\[ \Delta y=\beta_1\Delta x\] Rearranging the above equation yields \[ \beta_1 = \Delta y / \Delta x\]
\(\beta_1\) represents the change in y due to one unit increase in x. For example, if \(y=wage\) and \(x=YearsInEducation\), then \(\beta_1\) represents the change in the wage due to one additional year of schooling.
\[ \Delta log(y)=\beta_1\Delta x\] Since \(100*\Delta log(y) \approx \% \Delta y\) for small changes in \(y\), \[ (\% \Delta y) / 100=\beta_1\Delta x\] \[ \beta_1 = (\% \Delta y / 100)/(\Delta x) \quad \quad or \quad \quad 100 \beta_1=(\% \Delta y)/(\Delta x) \] \(100*\beta_1\) represents the percentage change in y due to one unit increase in x. For example, if \(y=wage\) and \(x=YearsInEducation\), then \(100*beta_1\) represents the percentage change in the wage due to one additional year of schooling. If one does not multiply \(beta_1\) by 100, one gets the change in the wage measured as a fraction.
\[ \Delta y=\beta_1\Delta log(x) = (\beta_1/100) [100*\Delta log(x)] \approx (\beta_1/100)(\%\Delta x) \] where we use the fact that \(100*\Delta log(x) \approx \% \Delta x\) for small changes in \(x\). So \(\beta_1/100\) is the (ceteris paribus) change in \(y\) when \(x\) increases by one percent.
\[ \beta_1/100 = (\Delta y)/(\%\Delta x) \] \(\beta_1/100\) represents the change in y due to one percent increase in x. For example, if \(y=NumberOfKidsInFamily\) and \(x=HouseholdIncome\), then \(beta_1/100\) represents the change in the number of kids a family is predicted to have due to one percent increase in houeshold income.
\[ \Delta log(y) = \beta_1 \Delta log(x) \] Since \(100*\Delta log(y) \approx \% \Delta y\) for small changes in \(y\), and \(100*\Delta log(x) \approx \% \Delta x\) for small changes in \(x\), \[ \% \Delta y / 100 = \beta_1 \% \Delta x / 100 \]
Simplification yields the following \[ \% \Delta y = \beta_1 \% \Delta x \quad \quad or \quad \quad \beta_1 = \% \Delta y / \% \Delta x\]
\(\beta_1\) represents the percent change in y due to one percent change in x. For example, if \(y=CEOSalary\) and \(x=FirmSales\), then \(\beta_1\) represents the percent change in the salary of the CEO due to percent change in the firm’s sales.