7.2 City system dynamics model
7.2.1 Purpose of the model
The mathematical model for calculating the integrated (system-dynamic) forecast of demographic and economic indicators (hereinafter - the Model) is intended for:
- Determining long-term trends (regression, approximating functions) of changes in basic socio-economic development (SED) indicators of the city;
- Forming a dynamic forecast (based on a dynamic function) of basic SED indicators, limited by the corridor of values of long-term trends;
- Calculating correlation functions describing the measure of mutual influence of indicators on each other and susceptibility to changes in external factors.
The model converts a set of actual, annual (for at least 6 years) values for 27 macroeconomic SED indicators and 4 additional macroeconomic indicators characterizing external factors into sets of the following values for each city:
- 27 macroeconomic SED indicators for the retrospective (until 2000) and prospective (until 2050) periods;
- Correlation matrices of dimension 31 indicators x 31 indicators x 50 annual periods, reflecting the degree of mutual influence of changes in SED indicators.
7.2.2 Basic form of the model
The basic differential equation of the Model is the damped oscillation equation, describing the change in system indicators \(x(t)\) over time: \[\begin{align*} \frac{d^{2} x(t)}{dt^{2}}+ \gamma \frac{dx(t)}{dt} + \omega^{2}x(t)=f(t), (1) \end{align*}\] where system parameters are:
\(\gamma\) - characteristic of the system’s oscillation damping due to resistance (“friction”) in the system; \(\omega\) - the natural frequency of the system’s oscillations in the absence of resistance (“friction”); \(f\) – the set of external influences.
The dependencies of indicators on time are represented by the sum of 42 basic functions with different oscillation periods:
\[\begin{align*} x(t)=\sum_{i=1}^{21}(C_{i}cos\omega_{i}t+S_{i}sin\omega_{i}t)e^{-a_{i}t}(2) \end{align*}\] To determine the coefficients \((C, S, \omega, a)\), the Laplace transform is applied to the dynamic function \(x(t)\): \[\begin{align*} L\{x(t)\}(s)=\int_{0}^{\infty}x(t) e^{-st} dt. (3) \end{align*}\] Or explicitly: \[\begin{align*} L\{x(t)\}(s)=\sum_{i=1}^{21}(C_{i}\frac{s}{(s+a_{i})^{2}+\omega_{i}^{2}}+S_{i}\frac{\omega_{i}}{(s+a_{i})^{2}+\omega_{i}^{2}}). (3.1) \end{align*}\] Based on available statistical data, the parameters \((C, S, \omega, a)\) of equation (3.1) are calculated, and then a dynamic function \(x(t)\) and the rate of change \(dx(t)/dt\) are built. For all analyzed SED indicators, oscillations with periods from 6 to 68 years make the greatest contribution.
7.2.3 Description of input variables
The Model uses retrospective values of the following indicators as input data (for at least 6 years):
| No | Code | Indicator | Unit |
|---|---|---|---|
| 1 | C013 | Population as of Jan 1 | people |
| 2 | C034 | Number of births | people |
| 3 | C035 | Number of deaths | people |
| 4 | C037 | Population inflow (migration) | people |
| 5 | C038 | Population outflow (migration) | people |
| 6 | C041 | Working-age population | people |
| 7 | C096 | Commissioning of apartment buildings | m2 |
| 8 | C149 | Total residential space | m2 |
| 9 | C221 | Average number of employees | people |
| 10 | C241 | Payroll fund | rubles |
| 11 | C698 | Taxable money income of individuals | rubles |
| 12 | C699 | Population income: social and other payments | rubles |
| 13 | C712 | Money income per capita | rub/year |
| 14 | C713 | Mandatory payments and contributions per capita | rub/year |
| 15 | C714 | Real disposable income per capita | rub/year |
| 16 | C757 | Integral provision of social infrastructure | % |
| 17 | C049 | Total city budget income | mln rub |
| 18 | C065 | Total city budget expenses | mln rub |
| 19 | C087 | Fixed capital investment | mln rub |
| 20 | C706 | Gross City Product (GCP) | mln rub |
| 21 | C708 | Tax revenues of all budget levels | mln rub |
| 22 | C709 | Local budget revenues (tax) | mln rub |
| 23 | C433 | Current assets | mln rub |
| 24 | C475 | Revenue | mln rub |
| 25 | C711 | Total cost (revenue - net profit) | mln rub |
| 26 | C707 | Gross value added of SMEs | mln rub |
| 27 | C409 | Fixed assets | mln rub |
| 28 | C715 | Regional Consumer Price Index (CPI) | units |
| 29 | C716 | Exchange rate | rub / $ |
| 30 | C717 | Balance of payments of the RF | dollars |
| 31 | C718 | Gross Domestic Product (GDP) of the RF | mln rub |
7.2.4 Complete description of the calculation algorithm
The calculation algorithm consists of the following steps:
Preparation of initial data including actual values of the model input variables specified in the List of model input variables.
Determination of the initial parameters of the regression model A, B, c, d by the least squares method, minimizing the error for all indicators simultaneously for each city and each group of socio-economic indicators of the city:
- Demographic (population, number of births and deaths, arrivals and departures, working-age population)
- Economic (average number of employees, payroll fund, revenue and profit, SME income, fixed and current assets, investment in fixed capital)
- City (Gross City Product, budget income and expenses, population income and expenses, tax revenues of all budget levels)
- Social (total residential space, commissioning of apartment buildings, commissioning of residential houses, social infrastructure provision)
Calculation of forecast and restoration of missing retrospective socio-economic development indicators based on the regression model.
Determination of calculation errors: \[\begin{align*} \varepsilon_{[20XX]} = x(20XX)-X_{[20XX]}, (6) \end{align*}\]
where:
\(\varepsilon_{[20XX]}\) – the error value for each indicator for the year 20XX; \(x(20XX)\) – the value of the indicator obtained based on the regression model in step 3; \(X_{[20XX]}\) – the actual value of the indicator.Refinement of parameters A, B, c, d using the Newton method of unconditional optimization of the sum of squared errors \(\sum\varepsilon^{2}\), obtained in step 4.
Iterative repetition of steps 2, 3, 4, 5 until the possible minimum of the sum of squared errors is reached.
Construction of the upper and lower boundaries of the regression model forecast. The approximation error \(err(t)\) includes two components: constant \(\varepsilon_{const}\) and variable \(\varepsilon_{var}(t-t_{0})\), equal to 0 at the starting point \(t_{0}\) (in the presented results \(t_{0}\) corresponds to January 1, 2019): \[\begin{align*} err(t)^{2}=\varepsilon_{const}^{2} + \varepsilon_{var}^{2}(t-t_{0}). (7) \end{align*}\]
To construct the upper \(H(t)\) and lower \(L(t)\) boundaries of the regression model indicators, the error \(err(t)\) is multiplied by the quantile \(z_{[P]}\), which depends on the confidence probability \(P\), the number of parameters, and the number of actual values (for the given example at \(P=90\%\), 16 parameters, and 90 actual values, \(z_{[90\%]} = 1.673\)): \[\begin{align*} H(t)=X(t)e^{z_{[P]}err(t)}, (8) L(t)=X(t)e^{-z_{[P]}err(t)}. (9) \end{align*}\]
Determination of the dynamic function parameters \((C, S, \omega, a)\) based on available statistical data, the upper and lower boundaries of the regression model indicators based on equation (2).
Formation of model values for each indicator for the forecast and retrospective periods based on the dynamic function built using parameters \((C, S, \omega, a)\).
Calculation of correlation matrices for each indicator by formula (4) based on the obtained model values.
7.2.5 Scope of permissible application of mathematical models
The scope (boundary) of permissible application of mathematical models is:
– formation of inertial medium-term and long-term forecasts of basic socio-economic development indicators of cities; – restoration of missing values for socio-economic development indicators of cities in the retrospective period no earlier than 2000; – creation of correlation matrices for the implementation of models for assessing the impact of investment project parameters on the socio-economic development of cities in the medium-term (1-5 years) and long-term (5-15 years) horizons.