Appendix E

A method for quantifying the respective contributions of population growth and changes in consumption per capita of any type of resource consumption was laid out in a landmark 1991 paper by Harvard physicist Prof. John Holdren.²⁰⁸  Although Dr. Holdren’s paper dealt specifically with the role of population growth in rising energy consumption, the method can be applied to many types of population/resource consumption analyses. In the case of sprawl, the resource under consideration is rural land, namely the expansion over time of the total acres of development in a state.

As stated in Appendix D, the total land area developed in a state can be expressed as:

(1)   A = P x a

Where:
                A = Area of total acres of development in state
                P = Population of that state
                a = area of state used by the average resident (per capita land use)

Following the logic in Holdren’s paper, if over a period of time st (e.g., a year or decade), the population grows by an increment sP and the per capita land use changes by sa, the total urbanized land area grows by sA which is given by substituting in equation. (1):

(2)   A + sA = (P + sP) x (a + sa)

Subtracting eqn. (1) from eqn. (2) and dividing through by A to compute the relative change (i.e., sA/A) in urbanized land area over time interval st yields:

(3)   sA/A = sP/P + sa/a + (sP/P) x (sa/a)

Now equation (3) is quite general and makes no assumption about the growth model or time interval. On a year-to-year basis, the percentage increments in P and a are small (i.e., single digit percentages), so the second order term in equation (3) can be ignored. Hence following the Holdren paradigm, eqn. (3) states that the percentage growth in urbanized land area (viz., 100 percent x sA/A ) is the sum of the percentage growth in the population ( 100 percent x sP/P) plus the percentage growth in the per capita land use (100 percent x sa/a). Stated in words, equation (3) becomes:

(4)  Overall percentage land area growth = Overall percentage population growth + Overall percentage per capita growth

In essence, the Holdren methodology quantifies population growth’s share of total land consumption (sprawl) by finding the ratio of the overall percentage change in population over a period of time to the overall percentage change in land area consumed for the same period. This can be expressed as:

(5)   Population share of growth = (Overall percentage population growth)
                                                         (Overall percentage land area growth)

The same form applies for per capita land use:

(6)   Per cap. land use share of growth = (Overall percentage per capita land use growth)
                                                                   (Overall percentage land area growth)

The above two equations follow the relationship based on Prof. Holdren’s equation (5) in his 1991 paper. A common growth model follows the form (say for population):

(7)   P(t) = P₀ (1 + g_P )t

Where P(t) is population at time t, P₀ is the initial population and g_P the growth rate over the interval. Solving for gP the growth rate yields:

(8)   ln (1 + g_P) = (1/t) ln (P(t)/P₀)

Since ln (1 + x) approximately equals x for small values of x, equation (8) can be written as:

(9)   g_P = (1/t) ln (P(t)/P₀).

The same form of derivation of growth rates can be written for land area (A) and per capita land use (a)

(10)   g_A = (1/t) ln (A(t)/A₀)

(11)   g_a = (1/t) ln (a(t)/a₀)

These three equations for the growth rates allow you to restate the Holdren result of equation (4) as:

(12)   g_P + g_a = g_A

Substituting the formulae (equations 9 through 11) for the growth rates and relating the initial and final values of the variables P, a and A over the period of interest into equation (12), the actual calculational relationship becomes:

(13)   ln (final population / initial population) +

         ln (final per capita land area / initial per capita land area) =

         ln (final total land area / initial total land area)

In other words, the natural logarithm (ln) of the ratio of the final to initial population, plus the logarithm of the ratio of the final to initial per capita land area (i.e., land consumption per resident), equals the logarithm of the final to the initial total land area.

In the case of Minnesota from 1982 to 1997, this formula would appear as:

(14)   ln (4,687,726 residents / 4,131,450 residents) +

         ln (0.466 acre per resident / 0.416 acre per resident) =

         ln (2,185,500 acres / 1,719,900 acres)

Computing the ratios yields:

(15)   ln (1.134) + ln (1.120) = ln (1.270)

         0.1258 + 0.1133 = 0.2390

Then applying equations (5) and (6), the percentage contributions of population growth and per capita land area growth are obtained by dividing (i.e., normalizing to 100 percent) each side by 0.250:

(16)   0.1258 + 0.1133 = 0.2390
         0.2390     0.2390    0.2390

Performing these divisions yields:

(17)   0.53 + 0.47 = 1.0

Thus, we note that in the case of the Minnesota from 1982 to 1997, the share of sprawl due to population growth was 53 percent [100 percent x (0.1258 / 0.2390)], while declining density (i.e., an increase in land area per capita) accounted for 47 percent [100 percent x (0.1133 / 0.2390)]. Note that the sum of both percentages equals 100 percent.
 

Endnote

²⁰⁸ John P. Holdren. 1991. "Population and the Energy Problem." Population and Environment, Vol. 12, No. 3, Spring 1991. Holdren is Teresa and John Heinz Professor of Environmental Policy and Director of the Program on Science, Technology, and Public Policy at Harvard University’s Kennedy School of Government, as well as Professor of Environmental Science and Public Policy in the Department of Earth and Planetary Sciences at Harvard University. Trained in aeronautics/astronautics and plasma physics at MIT and Stanford, he previously co-founded and co-led for 23 years the campus-wide interdisciplinary graduate degree program in energy and resources at the University of California, Berkeley. In 2000, he was awarded the Tyler Prize for Environmental Achievement at the University of Southern California, which administers the award. The Tyler Prize is the premier international award honoring achievements in environmental science, energy, and medical discoveries of worldwide importance.