From 12c8b6ffdde95ce33b55b33cefc7de1c301f27b8 Mon Sep 17 00:00:00 2001 From: dedwar65 Date: Thu, 26 Oct 2023 07:39:04 -0400 Subject: [PATCH 1/2] Notes on modeling bequests --- .DS_Store | Bin 0 -> 6148 bytes Notes/.#math.tex | 1 + Notes/.DS_Store | Bin 0 -> 6148 bytes Notes/BeqRefs.bib | 489 +++++++++++++++++++ Notes/Equations/#DeNardi_2004# | 1 + Notes/Equations/Abel_1985 | 8 + Notes/Equations/Barro_1974 | 9 + Notes/Equations/Cagetti_2003 | 8 + Notes/Equations/Cagetti_2006 | 7 + Notes/Equations/Cagetti_2008 | 9 + Notes/Equations/Carroll_1998 | 6 + Notes/Equations/DeNardi_2004 | 8 + Notes/Equations/DeNardi_2016 | 7 + Notes/Equations/Dynan_2004 | 7 + Notes/Equations/Gourinchas_2002 | 8 + Notes/Equations/Saez_2017 | 7 + Notes/Equations/Straub_2019 | 5 + Notes/auto/BeqRefs.el | 25 + Notes/auto/bequests_lit_rev.el | 42 ++ Notes/bequests_lit_rev-blx.bib | 11 + Notes/bequests_lit_rev.aux | 142 ++++++ Notes/bequests_lit_rev.bbl | 798 ++++++++++++++++++++++++++++++++ Notes/bequests_lit_rev.blg | 70 +++ Notes/bequests_lit_rev.out | 15 + Notes/bequests_lit_rev.pdf | Bin 0 -> 240238 bytes Notes/bequests_lit_rev.run.xml | 92 ++++ Notes/bequests_lit_rev.tex | 110 +++++ Paper/#math.tex# | 179 +++++++ Paper/.DS_Store | Bin 0 -> 6148 bytes Paper/auto/BeqRefs.el | 25 + Paper/auto/bequests_lit_rev.el | 28 ++ 31 files changed, 2117 insertions(+) create mode 100644 .DS_Store create mode 120000 Notes/.#math.tex create mode 100644 Notes/.DS_Store create mode 100755 Notes/BeqRefs.bib create mode 100644 Notes/Equations/#DeNardi_2004# create mode 100644 Notes/Equations/Abel_1985 create mode 100644 Notes/Equations/Barro_1974 create mode 100644 Notes/Equations/Cagetti_2003 create mode 100644 Notes/Equations/Cagetti_2006 create mode 100644 Notes/Equations/Cagetti_2008 create mode 100644 Notes/Equations/Carroll_1998 create mode 100644 Notes/Equations/DeNardi_2004 create mode 100644 Notes/Equations/DeNardi_2016 create mode 100644 Notes/Equations/Dynan_2004 create mode 100644 Notes/Equations/Gourinchas_2002 create mode 100644 Notes/Equations/Saez_2017 create mode 100644 Notes/Equations/Straub_2019 create mode 100644 Notes/auto/BeqRefs.el create mode 100644 Notes/auto/bequests_lit_rev.el create mode 100644 Notes/bequests_lit_rev-blx.bib create mode 100644 Notes/bequests_lit_rev.aux create mode 100644 Notes/bequests_lit_rev.bbl create mode 100644 Notes/bequests_lit_rev.blg create mode 100644 Notes/bequests_lit_rev.out create mode 100644 Notes/bequests_lit_rev.pdf create mode 100644 Notes/bequests_lit_rev.run.xml create mode 100644 Notes/bequests_lit_rev.tex create mode 100644 Paper/#math.tex# create mode 100644 Paper/.DS_Store create mode 100644 Paper/auto/BeqRefs.el create mode 100644 Paper/auto/bequests_lit_rev.el diff --git a/.DS_Store b/.DS_Store new file mode 100644 index 0000000000000000000000000000000000000000..861d0434faca505198e52dc3db55a254503ce75a GIT binary patch literal 6148 zcmeHKOHRWu5S@X7B6ZUxOJAWk2vz73A@u@~1_5d!N3`s73=YGVd-29Ipjbp=hYFgJ z>^Gi|KTnFCAtGMAG)tld5fx~HEXst4dD3-a!RJ6ubL^Y_wyk4qC#E@0lkEMBt|`)i z?kU^<$lN|GH{00tqce=WU976QjWxW*+wN)odU(Igr|n|%Lqz{LvBC}AQIBms|2bMm zM7in=I0MdrGjMDS;LR2(=8E1s1I~am@Wp`K4*^ZEG#nJu)qyUR0Kgf{NuWzDAvwXY zG#nJ+fv~0mHI=QzU`>ZTSX^m1C~7*f6(4LXe-%Ayg%g*G-a7-%K+eFa4j1zN zU*VIPEb{XdA2|cgz&~SvXVs=!;ZXLremkDLwE^uGO+?}bQ6SK>p8yQx9=S+P^#{=r YR~inAGK<(#I?#Uv5+U9>1HZt)2f3a#9{>OV literal 0 HcmV?d00001 diff --git a/Notes/.#math.tex b/Notes/.#math.tex new file mode 120000 index 0000000..0464b44 --- /dev/null +++ b/Notes/.#math.tex @@ -0,0 +1 @@ +dc@Decorys-MacBook-Pro.local.3903 \ No newline at end of file diff --git a/Notes/.DS_Store b/Notes/.DS_Store new file mode 100644 index 0000000000000000000000000000000000000000..9d80304f858abd067e85d5b45fbfafd1f9b79436 GIT binary patch literal 6148 zcmeHKOHRWu5S@V{1$9$)EPaLCAg$69^a7AZ0;B{NwcY0+y$lPk!Z~>38Bn5Hf)zq& zMzY`7^RfM;*cl??)v+y!7DQB_39={|k?E}I!kn*woN^rKi6TAIzTIaM{lzKS`!%iU zg^qMjqy2BYCdLlO%9*=|a 0$, the marginal utility of small bequests is bounded. \ No newline at end of file diff --git a/Notes/Equations/Dynan_2004 b/Notes/Equations/Dynan_2004 new file mode 100644 index 0000000..bd5f236 --- /dev/null +++ b/Notes/Equations/Dynan_2004 @@ -0,0 +1,7 @@ +\begin{equation} +\begin{aligned} +V(B_{is}) = \frac{\mu [(B_{is} + YL_{is}^{c})^{1-\gamma} -1]}{1-\gamma} \\ +\end{aligned} +\end{equation} + +where $\mu$ is the trade-off parameter between own consumption and bequests $B_{is}$ and $YL^{c}_{is}$ is the value of the next generation's lifetime earnings. diff --git a/Notes/Equations/Gourinchas_2002 b/Notes/Equations/Gourinchas_2002 new file mode 100644 index 0000000..5a9ba4b --- /dev/null +++ b/Notes/Equations/Gourinchas_2002 @@ -0,0 +1,8 @@ +\begin{equation} +\begin{aligned} +V_{\tau}(X_{\tau}, P_{\tau}, Z_{\tau}) \quad & = \max_{c_{\tau}, \ldots, c_{T}} \bigg[ \sum_{t=\tau}^{T} \beta^{t-\tau} v(Z_t) \frac{C_{t}^{1-\rho}}{1-\rho} + \beta^{T+1-\tau} \kappa v(Z_{T+1}) (X_{T+1} + h P_{T+1})^{1-\rho} \bigg] \\ +\text{s.t.} \quad & X_{t+1} = R (X_t - C_t) + Y_{t+1} \\ +\end{aligned} +\end{equation} + +where $X_{T+1}$ is cash on hand and $H_{T+1}$ is illiquid wealth in the first year of retirement. \ No newline at end of file diff --git a/Notes/Equations/Saez_2017 b/Notes/Equations/Saez_2017 new file mode 100644 index 0000000..444e636 --- /dev/null +++ b/Notes/Equations/Saez_2017 @@ -0,0 +1,7 @@ +\begin{equation} +\begin{aligned} +U_i(c_i, k_i, z_i) = c_i + a_i(k_i) - h_i(z_i) + \delta_i(k_{i}^{init} - k_i) +\end{aligned} +\end{equation} + +where utility for may be specified as $$ a_i(k) = \eta_i \frac{k_i^{\gamma}}{\gamma} $$. \ No newline at end of file diff --git a/Notes/Equations/Straub_2019 b/Notes/Equations/Straub_2019 new file mode 100644 index 0000000..336ee5b --- /dev/null +++ b/Notes/Equations/Straub_2019 @@ -0,0 +1,5 @@ +\begin{equation} +\begin{aligned} +U(a) = \frac{(\frac{a}{o})^{1-\Sigma} - 1}{1 - \Sigma} +\end{aligned} +\end{equation} diff --git a/Notes/auto/BeqRefs.el b/Notes/auto/BeqRefs.el new file mode 100644 index 0000000..63a6c7f --- /dev/null +++ b/Notes/auto/BeqRefs.el @@ -0,0 +1,25 @@ +(TeX-add-style-hook + "BeqRefs" + (lambda () + (LaTeX-add-bibitems + "Laitner2002-hu" + "Gokhale2001-dj" + "Abel1985-ef" + "Barro1974-mp" + "Saez2018-we" + "Auclert2023-ot" + "De_Nardi2010-tt" + "Michaillat2021-po" + "Straub_undated-fx" + "Straub_undated-gy" + "Dynan2004-bu" + "Dynan2002-za" + "De_Nardi2004-xs" + "De_Nardi2016-yi" + "Carroll1998-tz" + "Cagetti2008-cc" + "Cagetti2006-pu" + "Cagetti2003-yp" + "Gourinchas2002-lq")) + '(or :bibtex :latex)) + diff --git a/Notes/auto/bequests_lit_rev.el b/Notes/auto/bequests_lit_rev.el new file mode 100644 index 0000000..8463a31 --- /dev/null +++ b/Notes/auto/bequests_lit_rev.el @@ -0,0 +1,42 @@ +(TeX-add-style-hook + "bequests_lit_rev" + (lambda () + (TeX-add-to-alist 'LaTeX-provided-package-options + '(("inputenc" "utf8") ("biblatex" "backend=bibtex" "style=authoryear"))) + (add-to-list 'LaTeX-verbatim-macros-with-braces-local "path") + (add-to-list 'LaTeX-verbatim-macros-with-braces-local "url") + (add-to-list 'LaTeX-verbatim-macros-with-braces-local "nolinkurl") + (add-to-list 'LaTeX-verbatim-macros-with-braces-local "hyperbaseurl") + (add-to-list 'LaTeX-verbatim-macros-with-braces-local "hyperimage") + (add-to-list 'LaTeX-verbatim-macros-with-braces-local "href") + (add-to-list 'LaTeX-verbatim-macros-with-delims-local "path") + (TeX-run-style-hooks + "latex2e" + "./Equations/Abel_1985" + "./Equations/Cagetti_2008" + "./Equations/Barro_1974" + "./Equations/Cagetti_2006" + "./Equations/Carroll_1998" + "./Equations/Cagetti_2003" + "./Equations/Gourinchas_2002" + "./Equations/DeNardi_2004" + "./Equations/Dynan_2004" + "./Equations/DeNardi_2016" + "./Equations/Saez_2017" + "./Equations/Straub_2019" + "article" + "art10" + "color" + "hyperref" + "amsmath" + "amsfonts" + "dirtytalk" + "inputenc" + "biblatex") + (TeX-add-symbols + '("doi" 1) + '("doilink" 1)) + (LaTeX-add-bibliographies + "BeqRefs")) + :latex) + diff --git a/Notes/bequests_lit_rev-blx.bib b/Notes/bequests_lit_rev-blx.bib new file mode 100644 index 0000000..3134c23 --- /dev/null +++ b/Notes/bequests_lit_rev-blx.bib @@ -0,0 +1,11 @@ +@Comment{$ biblatex control file $} +@Comment{$ biblatex bcf format version 3.10 $} +% Do not modify this file! +% +% This is an auxiliary file used by the 'biblatex' package. +% This file may safely be deleted. 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It will be recreated as +% required. +% +\begingroup +\makeatletter +\@ifundefined{ver@biblatex.sty} + {\@latex@error + {Missing 'biblatex' package} + {The bibliography requires the 'biblatex' package.} + \aftergroup\endinput} + {} +\endgroup + +\datalist[entry]{nyt/global//global/global} + \entry{Abel1985-ef}{article}{} + \name{author}{1}{}{% + {{hash=AAB}{% + family={Abel}, + familyi={A\bibinitperiod}, + given={Andrew\bibnamedelima B}, + giveni={A\bibinitperiod\bibinitdelim B\bibinitperiod}, + }}% + } + \list{publisher}{1}{% + {American Economic Association}% + } + \strng{namehash}{AAB1} + \strng{fullhash}{AAB1} + \field{labelnamesource}{author} + \field{labeltitlesource}{title} + \field{labelyear}{1985} + \field{labeldatesource}{} + \field{sortinit}{A} + \field{sortinithash}{A} + \field{issn}{0002-8282} + \field{number}{4} + \field{pages}{777\bibrangedash 791} + \field{title}{{Precautionary Saving and Accidental Bequests}} + \verb{url} + \verb http://www.jstor.org/stable/1821355 + \endverb + \field{volume}{75} + \verb{file} + \verb All-Papers/A/Abel-1985-Precautionary-Saving-and-Accidental-Bequests.p + \verb df + \endverb + \field{journaltitle}{Am. Econ. Rev.} + \field{year}{1985} + \endentry + + \entry{Barro1974-mp}{article}{} + \name{author}{1}{}{% + {{hash=BRJ}{% + family={Barro}, + familyi={B\bibinitperiod}, + given={Robert\bibnamedelima J}, + giveni={R\bibinitperiod\bibinitdelim J\bibinitperiod}, + }}% + } + \list{publisher}{1}{% + {University of Chicago Press}% + } + \strng{namehash}{BRJ1} + \strng{fullhash}{BRJ1} + \field{labelnamesource}{author} + \field{labeltitlesource}{title} + \field{labelyear}{1974} + \field{labeldatesource}{} + \field{sortinit}{B} + \field{sortinithash}{B} + \field{issn}{0022-3808, 1537-534X} + \field{number}{6} + \field{pages}{1095\bibrangedash 1117} + \field{title}{{Are Government Bonds Net Wealth?}} + \verb{url} + \verb http://www.jstor.org/stable/1830663 + \endverb + \field{volume}{82} + \verb{file} + \verb All-Papers/B/Barro-1974-Are-Government-Bonds-Net-Wealth.pdf + \endverb + \field{journaltitle}{J. Polit. Econ.} + \field{year}{1974} + \endentry + + \entry{Cagetti2003-yp}{article}{} + \name{author}{1}{}{% + {{hash=CM}{% + family={Cagetti}, + familyi={C\bibinitperiod}, + given={Marco}, + giveni={M\bibinitperiod}, + }}% + } + \list{publisher}{1}{% + {[American Statistical Association, Taylor \& Francis, Ltd.]}% + } + \strng{namehash}{CM1} + \strng{fullhash}{CM1} + \field{labelnamesource}{author} + \field{labeltitlesource}{title} + \field{labelyear}{2003} + \field{labeldatesource}{} + \field{sortinit}{C} + \field{sortinithash}{C} + \field{abstract}{% + [This article constructs and simulates a life cycle model of wealth + accumulation and estimates the parameters of the utility function (the rate + of time preference and the coefficient of risk aversion) by matching the + simulated median wealth profiles with those observed in the Panel Study of + Income Dynamics and in the Survey of Consumer Finances. The estimates imply a + low degree of patience and a high degree of risk aversion. The results are + used to study the importance of precautionary savings in explaining wealth + accumulation. They imply that wealth accumulation is driven mostly by + precautionary motives at the beginning of the life cycle, whereas savings for + retirement purposes become significant only closer to retirement.]% + } + \field{issn}{0735-0015} + \field{number}{3} + \field{pages}{339\bibrangedash 353} + \field{title}{{Wealth Accumulation over the Life Cycle and Precautionary + Savings}} + \verb{url} + \verb http://www.jstor.org/stable/1392584 + \endverb + \field{volume}{21} + \verb{file} + \verb All-Papers/C/Cagetti-2003-Wealth-Accumulation-over-the-Life-Cycle-and + \verb -Precautionary-Savings.pdf + \endverb + \field{journaltitle}{J. Bus. Econ. Stat.} + \field{year}{2003} + \endentry + + \entry{Cagetti2006-pu}{article}{} + \name{author}{2}{}{% + {{hash=CM}{% + family={Cagetti}, + familyi={C\bibinitperiod}, + given={Marco}, + giveni={M\bibinitperiod}, + }}% + {{hash=DNM}{% + family={De\bibnamedelima Nardi}, + familyi={D\bibinitperiod\bibinitdelim N\bibinitperiod}, + given={Mariacristina}, + giveni={M\bibinitperiod}, + }}% + } + \list{publisher}{1}{% + {The University of Chicago Press}% + } + \strng{namehash}{CMDNM1} + \strng{fullhash}{CMDNM1} + \field{labelnamesource}{author} + \field{labeltitlesource}{title} + \field{labelyear}{2006} + \field{labeldatesource}{} + \field{sortinit}{C} + \field{sortinithash}{C} + \field{abstract}{% + This paper constructs and calibrates a parsimonious model of occupational + choice that allows for entrepreneurial entry, exit, and investment decisions + in the presence of borrowing constraints. The model fits very well a number + of empirical observations, including the observed wealth distribution for + entrepreneurs and workers. At the aggregate level, more restrictive borrowing + constraints generate less wealth concentration and reduce average firm size, + aggregate capital, and the fraction of entrepreneurs. Voluntary bequests + allow some high?ability workers to establish or enlarge an entrepreneurial + activity. With accidental bequests only, there would be fewer very large + firms and less aggregate capital and wealth concentration.% + } + \verb{doi} + \verb 10.1086/508032 + \endverb + \field{issn}{0022-3808} + \field{number}{5} + \field{pages}{835\bibrangedash 870} + \field{title}{{Entrepreneurship, Frictions, and Wealth}} + \verb{url} + \verb https://doi.org/10.1086/508032 + \endverb + \field{volume}{114} + \verb{file} + \verb All-Papers/C/Cagetti-and-De-Nardi-2006-Entrepreneurship,-Frictions,-a + \verb nd-Wealth.pdf + \endverb + \field{journaltitle}{J. Polit. Econ.} + \field{month}{10} + \field{year}{2006} + \endentry + + \entry{Cagetti2008-cc}{article}{} + \name{author}{2}{}{% + {{hash=CM}{% + family={Cagetti}, + familyi={C\bibinitperiod}, + given={Marco}, + giveni={M\bibinitperiod}, + }}% + {{hash=DNM}{% + family={De\bibnamedelima Nardi}, + familyi={D\bibinitperiod\bibinitdelim N\bibinitperiod}, + given={Mariacristina}, + giveni={M\bibinitperiod}, + }}% + } + \list{publisher}{1}{% + {Cambridge University Press}% + } + \keyw{Inequality; Savings; Life-Cycle Models} + \strng{namehash}{CMDNM1} + \strng{fullhash}{CMDNM1} + \field{labelnamesource}{author} + \field{labeltitlesource}{title} + \field{labelyear}{2008} + \field{labeldatesource}{} + \field{sortinit}{C} + \field{sortinithash}{C} + \field{abstract}{% + In the United States wealth is highly concentrated and very unequally + distributed: the richest 1\% hold one third of the total wealth in the + economy. Understanding the determinants of wealth inequality is a challenge + for many economic models. We summarize some key facts about the wealth + distribution and what economic models have been able to explain so far.% + } + \verb{doi} + \verb 10.1017/S1365100507070150 + \endverb + \field{issn}{1365-1005, 1469-8056} + \field{number}{S2} + \field{pages}{285\bibrangedash 313} + \field{title}{{Wealth Inequality: Data and Models}} + \verb{url} + \verb https://www.cambridge.org/core/journals/macroeconomic-dynamics/articl + \verb e/wealth-inequality-data-and-models/4C12213FDED94391870AD3D7BCCB7FC2 + \endverb + \field{volume}{12} + \verb{file} + \verb All-Papers/C/Cagetti-and-De-Nardi-2008-WEALTH-INEQUALITY-DATA-AND-MOD + \verb ELS.pdf + \endverb + \field{journaltitle}{Macroecon. Dyn.} + \field{month}{09} + \field{year}{2008} + \endentry + + \entry{Carroll1998-tz}{unpublished}{} + \name{author}{1}{}{% + {{hash=CCD}{% + family={Carroll}, + familyi={C\bibinitperiod}, + given={Christopher\bibnamedelima D}, + giveni={C\bibinitperiod\bibinitdelim D\bibinitperiod}, + }}% + } + \strng{namehash}{CCD1} + \strng{fullhash}{CCD1} + \field{labelnamesource}{author} + \field{labeltitlesource}{title} + \field{labelyear}{1998} + \field{labeldatesource}{} + \field{sortinit}{C} + \field{sortinithash}{C} + \field{abstract}{% + This paper considers several alternative explanations for the fact that + households with higher levels of lifetime income ( the rich') have higher + lifetime saving rates (Dynan, Skinner, and Zeldes (1996); Lillard and Karoly + (1997)). The paper argues that the saving behavior of the richest households + cannot be explained by models in which the only purpose of wealth + accumulation is to finance future consumption, either their own or that of + heirs. The paper concludes that the simplest model that explains the relevant + facts is one in which either consumers regard the accumulation of wealth as + an end in itself, or unspent wealth yields a flow of services (such as power + or social status) which have the same practical effect on behavior as if + wealth were intrinsically desirable.% + } + \verb{doi} + \verb 10.3386/w6549 + \endverb + \field{number}{6549} + \field{series}{Working Paper Series} + \field{title}{{Why Do the Rich Save So Much?}} + \verb{url} + \verb http://www.nber.org/papers/w6549 + \endverb + \list{institution}{1}{% + {National Bureau of Economic Research}% + } + \verb{file} + \verb All-Papers/C/Carroll-1998-Why-Do-the-Rich-Save-So-Much.pdf + \endverb + \field{month}{05} + \field{year}{1998} + \endentry + + \entry{De_Nardi2004-xs}{article}{} + \name{author}{1}{}{% + {{hash=DNM}{% + family={De\bibnamedelima Nardi}, + familyi={D\bibinitperiod\bibinitdelim N\bibinitperiod}, + given={Mariacristina}, + giveni={M\bibinitperiod}, + }}% + } + \list{language}{1}{% + {en}% + } + \list{publisher}{1}{% + {Oxford University Press (OUP)}% + } + \strng{namehash}{DNM1} + \strng{fullhash}{DNM1} + \field{labelnamesource}{author} + \field{labeltitlesource}{title} + \field{labelyear}{2004} + \field{labeldatesource}{} + \field{sortinit}{D} + \field{sortinithash}{D} + \verb{doi} + \verb 10.1111/j.1467-937x.2004.00302.x + \endverb + \field{issn}{0034-6527, 1467-937X} + \field{number}{3} + \field{pages}{743\bibrangedash 768} + \field{title}{{Wealth inequality and intergenerational links}} + \verb{url} + \verb https://users.nber.org/~denardim/research/denardi.pdf + \endverb + \field{volume}{71} + \verb{file} + \verb All-Papers/D/De-Nardi-2004-Wealth-inequality-and-intergenerational-li + \verb nks.pdf + \endverb + \field{journaltitle}{Rev. Econ. Stud.} + \field{month}{07} + \field{year}{2004} + \endentry + + \entry{De_Nardi2016-yi}{article}{} + \name{author}{2}{}{% + {{hash=DNM}{% + family={De\bibnamedelima Nardi}, + familyi={D\bibinitperiod\bibinitdelim N\bibinitperiod}, + given={Mariacristina}, + giveni={M\bibinitperiod}, + }}% + {{hash=YF}{% + family={Yang}, + familyi={Y\bibinitperiod}, + given={Fang}, + giveni={F\bibinitperiod}, + }}% + } + \keyw{Wealth inequality; Parental background; Estate taxation; Bequests; + Earnings shocks} + \strng{namehash}{DNMYF1} + \strng{fullhash}{DNMYF1} + \field{labelnamesource}{author} + \field{labeltitlesource}{title} + \field{labelyear}{2016} + \field{labeldatesource}{} + \field{sortinit}{D} + \field{sortinithash}{D} + \field{abstract}{% + This paper generates two main contributions. First, it provides a new + theory of wealth inequality that merges two empirically relevant forces + generating inequality: bequest motives and inheritance of ability across + generations; and an earnings process that allows for more earnings risk for + the richest. Second, it uses the resulting calibrated framework to study the + effects of changing estate taxation. Increasing the estate tax reduces the + wealth concentration in the hands of the richest few and the economic + advantage of being born to a rich and super-rich family at the cost of + reduced aggregate capital and output. However, all of these effects are quite + small. In contrast, increasing estate taxation can generate a significant + welfare gain to a newborn under the veil of ignorance, but this comes at a + large welfare cost for the super-rich.% + } + \verb{doi} + \verb 10.1016/j.jmoneco.2015.10.005 + \endverb + \field{issn}{0304-3932} + \field{pages}{130\bibrangedash 145} + \field{title}{{Wealth inequality, family background, and estate taxation}} + \verb{url} + \verb https://www.sciencedirect.com/science/article/pii/S0304393215001245 + \endverb + \field{volume}{77} + \verb{file} + \verb All-Papers/D/De-Nardi-and-Yang-2016-Wealth-inequality,-family-backgro + \verb und,-and-estate-taxation.pdf + \endverb + \field{journaltitle}{J. Monet. Econ.} + \field{month}{02} + \field{year}{2016} + \endentry + + \entry{Dynan2004-bu}{article}{} + \name{author}{3}{}{% + {{hash=DKe}{% + family={Dynan}, + familyi={D\bibinitperiod}, + given={Karen\bibnamedelima e}, + giveni={K\bibinitperiod\bibinitdelim e\bibinitperiod}, + }}% + {{hash=SJ}{% + family={Skinner}, + familyi={S\bibinitperiod}, + given={Jonathan}, + giveni={J\bibinitperiod}, + }}% + {{hash=ZSp}{% + family={Zeldes}, + familyi={Z\bibinitperiod}, + given={Stephen\bibnamedelima p}, + giveni={S\bibinitperiod\bibinitdelim p\bibinitperiod}, + }}% + } + \list{publisher}{1}{% + {The University of Chicago Press}% + } + \strng{namehash}{DKeSJZSp1} + \strng{fullhash}{DKeSJZSp1} + \field{labelnamesource}{author} + \field{labeltitlesource}{title} + \field{labelyear}{2004} + \field{labeldatesource}{} + \field{sortinit}{D} + \field{sortinithash}{D} + \field{abstract}{% + The question of whether higher--lifetime income households save a larger + fraction of their income was the subject of much debate in the 1950s and + 1960s, and while not resolved, it remains central to the evaluation of tax + and macroeconomic policies. We resolve this long‐standing question using + new empirical methods applied to the Panel Study of Income Dynamics, the + Survey of Consumer Finances, and the Consumer Expenditure Survey. We find a + strong positive relationship between saving rates and lifetime income and a + weaker but still positive relationship between the marginal propensity to + save and lifetime income. There is little support for theories that seek to + explain these positive correlations by relying solely on time preference + rates, nonhomothetic preferences, or variations in Social Security benefits. + There is more support for models emphasizing uncertainty with respect to + income and health expenses, bequest motives, and asset‐based means testing + or behavioral factors causing minimal saving rates among low‐income + households.% + } + \verb{doi} + \verb 10.1086/381475 + \endverb + \field{issn}{0022-3808, 1537-534X} + \field{number}{2} + \field{pages}{397\bibrangedash 444} + \field{title}{{Do the Rich Save More?}} + \verb{url} + \verb http://www.jstor.org/stable/10.1086/381475 + \endverb + \field{volume}{112} + \verb{file} + \verb All-Papers/D/Dynan-et-al.-2004-Do-the-Rich-Save-More.pdf + \endverb + \field{journaltitle}{J. Polit. Econ.} + \field{year}{2004} + \endentry + + \entry{Gokhale2001-dj}{article}{} + \name{author}{4}{}{% + {{hash=GJ}{% + family={Gokhale}, + familyi={G\bibinitperiod}, + given={Jagadeesh}, + giveni={J\bibinitperiod}, + }}% + {{hash=KLJ}{% + family={Kotlikoff}, + familyi={K\bibinitperiod}, + given={Laurence\bibnamedelima J}, + giveni={L\bibinitperiod\bibinitdelim J\bibinitperiod}, + }}% + {{hash=SJ}{% + family={Sefton}, + familyi={S\bibinitperiod}, + given={James}, + giveni={J\bibinitperiod}, + }}% + {{hash=WM}{% + family={Weale}, + familyi={W\bibinitperiod}, + given={Martin}, + giveni={M\bibinitperiod}, + }}% + } + \list{publisher}{1}{% + {Elsevier}% + } + \strng{namehash}{GJ+1} + \strng{fullhash}{GJKLJSJWM1} + \field{labelnamesource}{author} + \field{labeltitlesource}{title} + \field{labelyear}{2001} + \field{labeldatesource}{} + \field{sortinit}{G} + \field{sortinithash}{G} + \field{abstract}{% + This paper develops, calibrates, and simulates a dynamic 88-period OLG + model to study the intergenerational transmission of U.S. wealth inequality + via bequests. The model features marriage, realistic fertility patterns, + random death, assortative mating based on skills, heterogeneous skill + endowments, heterogeneous rates of return, skill inheritability, progressive + income taxation, and resource annuitization via social security. All bequests + arise from imperfect annuitization. Nonetheless, the model generates a + realistic ratio of aggregate wealth to aggregate labor income, a realistic + bequest flow relative to the stock of wealth, and a realistic wealth + distribution at retirement. Skill differences, assortative mating, social + security, and the time preference are the primary determinants of wealth + inequality. Bequests do propagate wealth inequality, but only in the presence + of social security, which disproportionately disinherits the lifetime poor. + Intergenerational wealth immobility, also considered here, is primarily + determined by the inheritance of skills from one's parents and the + magnification of the impact of this inheritance by marital sorting.% + } + \verb{doi} + \verb 10.1016/S0047-2727(00)00097-9 + \endverb + \field{issn}{0047-2727} + \field{number}{1} + \field{pages}{93\bibrangedash 128} + \field{title}{{Simulating the transmission of wealth inequality via + bequests}} + \verb{url} + \verb https://www.sciencedirect.com/science/article/pii/S0047272700000979 + \endverb + \field{volume}{79} + \verb{file} + \verb All-Papers/G/Gokhale-et-al.-2001-Simulating-the-transmission-of-wealt + \verb h-inequality-via-bequests.pdf + \endverb + \field{journaltitle}{J. Public Econ.} + \field{month}{01} + \field{year}{2001} + \endentry + + \entry{Gourinchas2002-lq}{article}{} + \name{author}{2}{}{% + {{hash=GPO}{% + family={Gourinchas}, + familyi={G\bibinitperiod}, + given={Pierre-Olivier}, + giveni={P\bibinithyphendelim O\bibinitperiod}, + }}% + {{hash=PJ}{% + family={Parker}, + familyi={P\bibinitperiod}, + given={Jonathan}, + giveni={J\bibinitperiod}, + }}% + } + \strng{namehash}{GPOPJ1} + \strng{fullhash}{GPOPJ1} + \field{labelnamesource}{author} + \field{labeltitlesource}{title} + \field{labelyear}{2002} + \field{labeldatesource}{} + \field{sortinit}{G} + \field{sortinithash}{G} + \field{issn}{0012-9682} + \field{number}{1} + \field{pages}{47\bibrangedash 89} + \field{title}{{Consumption Over the Life Cycle}} + \field{volume}{70} + \verb{file} + \verb All-Papers/G/Gourinchas-and-Parker-2002-Consumption-Over-the-Life-Cyc + \verb le.pdf + \endverb + \field{journaltitle}{Econometrica} + \field{year}{2002} + \endentry + + \entry{Laitner2002-hu}{article}{} + \name{author}{1}{}{% + {{hash=LJ}{% + family={Laitner}, + familyi={L\bibinitperiod}, + given={John}, + giveni={J\bibinitperiod}, + }}% + } + \list{publisher}{1}{% + {American Economic Association}% + } + \strng{namehash}{LJ1} + \strng{fullhash}{LJ1} + \field{labelnamesource}{author} + \field{labeltitlesource}{title} + \field{labelyear}{2002} + \field{labeldatesource}{} + \field{sortinit}{L} + \field{sortinithash}{L} + \field{issn}{0002-8282} + \field{number}{2} + \field{pages}{270\bibrangedash 273} + \field{title}{{Wealth Inequality and Altruistic Bequests}} + \verb{url} + \verb http://www.jstor.org/stable/3083415 + \endverb + \field{volume}{92} + \verb{file} + \verb All-Papers/L/Laitner-2002-Wealth-Inequality-and-Altruistic-Bequests.p + \verb df + \endverb + \field{journaltitle}{Am. Econ. Rev.} + \field{year}{2002} + \endentry + + \entry{Saez2018-we}{article}{} + \name{author}{2}{}{% + {{hash=SE}{% + family={Saez}, + familyi={S\bibinitperiod}, + given={Emmanuel}, + giveni={E\bibinitperiod}, + }}% + {{hash=SS}{% + family={Stantcheva}, + familyi={S\bibinitperiod}, + given={Stefanie}, + giveni={S\bibinitperiod}, + }}% + } + \list{publisher}{1}{% + {Elsevier}% + } + \keyw{Optimal taxation; Capital; Wealth} + \strng{namehash}{SESS1} + \strng{fullhash}{SESS1} + \field{labelnamesource}{author} + \field{labeltitlesource}{title} + \field{labelyear}{2018} + \field{labeldatesource}{} + \field{sortinit}{S} + \field{sortinithash}{S} + \field{abstract}{% + This paper develops a theory of optimal capital taxation that expresses + optimal tax formulas in sufficient statistics. We first consider a simple + model with utility functions linear in consumption and featuring + heterogeneous utility for wealth. In this case, there are no transitional + dynamics, the steady-state is reached immediately and has finite elasticities + of capital with respect to the net-of-tax rate. This allows for a tractable + optimal tax analysis with formulas expressed in terms of empirical + elasticities and social preferences that can address many important policy + questions. These formulas can easily be taken to the data to simulate optimal + taxes, which we do using U.S. tax return data on labor and capital incomes. + Second, we show how these results can be extended to the case with concave + utility for consumption. The same types of formulas carry over by + appropriately defining elasticities. We show that one can recover all the + results from the simpler model using a new and non standard steady state + approach that respects individual preferences even with a fully general + utility function.% + } + \verb{doi} + \verb 10.1016/j.jpubeco.2017.10.004 + \endverb + \field{issn}{0047-2727} + \field{pages}{120\bibrangedash 142} + \field{title}{{A simpler theory of optimal capital taxation}} + \verb{url} + \verb https://www.sciencedirect.com/science/article/pii/S0047272717301688 + \endverb + \field{volume}{162} + \verb{file} + \verb All-Papers/S/Saez-and-Stantcheva-2018-A-simpler-theory-of-optimal-cap + \verb ital-taxation.pdf + \endverb + \field{journaltitle}{J. Public Econ.} + \field{month}{06} + \field{year}{2018} + \endentry + + \entry{Straub_undated-gy}{article}{} + \name{author}{15}{}{% + {{hash=SL}{% + family={Straub}, + familyi={S\bibinitperiod}, + given={Ludwig}, + giveni={L\bibinitperiod}, + }}% + {{hash=AD}{% + family={Acemoglu}, + familyi={A\bibinitperiod}, + given={Daron}, + giveni={D\bibinitperiod}, + }}% + {{hash=AM}{% + family={Aguiar}, + familyi={A\bibinitperiod}, + given={Mark}, + giveni={M\bibinitperiod}, + }}% + {{hash=AM}{% + family={Angeletos}, + familyi={A\bibinitperiod}, + given={Marios}, + giveni={M\bibinitperiod}, + }}% + {{hash=AA}{% + family={Auclert}, + familyi={A\bibinitperiod}, + given={Adrien}, + giveni={A\bibinitperiod}, + }}% + {{hash=BM}{% + family={Bils}, + familyi={B\bibinitperiod}, + given={Mark}, + giveni={M\bibinitperiod}, + }}% + {{hash=DNM}{% + family={De\bibnamedelima Nardi}, + familyi={D\bibinitperiod\bibinitdelim N\bibinitperiod}, + given={Mariacristina}, + giveni={M\bibinitperiod}, + }}% + {{hash=FS}{% + family={Fanelli}, + familyi={F\bibinitperiod}, + given={Sebasti{\'a}n}, + giveni={S\bibinitperiod}, + }}% + {{hash=HM}{% + family={Holm}, + familyi={H\bibinitperiod}, + given={Martin}, + giveni={M\bibinitperiod}, + }}% + {{hash=HG}{% + family={Howard}, + familyi={H\bibinitperiod}, + given={Greg}, + giveni={G\bibinitperiod}, + }}% + {{hash=HE}{% + family={Hurst}, + familyi={H\bibinitperiod}, + given={Erik}, + giveni={E\bibinitperiod}, + }}% + {{hash=LE}{% + family={Liu}, + familyi={L\bibinitperiod}, + given={Ernest}, + giveni={E\bibinitperiod}, + }}% + {{hash=MB}{% + family={Moll}, + familyi={M\bibinitperiod}, + given={Ben}, + giveni={B\bibinitperiod}, + }}% + {{hash=RM}{% + family={Rognlie}, + familyi={R\bibinitperiod}, + given={Matthew}, + giveni={M\bibinitperiod}, + }}% + {{hash=ZN}{% + family={Zorzi}, + familyi={Z\bibinitperiod}, + given={Nathan}, + giveni={N\bibinitperiod}, + }}% + } + \strng{namehash}{SL+1} + \strng{fullhash}{SLADAMAMAABMDNMFSHMHGHELEMBRMZN1} + \field{labelnamesource}{author} + \field{labeltitlesource}{title} + \field{labelyear}{2019} + \field{labeldatesource}{} + \field{sortinit}{S} + \field{sortinithash}{S} + \field{howpublished}{\url{https://scholar.harvard.edu/files/straub/files/cons_ineq_rates.pdf}} + \field{note}{Accessed: 2023-10-25} + \field{title}{{Consumption, savings, and the distribution of permanent + income}} + \verb{url} + \verb https://scholar.harvard.edu/files/straub/files/cons_ineq_rates.pdf + \endverb + \verb{file} + \verb All-Papers/S/Straub-et-al.-Consumption,-savings,-and-the-distribution + \verb -of-permanent-income.pdf + \endverb + \field{year}{2019} + \endentry +\enddatalist +\endinput diff --git a/Notes/bequests_lit_rev.blg b/Notes/bequests_lit_rev.blg new file mode 100644 index 0000000..a4c8d67 --- /dev/null +++ b/Notes/bequests_lit_rev.blg @@ -0,0 +1,70 @@ +This is BibTeX, Version 0.99d (TeX Live 2023) +Capacity: max_strings=200000, hash_size=200000, hash_prime=170003 +The top-level auxiliary file: 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b/Notes/bequests_lit_rev.run.xml @@ -0,0 +1,92 @@ + + + + + + + + + + + + + + + + + + + + + + + + +]> + + + latex + + bequests_lit_rev.aux + bequests_lit_rev-blx.bib + + + bequests_lit_rev.bbl + + + blx-dm.def + blx-compat.def + blx-bibtex.def + biblatex.def + standard.bbx + authoryear.bbx + authoryear.cbx + biblatex.cfg + english.lbx + + + + bibtex + + bibtex + + bequests_lit_rev + + + bequests_lit_rev.aux + + + bequests_lit_rev.bbl + + + bequests_lit_rev.bbl + + + bequests_lit_rev.aux + bequests_lit_rev-blx.bib + + + BeqRefs.bib + + + biblatex.bst + + + diff --git a/Notes/bequests_lit_rev.tex b/Notes/bequests_lit_rev.tex new file mode 100644 index 0000000..cce9f94 --- /dev/null +++ b/Notes/bequests_lit_rev.tex @@ -0,0 +1,110 @@ +\documentclass{article} + +\usepackage{color,hyperref} +% Uses hyperref to link DOI +\newcommand\doilink[1]{\href{http://dx.doi.org/#1}{#1}} +\newcommand\doi[1]{doi:\doilink{#1}} + +\usepackage{amsmath} +\usepackage{amsfonts} + +\usepackage{dirtytalk} + +% Packages for citations +\usepackage[utf8]{inputenc} +\usepackage[backend=bibtex, style=authoryear]{biblatex} % You can change the style if you prefer a different citation style +\addbibresource{BeqRefs.bib} + +% Preamble +\title{Notes on Modeling Bequests} +\author{Decory Edwards} +\date{\today} + + +\begin{document} + +\maketitle + +Many consumption-saving models incorporate a probability of death in the given period as a way to ensure a stationary distribution of wealth. A common accompanying assumption of these models is that, upon death, assets are divided equally amongst all remaining survivors. This receipt of assets is a \say{surprise}, since these assets were accumulated by a previous consumer due to their desire to finance future consumption via saving. + +This rationale can be applied to incorporate bequests into the consumption-saving framework. Traditionally, this is done by extending the model to include (i) some overlapping generation component and/or (ii) some component to link current and future households (i.e. parent and descendants). \footnote{However, the most general specification dealing with bequests only requires this \say{surprise} event of death in a given period.} + +With this in mind, there are three broadly defined classes of bequest models: + +\begin{enumerate} +\item Accidental bequests - Households have some number of descendants who, upon death, receive their assets as bequests. +\item Dynastic (or altruistic) models - Households have some number of descendants \textit{whose consumption they care about}. Upon death, those descendants receive assets of their predecessor as bequests. +\item Wealth in utility models - Households have some desire to hold assets \textit{beyond} potentially fianancing their future consumption via saving. +\end{enumerate} + +Two surveys which provide a useful way of understanding the different types of bequests models are \cite{Laitner2002-hu} and \cite{Cagetti2008-cc}. A further discussion on the implications of these different model assumptions is detailed in \cite{Carroll1998-tz}. + +\section{Accidental Bequests} + +Not only are these models are natural extension of the consumption-saving framework after incorporating a probablility of death, they are also more consistent with the empirical literature on the nature of bequests. Indeed, there is little support for the idea that households accumulate resources for altrusitic considerations regarding their descendants. \cite{Gokhale2001-dj} provides a thorough discussion of this counter-evidence and states that, in general, the empirical evidence suggests that bequests are \say{unintended or motivated by non-altruistic considerations}. + +\subsection{Model from \cite{Abel1985-ef}} + +\input{./Equations/Abel_1985} + +\subsection{Model from \cite{Cagetti2008-cc}} + +\input{./Equations/Cagetti_2008} + +\section{Dynasitc or altruistic models} + +Models where current households care about the utility of their future descendants have the distinct implication that the entire dynasty's wealth should matter to the current household making the consumption-saving decision. Thus, there should be evidence of consumption smoothing across the dynasties. When compared to empirical evidence, this prediction does not hold up for most households (though it may be plausible for the extremely wealthy). + +\subsection{Model from \cite{Barro1974-mp}} + +\input{./Equations/Barro_1974} + +\subsection{Model from \cite{Cagetti2006-pu}} + +\input{./Equations/Cagetti_2006} + +\section{Bequests and wealth in utility function} + +The most general class of models accomodating bequests are those that assume households receive direct utility from holding assets (i.e. other than the utility earned from being able to finance future consumption with saving today). This includes \say{warm glow} bequest models as a special case. + +\subsection{Model from \cite{Carroll1998-tz}} + +\input{./Equations/Carroll_1998} + +\subsection{Model from \cite{Cagetti2003-yp}} + +\input{./Equations/Cagetti_2003} + +\subsection{Model from \cite{Gourinchas2002-lq}} + +\input{./Equations/Gourinchas_2002} + +Now that we see how the wealth in the utility function enters the recursive formulation of the household's problem, form this point we include only the component of the utility function which captures the preference for wealth holdings beyond financing future consumption. + +\subsection{Model from \cite{De_Nardi2004-xs}} + +\input{./Equations/DeNardi_2004} + +\subsection{Model from \cite{Dynan2004-bu}} + +\input{./Equations/Dynan_2004} + +\subsection{Model from \cite{De_Nardi2016-yi}} + +\input{./Equations/DeNardi_2016} + +\subsection{Model from \cite{Saez2018-we}} + +\input{./Equations/Saez_2017} + +\subsection{Model from \cite{Straub_undated-gy}} + +\input{./Equations/Straub_2019} + +%\subsection{Stone-geary and related extensions} + +%\subsection{Non-separable wealth in utility} + +\printbibliography + +\end{document} diff --git a/Paper/#math.tex# b/Paper/#math.tex# new file mode 100644 index 0000000..0656cb3 --- /dev/null +++ b/Paper/#math.tex# @@ -0,0 +1,179 @@ +5;42;26M--- +title: Structural Estimation of Life Cycle Models with Wealth in the Utility Function +subject: Economics +# subtitle: Evolve your markdown documents into structured data +short_title: Structural Estimation +authors: +- name: Alan Lujan +affiliations: +- Ohio State University +- Econ-ARK +email: alanlujan91@gmail.com +license: CC-BY-4.0 +keywords: structural estimation, life cycle, wealth in the utility +exports: +- format: tex+pdf +template: arxiv_nips +output: structural_estimation.pdf +show_date: true +line_spacing: doublespacing +--- + ++++ {"part": "abstract"} + +Heterogeneous Agent Models (HAM) are a powerful tool for understanding the effects of monetary and fiscal policy on the economy. However, state of the art frameworks such as Heterogeneous Agent New Keynsian (HANK) models have been unable to replicate the observed hoarding of wealth at the very top of the distribution and generally lack important life cycle properties such as time-varying mortality and income risk. On the one hand, the inability to pin down wealth at the tail of the distribution has been a problem for HANK models precisely because it has implications for the transmission of monetary and fiscal policy. On the other hand, agents in HANK are generally conceived as perpetual youth with infinite horizons and without age-specific profiles of mortality and income risk. This is problematic as it ignores the effects of these policies on potentially more affected communities, such as young families with children or the low-wealth elderly. In this paper, I investigate the effects of both life cycle considerations as well as wealth in the utility on the structural estimation of HAMs. Structural estimation is the first step in evaluating the effect of monetary and fiscal policies in a HANK framework, and my hope is that this paper will lead to better models of the economy that can be used to inform policy.. + ++++ + ++++ {"part": "acknowledgements"} + +I would like to thank my advisor, Chris Carroll, for his guidance and support throughout this project, as well as the members of the Econ-ARK team for providing a great collaborative community to work in. + ++++ + +# Introduction + +# Life Cycle Models + +## The Baseline Model + +The agent's objective is to maximize present discounted utility from consumption over a last cycle with a terminal period of $T$: + +\begin{equation}\label{eq:MaxProb} + \max ~ \uFunc({\cLvl}_{t}) + \Ex_{t}\left[ \sum_{n=1}^{T-t} {\beth}^{n} \Alive_{t}^{t+n} + \hat{\DiscFac}_{t}^{t+n} \uFunc({\cLvl}_{t+n})\right] +\end{equation} + +where + +\begin{align} + \beth & : & \text{time-invariant `pure' discount factor} + \\ \Alive _{t}^{t+n} & : & \text{probability to }\Alive\text{ive until age $t+n$ given alive at age $t$} + \\ \hat{\DiscFac}_{t}^{t+n} & : & \text{age-varying discount factor between ages $t$ and $t+n$} +\end{align} + + + +\begin{align} + {\vFunc}_{t}({m}_{t}) & = \max_{\cNrm_{t}} ~ \uFunc(\cNrm_{t})+\beth\Alive_{t+1}\hat{\DiscFac}_{t+1} + \Ex_{t}[(\PermShk_{t+1}\PermGroFac_{t+1})^{1-\CRRA}{\vFunc}_{t+1}({m}_{t+1})] + \\ & \text{s.t.} & \nonumber + \\ \aNrm_{t} & = {m}_{t}-\cNrm_{t} \nonumber + \\ {m}_{t+1} & = \aNrm_{t}\underbrace{\left(\frac{\Rfree}{\PermShk_{t+1}\PermGroFac_{t+1}}\right)}_{\equiv \RNrm_{t+1}} + + ~\TranShkEmp_{t+1} +\end{align} + +\begin{align} + \Psi_{t} & : & \text{mean-one shock to permanent income} +\end{align} + + + +\begin{align} + \Xi_{s} & = + \begin{cases} + 0\phantom{/\pZero} & \text{with probability $\pZero>0$} + \\ \TranShkEmp_{s}/\pZero & \text{with probability $(1-\pZero)$, where + $\log \TranShkEmp_{s}\thicksim \mathcal{N}(-\sigma_{\TranShkEmp}^{2}/2,\sigma_{\TranShkEmp}^{2})$} + \end{cases} + \\ \log \PermShk_{s} & \thicksim \mathcal{N}(-\sigma_{\PermShk}^{2}/2,\sigma_{\PermShk}^{2}) +\end{align} + +## Wealth in the Utility Function + +\begin{align} + {\vFunc}_{t}({m}_{t}) & = \max_{\cNrm_{t}} ~ \uFunc(\cNrm_{t}, \aNrm_{t})+\beth\Alive_{t+1}\hat{\DiscFac}_{t+1} + \Ex_{t}[(\PermShk_{t+1}\PermGroFac_{t+1})^{1-\CRRA}{\vFunc}_{t+1}({m}_{t+1})] + \\ & \text{s.t.} & \nonumber + \\ \aNrm_{t} & = {m}_{t}-\cNrm_{t} \nonumber + \\ {m}_{t+1} & = \aNrm_{t}\RNrm_{t+1}+ ~\TranShkEmp_{t+1} +\end{align} + +### Separable Utility + +\begin{equation} + \uFunc(\cNrm_{t}, \aNrm_{t}) = \frac{\cNrm_{t}^{1-\CRRA}}{1-\CRRA} + + \kapShare_{t} \frac{(\aFunc_{t} - \underline\aNrm)^{1-\wealthShare}}{1-\wealthShare} +\end{equation} + +### Non-separable Utility + +\begin{equation} + \uFunc(\cNrm_{t}, \aNrm_{t}) = \frac{(\cNrm_{t}^{1-\wealthShare} (\aNrm_{t} - \underline\aNrm)^\wealthShare)^{1-\CRRA}}{(1-\CRRA)} +\end{equation} + +# Solution Methods + +For a brief departure, let's consider how we solve these problems generally. Define the post-decision value function as: + +\begin{align} + \DiscFac_{t+1} \wFunc_{t}(\aNrm_{t}) & = \beth\Alive_{t+1}\hat{\DiscFac}_{t+1} + \Ex_{t}[(\PermShk_{t+1}\PermGroFac_{t+1})^{1-\CRRA}{\vFunc}_{t+1}({m}_{t+1})] + \\ & \text{s.t.} + \\ {m}_{t+1} & = \aNrm_{t}\RNrm_{t+1}+ ~\TranShkEmp_{t+1} +\end{align} + +For our purposes, it will be useful to simplify the notation a bit by dropping time subscripts. The recursive problem can then be written as: + +\begin{align} + \vFunc(\mRat) & = \max_{\cNrm} ~ \uFunc(\cNrm, \aNrm) + \DiscFac \wFunc(\aRat) + \\ & \text{s.t.} + \\ \aNrm & = \mRat-\cNrm +\end{align} + +## Endogenous Grid Method, Abridged + +In the standard incomplete markets (SIM) model, the utility function is simply $\uFunc(\cNrm)$ and the Euler equation is $\uFunc'(\cNrm) = \DiscFac \wFunc'(\aNrm)$, which is a necessary and sufficient condition for an internal solution of the optimal choice of consumption. If $\uFunc(\cNrm)$ is differentiable and its marginal utility is invertible, then the Euler equation can be inverted to obtain the optimal consumption function as $\cNrm(\aNrm) = \uFunc'^{-1}(\DiscFac \wFunc'(\aNrm))$. Using an _exogenous_ grid of post-decision savings $\aMat$, we can obtain an _endogenous_ grid of market resources $\mMat$ by using the budget constraint $\mNrm(\aMat) = \aMat + \cNrm(\aMat)$ such that this collection of grids satisfy the Euler equation. This is the endogenous grid method (EGM) of [](doi:10.1016/j.econlet.2005.09.013). + +In the presence of wealth in the utility function, the Euler equation is more complicated and may not be invertible in terms of optimal consumption. Consider the first order condition for an optimal combination of consumption and savings, denoted by $^*$: + +\begin{equation} + \uFunc_{c}'(\cNrm^*, \aNrm^*) - \uFunc_{a}'(\cNrm^*, \aNrm^*) = \DiscFac \wFunc'(\aNrm^*) +\end{equation} + +If the utility of consumption and wealth is additively separable, then the Euler equation can be written as $\uFunc_{c}'(\cNrm) = \uFunc_{a}'(\aNrm) + \DiscFac \wFunc'(\aNrm)$. This makes sense, as the agent will equalize the marginal utility of consumption with the marginal utility of wealth today plus the discounted marginal value of wealth tomorrow. In this case, the EGM is simple: we can invert the Euler equation to obtain the optimal consumption policy as $\cNrm(\aNrm) = \uFunc_{c}'^{-1}\big(\uFunc_{a}'(\aNrm) + \DiscFac \wFunc'(\aNrm)\big)$. We can proceed with EGM as usual, using the budget constraint to obtain the endogenous grid of market resources $\mNrm(\aMat) = \aMat + \cNrm(\aMat)$. + +## Generalized Endogenous Grid Method + +When the utility of consumption and wealth is not additively separable, the Euler equation is not analytically invertible for the optimal consumption policy. The usual recourse is to use a root-finding algorithm to obtain the optimal consumption policy for each point on the grid of market resources, which turns out to be more efficient than grid search maximization. As far as I am aware, this is the first paper to avoid root-finding or grid search maximization by using an EGM-like approach to solve a problem with a non-invertible Euler equation. + +Holding $\aNrm$ constant, define a function $f_{a}$ as the difference between the marginal utility of consumption and the marginal utility of wealth, and using an _exogenous_ grid of $\cMat$, label the difference as $[\xFer_{a}]$: + +\begin{equation} + f_{a}(\cMat) = \uFunc_{c}'(\cMat, \aNrm) - \uFunc_{a}'(\cMat, \aNrm) = [\xFer_{a}] +\end{equation} + +So far, we haven't done much, as $f_a$ is still not analytically invertible. However, as we have held $\aNrm$ constant, this is now a single variable function with a single output. We can easily create a linear interpolator for the inverse of this function by reversing output with input appropriately. This is the key step that allows us to avoid root-finding or grid search maximization. We will call this function $\hat{f}_{a}^{-1}$, as it is an approximation of the inverse of $f_{a}$ which could still introduce some error[^f_error]. + +[^f_error]: A main difference between EGM and GEGM is that EGM uses the exact inverse of the Euler equation, thereby giving the unique optimal consumption policy for each point on the grid of post-decision savings that exactly satisfies the Euler equation. GEGM uses an approximation of the inverse of the Euler equation, thereby giving an approximate optimal consumption policy for each point on the grid of market resources. With careful grid choice, the approximation error can be made arbitrarily small. + +\begin{equation} + \hat{f}_{a}^{-1}([\xFer_{a}]) = \cMat +\end{equation} + +We can now construct the function $g(\aNrm)$ as the composition of $\hat{f}_{a}^{-1}$ and discounted marginal value of wealth $\DiscFac \wFunc'$, which should provide an approximation of the optimal consumption policy for each point on the grid of post-decision savings: + +\begin{equation} + g(\aNrm^*) = \hat{f}_{a^*}^{-1} \big( \DiscFac \wFunc'(\aNrm^*) \big) = \cNrm^* +\end{equation} + +This completes the modified step in the Generalized Endogenous Grid Method (GEGM). If this looks familiar, it is because it exactly parallels the EGM method. In the SIM model, $f_a(\cNrm,\aNrm) = \uFunc'(\cNrm)$, which does not depend on $\aNrm$, so we can drop it. The inverse of this function is exactly $f^{-1}(\xFer) = \uFunc'^{-1}(\xFer)$, so we don't need an approximating interpolator. The composition of this function with the discounted marginal value of wealth exactly provides the consumption policy as $g(\aNrm) = \uFunc'^{-1} \big( \DiscFac \wFunc'(\aNrm) \big) = \cNrm$. GEGM is a generalization of EGM to the case where the Euler equation is not analytically invertible. + +An additional feature of GEGM is that the inverse interpolating function $\hat{f}_{a}^{-1}$ only needs to be constructed once and can be used for all backward iterations, as long as the utility function isn't time-varying. This is because the inverse interpolating function only depends on the marginal utilities of consumption and wealth, whose parameters are constant in our specification. This also means that we can construct this interpolating function from the onset of our process to have arbitrary precision by optimally choosing the grid of $\cMat$. Computationally, this adds just one additional step to the standard EGM at the beginning of the process, inheriting its substantial improvements in speed and accuracy relative to root finding and grid search maximization[^speed]. + +[^speed]: If additional precision is needed, the resulting $\hat{\cNrm}$ can be used as the initial guess for a root-finding algorithm with arbitrary precision, which should converge in few iterations due to the proximity of the initial guess to the true solution. + +# Calibration and Estimation + +# Conclusion + +# References + + [](doi:10.3386/w7826) +[](doi:10.3386/w6549) +[](doi:10.1162/rest_a_00893) +[](doi:10.3982/ECTA17434) +[](doi:10.3386/w26941) +[](doi:10.1257/aer.20160042) +[](doi:10.3386/w26647) +[](doi:10.1198/073500103288619007) \ No newline at end of file diff --git a/Paper/.DS_Store b/Paper/.DS_Store new file mode 100644 index 0000000000000000000000000000000000000000..973975f7841d9044ce7b22bcce116379190edc6d GIT binary patch literal 6148 zcmeHK%}N6?5T4N%7QFQ6F|W`!h-LNYr7s}c6%}_$q4l1}@L|0AUVf7qWkVOdh=|NU z@=fL^*?h1`hKP9c(yxhDMAV=OvM4hm=2_R71uubI<~a0+-O!~p%uMq)P4ewax}`)% zx~J;S@Q+e-p8PJXv9~n5)i!GvEw3 z1J1zDF@QT;q`6S^-WhNPoPjR}u@Fa|22M@*&@GA z@sTs&4E!?&c+qa#4K`(W>&f=yt_^5+Xd)7qMS(!C9swB0IdWN?>JOqLE*fS Date: Thu, 26 Oct 2023 07:50:51 -0400 Subject: [PATCH 2/2] small edit --- Notes/Equations/Barro_1974 | 2 +- Notes/bequests_lit_rev.pdf | Bin 240238 -> 240241 bytes 2 files changed, 1 insertion(+), 1 deletion(-) diff --git a/Notes/Equations/Barro_1974 b/Notes/Equations/Barro_1974 index d341568..a9d4413 100644 --- a/Notes/Equations/Barro_1974 +++ b/Notes/Equations/Barro_1974 @@ -2,7 +2,7 @@ The utility function of the member of the $i$th generation has the form \begin{equation} \begin{aligned} -U_i \quad & U(c_{i}^{y}, c_{i}^{o}, U_{i+1}^{*} ) \\ +U_i = U(c_{i}^{y}, c_{i}^{o}, U_{i+1}^{*} ) \\ \end{aligned} \end{equation} diff --git a/Notes/bequests_lit_rev.pdf b/Notes/bequests_lit_rev.pdf index 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