Byran Caplan asks

Suppose half of the sectors of the economy grow forever at 4%, while the rest completely stagnate.  I’m strongly tempted to say that this economy’s growth rate equals 2% forever.  Anyone tempted to disagree?  If so, why?

Before you answer: Would it matter if the 4%-growth sectors were all in "virtual reality" – and the stagnant sectors were for actual food, shelter, cars, etc.?  If it matters, why does it matter?

At first I was puzzled because the answer seemed so obviously to be 4%.

However, I suppose it depends on what you mean by “grows at 4%” 

If you mean that when you look up the real output of this sector in the NIPA tables its four percent larger each year than the year before, while the other sector is always equal to what is was the year before then of course the growth rate of total GDP will converge to 4%

If you mean that the quantity index of one sector grows at 4% while the quantity index of the other sector grows at 0% then the answer could range between 0 and 4% depending on the fraction of budget shares that went into each sector.

For example, suppose that the total quantity of virtual reality grew at 4% but total spending on virtual reality fell by 3% each year because productivity was growing at 7%. Then the growth rate of the economy would converge to zero as virtual reality became a trivial expenditure share.

To get Bryan’s answer then you would need spending on VR and the rest of the economy to remain in equal proportion.