Revise model ,add appendix with end commuting shares, compile #267 #263

paper_slides/biblio.bib

@@ -174,15 +174,22 @@ @article{CegnizEtAl2019
   publisher={Oxford University Press}
 }
 
-@article{Aaronson2001,
-  title={Price Pass-Through and the Minimum Wage},
-  author={Aaronson, Daniel},
-  journal={Review of Economics and Statistics},
-  volume={83},
-  number={1},
-  pages={158--169},
-  year={2001},
-  publisher={MIT Press}
+@techreport{AhlfeldtEtAl2022,
+  title={Optimal minimum wages},
+  author={Ahlfeldt, Gabriel M and Roth, Duncan and Seidel, Tobias},
+  year={2022},
+  institution={Center for Economic and Policy Research},
+  type={CEPR Discussion Paper},
+  number={DP16913}
+}
+
+@techreport{BergerHerkenhoffMongey2022,
+  title={Minimum wages, efficiency and welfare},
+  author={Berger, David W and Herkenhoff, Kyle F and Mongey, Simon},
+  year={2022},
+  institution={National Bureau of Economic Research},
+  type = {NBER Working Papers},
+  number={29662}
 }
 
 @article{Alonso2020,
@@ -382,6 +389,31 @@ @article{Leung2021
   year={2021}
 }
 
+@article{CooperEtAl2020,
+    author = {Cooper, Daniel and Luengo-Prado, María José and Parker, Jonathan A.},
+    title = {The Local Aggregate Effects of Minimum Wage Increases},
+    journal = {Journal of Money, Credit and Banking},
+    volume = {52},
+    number = {1},
+    pages = {5-35},
+    doi = {https://doi.org/10.1111/jmcb.12684},
+    url = {https://onlinelibrary.wiley.com/doi/abs/10.1111/jmcb.12684},
+    eprint = {https://onlinelibrary.wiley.com/doi/pdf/10.1111/jmcb.12684},
+    abstract = {Abstract Using variation in minimum wages across cities and controlling for differences in business-cycle factors and long-run local economic trends, we find that following minimum wage increases, both, prices and nominal spending rise modestly. These gains are larger for certain subcategories of goods such as food away from home and in locations where low-wage workers account for a larger share of employment. Further, minimum wage increases are associated with reduced total debt among households with low credit scores, higher auto debt, and increased access to credit.},
+    year = {2020}
+}
+
+@article{Aaronson2001,
+  title={Price Pass-Through and the Minimum Wage},
+  author={Aaronson, Daniel},
+  journal={Review of Economics and Statistics},
+  volume={83},
+  number={1},
+  pages={158--169},
+  year={2001},
+  publisher={MIT Press}
+}
+
 @article{AllegrettoReich2018,
   title={Are Local Minimum Wages Absorbed by Price Increases? Estimates From Internet-Based Restaurant Menus},
   author={Allegretto, Sylvia and Reich, Michael},
@@ -401,24 +433,6 @@ @article{RenkinEtAl2020
   year={2020}
 }
 
-@techreport{BergerHerkenhoffMongey2022,
-  title={Minimum wages, efficiency and welfare},
-  author={Berger, David W and Herkenhoff, Kyle F and Mongey, Simon},
-  year={2022},
-  institution={National Bureau of Economic Research},
-  type = {NBER Working Papers},
-  number={29662}
-}
-
-@techreport{AhlfeldtEtAl2022,
-  title={Optimal minimum wages},
-  author={Ahlfeldt, Gabriel M and Roth, Duncan and Seidel, Tobias},
-  year={2022},
-  institution={Center for Economic and Policy Research},
-  type={CEPR Discussion Paper},
-  number={DP16913}
-}
-
 %%%%%%%%%%%%%% LABOR %%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 

paper_slides/paper/appendix.tex

@@ -2,7 +2,82 @@
 %%%%%                              APPENDIX                                 %%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-\section{A Dynamic Supply and Demand Model}\label{sec:dyn_theory_model}
+\section{Model Appendix}
+
+\subsection{Proofs}\label{sec:proofs}
+
+\begin{proof}[Proof of Proposition \ref{prop:comparative_statics}]
+    Fully differentiate the market clearing condition with respect to $\ln R_i$ 
+    and $\ln \MW_z$ for all $z\in\Z(i)$.
+    Using \eqref{eq:equilibrium} and appropriate algebraic manipulations, 
+    one can show that
+    \begin{equation}\label{eq:diff_equilibrium}
+        \Big(\eta_i - \sum_z \pi_{iz} \xi^R_{iz} \Big) d \ln R_i
+        = 
+        \sum_z \pi_{iz} \left(\xi^P_{iz} \epsilon_{i}^P d \ln \MW_i 
+                            + \xi^Y_{iz} \epsilon_{iz}^Y d \ln \MW_z \right) ,
+    \end{equation}
+    where
+    $\xi_{iz}^x = \frac{d h_{iz}}{d x_i} \frac{x_i}{\sum_z \pi_{iz} h_{iz}}$ for
+    $x\in\{R,P\}$ is the elasticity of the per-capita housing demand with respect
+    to $x$ evaluated at the average per-capita demand of ZIP code $i$,
+    $\xi_{iz}^Y = \frac{d h_{iz}}{d Y_z} \frac{Y_z}{\sum_z \pi_{iz} h_{iz}}$ 
+    represents the analogous elasticity with respect to income $Y$ from each 
+    workplace $z$,
+    $\epsilon_{i}^P = \frac{d P_i}{d \MW_i} \frac{\MW_i}{P_i}$ and 
+    $\epsilon_{iz}^Y = \frac{d Y_z}{d \MW_z} \frac{\MW_z}{Y_z}$ are
+    elasticities of prices and income to the MW, and
+    $\eta_i = \frac{d S_i}{d R_i} \frac{R_i}{S_i}$ is the elasticity 
+    of housing supply in ZIP code $i$.
+
+    For any $z'\in\Z_0\setminus\{i\}$ the partial effect on rents of the policy
+    is given by
+    $$
+    \frac{d\ln R_i}{d\ln\MW_{z'}} 
+      = \left(\eta_i - \sum_z \pi_{iz} \xi^R_{iz}\right)^{-1} 
+              \pi_{iz'}\xi^Y_{iz}\epsilon_{iz'}^Y.
+    $$
+    Because $\eta_i\geq0$ and $\xi^R_{iz} < 0$ for all $z\in\Z(i)$, 
+    the first factor is positive.
+    From Assumptions \ref{assu:mws} and \ref{assu:housing_demand},
+    $\epsilon_{iz}^Y\geq0$ and $\xi^Y_{iz}>0$.
+    Therefore, the effect is positive if for $z'$ we have 
+    $\frac{d Y_{z'}}{d \MW_{z'}}>0$ (or $\epsilon_{iz'}^Y>0$), 
+    and the effect is zero otherwise.
+
+    For ZIP code $i$ the partial effect is given by
+    $$
+    \frac{d\ln R_i}{d\ln\MW_{i}} 
+      = \left(\eta_i - \sum_z \pi_{iz} \xi^R_{iz}\right)^{-1} 
+        \left(\epsilon_{i}^P \sum_z \pi_{iz}\xi^P_{iz} 
+             + \pi_{ii}\xi^Y_{ii}\epsilon_{ii}^Y \right) .
+    $$
+    By Assumption \ref{assu:mws} we have that $\epsilon_{i}^P>0$ and that 
+    $\epsilon_{ii}^Y\geq0$.
+    By Assumption \ref{assu:housing_demand} we have that $\xi^Y_{ii}>0$ and that, 
+    for all $z\in\Z(i)$, $\xi^P_{iz}<0$.
+    Then, the second parenthesis has an ambiguous sign.
+    The third statement of the Proposition follows directly.
+\end{proof}
+
+\begin{proof}[Proof of Proposition \ref{prop:representation}]
+    Under the stated assumptions we can manipulate \eqref{eq:diff_equilibrium} 
+    to write
+    $$
+    d r_{i} = \beta_i d \mw_{i}^{\wkp} + \gamma_i d \mw^{\text{res}}_i
+    $$
+    where
+    $\beta_i = \frac{\xi_{i}^{Y}\epsilon_i^{Y}}
+                     {\eta_{i} - \sum_z \pi_{iz} \xi_{iz}^R} 
+              > 0$ and
+    $\gamma_i = \frac{\sum_{z\in\Z(i)}\pi_{iz}\xi_{iz}^{P}\epsilon_{i}^{P}}
+                    {\eta_{i} - \sum_z \pi_{iz} \xi_{iz}^R} 
+              < 0$
+    are parameters, which signs can be verified using
+    Assumptions \ref{assu:mws} and \ref{assu:housing_demand}.
+\end{proof}
+
+\subsection{A Dynamic Supply and Demand Model}\label{sec:dyn_theory_model}
 
 The geography is represented by a set of ZIP codes $\Z$.
 There is an exogenously given distribution of workers with differing 
@@ -43,7 +118,7 @@ \section{A Dynamic Supply and Demand Model}\label{sec:dyn_theory_model}
 We assume that all square feet are homogeneous, and so they have the same price 
 in the market.
 
-\subsection*{Within-period equilibrium}
+\subsubsection*{Within-period equilibrium}
 
 We assume the following timing: 
 (1) At the beginning of period $t$, a share $\lambda_{it}$ of contracts 
@@ -96,6 +171,99 @@ \subsection*{Within-period equilibrium}
 The results in Section \ref{sec:model} can be extended to a dynamic setting if
 the demand and supply functions in $t$ only depend on MW levels in $t$.
 
+\subsection{A Static Model with Flexible Commuting Shares}\label{sec:model_endogenous_shares}
+
+In this section we use a simplified version of the partial-equilibrium model 
+in Section \ref{sec:model}.
+We do so to focus on the implications of relaxing the assumption of fixed 
+commuting shares.
+
+Assume that housing demand depends directly on the workplace and residence MW,
+abstracting away from the mediation channels of prices and income.
+Let commuting shares depend on the MW at the respective workplace location.
+Then, the housing market equilibrium can be written as
+\begin{equation}\label{eq:equilibrium_simplified}
+	L_i \sum_{z\in\Z(i)} \pi_{iz} (\MW_z) h_{iz} \left(R_i, \MW_i, \MW_z\right) = S_i(R_i) .
+\end{equation}
+Following empirical results in \textcite{PerezPerez2021}, we assume that 
+an increase in the workplace MW may decrease the share of workers traveling to 
+a given destination.
+
+\begin{assu}[Endogenous commuting shares]\label{assu:commuting_shares}
+    Commuting shares of location $i$'s residents are given by 
+    $\{\pi_{iz}\left(\MW_z\right)\}_{z\in\Z(i)}$, where $d\pi_{iz}/d\MW_z \leq 0$.
+\end{assu}
+
+With this assumption we are ready to prove the following result.
+
+\begin{prop}[Representation with endogenous shares]\label{prop:representation_endog_shares}
+    Assume that for all ZIP codes $z\in\Z(i)$ we have
+    (a) homogeneous elasticity of per-capita housing demand to the MW 
+    at workplace locations,
+    $\xi^{\wkp}_{iz}=\xi^{\wkp}_{i}$, and
+    (b) homogeneous elasticity of commuting shares to the MW,
+    $\zeta_{iz}=\zeta_{i}$.
+    Then, approximating $\pi_{iz} h_{iz} / \sum_{z'}\pi_{iz'} h_{iz'}\approx\pi_{iz}$,
+    we can write
+    $$
+    d r_{i} = \left(\beta_i + \zeta_i\right) d \mw_{i}^{\wkp} 
+            + \gamma_i d \mw^{\text{res}}_i
+    $$
+    where 
+    $r_{i} = \ln R_i$,
+    $\mw_{i}^{\wkp} = \sum_{z\in\Z(i)} \pi_{iz} \ln \MW_z$ 
+    is ZIP code $i$'s \textbf{workplace MW}, 
+    $\mw^{\res}_i = \ln \MW_i$ 
+    is ZIP code $i$'s \textbf{residence MW}, and 
+    $\beta_i > 0$, $\zeta_i<0$, and $\gamma_i < 0$ are parameters.
+\end{prop}
+
+\begin{proof}    
+    Fully differentiate the market clearing condition with respect to $\ln R_i$ 
+    and $\ln \MW_z$ for all $z\in\Z(i)$.
+    Using \eqref{eq:equilibrium_simplified} and appropriate algebraic 
+    manipulations, one can show that
+    \begin{equation}\label{eq:diff_equilibrium_endog_shares}
+        \Big(\eta_i - \sum_z \pi_{iz} \xi^R_{iz} \Big) d \ln R_i
+        = 
+        \left(\sum_z \pi_{iz} \xi^{\res}_{iz} \right)d \ln \MW_i 
+        + \sum_z \pi_{iz} (\xi^{\wkp}_{iz} + \zeta_{iz}) d \ln \MW_z  ,
+    \end{equation}
+    where we also impose the approximation that 
+    $\pi_{iz} h_{iz} / \sum_{z'}\pi_{iz'} h_{iz'}\approx\pi_{iz}$.
+    In this expression
+    $\xi^R_{iz} = \frac{d h_{iz}}{d R_i} \frac{R_i}{\sum_z \pi_{iz} h_{iz}}$
+    is the elasticity of housing demand to rents,
+    $\xi^{\res}_{iz} = \frac{d h_{iz}}{d \MW_i} \frac{\MW_i}{\sum_z \pi_{iz} h_{iz}}$
+    is the elasticity of housing demand to the MW at $i$, 
+    $\xi^{\wkp}_{iz} = \frac{d h_{iz}}{d \MW_z} \frac{\MW_z}{\sum_z \pi_{iz} h_{iz}}$
+    is the elasticity of housing demand to the MW at workplace $z$, and
+    $\zeta_{iz} = \frac{d \pi_{iz}}{d \MW_z} \frac{\MW_z}{\sum_z \pi_{iz} h_{iz}}$
+    is the elasticity of commuting shares to the MW at $z$.
+
+    Under the stated assumptions we can manipulate 
+    \eqref{eq:diff_equilibrium_endog_shares} to write
+    \begin{equation} \label{eq:theory_representation_endog_shares}
+        d r_i = (\beta_i + \zeta_i)  d \mw^{\wkp}_i
+              + \gamma_i d \mw^{\res}_i
+    \end{equation}
+    where
+    $\beta_i = \frac{\xi^{\wkp}_i}
+                    {\eta_{i} - \sum_z \pi_{iz} \xi_{iz}^R}
+              > 0$,
+    $\zeta_i = \frac{\zeta_i}
+                    {\eta_{i} - \sum_z \pi_{iz} \xi_{iz}^R} \leq 0$, and
+    $\gamma_i = \frac{\sum_z \pi_{iz} \xi^{\res}_{iz}}
+                     {\eta_{i} - \sum_z \pi_{iz} \xi_{iz}^R} 
+              < 0$
+    are parameters.
+    The sign of $\zeta_i$ is given by Assumption \ref{assu:commuting_shares}.
+\end{proof}
+
+This result implies that, up to an approximation, the response of rents to 
+the workplace MW includes the negative effect of the MW on commuting shares.
+
+
 \clearpage
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Data appendix}