tcilatex Estimating the Pricing Kernelthanks: We thank cool dudes

AbstractWe do cool stuff.

tcilatex This version: November 19, 2017

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Introduction

Parameters

Discount factor of type 1 \(\beta_{1}\)
Discount factor of type 2 \(\beta_{2}\)
Survival probability \(\pi\)
Fraction of type \(i\) \(\mu_{i}\)

Basic variables

Aggregate consumption of all type \(i\) persons \(C_{i}\)
Consumption of all type \(i\) young persons \(C_{i}^{Y}\)
Consumption of all type \(i\) old persons \(C_{i}^{O}\)
Nominal bonds held by all type \(i\) in dollars \(a_{i}\)
Real pricing Kernel \(Q\)
Nominal pricing Kernel \(Q^{N}\)
Dollar price of an apple \(p\)
Apple price of a tree \(p_{k}\)

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Value Function Derivation

Problem of type \(i\)

\begin{equation} \label{VF} \label{VF}V^{i}\left(\frac{a_{i}}{p}\right)=\max_{a_{i}^{\prime}}\left\{\log\left(\frac{a_{i}}{p}-\frac{\pi Q^{N}a_{i}^{\prime}}{p}\right)+E\left[\pi\beta_{i}V^{i}\left(\frac{a_{i}^{\prime}}{p^{\prime}}\right)\right]\right\}\\ \end{equation}

Budjet constraint

\begin{equation} \label{BC} \label{BC}c_{i}=\frac{a_{i}}{p}-\frac{\pi Q^{N}a_{i}^{\prime}}{p}.\\ \end{equation}

Conjecture

\begin{equation} \label{CONJ} \label{CONJ}V\left(\frac{a_{i}}{p}\right)=\psi_{i}\log\left(\frac{a_{i}}{p}\right),\ \ \ \ \ c_{i}=\Gamma_{i}\frac{a_{i}}{p}\\ \end{equation}

Envelope condition

\begin{equation} \frac{\psi_{i}}{a_{i}}=\frac{1}{pc_{i}}=\frac{1}{\Gamma_{i}a_{i}}\nonumber \\ \end{equation}

which implies that

\begin{equation} \label{c1} \label{c1}\Gamma_{i}\psi_{i}=1.\\ \end{equation}

Euler equation

\begin{equation} \label{c10} \label{c10}\frac{\pi Q^{N}}{pc_{i}}=E\left[\frac{\beta_{i}\pi\psi_{i}}{a_{i}^{\prime}}\right]\\ \end{equation}

which implies that

\begin{equation} \label{c2} \label{c2}a_{i}^{\prime}=\frac{\beta_{i}}{Q^{N}}a_{i}.\\ \end{equation}

Using (\ref{BC}), (\ref{c1}), (\ref{CONJ}) and (\ref{c2}

\begin{equation} \Gamma_{i}\frac{a_{i}}{p}=\frac{a_{i}}{p}-\frac{\pi\beta a_{i}}{p}\nonumber \\ \end{equation}

Canceling \(a_{i}\) and \(p\) gives

\begin{equation} \Gamma_{i}=1-\beta_{i}\pi.\nonumber \\ \end{equation}