# Model-Based Dynamic Forecasting for Residential Construction Market Demand: A Systemic Approach
## 4 Discussion
### Stock and Flow Diagram
#### 4.2.4 Formulae of Modules  
The process of conceptualizing and formulating the model structure's formulae and converting it into a stock-flow form that can be computer-simulated through formalization is presented below. The construction stock (present stock value, \(PV_{t}\)) at the time \(t\) is composed of the total amount of input \(IV_{t-s}\) in the past year (\(t\)-\(s\)) that reflects the depreciation rate (\(r_{t}\)) in the input value (\(IV_{t}\)) of the construction year. The estimate of the construction capital stock amount can be expressed as Equation (1).  
\[PV_{t}=IV_{t}+(1-r_{t})IV_{t-1}+(1-r_{t})(1-r_{t-1})IV_{t-2}+\cdots+(1-r_{t}) \cdots(1-r_{t-s+1})IV_{t-s} \tag{1}\]  
\(PV_{t}\): Construction capital stock amount  
\(IV_{t}\): Construction capital input  
\(r_{t}\): depreciation rate  
Assuming that the depreciation form follows the declining rate method since the depreciation rate is constant every year, Equation (1) can be expressed simply as Equation (2) below, and the left side is defined as the right side.  
\[PSV_{t-1}=IV_{t-1}+(1-r_{t-1})IV_{t-2}+(1-r_{t-1})^{2}IV_{t-3}+\cdots(1-r_{t-1 })^{s}IV_{t-(s+1)} \tag{2}\]  
The time for large-scale maintenance of residential buildings is set to 20 years after completion, and the service life of facilities is 49.84 years [44]. The construction stock amount consisting of the new amount, maintenance amount, and demolition amount of the residential sector is expressed as Equation (3).  
\[RPSV_{t}=NV_{t}+\int_{t_{0}}^{t}[(1-r_{t})^{t}PSV_{t-1}]dt+MV_{t}+DV_{t} \tag{3}\]  
The amount of new construction is generated through optimal parameter estimation using inputs of changes in explanatory environmental variables such as various societies, economies, and institutions, and the parameters are searched to optimize the results of the simulation model. The formula for the amount of new residential construction is presented in Equation (4).  
\[NV_{t}=\alpha_{1}x_{1t}+\alpha_{2t}x_{2t}+\cdots+\alpha_{nt}x_{nt}+c_{it}=\sum _{i=1}^{n}\alpha_{it}x_{it}+c_{it} \tag{4}\]  
\(\alpha_{i}\): parameter(d), \(x_{i}\): variable(d), \(c_{i}\): constant(d)  
It is difficult to say that the planned maintenance amount will be converted into the actual maintenance amount due to realistic constraints such as a limited budget [45]. The actual amount of maintenance construction is estimated through the additional correction process of the maintenance pattern coefficient and the conversion coefficient to apply the ratio of the cost of maintenance to the new construction cost.  
The ratio of the cost of maintenance to new construction ranges from 95% to 100% for horizontal expansion, from 85% to 90% for vertical expansion, is about 80% for houses undergoing full-scale remodeling, and from 60% to 70% for the non-housing stock. The residential construction market corrects the cost conversion factor to 80% (Equation (5)).  
\[MV_{t}=\frac{1}{1-\alpha e^{bt}}\times PSV_{t-10},\ \alpha=0.16,\ b=0.1(49.84 \geq t-to\geq 20) \tag{5}\]As with the maintenance amount, it is difficult to say that the planned demolition amount will be converted to the actual demolition amount due to realistic restrictions. The actual amount of demolition construction is estimated through the additional correction process of the demolition pattern coefficient and the conversion coefficient to apply the ratio of the cost of demolition to the new construction cost. The ratio of the cost of demolition is generally within 3% of the cost of repair work, 3.48% in the Public Procurement Service (2017), and 3.2% in the Korea Institute of Construction Technology (2013). This study will correct the conversion ratio of 3.2% to the demolition scale (Equation (6)).  
\[DV_{t}=\frac{1}{1+\alpha e^{bt}}\times PSV_{t-10},\;\alpha=0.02,\;b=0.06(t-to \geq 49.84) \tag{6}\]