# Intertemporal decision making

## Introduction

### Intertemporal decision theory

\(U_T=\sum_[t=T]d_tU(x_t)\)

## Types of discounting

### Exponential discounting

#### Introduction

We have:

\(U_T=\sum_[t=T]^{\infty }d_tU(x_t)\)

#### Exponential discounting

\(d_t=(1+\delta )^t\)

\(U_T=\sum_[t=T]^{\infty }(1+\delta )^tU(x_t)\)

\(\delta \) is the discount rate.

### Hyperbolic discounting

#### Introduction

We have:

\(U_T=\sum_[t=T]^{\infty }d_tU(x_t)\)

#### Hyerbolic discounting

\(d_t=\dfrac{1}{1+kt}\)

\(U_T=\sum_[t=T]^{\infty }\dfrac{1}{1+kt}U(x_t)\)

\(k\) is the discount parameter.

### Quasi-hyperbolic discounting

#### Introduction

We have:

\(U_T=\sum_[t=T]^{\infty }d_tU(x_t)\)

#### Quasi-hyperbolic discounting

\(d_0=1\)

\(d_t=\beta \delta^t\)

\(U_T=U(x_0)+\sum_[t=T+1]^{\infty }\beta \delta ^tU(x_t)\)

\(\delta \) is the discount rate.

## Intertemporal economics

### Intertemporal discounting

\(u_{t}=f(x_t)\)

\(U_{t}=\sum_{i=t}u(x_i)d^i\)

### Eulerâ€™s equation

### Hyperbolic discounting