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?
He received multiple answers, and I can see why. I think that the answer depends on the behavior of demand.
Suppose we think of 100 units of labor producing goods, and 100 units of labor producing services. Initially, each unit of labor can produce one unit of output in each sector.
Next, let productivity growth in the goods sector be 4%, with 0% productivity growth in services. Now think about various configurations of demand.
Case 1: satiation in services. No matter what, people only want 100 units of services. In that case, the goods-producing sector gets to be a larger and larger share of the economy, so that the overall growth rate approaches 4 percent.
Case 2: satiation in goods. No matter what, people only want 100 units of goods. In that case, about 4 percent of labor will be re-allocated to services each year, and the overall growth rate will approach 0 percent.
Case 3: balanced demand. No matter what, people allocate half their incomes to goods and half to services. In that case, about 2 percent of labor will be re-allocated to services each year, and the overall growth rate will be about 2 percent. That is Bryan's answer.
I think that the real world lies somewhere between case 2 and case 3. That is, as Robert Fogel has pointed out, the income elasticity of the demand for goods is about one-half, and that for services is about one and one-half.
Arithmetically, this implies stagnation. The overall growth rate will slow, notwithstanding productivity gains in goods production.
However, I do not think that we are doomed to have low productivity growth forever in education and health care. I think that efficiency in both sectors is held back by cultural norms and government intervention.