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The given information to find the rate of growth of per capita real GDP is as follows:

The annual rate of real GDP growth is$0.3\%$

The monetary market worth of all final goods and services produced within a nation's domestic territory during a specific period of time is referred to as GDP.

From the base period to the current time, real GDP is the nominal GDP adjusted by rising price levels (inflation).

The rate of growth of a variable is the rate at which that variable expands over time.

The following is the link between population growth rate, real GDP growth rate, and per-capita real GDP growth rate:$Populationgrowthrate=\left(RealGDPgrowthrate\right)-\left(RateofgrowtrhofpercapitalRealGDP\right)$

Therefore,

$RateofgrowtrhofpercapitalRealGDP=\left(RealGDPgrowthrate\right)-\left(Populationgrowthrate\right)$

To determine the rate of growth of per-capita real GDP in the absence of such a change in the population growth rate, substitute the values of real GDP growth rate $0.3$percent. This is accomplished as follows:

$RateofgrowtrhofpercapitalRealGDP=\left(RealGDPgrowthrate\right)-\left(Populationgrowthrate\right)$

$$\begin{gathered} =0.3\%-0\% \\ =0.3\% \end{gathered}$$

Therefore, the Rate of growth of per capita real GDP before the increase in population growth is$0.3\%$

The following are the values to find the new rate of growth of per capita real GDP is,

Real GDP Growth rate is $0.3\%$

Population Growth rate is$0.5\%$

Assume that the country's birth rate has boosted the population growth rate to $0.5$percent. The real GDP growth rate remained constant at $0.3$percent.

Following an increase in population growth, the rate of growth of per-capital real GDP is computed as follows:

$RateofgrowtrhofpercapitalRealGDP=\left(RealGDPgrowthrate\right)-\left(Populationgrowthrate\right)$$$\begin{gathered} =0.3\%-0.5\% \\ =-0.2\% \end{gathered}$$

Hence, the rate of growth of per capital real GDP following the increase in the birthrate is$-0.2\%$