Country Healthy-living in a Malthusian world has the following production function: Y = 560 D^ (0.4) N^ (0.6), where Y is measured in pounds of food and D acres of land. (a) Suppose that the sustainable consumption is 600 pounds of food, i. e. c = 600. What is the country's population density at the steady-state? (b) Suppose that an advance in medicine in the country Healthy-living makes its citizens healthier; hence they could sustain life and reproduction at a lower level of consumption, say 550 pounds. What will the country's population density be in the long run after the medical advance? How would its standard of living, measured by GDP per capita, have changed?

Y = 560D0.4N0.6,

When sustainable consumption is 600 pounds of food, we have to find population density. The density of population is the number of people living in a given area, or N / D, per sqkm. We first divide the production function with N to get land per person

Y/N = (560D0.4N0.6) N

y = 560 (D/N) 0.4 (N/N)

y = 560d0.4

We now have a sustainable or stable consumption level of 600. Hence

600 = 560d0.4

d = 1.189

Hence land per person is 1.189

Now population density is N/D or 1/d so it becomes 1/1.189 = 0.84

When sustainable consumption falls to 550 pounds

550 = 560d0.4

d = 0.95

A - So, when sustained consumption is 550 pounds, land falls to 0.95 per person. Again population density is N/D or 1/d so it becomes 1/0.95 = 1.04

B - GDP per capita is expressed in living standards. The living standards have deteriorated because they were decreasing from 600 to 550.

Given the production function

Y = 560D0.4N0.6,

We need to find population density when sustainable consumption is 600 pounds of food. Population density is the number of people living per sq km, or in a given area of land which here means N/D. Here first divide the production function with N to get land per person

Y/N = (560D0.4N0.6) N

y = 560 (D/N) 0.4 (N/N)

y = 560d0.4

Now that sustainable or steady state level of consumption is 600, we have

600 = 560d0.4

d = 1.189

Hence land per person is 1.189

Now population density is N/D or 1/d so it becomes 1/1.189 = 0.84

When sustainable consumption falls to 550 pounds

550 = 560d0.4

d = 0.95

Hence land per person falls to 0.95 when sustainable consumption falls to 550 pounds. Again population density is N/D or 1/d so it becomes 1/0.95 = 1.04

Standard of living is measured by GDP per capital or here, y. Since y has fallen from 600 to 550, standards of living have fallen.