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Let $Div_t$ be the dividend per share at date $t$.

$\displaystyle Div_t=\frac{\text{Earnings}_t}{\text{Shares Outstanding}_t}\times\text{Dividend Payout Rate}_t$ ------------------- (1)

My textbook says that if $\text{Shares Outstanding}_t$ is constant and $\text{Dividend Payout Rate}_t$ is constant, growth in dividends will equal growth of earnings.

However, differentiating (1), we get

$\displaystyle\frac{d}{dt}Div_t=\frac{\text{Dividend Payout Rate}_t}{\text{Shares Outstanding}_t}\frac{d}{dt}\text{Earnings}_t$

Clearly, $\displaystyle\frac{d}{dt}Div_t\ne\frac{d}{dt}\text{Earnings}_t$

Where have I gone wrong?

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The textbook probably means growth rate and not absolute growth. Given your assumptions $$ \frac{\frac{d}{dt}Div_t}{Div_t}= \frac{\frac{d}{dt}\text{Earnings}_t}{\text{Earnings}_t} $$ does hold.

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