• closed class words - words with fixed memberships (prepositions, conjunctivas, etc.); not being created or added much, used for grammatical function
• open class words - words that are set as content, and are focused on content

For some \(a \in \mathbb{F}\), we define \(a^m\) to be \(a\) multiplied with itself \(m\) times.

additional information

• \((a^m)^n = a^{mn}\)
• \((ab)^m = a^mb^m\)

We can now use power series to also solve differential equations.

\begin{equation} \dv{x}{t} = 0; x(0)=1 \end{equation}

We wish to have a power-series solution of shape:

\begin{equation} x(t) = \sum_{k=0}^{\infty }a_{k}t^{k} \end{equation}

We want to find the coefficients \(a_{k}\). If you can find such a function that fits this form, they both 1) converge and 20 behave the same way as \(e^{x}\) does in Simple Differential Equations.

analytic functions

Functions which can be described with a power series are called analytic functions.

the power set of \(A\) is the set of all subsets of \(A\). that is, the power set of \(\qty {0,1}\) is \(\qty {\emptyset, \qty {0}, \qty {1}, \qty {0,1}}\).

power utility, or isoelastic utility, is a financial econometric is a utility that results absolute, constant relative risk aversion. i.e.: you tell me how risk averse you are exogenously, I tell you how much utility some consumption is.

constituents

• some relative risk coefficient \(\gamma \in (0,1)\), higher more risk averse
• consumption of some asset \(C\)

requirements

Utility \(U( C)\) is defined by:

\begin{equation} U( C) = \frac{c^{1-\gamma}-1}{1-\gamma} \end{equation}

additional information

As you can see, the higher \(\gamma\), the lower utility some consumption brings.