Lecture 44 Discounted Utility Model

Comparison of intertemporal trade-offs required a cardinal (not ordinal) measure of utility.
Intertemporal utility function
- $U_{t}(c_{t},\dots,c_{T})$ =
- describing utility at $t$ of the consumption profile $c_{t},c_{t+1},\dots$ to period $T$
$D(k) = \left( \dfrac{1}{1+\rho}\right)^k$ is the discount factor.

\[ U_{t}(c_{t},\dots,c_{T})$ = \sum_{k=0}^{T-t}D(k)u(c_{t+k}) \]

Mathematics of the Exponential Utility Modej¶

\[ U(C_{0},C_{1}) = u(C_{0}) + \delta u(C_{1}) \]

$\delta$ is the discount factor.
Let $u'(C_{0})$ be the partial derivative of $U$ wrt $C_{0}$
similarly, $\dfrac{\partial U}{\partial C_{1}} = \delta u'(C_{1})$, so the Euler equation would be

\[ \dfrac{u'(C_{0})}{\delta u'(C_{1})}= 1+r \]

We call it "exponential" because utility from period $t$ is discounted by exponential function $\delta^t$

MRS of consumption in period $t$ and that in $t+k$ to their relative price is given by

\[ \dfrac{u'(C_{t})}{\delta ^ku'(C_{t+k})}= (1+r)^k \]