Assume that GDP per capita for two countries is displayed in a plot with a ratio scale on the y-axis and a linear time scale (in years) on the x-axis. If the two times series are straight lines in this plot, then the growth rates are constant over time. If in addition, the two lines are parallel and upward-sloping, then the income gap (where the vertical bars are the symbol for "absolute value") is increasing over time.

Question

Assume that GDP per capita for two countries is displayed in a plot with a ratio scale on the y-axis and a linear time scale (in years) on the x-axis. If the two times series are straight lines in this plot, then the growth rates are constant over time. If in addition, the two lines are parallel and upward-sloping, then the income gap  (where the vertical bars are the symbol for "absolute value") is increasing over time.

BlackTom AI · Accepted Answer

The question presents a scenario where GDP per capita for two countries is plotted with a ratio scale on the y-axis and a linear time axis on the x-axis. It then makes two claims about the shape of the two time series and what that implies about growth and the income gap. We will evaluate each statement in turn.

Option 1: The statement that the growth rates are constant over time when the two time series are straight lines on this plot.
- If the plotted relationship is a straight line when using a ratio scale on the y-axis, the mathematical interpretation is that the ratio (or proportional relationship) between the two GDP per capita series changes at a constant rate with respect to time. A straight line in this setting indicates that the growth in GDP per capita behaves exponentially over time with a constant percentage (or growth) rate, i.e., a constant growth rate over time. This aligns with standard interpretations of log/ratio-scale plots: linearity in time corresponds to constant proportional growth, so the first claim that growth rates are constant is correct.
- A potential point of confusion is to think in terms of an absolute difference growing linearly; however, on a ratio scale the emphasis is on proportional change rather than an additive change. The key is that straight-line behavior in a ratio-scale plot corresponds to a constant growth rate, not a changing one.

Option 2: The statement that the income gap is increasing over time if the two lines are parallel and upward-sloping.
- When two lines are parallel and upward-sloping on a plot with a ratio scale, they share the same slope but are offset from each other by a constant vertical distance. If we are looking at a ratio between the two countries’ GDP per capita, parallel lines imply that the ratio itself changes at a constant rate over time, and the absolute difference (the gap on the y-axis) between the two series grows at a constant pace as time increases.
- Specifically, since the lines are upward-sloping, the absolute difference on the vertical axis between the two series increases as time progresses. Thus, the income gap, interpreted as the absolute vertical distance between the two curves, indeed increases over time.
- A common misconception is to think parallel lines imply a constant gap; but constant slope with upward direction means the gap widens in absolute terms as time advances, given that both lines rise together but are offset. This supports the claim that the income gap grows over time, assuming the gap is measured in absolute (not percentage) terms on the ratio scale.

Summary for both options:
- For the first option, the straight-line relationship on a ratio-scale plot suggests constant proportional growth (a constant growth rate) over time.
- For the second option, parallel upward-sloping lines indicate that the absolute income gap between the two series increases over time, since the vertical separation, measured on the ratio scale, grows as time progresses.

Therefore, the two selected options—constant for the growth rate and increasing for the income gap—are consistent with the described plotting setup and the implied interpretations of linearity on a ratio scale.

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