}{\check{v}^{e}}
\providecommand{\VE}{{V}^{e}}
\providecommand{\vE}{{v}^{e}}
\providecommand{\vFirm}{\ensuremath{\mathrm{e}}}
\providecommand{\VFunc}{\ensuremath{\mathrm{V}}}
\providecommand{\vFunc}{\ensuremath{\mathrm{v}}}
\providecommand{\vLevBF}{\ensuremath{\mathbf{v}}}
\providecommand{\VLevFunc}{\ensuremath{\pmb{\mathrm{V}}}}
\providecommand{\vLevFunc}{\ensuremath{\pmb{\mathrm{v}}}}
\providecommand{\VLev}{\ensuremath{\pmb{V}}}
\providecommand{\vLev}{\ensuremath{\pmb{v}}}
\providecommand{\vNorm}{\ensuremath{\mathrm{w}}} % end-of-period value function
\providecommand{\VRat}{\ensuremath{V}}
\providecommand{\vRat}{\ensuremath{v}}
\providecommand{\VNum}{\ensuremath{V}}
\providecommand{\vNum}{\ensuremath{v}}
\providecommand{\vTarg}{\Target{\vRat}}
\providecommand{\VU}{{V}^{u}}
\providecommand{\vU}{{v}^{u}}
\providecommand{\vInv}{\ensuremath{\scriptstyle \Lambda \displaystyle}}
\providecommand{\VInv}{\ensuremath{\Lambda}}
\providecommand{\Wage}{\ensuremath{\mathsf{W}}}
\providecommand{\wage}{\ensuremath{\mathsf{w}}}
\providecommand{\WAllLev}{\ensuremath{\mathbf{O}}}
\providecommand{\wAllLev}{\ensuremath{\mathbf{o}}}
\providecommand{\WAllRat}{\ensuremath{O}}
\providecommand{\wAllRat}{\ensuremath{o}}
\providecommand{\WAll}{{O}}
\providecommand{\wAll}{{o}}
\providecommand{\kPrice}{\ensuremath{\mathsf{P}}}
\providecommand{\WBeg}{\KLev} % Wealth as of the beginning of the period (before R is received, not including Y)
\providecommand{\wBeg}{\kLev} % wealth as of the beginning of the period (before R is received, not including Y)
\providecommand{\Wealth}{\ensuremath{O}}
\providecommand{\wealth}{\ensuremath{o}}
\providecommand{\WEnd}{\ALev} % Wealth as of the end of the period (after C has been chosen)
\providecommand{\wEnd}{\aLev} % wealth as of the end of the period (after C has been taken)
\providecommand{\Wend}{\ARat} % Wealth as of the end of the period (after C has been chosen)
\providecommand{\wend}{\aRat} % wealth as of the end of the period (after C has been taken)
\providecommand{\WEndRat}{\ARat} % 
\providecommand{\wEndRat}{\aRat} % 
\providecommand{\wFunc}{\mathrm{w}}
\providecommand{\WGro}{\ensuremath{\mathsf{G}}}
\providecommand{\WGroPF}{\ensuremath{\mathsf{G}}}
\providecommand{\wGro}{\ensuremath{\mathsf{g}}}
\providecommand{\WHum}{\HLev} % Human wealth -- aggregate 
\providecommand{\wHum}{\hLev} % human wealth -- individual
\providecommand{\Whum}{\HRat} % Human wealth -- aggregate 
\providecommand{\whum}{\hRat} % human wealth -- individual
\providecommand{\whumMin}{\ushort{\hRat}} % human wealth -- individual
\providecommand{\wLev}{\pmb{w}}
\providecommand{\WMid}{\BLev} % Wealth as of the middle of the period (after R is received, not including Y)
\providecommand{\WMid}{\BLev} % Wealth as of the middle of the period (after R is received, not including Y)
\providecommand{\Wmid}{\BRat} % Wealth as of the middle of the period (after R is received, not including Y)
\providecommand{\wMid}{\bLev} % wealth as of the middle of the period (after R is received, not including Y)
\providecommand{\wmid}{\bRat} % wealth as of the middle of the period (after R is received, not including Y)
\providecommand{\WMkt}{\MLev} % Wealth as of the middle of the period (after R is received, including Y)
\providecommand{\wMkt}{\mLev} % wealth as of the middle of the period (after R is received, including Y)
\providecommand{\wmkt}{\mLev} % wealth as of the middle of the period (after R is received, including Y)
\providecommand{\WPre}{{K}}
\providecommand{\wPre}{{k}}
\providecommand{\wTot}{\ensuremath{\mathbf{o}}} % 
\providecommand{\WTot}{\ensuremath{\mathbf{O}}} % Total wealth
\providecommand{\wtot}{\ensuremath{o}} % 
\providecommand{\Wtot}{\ensuremath{O}} % Total wealth
\providecommand{\wRat}{\ensuremath{o}} % 
\providecommand{\WRat}{\ensuremath{O}} % Ratio to permanent income
\providecommand{\wNet}{\ensuremath{x}} % 
\providecommand{\WNet}{\ensuremath{X}} % Total wealth
\providecommand{\XFer}{X}
\providecommand{\xFer}{\chi}
\providecommand{\xFunc}{\mathrm{x}}
\providecommand{\XLev}{\ensuremath{\pmb{X}}}
\providecommand{\xLev}{\ensuremath{\pmb{x}}}
\providecommand{\XperGro}{\ensuremath{\mathsf{X}}}
\providecommand{\xperGro}{\ensuremath{\mathsf{x}}}
\providecommand{\XRat}{\ensuremath{X}}
\providecommand{\xRat}{\ensuremath{x}}
\providecommand{\yFunc}{\mathrm{y}}
\providecommand{\yLevBF}{\ensuremath{\mathbf{y}}}
\providecommand{\YLev}{\ensuremath{\pmb{Y}}}
\providecommand{\yLev}{\ensuremath{\pmb{y}}}
\providecommand{\YRat}{\ensuremath{Y}}
\providecommand{\yRat}{\ensuremath{y}}
\providecommand{\yTarg}{\ensuremath{\check{y}}}
\providecommand{\yTargE}{\ensuremath{\check{y}^{e}}}
\providecommand{\zAgg}{\ensuremath{\pmb{Z}}}
\providecommand{\zFunc}{\mathrm{z}}
\providecommand{\ZLevBF}{\ensuremath{\mathbf{Z}}}
\providecommand{\zLevBF}{\ensuremath{\mathbf{z}}}
\providecommand{\ZLev}{\ensuremath{\pmb{Z}}}
\providecommand{\zLev}{\ensuremath{\pmb{z}}}
\providecommand{\ZRat}{\ensuremath{Z}}
\providecommand{\zRat}{\ensuremath{z}}
\providecommand{\vOpt}{\ensuremath{\mathfrak{w}}}

\providecommand{\NFALev}{\NLev}
\providecommand{\NFARat}{\NRat}
\providecommand{\NI}{\ensuremath{Z}}
\providecommand{\GDPLev}{\pmb{P}}
\providecommand{\GDPRat}{P}
\providecommand{\GDPGro}{\gimel}
\providecommand{\gdpLev}{\pmb{p}}
\providecommand{\gdpRat}{p}
\providecommand{\weight}{\omega}

\providecommand{\bi}{}
\renewcommand{\bi}{\begin{itemize}}
\providecommand{\ei}{}
\renewcommand{\ei}{\end{itemize}}
\providecommand{\reqd}{}
\renewcommand{\reqd}{\item[$^{*}$]}
\providecommand{\recm}{}
\renewcommand{\recm}{\item[\phantom{$^{*}$}]}
