gordo

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What Do GDP Growth Curves Really Mean?

Fair enough - my point was not to come up with the exact quantitative growth rate, but to show some of the assumptions that the original post glossed over. 

What Do GDP Growth Curves Really Mean?

I just want to formalize what others have mostly already commented in the context of your example, which will help draw out some of the subtleties that are missed. In this two sector economy we maximize utility from apples and brass, subject to the constraint that our spending on these items is less than or equal to our income.

This results in the condition that our expenditure on the two items is equal to the ratio of our marginal utilities between them.

In the first period you have this ratio as 1, and in the second period it equals 2. So something has changed in our preferences between the periods. If you wanted to hold preferences constant, then given the price changes you pose, brass consumption should be 10x apple consumption not 5x. This isn't necessarily a problem, technology can change preferences, let's go with that. 

Another implication of our model is that if we are maximizing utility, our income will equal our total expenditure. So in the first period total income is 6, but in the second period income is 4.5. So nominal income has actually gone down in your example. What about utility? If we posit a relatively standard functional form for utility:

To make this consistent with your example,  in period 1 and .667 in period 2. If we do the calculations, we find that U = 3 in period 1 and U = 5.13 in period 2, giving real GDP growth of 71% - somewhere in between your two numbers.

In a real economy, one issue is that new goods appear all the time, and how do we map those goods onto past preferences. I'm not sure, but I think that when economists use the more recent numbers as deflators, what they are attempting to do is get more accurate measures of the 'expenditure shares' on different types of goods in order to calibrate the utility function. Obviously this work is assumption driven, and full of pitfalls, but it's not nearly as straightforward as your example. Hal Varian has some interesting discussion of these issues:

https://www.brookings.edu/wp-content/uploads/2016/08/varian.pdf