Final answer:
Initially, Cape Despair had a real GDP of $15 billion and labor productivity of $15 per hour. Post-technological advancement, labor productivity rises by 50%, leading to an increase in population growth rate. The eventual levels of real GDP and real GDP per hour will balance out at a higher subsistence level due to the productivity gains.
Step-by-step explanation:
To answer the question posed about the economy of Cape Despair, let's consider the initial levels of real GDP and labor productivity based on the provided production function, the effects of a technological advance, and its impact on population growth and real GDP per hour of labor.
Initial Levels of Real GDP and Labor Productivity
The initial level of real GDP is given as $15 billion, which occurs when labor is 1.0 billion hours per year, making labor productivity (real GDP per hour of labor) $15 per hour. This meets the subsistence real wage rate, indicating that initially, the population of Cape Despair is constant.
Labor Productivity Post-Technological Advance
Immediately following the technological advance, labor productivity rises by 50 percent at each level of labor. If we calculate this increase for the 1.0 billion labor hour mark, the new labor productivity figure would be $22.50 per hour (1.5 times the original productivity of $15).
Population Growth Rate After Technological Advance
With the increase in labor productivity above the subsistence level, the population growth rate will increase, as per the conditions described for Cape Despair's economy.
Eventual Levels of Real GDP and Real GDP Per Hour of Labor
Eventually, the levels of real GDP and real GDP per hour of labor will likely balance out at a new higher subsistence level. For example, if at the increased productivity rate the economy operates where labor is 1.0 billion hours, the new real GDP would be $22.50 billion, and the real GDP per hour would also be $22.50.