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

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

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

Euler’s equation

Hyperbolic discounting