year	population in thousands
0000 . . .	300,000
+1,750	+164 %
1750	791,000
+100	+60 %
1850	1,262,000
+100	+100 %
1950	2,518,629
+50	+141 %
2000	6,070,581

        World population has been increasing faster and faster. Two thousand years ago there were only 300 million people on our planet (Wales et al., 2006a). It took 1750 years for the population to reach 791 million. Only 100 years later the population had increased by 60% to reach 1,262 million. In the next hundred years it doubled, becoming 2,519 million. Then it took only 50 years to increase by 141% to 6,071 million. The present world population of more than 6.5 billion is expected to reach 8.9 billion by 2050.

        There are three basic parameters that control population growth: (a) fertility rate, (b) survival rates, and (c) the length of the interval between generations. Let's consider a mathematical model of population in which there are only three generations. We can think of them as (1) children, (2) parents, or (3) grandparents. In our simplified model after a fixed interval--a generation of 25 years, for example--everyone moves up one generation. All the grandparents die and a new generation of children is born. That is to say that the survival rates for children and parents are 100%, but for grandparents is 0%. The life span is 75 years. With two of the parameters set, we can look at the effects of the third--fertility rate.

        If each pair of parents has two children, then the population will be stable. The number of children will be the same as the number of parents and of grandparents, let's say 100 (million). Then the total population would always remain 100 children + 100 parents + 100 grandparents = 300 (like it was 2,000 years ago). There would be two children in each family of four (or six, if one set of grandparents lives with them). This 2-Child Population Model produces a stable population. Now let's take a look what happens when the fertility rate rises to four children per couple. After 25 years there will be an increase of 100 children (total population 400). A generation later there will be 400 children and 200 parents (total population 700). From then on the population will double every generation. The population will rise so quickly

that the increase can be described as an explosion. Starting with a population the size of America's (twice the size of Japan's) we get close to the world population after only six generations (about 150 years). Furthermore, the population is still continuing to double every 25 years. Fertility rates are what cause populations to explode in a geometrical progression.

        Next let's consider the effects of the two other parameters. In the long run each indivitual's survival rate is 0%. Everyone dies; no one "gets out" alive. In our 3-Generation Population Model there are three survival rates, one for each transition: child to parent (C-P), parent to grandparent (P-G), and grandparent to great grandparent (G-GG). Survival rates determine life span. Because we are assuming survival rates of 100%, 100%, and 0%, the life span is three full generations or 75 years. If the G-GG survival rate rises above 0 for any reason--economic prosperity and medical advances, for example, we add a new generation. If rate goes to 20%, then life span increases by that percent of a generation.

That will cause the stable 2-Child Model to experience a one-time increase of 20 great grandparents and then stabilize at this new equilibrium. The exploding 4-Child Model will experience a one-time increase above its geometric increases and then settle back into its doubling mode for all four generations.
        If the C-P survival rate falls below 100% the major effect on population will be the same as a corresponding drop in the fertility rate, as if they had never been born. Although a lower P-G survival rate in our simple model would have only a minor effect on population, similar to that of the G-GG survival rate, in the real world the death of a parent can negatively affect a child's chances of survival.

        Increasing the length of a generation, our third and final parameter, will leave the characteristics of population growth unchanged except for the time scale. The population in our 2-Child Population Model remains stable and at the same level, but with an increased life span of 90 years, rather than 75. The population explosion of our 4-Child Population Model will still explode--a little more slowly, but just as surely. The important point here is that fertility rates and the C-P survival rate are what drive a population explosion. Longevity and the length of a generation have only minor roles to play.

        Malthus' concern at the turn of the 19th century was that agricultural production could not keep pace with an exploding population. In order to consider why that might be, we need to assign economic roles--the dual roles of consumer and producer--to the members of our population model. To keep the model simple let us postulate that everyone consumes the same fixed amount, so that consumption is directly proportional to population. Parents only, however, are assumed to be productive members of society--an agricultural society--so they produce food. From Table 3 (above) we can see what happens as a rise in fertility creates a population explosion. The portion of the population that is productive initially drops from 33.3% to 25%, then reaches an equilibrium at 28.6%. After that, both population and food production should expand at the same exponential rate allowing a fixed level of per capita consumption. Two basic economic forces, however, often cause efficiency to deviate from any such fixed level: the Law of Diminishing Returns and economies of scale.

        The Law of Diminishing Returns predicts that efficiency will decrease as production increases. That is because the most efficient factors of production will be employed first. In an agricultural society the most productive land would be cultivated first. As population and the demand for food grows, less and less productive land comes under cultivation. Nutrients in the soil might also become depleted with continued cultivation causing further decreases in productivity. Eventually you might even run out of a fixed resource such as land. Thus per capita production would be expected to decline with a rise in population if not for the compensating increases in efficiency for large scale production.

        Advances in agricultural technology, transportation, and communication coupled with mass production and economies of scale have allowed the more advanced and densely populated cultures to produce and distribute food more and more efficiently (Diamond, 1999). It all began with the domestication of plants and animals during the Neolithic Revolution. Hunter-gatherers discovered that they could get more food for less effort by farming than nature provided in fields and forests. As farm land and population increased surplus labor migrated to big cities where a money-based economy allowed division and specialization of labor in factories, which produced farming machinery, chemicals, and pesticides. Thus industrialization caused a second agricultural revolution. Family farms became large-scale agribusinesses. Advances in transportation and communication allowed mass marketing and distribution of mass produced agricultural products. It also allowed diffusion of technological advances, such as Norman Borlaug's high-yield strains of wheat and rice that came to be known as the Green Revolution [1943-1960s], bringing the world's supply of food per capita to at an all time high.

        During the last 200 years as the population went from 1 billion to 6 billion, the surface of the Earth has remained fixed at 510 million square kilometers with about 70 percent of that area under salt water. At the time Malthus was writing his theory, the excess population of the British Isles was still pouring into North America and Australia. Africa and the Americas were largely covered by forests teeming with wildlife, so that the concern was with the ability of society to produce food, shelter, clothing, and, later, industrial machinery that would further accelerate production and transport the supply of goods to wherever the demand was greatest. The focus of concern has now shifted from production and supply to the availability of a range of natural resources (see Table 4).