Intertemporal decision making

Introduction

Intertemporal decision theory

$U_T=\sum _{[}t=T]d_{t}U(x_t)$

Types of discounting

Exponential discounting

Introduction

We have:

$U_T=\sum _{[}t=T{]}^{\mathrm{\infty }}{d}_{t}U(x_t)$

Exponential discounting

$d_t=(1+\delta {)}^t$

$U_T=\sum _{[}t=T{]}^{\mathrm{\infty }}(1+\delta {)}^{t}U(x_t)$

$\delta$ is the discount rate.

Hyperbolic discounting

Introduction

We have:

$U_T=\sum _{[}t=T{]}^{\mathrm{\infty }}{d}_{t}U(x_t)$

Hyerbolic discounting

$d_t={\displaystyle \frac{1}{1+kt}}$

$U_T=\sum _{[}t=T{]}^{\mathrm{\infty }}{\displaystyle \frac{1}{1+kt}}U(x_t)$

$k$ is the discount parameter.

Quasi-hyperbolic discounting

Introduction

We have:

$U_T=\sum _{[}t=T{]}^{\mathrm{\infty }}{d}_{t}U(x_t)$

Quasi-hyperbolic discounting

$d_0=1$

$d_t=\beta {\delta }^t$

$U_T=U(x_0)+\sum _{[}t=T+1{]}^{\mathrm{\infty }}\beta {\delta }^{t}U(x_t)$

$\delta$ is the discount rate.

Intertemporal economics

Intertemporal discounting

$u_t=f(x_t)$

$U_t=\sum _{i=t}u(x_i)d^i$

Euler’s equation

Hyperbolic discounting