The Textbook Shows That $\ln ( x + \Delta x ) - \ln ( x ) \cong \frac { \Delta x } { x }$

question 42

Essay

The textbook shows that $\ln ( x + \Delta x ) - \ln ( x ) \cong \frac { \Delta x } { x }$ Show that this is equivalent to the following approximation $\ln ( 1 + y ) \cong y$ if y is small. You use this idea to estimate a demand for money function, which is of the form $m = \beta _ { 0 } \times G D P ^ { \beta _ { 1 } } \times ( 1 + R ) ^ { \beta _ { 2 } } \times e ^ { u } ,$ where m is the quantity of (real) money, G D P is the value of (real) Gross Domestic Product, and R is the nominal interest rate. You collect the quarterly data from the Federal Reserve Bank of St. Louis data bank ("FRED"), which lists the money supply and GDP in billions of dollars, prices as an index, and nominal interest rates in percentage points per year You generate the variables in your regression program as follows: m= (money supply)/price index; GDP = (Gross Domestic Product/Price Index), and R= nominal interest rate in percentage points per annum. Next you perform the log-transformations on the real money supply, real G D P , and on (1+R) . Can you for see a problem in using this transformation?

---

Definitions:

Real Rate of Return

The annual percentage profit earned on an investment, adjusted for changes in the price level due to inflation or deflation.

Annual Interest Rate

The percentage increase in money per year when it is lent or invested, excluding the effects of compounding.

Rate of Inflation

The percentage increase in prices for goods and services over a period, decreasing the purchasing power of money.

Risk Premium

The additional yield beyond the risk-free rate that investors demand as a reward for taking on the risk associated with an investment.