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At first, it may seem obvious that consumption will rely on Y. If GDP is doubled in real terms over a number of years, government consumption, private consumption and investment will also each roughly be doubled. If you draw a graph of GDP and consumption over time you see that consumption does grow by about the same rate as GDP.
Though from this reasoning, we can't conclude that C relies on the Y since growth has been removed from our variables C and Y. We need to think of Y as a variable which varies over time around some average. Sometimes it is above the average and sometimes it's below the average however there is now upward trend. The same is true for C.
The crucial question then is whether consumption is above its average in periods when GDP is above its average and vice versa (technically, if detrended variables are correlated over time).
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