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Paper 1: "Monetary Policy, Heterogeneity, and the Housing Channel"
+

+
Hedlund, Karahan, Mitman, Ozkan (2017)
 +

+
Nino Kodua
+

+
Johns Hopkins University
+

+
February 16, 2020

+ +
+
+
+
+
+
+

Overview

    +
  • Research Question: What is the extent to which housing and mortgage debt affect the transmission and effectiveness of monetary policy?
    • +
  • Channels to consider:
      +
    • Direct intertemporal substitution effect
      • +
    • House price channel
      • +
    • Liquidity channel
      • +
    • Cash-flow channel
      • +
    • Redistribution channel
      • +
    +
    • +
  • HANK model with nominal rigidities and a frictional housing market
    • +
+ +
+
+
+
+
+
+

Households

    +
  • Infinitely lived households with uninsurable idiosyncratic labor productivity risk
    • +
+ +
+
+
+
+
+
+

Selling decision of a household:


+ + + \begin{aligned} +V_{O W N}\left(a_{t}, M_{t}, h_{t}, z_{t}\right)=\max _{\varsigma_{t} \in\{0,1\}} \varsigma_{t} V_{S E L L}\left(a_{t}, M_{t}, h_{t}, z_{t}\right)+\left(1-\varsigma_{t}\right) V_{N S e l l}\left(a_{t}, M_{t}, h_{t}, z_{t}\right) + \end{aligned} + +

+
    +
  • $\varsigma_{t}$ - Mortgage initiation cost;
    • +
  • $M_{t}$ - Mortgage size;
    • +
  • $h_{t}$ - House size;
    • +
  • $z_{t}$ - Idiosyncratic labor productivity;
    • +
+ +
+
+
+
+
+
+

A household wanting to sell the house chooses a list price ${x_{s}}$


+ + +\begin{aligned} + V_{S E L L}\left(a_{t}, M_{t}, h_{t}, z_{t}\right) &=\max _{x_{s t}}-\xi+\tilde{p}_{t}\left(\theta_{t}\left(x_{s}, h\right)\right) V_{N O w n}^{t}\left(a_{t}+x_{s t}-M_{t}, z_{t}\right) \\ &+\tilde{p}_{t}\left(\theta_{t}\left(x_{s}, h\right)\right) V_{N S e l l}\left(a_{t}, M_{t}, h_{t}, z_{t}\right) + \end{aligned} + +

+
    +
  • $\tilde{p}_{t}\left(\theta_{t}\left(x_{s}, h\right)\right)$ - Probability of a meeting between a real estate broker and a seller;
    • +
  • $\theta_{t}\left(x_{s}, h\right)$ - tightness of the submarket for house size $h$ and a price ${x_{s}}$
    • +

  • $\xi$ - utility cost if homeowner tries but fails to sell the house +

    +
    • +

  • Free entry condition for brokers: + + +\begin{aligned} \kappa h=\overbrace{\alpha_{t}\left(\theta_{t}\left(x_{s}, h\right)\right)}^{\text {prob of match }} \overbrace{\left(p_{t}^{H} h-x_{s}\right)}^{\text {broker revenue }}\end{aligned} + +

    +
    • +
+

Buying decision of a household who does not own a house (recently sold or never owned)


+ + +\begin{aligned} +V_{N O w n}^{t}\left(a_{t}, z_{t}\right)=\max _{B u y_{t} \in\{0,1\}} B u y_{t} V_{B u y}^{t}\left(a_{t}, z_{t}\right)+\left(1-B u y_{t}\right) V_{R e n t}^{t}\left(a_{t}, z_{t}\right) +\end{aligned} + +

+ +
+
+
+
+
+
+

Household who buys a house


+ + +\begin{aligned} + V_{B u y}^{t}\left(a_{t}, z_{t}\right)=\max _{h_t, M_{t+1}, b_{t+1}, c_{t}, l_{t} \geq 0} u\left(c_{t}, h_{t}, l_{t}\right)+\beta_{L} \mathbb{E}\left[V_{o w n}^{t+1}\left(a_{t+1}, M_{t+1}, h_{t+1}, z_{t+1}\right)\right]\end{aligned} +\begin{aligned} s.t. P_{t} c_{t}+q_{t}^{B} b_{t+1}+p_{t}^{H} h_{t} & \leq a_{t}+q_{m t}^{0} M_{t+1} \\ a_{t+1} &=P_{t+1} w_{t+1} z_{t+1} l_{t+1}+b_{t+1} \end{aligned} + +

+
    +
  • involves decisions of house price, mortgage debt payment, bond holdings, consumption and labor supply
    • +
  • Banks issue long-term, adjustable rate mortgage contracts.
      +
    • Borrower received nominal resources $q_{m t}^{0} M_{t+1}$ ($q_{m t}^{0}$ - mortgage price at origination).
      • +
    • Interest rate on the loan $r_{m t}$ adjusts each period:
      + +\begin{aligned}1+r_{m t}=\underbrace{(1+\phi)}_{\text {spread }} \underbrace{\left(1+r_{t}\right)\left(1+\pi_{t}\right)}_{\text {nominal risk-free rate }}\end{aligned} +
      • +
    +
    • +
+

Household who rents a house


+ + +\begin{aligned} +V_{\text {Rent }}^{t}\left(a_{t}, z_{t}\right)=\max _{b_{t+1}, s_{t}, c_{t}, l_{t} \geq 0} u\left(c_{t}, h_{t}, l_{t}\right)+\beta_{L} \mathbb{E}\left[V_{N O w n}^{t+1}\left(a_{t+1}, z_{t+1}\right)\right] +\end{aligned} +\begin{aligned} s.t. P_{t} c+q_{t}^{B} b_{t+1}+P_{t} r_{s} s_{t} & \leq a_{t} \\ s_{t} & \leq \bar{s} \\ a_{t+1} &=P_{t+1} w_{t+1} z_{t+1} l_{t+1}+b_{t+1} \end{aligned} + +
+

+
    +
  • $s_t$ - service flow from renting a house
    • +
+ +
+
+
+
+
+
+

Mortgages and Banks - two types of risks when issuing loans

1. Borrowers default - foreclosure

+
    +
  • Bank sells the repossessed house (REO)
      +
    • in a frictional decentralized housing market as individual sellers to
      • +
    • incurs a loss $\gamma^{ROE}$
      • +
    +
    • +

  • Value to banks of repossessing a house size $h$:

    +

    + + \begin{aligned} J_{R E O}^{t}(h)=R_{R E O}^{t}(h)-\eta h+\frac{1}{1+r_{t+1}} J_{R E O}^{t+1}(h) + \end{aligned} + +

    +
    • +
+

+ +\begin{aligned} R_{R E O}^{t}(h)=\max \left\{0, \max _{x_{s} \geq 0} \tilde{p}_{t}\left(\theta_{t}\left(x_{s}, h\right)\right)\left[\left(1-\gamma^{R E O}\right) x_{s}-\left(-\eta h+\frac{1}{1+r_{t+1}} J_{R E O}^{t+1}(h)\right)\right]\right\} + \end{aligned} + +

+
    +
  • $\eta$ - the cost of holding onto the house (maintenance, property taxes, etc.)
    • +
  • $R_{R E O}^{t}(h)$ - the option value of trying to sell the house in period $t$
    • +
+

+

2. Prepayment and refinancing the loan by households

+ +
+
+
+
+
+
+

Mortgages and Banks - two types of risks when issuing loans


+
    +
  • Banks take these risks into account and determine the mortgage price $q_{m t}^{0}$ accordingly:
    • +
+


+ + + \begin{aligned} q_{m t}\left(M_{t+1}, b_{t+1}, h_{t}, z_{t}\right) M_{t+1}=\frac{1}{\left(1+r_{m t}\right)} \mathbb{E}\{\overbrace{\tilde{p}_{t+1}\left(\theta_{t+1}\left(x_{s t+1}, h_{t}\right)\right) M_{t+1}}^{\text {sell }+\text { repay }}+\overbrace{\left[1-\tilde{p}_{t+1}\left(\theta_{t+1}\left(x_{s t+1}, h_{t}\right)\right)\right]}^{\text {no sale (do not try } / \text { fail })} +\times[\underbrace{d_{t+1} \min \left\{P_{t+1} J_{R E O}(h), M_{t+1}\right\}}_{\text {default + repossession }}+\operatorname{Refi}_{t+1} M_{t+1} +\left.\left.+\left(1-d_{t+1}-\operatorname{Refi}_{t+1}\right)(\underbrace{M_{t+1}-\frac{M_{t+2}}{\left(1+r_{m t+1}\right)}}_{\text {borrower payment net of servicing costs }}+\underbrace{q_{m t+1}\left(M_{t+2}, b_{t+2}, h_{t}, z_{t+1}\right) M_{t+2}}_{\text {continuation value of new } M^{\prime \prime}})\right]\right\} + \end{aligned} + +

+ +
+
+
+
+
+
+

Additional Ingredients:


+

+
    +
  • To close the model:
      +
    • Government sponsored enterprises
        +
      • provide insurance to the banks against the default risk of mortgages
        • +
      +
      • +
    • Intermediate goods produces - monopolistically competitive
      • +
    • A representative final goods produce
        +
      • intermediate goods into the final consumption good
        • +
      +
      • +
    • Government - sets fiscal policy
      • +
    • Monetary authority - sets the nominal interest rate
      • +
    +
    • +
+ +
+
+
+
+
+
+

Results:

    +
  • The calibrated steady state of the model
      +
    • matches heterogeneity in home ownership, leverage and MPC across households
        +
      • MPC differs between low- and high-loan-to-value (LTV) households)
        • +
      +
      • +
    • the US microeconomic & macroeconomic data over the past 20 years
      • +
    +
    • +
  • Effectiveness of the monetary policy depends on mortgage debt distribution:
      +
    • In a high-LTV economy expansionary monetary policy is more effective
      • +
    +
    • +
+ +
+
+
+
+
+ + + + + + diff --git a/models/We-Would-Like-In-Econ-ARK/HKMOHousingChannelMP/HKMOHousingChannelMP.ipynb b/models/We-Would-Like-In-Econ-ARK/HKMOHousingChannelMP/HKMOHousingChannelMP.ipynb new file mode 100644 index 00000000..e63643a1 --- /dev/null +++ b/models/We-Would-Like-In-Econ-ARK/HKMOHousingChannelMP/HKMOHousingChannelMP.ipynb @@ -0,0 +1,295 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "

Paper 1: \"Monetary Policy, Heterogeneity, and the Housing Channel\"
\n", + "

\n", + "
Hedlund, Karahan, Mitman, Ozkan (2017)
\n", + "

\n", + "
Nino Kodua
\n", + "

\n", + "
Johns Hopkins University
\n", + "

\n", + "
February 16, 2020
" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "# Overview\n", + "\n", + "* **Research Question: What is the extent to which housing and mortgage debt affect the transmission and effectiveness of monetary policy?** \n", + "* **Channels to consider:** \n", + " * **Direct intertemporal substitution effect**\n", + " * **House price channel**\n", + " * **Liquidity channel**\n", + " * **Cash-flow channel**\n", + " * **Redistribution channel**\n", + "* **HANK model with nominal rigidities and a frictional housing market** \n", + "\n" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "# Households\n", + "\n", + "* **Infinitely lived households with uninsurable idiosyncratic labor productivity risk**\n", + "
" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ + "## Selling decision of a household: \n", + "
\n", + "\n", + " \n", + " \\begin{aligned}\n", + "V_{O W N}\\left(a_{t}, M_{t}, h_{t}, z_{t}\\right)=\\max _{\\varsigma_{t} \\in\\{0,1\\}} \\varsigma_{t} V_{S E L L}\\left(a_{t}, M_{t}, h_{t}, z_{t}\\right)+\\left(1-\\varsigma_{t}\\right) V_{N S e l l}\\left(a_{t}, M_{t}, h_{t}, z_{t}\\right)\n", + " \\end{aligned}\n", + " \n", + "\n", + "\n", + "* ***$\\varsigma_{t}$ - Mortgage initiation cost;*** \n", + "* ***$M_{t}$ - Mortgage size;*** \n", + "* ***$h_{t}$ - House size;*** \n", + "* ***$z_{t}$ - Idiosyncratic labor productivity;***\n", + "\n" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ + "## A household wanting to sell the house chooses a list price ${x_{s}}$\n", + "
\n", + "\n", + " \n", + "\\begin{aligned} \n", + " V_{S E L L}\\left(a_{t}, M_{t}, h_{t}, z_{t}\\right) &=\\max _{x_{s t}}-\\xi+\\tilde{p}_{t}\\left(\\theta_{t}\\left(x_{s}, h\\right)\\right) V_{N O w n}^{t}\\left(a_{t}+x_{s t}-M_{t}, z_{t}\\right) \\\\ &+\\tilde{p}_{t}\\left(\\theta_{t}\\left(x_{s}, h\\right)\\right) V_{N S e l l}\\left(a_{t}, M_{t}, h_{t}, z_{t}\\right) \n", + " \\end{aligned}\n", + " \n", + " \n", + "\n", + "* ***$\\tilde{p}_{t}\\left(\\theta_{t}\\left(x_{s}, h\\right)\\right)$ - Probability of a meeting between a real estate broker and a seller;***\n", + "* ***$\\theta_{t}\\left(x_{s}, h\\right)$ - tightness of the submarket for house size $h$ and a price ${x_{s}}$***\n", + "* ***$\\xi$ - utility cost if homeowner tries but fails to sell the house***\n", + "
\n", + "\n", + "* **Free entry condition for brokers:**\n", + "\n", + " \n", + "\\begin{aligned} \\kappa h=\\overbrace{\\alpha_{t}\\left(\\theta_{t}\\left(x_{s}, h\\right)\\right)}^{\\text {prob of match }} \\overbrace{\\left(p_{t}^{H} h-x_{s}\\right)}^{\\text {broker revenue }}\\end{aligned}\n", + " \n", + " \n", + "\n", + "\n", + "## Buying decision of a household who does not own a house (recently sold or never owned)\n", + "
\n", + "\n", + " \n", + "\\begin{aligned} \n", + "V_{N O w n}^{t}\\left(a_{t}, z_{t}\\right)=\\max _{B u y_{t} \\in\\{0,1\\}} B u y_{t} V_{B u y}^{t}\\left(a_{t}, z_{t}\\right)+\\left(1-B u y_{t}\\right) V_{R e n t}^{t}\\left(a_{t}, z_{t}\\right)\n", + "\\end{aligned}\n", + " \n", + " " + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ + "## Household who buys a house\n", + "
\n", + "\n", + " \n", + "\\begin{aligned} \n", + " V_{B u y}^{t}\\left(a_{t}, z_{t}\\right)=\\max _{h_t, M_{t+1}, b_{t+1}, c_{t}, l_{t} \\geq 0} u\\left(c_{t}, h_{t}, l_{t}\\right)+\\beta_{L} \\mathbb{E}\\left[V_{o w n}^{t+1}\\left(a_{t+1}, M_{t+1}, h_{t+1}, z_{t+1}\\right)\\right]\\end{aligned} \n", + "\\begin{aligned} s.t. P_{t} c_{t}+q_{t}^{B} b_{t+1}+p_{t}^{H} h_{t} & \\leq a_{t}+q_{m t}^{0} M_{t+1} \\\\ a_{t+1} &=P_{t+1} w_{t+1} z_{t+1} l_{t+1}+b_{t+1} \\end{aligned} \n", + " \n", + " \n", + "\n", + "\n", + "* **involves decisions of house price, mortgage debt payment, bond holdings, consumption and labor supply** \n", + "* **Banks issue long-term, adjustable rate mortgage contracts.** \n", + " * **Borrower received nominal resources $q_{m t}^{0} M_{t+1}$ ($q_{m t}^{0}$ - mortgage price at origination).**\n", + " * **Interest rate on the loan $r_{m t}$ adjusts each period:** \n", + " \n", + " \\begin{aligned}1+r_{m t}=\\underbrace{(1+\\phi)}_{\\text {spread }} \\underbrace{\\left(1+r_{t}\\right)\\left(1+\\pi_{t}\\right)}_{\\text {nominal risk-free rate }}\\end{aligned} \n", + " \n", + "\n", + "\n", + "## Household who rents a house\n", + "
\n", + "\n", + " \n", + "\\begin{aligned} \n", + "V_{\\text {Rent }}^{t}\\left(a_{t}, z_{t}\\right)=\\max _{b_{t+1}, s_{t}, c_{t}, l_{t} \\geq 0} u\\left(c_{t}, h_{t}, l_{t}\\right)+\\beta_{L} \\mathbb{E}\\left[V_{N O w n}^{t+1}\\left(a_{t+1}, z_{t+1}\\right)\\right]\n", + "\\end{aligned} \n", + "\\begin{aligned} s.t. P_{t} c+q_{t}^{B} b_{t+1}+P_{t} r_{s} s_{t} & \\leq a_{t} \\\\ s_{t} & \\leq \\bar{s} \\\\ a_{t+1} &=P_{t+1} w_{t+1} z_{t+1} l_{t+1}+b_{t+1} \\end{aligned}\n", + " \n", + " \n", + "
\n", + "\n", + "* **$s_t$ - service flow from renting a house**" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "## Mortgages and Banks - two types of risks when issuing loans\n", + "\n", + "**1. Borrowers default - foreclosure**\n", + " * **Bank sells the repossessed house (REO)** \n", + " * **in a frictional decentralized housing market as individual sellers to**\n", + " * **incurs a loss $\\gamma^{ROE}$**\n", + " * **Value to banks of repossessing a house size $h$:** \n", + " \n", + " \n", + " \n", + " \\begin{aligned} J_{R E O}^{t}(h)=R_{R E O}^{t}(h)-\\eta h+\\frac{1}{1+r_{t+1}} J_{R E O}^{t+1}(h)\n", + " \\end{aligned}\n", + " \n", + " \n", + " \n", + "\n", + " \n", + "\\begin{aligned} R_{R E O}^{t}(h)=\\max \\left\\{0, \\max _{x_{s} \\geq 0} \\tilde{p}_{t}\\left(\\theta_{t}\\left(x_{s}, h\\right)\\right)\\left[\\left(1-\\gamma^{R E O}\\right) x_{s}-\\left(-\\eta h+\\frac{1}{1+r_{t+1}} J_{R E O}^{t+1}(h)\\right)\\right]\\right\\}\n", + " \\end{aligned}\n", + " \n", + " \n", + " \n", + "* ***$\\eta$ - the cost of holding onto the house (maintenance, property taxes, etc.)*** \n", + "* ***$R_{R E O}^{t}(h)$ - the option value of trying to sell the house in period $t$***\n", + "\n", + "\n", + " \n", + "**2. Prepayment and refinancing the loan by households**\n", + " \n", + "\n" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ + "## Mortgages and Banks - two types of risks when issuing loans\n", + "
\n", + "\n", + "* **Banks take these risks into account and determine the mortgage price $q_{m t}^{0}$ accordingly:**\n", + "\n", + "
\n", + " \n", + " \n", + " \\begin{aligned} q_{m t}\\left(M_{t+1}, b_{t+1}, h_{t}, z_{t}\\right) M_{t+1}=\\frac{1}{\\left(1+r_{m t}\\right)} \\mathbb{E}\\{\\overbrace{\\tilde{p}_{t+1}\\left(\\theta_{t+1}\\left(x_{s t+1}, h_{t}\\right)\\right) M_{t+1}}^{\\text {sell }+\\text { repay }}+\\overbrace{\\left[1-\\tilde{p}_{t+1}\\left(\\theta_{t+1}\\left(x_{s t+1}, h_{t}\\right)\\right)\\right]}^{\\text {no sale (do not try } / \\text { fail })}\n", + "\\times[\\underbrace{d_{t+1} \\min \\left\\{P_{t+1} J_{R E O}(h), M_{t+1}\\right\\}}_{\\text {default + repossession }}+\\operatorname{Refi}_{t+1} M_{t+1}\n", + "\\left.\\left.+\\left(1-d_{t+1}-\\operatorname{Refi}_{t+1}\\right)(\\underbrace{M_{t+1}-\\frac{M_{t+2}}{\\left(1+r_{m t+1}\\right)}}_{\\text {borrower payment net of servicing costs }}+\\underbrace{q_{m t+1}\\left(M_{t+2}, b_{t+2}, h_{t}, z_{t+1}\\right) M_{t+2}}_{\\text {continuation value of new } M^{\\prime \\prime}})\\right]\\right\\}\n", + " \\end{aligned}\n", + " \n", + " " + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ + "## Additional Ingredients: \n", + "
\n", + "\n", + " \n", + "* **To close the model:**\n", + " * **Government sponsored enterprises**\n", + " * **provide insurance to the banks against the default risk of mortgages**\n", + " * **Intermediate goods produces - monopolistically competitive**\n", + " * **A representative final goods produce**\n", + " * **intermediate goods into the final consumption good**\n", + " * **Government - sets fiscal policy**\n", + " * **Monetary authority - sets the nominal interest rate**\n", + " " + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "## Results:\n", + "\n", + "* **The calibrated steady state of the model** \n", + " * **matches heterogeneity in home ownership, leverage and MPC across households**\n", + " * **MPC differs between low- and high-loan-to-value (LTV) households)**\n", + " * **the US microeconomic & macroeconomic data over the past 20 years**\n", + "* **Effectiveness of the monetary policy depends on mortgage debt distribution:**\n", + " * **In a high-LTV economy expansionary monetary policy is more effective**" + ] + } + ], + "metadata": { + "celltoolbar": "Slideshow", + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.7.3" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/models/We-Would-Like-In-Econ-ARK/PopAgingMPtransmission/PopAgingMPtransmission.ipynb b/models/We-Would-Like-In-Econ-ARK/PopAgingMPtransmission/PopAgingMPtransmission.ipynb new file mode 100644 index 00000000..f8389b34 --- /dev/null +++ b/models/We-Would-Like-In-Econ-ARK/PopAgingMPtransmission/PopAgingMPtransmission.ipynb @@ -0,0 +1,207 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "

Paper 2: \"Population Aging and the Transmission of Monetary Policy to Consumption\"
\n", + "

\n", + "
Arlene Wond (2015)
\n", + "

\n", + "
Nino Kodua
\n", + "

\n", + "
Johns Hopkins University
\n", + "

\n", + "
February 16, 2020
" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "# Overview\n", + "\n", + "* **Research Question: What are the effects of demographic changes on the transmission of monetary policy to consumption?**\n", + "* **Part I: Empirical estimate**\n", + " * **Age-specific consumption elasticities to interest rate**\n", + " * **Consumption of young people is two-three times more responsive to interest rate shocks**\n", + " * **Consumption elasticities decline with age**\n", + " * **Consumption response driven by homeowners**\n", + " * **refinance or new loans following interest rate decline**\n", + "* **Part II: Life-cycle model - partial equilibrium overlapping generations economy**\n", + " * **Uninsurable labor income risk, life-cycle saving motive, fixed-rate mortgage structure**\n", + " * **Fixed costs to adjust long-term assets**\n", + " * **housing and fixed-rate mortgage**" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "# Household's maximization problem\n", + "\n", + "* **Each period households choose**\n", + " * **1. rent**\n", + " * **2. continue owning a house & not adjust existing mortgage**\n", + " * **3. adjust mortgage and housing stock**\n", + "* **Conditional on the adjustment, households choose**\n", + " * **non-durable consumption, savings in bonds and mortgage debt**\n", + "
\n", + "\n", + " \n", + "\\begin{aligned}V_{j a t}\\left(z_{j a t}\\right)=\\max \\left\\{V_{j a t}\\left(z_{j a t}\\right)^{\\mathrm{rent}}, V_{j a t}\\left(z_{j a t}\\right)^{\\mathrm{own} \\& \\text { no-adjust }}, V_{j a t}\\left(z_{j a t}\\right)^{\\text {own } \\& \\text { adjust }}\\right\\}\\end{aligned} \n", + " \n", + " \n", + "\n", + "\n", + " \n", + " \\begin{aligned}\n", + "z_{j a t}=\\left\\{S_{t}, y_{j a t}, \\operatorname{assets}_{j, a-1, t-1}\\right\\}\n", + "\\end{aligned} \n", + " \n", + " \n", + "\n", + "* **$S_t$ - uncertainty from aggregate state variables**\n", + "* **$y_{j a t}$ - uncertainty from idiosyncratic labor income**" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ + "# 1. Rent Case\n", + "\n", + "\n", + " \n", + "\\begin{aligned}V_{j a t}\\left(z_{j a t}\\right)^{\\mathrm{rent}}=\\max _{c_{j a t}, h_{j a t}^{\\mathrm{rat}}, s_{j a t}} \\frac{\\left(c_{j a t}^{\\alpha} \\cdot h_{j a t}^{1-\\alpha}\\right)^{1-\\sigma}-1}{1-\\sigma}+E_{j a t}\\left[V_{j, a+1, t+1}\\left(z_{j, a+1, t+1}\\right)\\right]\\end{aligned} \n", + " \n", + " \n", + "\n", + "\n", + " \n", + " \\begin{aligned} s.t. h_{j a t} &=h_{j a t}^{\\text {rent }} \\\\ c_{j a t}+s_{j a t}+p_{t}^{r} h_{j a t}^{\\text {rent }} &=y_{j a t}+(1-\\delta) p_{t} h_{j, a-1, t-1}^{\\text {own }}+\\left(1+r_{t}\\right) s_{j, a-1, t-1}-b_{j, a-1, t-1}\\left(1+R_{j, a-1, t-1}\\right) \\\\ h_{j a t}^{\\text {own }} &=b_{j a t}=0 \\\\ s_{j a t} & \\geq-\\underline{s} \\\\ \\log \\left(y_{j a t}\\right) &=\\chi_{a}+\\eta_{j a t}+\\phi_{a}\\left(y_{t} / y\\right) \\\\ S_{t} &=A_{0}+A_{1} S_{t-1}+u_{t} \\end{aligned}\n", + " \n", + " " + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ + "# 2. Own & No-adjust Case\n", + "\n", + "\n", + " \n", + "\\begin{aligned}V_{j a t}\\left(z_{j a t}\\right)^{\\text {own \\& no-adjust }}=\\max _{c_{j a t}, s_{j a t}} \\frac{\\left(c_{j a t}^{\\alpha} \\cdot h_{j a t}^{1-\\alpha}\\right)^{1-\\sigma}-1}{1-\\sigma}+E_{j a t}\\left[V_{j, a+1, t+1}\\left(z_{j, a+1, t+1}\\right)\\right]\\end{aligned} \n", + " \n", + " \n", + "\n", + "\n", + " \n", + " \\begin{aligned} s.t. h_{j a t} &=(1-\\delta) h_{j, a, t-1}^{\\mathrm{own}} \\\\ c_{j a t}+s_{j a t} &=y_{j a t}+\\left(1+r_{t}\\right) s_{j, a-1, t-1}-M_{j a t} \\\\ s_{j a t} & \\geq-\\underline{s} \\\\ \\log \\left(y_{j a t}\\right) &=\\chi_{a}+\\eta_{j a t}+\\phi_{a}\\left(y_{t} / y\\right) \\\\ S_{t} &=A_{0}+A_{1} S_{t-1}+u_{t} \\end{aligned}\n", + " \n", + " " + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ + "# 3. Own & Adjust Case\n", + "\n", + "\n", + " \n", + "\\begin{aligned}V_{j a t}\\left(z_{j a t}\\right)^{\\text {own \\& adjust }}=\\max _{c_{j a t}, s_{j a t}, h_{j a t}^{\\text {own }}, b_{j a t}} \\frac{\\left(c_{j a t}^{\\alpha} \\cdot h_{j a t}^{1-\\alpha}\\right)^{1-\\sigma}-1}{1-\\sigma}+E_{j a t}\\left[V_{j, a+1, t+1}\\left(z_{j, a+1, t+1}\\right)\\right] \\end{aligned} \n", + " \n", + " \n", + "\n", + "\n", + " \n", + " \\begin{aligned} s.t. h_{j a t} &=h_{j, a}^{\\mathrm{own}} \\\\ b_{j a t} & \\leq(1-\\phi) p_{t} h_{j a t}^{\\mathrm{own}} \\\\ c_{j a t}+s_{j a t}+p_{t} h_{j a t}^{\\mathrm{own}}-b_{j a t}-F &=y_{j a t}+(1-\\delta) p_{t} h_{j, a-1, t-1}^{\\mathrm{own}}+\\left(1+r_{t}\\right) s_{j, a-1, t-1}-b_{j, a-1, t-1}\\left(1+R_{j, a-1, t-1}\\right) \\\\ s_{j a t} & \\geq-\\underline{s} \\\\ \\log \\left(y_{j a t}\\right) &=\\chi_{a}+\\eta_{j a t}+\\phi_{a}\\left(y_{t} / y\\right) \\\\ S_{t} &=A_{0}+A_{1} S_{t-1}+u_{t} \\end{aligned} \n", + " \n", + " " + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ + "# Retirement and Death\n", + "\n", + "* **Upon death, agent bequasts total new wealth:**\n", + "\n", + "\n", + " \n", + "\\begin{aligned}W_{j a t}=(1-\\delta) p_{t} h_{j, a-1, t-1}^{\\mathrm{own}}+\\left(1+r_{t}\\right) s_{j, a-1, t-1}\n", + " \\end{aligned} \n", + " \n", + " " + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "# Results\n", + "\n", + "* **The refinancing and new lending channel** \n", + " * **the difference in the consumption responses of the young-old to interest rate shocks**\n", + "* **Aging population can significantly dampen the transmission of monetary policy** " + ] + } + ], + "metadata": { + "celltoolbar": "Slideshow", + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.7.3" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/models/We-Would-Like-In-Econ-ARK/PopAgingMPtransmission/PopAgingMPtransmission.slides.html b/models/We-Would-Like-In-Econ-ARK/PopAgingMPtransmission/PopAgingMPtransmission.slides.html new file mode 100644 index 00000000..16cfaaf2 --- /dev/null +++ b/models/We-Would-Like-In-Econ-ARK/PopAgingMPtransmission/PopAgingMPtransmission.slides.html @@ -0,0 +1,13474 @@ + + + + + + + + + + + + +PopAgingMPtransmission slides + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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Paper 2: "Population Aging and the Transmission of Monetary Policy to Consumption"
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+
Arlene Wond (2015)
 +

+
Nino Kodua
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+
Johns Hopkins University
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+
February 16, 2020

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+

Overview

    +
  • Research Question: What are the effects of demographic changes on the transmission of monetary policy to consumption?
    • +
  • Part I: Empirical estimate
      +
    • Age-specific consumption elasticities to interest rate
        +
      • Consumption of young people is two-three times more responsive to interest rate shocks
        • +
      • Consumption elasticities decline with age
        • +
      +
      • +
    • Consumption response driven by homeowners
        +
      • refinance or new loans following interest rate decline
        • +
      +
      • +
    +
    • +
  • Part II: Life-cycle model - partial equilibrium overlapping generations economy
      +
    • Uninsurable labor income risk, life-cycle saving motive, fixed-rate mortgage structure
      • +
    • Fixed costs to adjust long-term assets
        +
      • housing and fixed-rate mortgage
        • +
      +
      • +
    +
    • +
+ +
+
+
+
+
+
+

Household's maximization problem

    +
  • Each period households choose
      +
    • 1. rent
      • +
    • 2. continue owning a house & not adjust existing mortgage
      • +
    • 3. adjust mortgage and housing stock
      • +
    +
    • +
  • Conditional on the adjustment, households choose
      +
    • non-durable consumption, savings in bonds and mortgage debt +
      + + +\begin{aligned}V_{j a t}\left(z_{j a t}\right)=\max \left\{V_{j a t}\left(z_{j a t}\right)^{\mathrm{rent}}, V_{j a t}\left(z_{j a t}\right)^{\mathrm{own} \& \text { no-adjust }}, V_{j a t}\left(z_{j a t}\right)^{\text {own } \& \text { adjust }}\right\}\end{aligned} + +
      • +
    +
    • +
+

+ + \begin{aligned} +z_{j a t}=\left\{S_{t}, y_{j a t}, \operatorname{assets}_{j, a-1, t-1}\right\} +\end{aligned} + +

+
    +
  • $S_t$ - uncertainty from aggregate state variables
    • +
  • $y_{j a t}$ - uncertainty from idiosyncratic labor income
    • +
+ +
+
+
+
+
+
+

1. Rent Case

+ +\begin{aligned}V_{j a t}\left(z_{j a t}\right)^{\mathrm{rent}}=\max _{c_{j a t}, h_{j a t}^{\mathrm{rat}}, s_{j a t}} \frac{\left(c_{j a t}^{\alpha} \cdot h_{j a t}^{1-\alpha}\right)^{1-\sigma}-1}{1-\sigma}+E_{j a t}\left[V_{j, a+1, t+1}\left(z_{j, a+1, t+1}\right)\right]\end{aligned} + +

+

+ + \begin{aligned} s.t. h_{j a t} &=h_{j a t}^{\text {rent }} \\ c_{j a t}+s_{j a t}+p_{t}^{r} h_{j a t}^{\text {rent }} &=y_{j a t}+(1-\delta) p_{t} h_{j, a-1, t-1}^{\text {own }}+\left(1+r_{t}\right) s_{j, a-1, t-1}-b_{j, a-1, t-1}\left(1+R_{j, a-1, t-1}\right) \\ h_{j a t}^{\text {own }} &=b_{j a t}=0 \\ s_{j a t} & \geq-\underline{s} \\ \log \left(y_{j a t}\right) &=\chi_{a}+\eta_{j a t}+\phi_{a}\left(y_{t} / y\right) \\ S_{t} &=A_{0}+A_{1} S_{t-1}+u_{t} \end{aligned} + +

+ +
+
+
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2. Own & No-adjust Case

+ +\begin{aligned}V_{j a t}\left(z_{j a t}\right)^{\text {own \& no-adjust }}=\max _{c_{j a t}, s_{j a t}} \frac{\left(c_{j a t}^{\alpha} \cdot h_{j a t}^{1-\alpha}\right)^{1-\sigma}-1}{1-\sigma}+E_{j a t}\left[V_{j, a+1, t+1}\left(z_{j, a+1, t+1}\right)\right]\end{aligned} + +

+

+ + \begin{aligned} s.t. h_{j a t} &=(1-\delta) h_{j, a, t-1}^{\mathrm{own}} \\ c_{j a t}+s_{j a t} &=y_{j a t}+\left(1+r_{t}\right) s_{j, a-1, t-1}-M_{j a t} \\ s_{j a t} & \geq-\underline{s} \\ \log \left(y_{j a t}\right) &=\chi_{a}+\eta_{j a t}+\phi_{a}\left(y_{t} / y\right) \\ S_{t} &=A_{0}+A_{1} S_{t-1}+u_{t} \end{aligned} + +

+ +
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3. Own & Adjust Case

+ +\begin{aligned}V_{j a t}\left(z_{j a t}\right)^{\text {own \& adjust }}=\max _{c_{j a t}, s_{j a t}, h_{j a t}^{\text {own }}, b_{j a t}} \frac{\left(c_{j a t}^{\alpha} \cdot h_{j a t}^{1-\alpha}\right)^{1-\sigma}-1}{1-\sigma}+E_{j a t}\left[V_{j, a+1, t+1}\left(z_{j, a+1, t+1}\right)\right] \end{aligned} + +

+

+ + \begin{aligned} s.t. h_{j a t} &=h_{j, a}^{\mathrm{own}} \\ b_{j a t} & \leq(1-\phi) p_{t} h_{j a t}^{\mathrm{own}} \\ c_{j a t}+s_{j a t}+p_{t} h_{j a t}^{\mathrm{own}}-b_{j a t}-F &=y_{j a t}+(1-\delta) p_{t} h_{j, a-1, t-1}^{\mathrm{own}}+\left(1+r_{t}\right) s_{j, a-1, t-1}-b_{j, a-1, t-1}\left(1+R_{j, a-1, t-1}\right) \\ s_{j a t} & \geq-\underline{s} \\ \log \left(y_{j a t}\right) &=\chi_{a}+\eta_{j a t}+\phi_{a}\left(y_{t} / y\right) \\ S_{t} &=A_{0}+A_{1} S_{t-1}+u_{t} \end{aligned} + +

+ +
+
+
+
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+

Retirement and Death

    +
  • Upon death, agent bequasts total new wealth:
    • +
+

+ +\begin{aligned}W_{j a t}=(1-\delta) p_{t} h_{j, a-1, t-1}^{\mathrm{own}}+\left(1+r_{t}\right) s_{j, a-1, t-1} + \end{aligned} + +

+ +
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Results

    +
  • The refinancing and new lending channel
      +
    • the difference in the consumption responses of the young-old to interest rate shocks
      • +
    +
    • +
  • Aging population can significantly dampen the transmission of monetary policy
    • +
+ +
+
+
+
+
+ + + + + + +