The data for past populations

LINEAR GROWTH

Projection Equation

POP_PROJ = POP_LAST + B (PROJ.YEAR-LAST.YEAR)
 

B =(POP_LAST - POP_FIRST)/(LAST.YEAR-FIRST.YEAR)

SAN JOSE

B = (1,477,00-1,072,600)/(1990-1970)=(404400)/(20) = 20,220 PER YEAR

PROJECTION FOR 2000 AD = 1,477,000 + (20,220)(10) = 1,679,200

CALIFORNIA

B =(29,839,250-20,039,000)/20 = 490,012.5 PER YEAR

PROJECTION FOR 2000 AD = 29,839,250 + 4,900,125 = 34,739,375

EXPONENTIAL GROWTH

Projection Equation

POP_PROJ = POP_LAST (1+GROWTH.RATE)^{(PROJ.YEAR-LAST.YEAR)}
 

(1+GROWTH.RATE) = (POP_LAST/POP_FIRST)^{1/(LAST.YEAR-FIRST.YEAR)}

SAN JOSE

(1+GROWTH.RATE) = (1,477,000/1,072,600) ^1/20

= (1.377) ^.05 = 1.016125

POP₂₀₀₀ = (1,477,000)(1.016125) ¹⁰

= (1,477,000)(1.17468) = 1,733,213

CALIFORNIA

(1+GROWTH.RATE) = (1.48906) ^.05 = 1.0201

POP₂₀₀₀ = (29,839,250)(1.0201) ¹⁰

= (29,839,250)(1.2203) = 36,411,940

MODIFIED EXPONENTIAL

Projection Equation

POP = C + AB^TIME
 

FOR THREE EQUALLY SPACED YEARS

Let TIME be coded as:

FIRST YEAR= -1, MIDDLE YEAR = 0, and LAST YEAR = +1.

Then:

• POP_FIRST = C + AB^-1
• POP_MIDDLE = C + AB⁰ = C + A
• POP_LAST = C + AB¹ = C + AB

Subtracting the second equation from the third yields:

(1)

(POP_LAST - POP_MIDDLE) = A(B-1)

Subtracting the first equation from the second yields:

(2)

(POP_MIDDLE - POP_FIRST) = A(1 - 1/B) = A(B-1)/B.

IF the previous equation (2) is divided into the equation before that (1) the result, after the cancellation of the factor A(B-1) is just B.

Thus:

B = (POP_LAST - POP_MIDDLE)/(POP_MIDDLE - POP_FIRST)

Since B is now known the value of A can be determined from equation (1):

A = (POP_LAST - POP_MIDDLE)/(B-1)

With a knowledge of A the value of C can be determined from the equation for P_MIDDLE; i.e.,

C = P_MIDDLE - A

MODIFIED EXPONENTIAL PROJECTIONS FOR:

SAN JOSE

B = (1,477,000 - 1,299,700)/(1,299,700 - 1,072,600)

= 0.7807

A = -808,530

C = 2,108,200

Since time is coded: YEAR 1970 = -1, YEAR 1980 = 0, and YEAR 1990 = +1, this means that YEAR 2000 = +2.

Therefore:

POP₂₀₀₀ = 2,108,200 - (808,530)(0.7807)²

= 2,108,200 - (808,530)(0.6095) = 1,615,407

The years 2010 and 2020 correspond to TIME=+3 and TIME=+4, respectively.

MODIFIED EXPONENTIAL PROJECTIONS FOR:

CALIFORNIA

B = (248718291 - 23,780,100)/(23,780,100 - 20,039,000)

= 1.6196

A = 9,779,132

C = 14,000,968

Therefore:

POP₂₀₀₀ = 14,000,968 + (9,779,132)(1.6196)²

= 14,000,968 + (9,779,132)(2.6231)

= 39,652,650

MODIFIED EXPONENTIAL PROJECTIONS FOR:

UNITED STATES

B = (29,839,250 - 226,542,199)/(226,542,199 - 203,402,031)

= 0.95421

A = -484,000,000

C = 710,883,640

Therefore:

POP₂₀₀₀ = 710,883,640 - (484,000,000)(0.95421)²

= 710,883,640 - (484,000,000)(0.91052)

= 269,879,027

The projections are compiled in the following table:

There are two other extrapolation curves that are like the modified exponential but more complicated mathematically:

• The Logistic Curve:

  
  P = 1/(C+AB^TIME)
   

• The Gompertz Curve:

  
  P = CA^(BTIME)
   

The general shape of the Gompertz curve is the same as the Logistic curve.

If the logistics curve is expressed in terms of reciprocal population; i.e.,

1/P = C+AB^TIME,
 

the form is the same as that of the modified exponential and the same method used for the modified exponential can be used to get the projection of the reciprocal population.

If the logarithms are taken of both sides of the Gompertz equation the result is

log(P) = log(C) + (B^TIME)log(A).
 

This also is mathematically the same form as the modified exponential and the same method can be used to project the logarithm of population.