I am starting to have difficulty believing that it indicates more productivity. I considered two cases where productivity of the worker doubles, and consider its long term influence on the GDP.
Case 1: Limited demand for item: In this extreme, the demand for the item is completely limited at the current or lower price. Influence of higher price is not relevant here. Let the demand be satisfied by 100 people making $50k average. Assuming wages match product cost, together, they contribute $5M to the GDP.
Now, imagine productivity of the workers producing this item suddenly doubled. Instead of needing 100 people being paid $50k, we need just 50 people making $50k to satisfy demand. What is the influence on the GDP?
50 people suddenly are unemployed because we assume constant demand. We assume the 50 unemployed people find other jobs paying the same amount. Thus they contribute $2.5M to the GDP from the outside.
In this industry, the same 50 people now are able to produce twice as much leading to a halving of price. As demand does not change, same number of items contribute half as much to the GDP: $2.5M.
So the total GDP should not change despite a doubling of efficiency for a particular item.
Case 2: Unlimited demand, limited money: This is the other extreme where the people will pay a limited amount of money for the item, but can consume it in unlimited quantities. Again, consider 100 people working for $50k average.
After the productivity increase, 100 people can make twice as much goods for the same half price adding up to the same total - $5M, all of which will be produced and consumed. The contribution to the GDP by this 100 people again remains the same - $5M.
I think most practical situations will fall between these two extremes. As there should be no GDP growth in either of the extremes, I find it hard to believe the combined situation would increase the GDP.
When can GDP increase?
I can also provide a scenario where GDP increases after being adjusted for inflation.
Inflation is measured based on the purchasing habits of a typical consumer. However, GDP is the total product, most of which is consumed by the rich consumer, given the current state of wide income and spending disparity.
If you print money, and give it to rich people, it would influence your inflation metric only a bit because the typical consumer does not have more money and cannot drive up the prices on the things he needs. But there would be a substantial increase in prices and product for the rich people contributing substantially to the overall GDP.
For example, Japan has had languishing GDP growth for over a decade. But they increased their productivty over time along with other nations as they have remained competitive.
I am trying to understand what is the flaw in my reasoning. I will update with interesting answers.
TL;DR: May be productivity increase does not cause GDP growth. Increase in spending inequality does.
EDIT: By real GDP, I mean nominal GDP after adjusting for inflation. I am using it as a technical term and I think that is its definition.