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>>If you have 10 jobs, and ship 5 overseas, how does that result in rising productivity?<<

It doesn't if numbers were reported accurately.

However, if I outsource the labor intensive bits of a process and buy a more complete "raw material" and then finish it or package it or put a sticker on it it now looks like I've produced the same thing as before with fewer people and for a lower cost. Productivity doesn't take into account the additional work produced by upstream providers. It just takes into account the cost of raw materials.

If I lay off 20 people here who get $10/hour for $200/hour labor cost and replace them with 100 people making 50 cents an hour in China, my resulting labor cost is now $50/hour. The difference is that those 100 people aren't counted as employees because they don't work in the US.

Before outsourcing/offshoring I would spend $200 in labor, $50 in material and all other costs for $250/hour total production cost.

After outsourcing/offshoring, I buy a mostly or completely finished good that may cost me $75 for an hour's production cost.

Instead of having 30 employees, now I've got 10. The bean counters call it "productivity" to do the same final output with 10 employees instead of 30 and I'm doing it for about 1/3 the cost. This is where all that miraculous productivity is coming from.