If a society's population doubles does its productivity automatically double?

Answer 1

No. Production may double, or it may grow more or less quickly: it may quadruple or grow even faster than that. Productivity per worker may double, or it may grow more or less quickly, or it may fall. There's no automatic link between population and productivity: both are constrained by resource availability (traditionally agricultural land in the case of population, but also energy and raw materials) and productivity is affected by the supply of capital (buildings, infrastructure, equipment). Over much of the last millennium world population tended to grow faster than labour productivity (thoughn sometimes population fell and labour productivity is likely to have risen slightly because of the resulting per capita land windfall); in the last century or so productivity has tended to rise faster than population, though not everywhere.

Answer 2

No, a society's productivity does not automatically double if its population doubles. Productivity is influenced by various factors such as technology, education, infrastructure, and efficiency of labor. Simply increasing the population does not guarantee a proportional increase in productivity.