From b6fbcedfa66e8c88ecefe46130f789dc04ca3b02 Mon Sep 17 00:00:00 2001 From: JingkunZhao Date: Mon, 18 Mar 2024 12:26:30 +1100 Subject: [PATCH] [geom_series] Update editorial suggestions Resolve most of the comments --- lectures/geom_series.md | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/lectures/geom_series.md b/lectures/geom_series.md index a18ed010..7a7d4c9d 100644 --- a/lectures/geom_series.md +++ b/lectures/geom_series.md @@ -47,7 +47,7 @@ Among these are These and other applications prove the truth of the wise crack that ```{epigraph} -"in economics, a little knowledge of geometric series goes a long way " +"In economics, a little knowledge of geometric series goes a long way." ``` Below we'll use the following imports: @@ -171,7 +171,7 @@ The right side records bank $i$'s liabilities, namely, the deposits $D_i$ held by its depositors; these are IOU's from the bank to its depositors in the form of either checking accounts or savings accounts (or before 1914, bank notes issued by a -bank stating promises to redeem note for gold or silver on demand). +bank stating promises to redeem notes for gold or silver on demand). Each bank $i$ sets its reserves to satisfy the equation @@ -573,7 +573,7 @@ Recall that $R = 1+r$ and $G = 1+g$ and that $R > G$ and $r > g$ and that $r$ and $g$ are typically small numbers, e.g., .05 or .03. -Use the Taylor series of $\frac{1}{1+r}$ about $r=0$, +Use the [Taylor series](https://en.wikipedia.org/wiki/Taylor_series) of $\frac{1}{1+r}$ about $r=0$, namely, $$ @@ -641,7 +641,7 @@ $$ Expanding: $$ -\begin{aligned} p_0 &=\frac{x_0(1-1+(T+1)^2 rg -r(T+1)+g(T+1))}{1-1+r-g+rg} \\&=\frac{x_0(T+1)((T+1)rg+r-g)}{r-g+rg} \\ &\approx \frac{x_0(T+1)(r-g)}{r-g}+\frac{x_0rg(T+1)}{r-g}\\ &= x_0(T+1) + \frac{x_0rg(T+1)}{r-g} \end{aligned} +\begin{aligned} p_0 &=\frac{x_0(1-1+(T+1)^2 rg +r(T+1)-g(T+1))}{1-1+r-g+rg} \\&=\frac{x_0(T+1)((T+1)rg+r-g)}{r-g+rg} \\ &= \frac{x_0(T+1)(r-g)}{r-g + rg}+\frac{x_0rg(T+1)^2}{r-g+rg}\\ &\approx \frac{x_0(T+1)(r-g)}{r-g}+\frac{x_0rg(T+1)}{r-g}\\ &= x_0(T+1) + \frac{x_0rg(T+1)}{r-g} \end{aligned} $$ We could have also approximated by removing the second term