# tcilatex Estimating the Pricing Kernel††thanks: We thank cool dudes

AbstractWe do cool stuff.

tcilatex This version: November 19, 2017

Writing stuff here

tcilatex

## 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 $$p_{k}$$

tcilatex

## Value Function Derivation

Problem of type $$i$$

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

Budjet constraint

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

Conjecture

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

Envelope condition

$$\frac{\psi_{i}}{a_{i}}=\frac{1}{pc_{i}}=\frac{1}{\Gamma_{i}a_{i}}\nonumber \\$$

which implies that

$$\label{c1} \label{c1}\Gamma_{i}\psi_{i}=1.\\$$

Euler equation

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

which implies that

$$\label{c2} \label{c2}a_{i}^{\prime}=\frac{\beta_{i}}{Q^{N}}a_{i}.\\$$

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

$$\Gamma_{i}\frac{a_{i}}{p}=\frac{a_{i}}{p}-\frac{\pi\beta a_{i}}{p}\nonumber \\$$

Canceling $$a_{i}$$ and $$p$$ gives

$$\Gamma_{i}=1-\beta_{i}\pi.\nonumber \\$$