Assume the Inada conditions hold for F, then with θ fixed it is immediate that starting from
any k(0) > 0, the economy converges to a unique k∗ as in the Solow model. As the economy
approaches its balanced growth path, aggregate output, consumption and capital all grow at
rate gB +n while their corresponding per capita magnitudes grow at rate gB. Using standard
notation for growth in per capita magnitudes, along the balanced growth path we must have
gy = gk = gc = gB > 0. A potentially worsening environment however threatens this
happy existence. Since k approaches the constant k∗ we can infer from (5) that the growth
rate of aggregate emissions along the balanced growth path, gE, is given by:
gE = gB + n − gA (6)
The first two terms in (6) represent the scale effect of growth on emissions since aggregate
output grows at rate gB +n along the balanced growth path. The second term is a technique
effect created by technological progress in abatement.
Define sustainable growth as a balanced growth path generating both rising consumption
per capita and an improving environment. Sustainable growth is guaranteed by:
gB >0 and gA > gB + n (7)
Technological progress in goods production is necessary to generate per capita income
growth. Technological progress in abatement must exceed growth in aggregate output in
order for pollution to fall and the environment to improve.