Population growth is described as a function of time usually by dynamic models based on differential equations.

The most common practical methods are component, exponential and logistic models (Enkhbat & Tungalag, 2006).

On the other hand, the population growth can be considered stochastic variable since population depends on social and economic policies, political stability and so on.

In this paper, first we examine the existing growth models such as exponential, logistic and stochastic models. Then we deal with two classic models. It is well known that exponential and logistic functions have some drawbacks in terms of too fast growth or slow growth in a long term prediction in economy. For this purpose, we introduce a new model called Exlog Weighted Sum Model for predicting macroeconomic indicators and focus on the parameter estimation problem of the exponential and logistic models. Finding a parameter of the exponential model reduces to unconstrained minimization problem which has been solved analytically. As far as the parameter of logistic model is concerned, the least square method reduces to difficult nonconvex optimization problem but we have proposed simple formula for finding parameters of a new model. The parameters of both exponential and logistic models have been estimated for Mongolian population data. We also propose a new method for estimating optimal weights of the Exlog model using statistical data. The new model was tested on Mongolian population forecasting.

The general form to model the growth of population is

d y d t = f ( y ) ,

where y is equal to the number of individuals at timet, i.e. the population.

Many different forms of f ( y ) exist. Two simple models are exponential and logistic models which are the most popular models (Panik, 2014) for predicting economic indicators.

The exponential model is given by the following differential equation:

{ d y d t = r y , y ( 0 ) = y 0 .

where r-growth rate, t-period, y₀-population for the base year.

This model plays also an important role in Solow growth theory (Solow, 1956) as a function of population growth.

By solving this equation, we get

y ( t ) = y 0 e r t (1)

Using the least square approximation, we estimated the growth rate r based on data in Table 1.

For estimating parameter r using statistical data, we apply the least square method reducing this problem to an unconstrained convex minimization problem after taking logarithm from the expression (1).

In order to estimate parameter r of the exponential model

y = y 0 e r t

We use the least square method after reducing it to linear model

ln y = ln y 0 + r t

F ( r ) = ∑ i = 1 n [ ln y 0 + r t i − y i ] 2 → min (2)

Denote by y ¯ 0 = ln y 0 and solve problem (2)

Source: http://www.1212.mn/ (accessed on 14 January, 2022).

F ′ ( r ) = 2 ∑ i = 1 n ( y ¯ 0 + r t i − y i ) t i = 0

∑ i = 1 n y ¯ 0 t i + r ∑ i = 1 n t i 2 − ∑ i = 1 n y i t i = 0

From the last equation, we find

r = ∑ i = 1 n y i t i ∑ i = 1 n t i 2 (3)

For our data r is computed as 0.0167.

On the other hand, the logistic model is given by the following differential equation:

y ′ = k y ( M − y ) y ( 0 ) = y 0

The solution to this equation is

y ( t ) = M y 0 y 0 + ( M − y 0 ) e − M k t

This function called the logistic curve which is more suitable for modeling long-term growth.

The parameters M and k of the model are estimated usually by the least square method. For this purpose, the following type of a statistical data is used.

F ( M , k ) = ∑ i = 1 m [ M y 0 y 0 + ( M − y 0 ) e − M k t i − y i ] 2 → min ,       M > 0 , k > 0 (4)

Problem (4) is a constrained minimization problem has to be solved with respect to variables M and k. From a view point of optimization theory and methods, it is well known that this is very difficult nonconvex optimization problem. F(M, k) is a differentiable nonlinear function. In general, problem (4) can be solved numerically by the gradient method. The gradient of the function is computed as follows.

∂ F ∂ M = 2 ∑ i = 1 m [ ( p i − f ( M , k , t i ) ) ⋅ e − M k t i [ 1 − ( M − p 0 ) k t i − p 0 ( M − p 0 ) − p 0 2 ] ( p 0 + ( M − p 0 ) e − M k t 1 ) 2 ] ∂ F ∂ k = 2 ∑ i = 1 m [ p i − f ( M , k , t i ) ] ( e − M k t i [ 1 − ( M − p 0 ) ( k t i + p 0 ) ] − p 0 2 ( p 0 + ( M − p 0 ) e − M k t i ) 2 )

g r a d F ( M , k ) = ( ∂ F ∂ M , ∂ F ∂ k )

The maximum value of M can be estimated by exponential growth in advance. If the maximum value M is given, problem (4) reduces to one dimensional global minimization problem with respect to k.

φ ( k ) = ∑ i = 1 m [ M y 0 y 0 + ( M − y 0 ) e − M k t i − y i ] 2 → min ,       k > 0 (5)

In order to estimate M and k by the least square method, we propose the following simple method. The first order differential equation of the model should be replaced by the difference scheme.

y ′ ≈ y ( t + 1 ) − y ( t ) ; y ( 0 ) = y 0

The equation has the following form.

y ( t + 1 ) − y ( t ) = k y ( M − y ) y ( t + 1 ) = M k y − k y 2 + y ( t )

We have to find k and M models, using the least square method. In this case, the following function has to be minimized:

F ( M , k ) = ∑ i = 1 m [ M k y i − k y i 2 + y i − y i + 1 ] 2

This problem is two variables’ minimization problem, and optimal conditions for extremum is written as follows:

∂ F ∂ M = 2 ∑ i = 1 m [ M k y i − k y i 2 + y i − y i + 1 ] k y i = 0 ∂ F ∂ k = 2 ∑ i = 1 m [ M k y i − k y i 2 + y i − y i + 1 ] ( M y i − y i 2 ) = 0

If it is simplified:

∂ F ∂ M = ∑ i = 1 n ( M k y i 2 − k y i 3 + y i 2 − y i y i + 1 ) = 0 ; ∂ F ∂ k = ∑ i = 1 m [ ( M k y i − k y i 2 + y i − y i + 1 ) M y i − ( M k y i − k y i 2 + y i − y i + 1 ) y i 2 ] = 0

Simplification of the system’s second equation is:

∑ i = 1 m ( M 2 k y i 2 − M k y i 3 + M y i 2 − M y i y i + 1 − M k y i 3 + k y i 4 − y i 3 + y i 2 y i + 1 ) = 0

which gives the following equation:

M 2 k ∑ i = 1 m y i 2 − 2 M k ∑ i = 1 m y i 3 + M ∑ i = 1 m ( y i 2 − y i y i + 1 ) + k ∑ i = 1 m y i 4 = ∑ i = 1 m ( y i 3 − y i 2 y i + 1 ) .

Then the system becomes as following:

M k ∑ i = 1 m y i 2 − k ∑ i = 1 m y i 3 = ∑ i = 1 m ( y i y i + 1 − y i 2 )     ; M 2 k ∑ i = 1 m y i 2 − 2 M k ∑ i = 1 m y i 3 + M ∑ i = 1 m ( y i 2 − y i y i + 1 ) + k ∑ i = 1 m y i 4 = ∑ i = 1 m ( y i 3 − y i 2 y i + 1 )

This system of nonlinear equations should be solved with respect to M and k. If M is given in advance, the system reduces to one variable equation:

∂ F ∂ k = ∑ i = 1 m [ M k y i − k y i 2 + y i − y i + 1 ] ( M y i − y i 2 ) = 0.

We find k from the equation:

k [ M ¯ 2 ∑ i = 1 m y i 2 − 2 M ¯ ∑ i = 1 m y i 3 + ∑ i = 1 m y i 4 ] = ∑ i = 1 m ( y i 3 − y i 2 y i + 1 ) − M ¯ ∑ i = 1 m ( y i 2 − y i y i + 1 )     , k ¯ = ∑ i = 1 m ( y i 3 − y i 2 y i + 1 ) − M ¯ ∑ i = 1 m ( y i 2 − y i y i + 1 ) M ¯ 2 ∑ i = 1 m y i 2 − 2 M ¯ ∑ i = 1 m y i 3 + ∑ i = 1 m y i 4 .

In practice, the maximum value of M can be estimated by the exponential growth in advance. And then use an approximation formula of y ′ .

y i + 1 − y i = k y i ( M − y i ) , y i = y ( t i ) ,   i = 1 , 2 , ⋯ , n

k i = y i + 1 − y i y i ( M − y i ) ,   i = 1 , 2 , ⋯ , n

k ¯ = ∑ i = 1 n k i r (6)

We introduce uncertainty in the population growth. Due to the unpredictable nature of population growth, we add a random element to the exponential growth equation (Enkhbat, Enkhbayar, & Bayanjargal, 2018).

d y ( t ) = r y ( t ) d t + σ y ( t ) d W

where r is the growth rate, σ is number called diffusion coefficient and W is a Wiener process.

Let the number of population for base year be y₀ at time t = 0 and t i = i Δ t , so the numbers of population are to be determined at discrete points { t i } .

Then our discrete-time model is

y ( t i + 1 ) = y ( t i ) + r Δ t y ( t i ) + σ Δ t Z i y ( t i ) ,

where Z i ( i = 0 , 1 , 2 , ⋯ ) are i.i.d N(0,1).

Based on the Central Limit theorem and Z i ( i = 0 , 1 , 2 , ⋯ ) are i.i.d N(0,1), we take a limit Δ t → 0 to get continuous population model. Continuous time expression for the model is

y ( t ) = y 0 e ( r − 1 2 σ 2 ) t + σ t Z ,

where Z ~ N ( 0 , 1 ) .

A random variable y ( t ) of the form has a so-called lognormal distribution, that is, its log is normally distributed.

So we can describe the evolution of the population over any sequence of time points 0 = t 0 < t 1 < t 2 < ⋯ < t M by

y ( t i + 1 ) = y ( t i ) e ( r − 1 2 σ 2 ) ( t i + 1 − t i ) + σ t i + 1 − t i Y i ,

for i.i.d Y i ~ N ( 0 , 1 ) .

Using the exponential and logistic models, we introduce so-called Exlog Weighted Sum Model given by the following formula:

y ( t ) = λ 1 у 0 e r t + λ 2 M y 0 y 0 + ( M − y 0 ) e − M k t (7)

where,

y(t):Macroeconomic indicator;

y₀:initial value of y at the moment t = 0;

λ 1 , λ 2 are parameters of the model,and λ 1 + λ 2 = 1 , λ 1 > 0 , λ 2 ≥ 0 .

Using the above data, we can estimate the parameters of the model by solving the following constrained minimization problem:

F ( λ 1 , λ 2 ) = ∑ i = 1 m [ λ 1 y 0 e r t + λ 2 M y 0 y 0 + ( M − y 0 ) e − M k t − y i ] 2 → min (8)

λ 1 + λ 2 = 1 ,     λ 1 > 0 ,     λ 2 ≥ 0 (9)

In general, problem (8)-(9) is convex optimization problem (Enkhbat, 2018). For simplicity, we assume that we have the following models

y e x = y 0 e r t ,   y l o g = M y 0 y 0 + ( M − y 0 ) e − M k t

and data:

Formula (7) has the form

y = y e x + y l o g .

Here, values of y i e x and y i l o g have been estimated by exponential and logistic functions.

Then finding parameters of the Exlog Weighted Sum Model reduces to a constrained minimization problem:

F ( λ 1 , λ 2 ) = ∑ i = 1 m [ λ 1 у i e x + λ 2 y i l o g − y i ] 2 → min (10)

λ 1 + λ 2 = 1 ,     λ 1 ≥ 0 ,     λ 2 ≥ 0 (11)

Since the problem is convex, then we apply Lagrange method. The Lagrangean function is

F ( λ 1 , λ 2 , λ ) = ∑ i = 1 m [ λ 1 у i e x + λ 2 y i l o g − y i ] 2 + λ ( λ 1 + λ 2 − 1 )

where λ is a Lagrange multiplier.

If we take derivatives of Lagrangean function with respect to variables and equal to zero, we obtain the following system of equations:

{ ∂ L ∂ λ 1 = 2 ∑ i = 1 m ( λ 1 y i e x + λ 2 y i l o g − y i ) y i e x + λ = 0 ∂ L ∂ λ 2 = 2 ∑ i = 1 m ( λ 1 y i e x + λ 2 y i l o g − y i ) y i l o g + λ = 0 ∂ L ∂ λ = λ 1 + λ 2 = 1

We simplify the above system as follows:

{ 2 λ 1 ∑ i = 1 m ( y i e x ) 2 + 2 λ 2 ∑ i = 1 m ( y i l o g y i e x ) = 2 ∑ i = 1 m ( y i y i e x ) − λ 2 λ 1 ∑ i = 1 m ( y i l o g y i e x ) + 2 λ 2 ∑ i = 1 m ( y i l o g ) 2 = 2 ∑ i = 1 m ( y i y i log ) − λ λ 1 + λ 2 = 1

Consequently we have,

{ 2 λ 1 ∑ i = 1 m ( y i e x ) 2 + 2 ( 1 − λ 1 ) ∑ i = 1 m ( y i l o g y i e x ) = 2 ∑ i = 1 m ( y i y i e x ) − λ 2 λ 1 ∑ i = 1 m ( y i l o g y i e x ) + 2 ( 1 − λ 1 ) ∑ i = 1 m ( y i l o g ) 2 = 2 ∑ i = 1 m ( y i y i l o g ) − λ λ 2 = 1 − λ 1

Hence, we obtain the following system of equations:

{ 2 λ 1 [ ∑ i = 1 m ( y i e x ) 2 − ∑ i = 1 m ( y i l o g y i e x ) ] = 2 ∑ i = 1 m ( y i y i e x − y i l o g y i e x ) − λ 2 λ 1 [ ∑ i = 1 m ( y i e x y i l o g ) − ∑ i = 1 m ( y i l o g ) 2 ] = 2 ∑ i = 1 m ( y i y i l o g − ( y i l o g ) 2 ) − λ

Substituting a value of

λ 2 = 1 − λ 1

into last equation, we find λ 1 , λ 2 and λ .

2 λ 1 ∑ i = 1 m ( y i e x ) 2 − ∑ i = 1 m ( y i l o g y i e x ) + ∑ i = 1 m ( y i l o g ) 2 = 2 ∑ i = 1 m ( y i y i e x − y i l o g y i e x − y i y i l o g + ( y i l o g ) 2 )

{ λ 1 = ∑ i = 1 m [ у i y i e x − y i log y i e x − у i y i l o g + ( y i l o g ) 2 ] ∑ i = 1 m ( y i e x − y i l o g ) 2 λ 2 = 1 − λ 1 (12)

The values of λ 1 , and λ 2 were computed by (12) and found as λ 1 = 0.8 , and λ 2 = 0.2 .

Formula (12) can be used in predicting some other macroeconomic indicators.

Traditional exponential model provides too fast growth, on the other hand, the logistic model ensures too slow growth which are not suitable for long term growth of considered indicators. We illustrate Exlog Weighted Sum Model on Mongolian Population data. Data covers period between 2000 and 2020. In model (7) we take values of M = 5000.0 and k,r computed by formulas (3), (6) as follows:

k = 0.0000079 , r = 0.0167 .

Based on survey for the period 2000-2020 (Tungalag et al., 2015), we predict the Mongolian population by Exlog Weighted Sum Model:

y = 1926 e 0.0167 t + 2407500 2407.5 + 2592.5 e − 0.0395 t

We forecast the Mongolian population up to the 2040 year in Table 2.

We examine the existing population growth models such as exponential and logistic models and propose a method for finding parameters of the models by

solving corresponding convex and nonconvex optimization problems.

We propose also so-called Exlog Weighted Sum Model combining well known classical models such as exponential and logistic for predicting macroeconomic indicators of Mongolia. We derive formulas for estimating weights of the proposed model by solving a constrained convex minimization problem. The proposed model was tested for predicting Mongolian population up to 2040. The proposed approach can be applied to forecasting any macroeconomic indicators.

The authors declare no conflicts of interest regarding the publication of this paper.

Ouyntsetseg, L., Tungalag, N., & Enkhbat, R. (2022). Exlog Weighted Sum Model for Long Term Forecasting. iBusiness, 14, 31-40. https://doi.org/10.4236/ib.2022.142003