# American Institute of Mathematical Sciences

September  2020, 16(5): 2065-2086. doi: 10.3934/jimo.2019043

## Optimal expansion timing decisions in multi-stage PPP projects involving dedicated asset and government subsidies

 1 Department of Mathematics, Tianjin University of Commerce, Tianjin 300134, China 2 Coordinated Innovation Center for Computable Modeling in Management Science, Tianjin University of Finance and Economics, Tianjin 300222, China

* Corresponding author: Jinghuan Li

Received  September 2018 Revised  January 2019 Published  May 2019

Fund Project: This project was supported in part by the the Major Research Plan of the National Natural Science Foundation of China (91430108), the National Natural Science Foundation of China (11771322, 71471132, 71573189), and Tianjin Education Commission Scientific Research Plan(2017SK076, 2017KJ236)

The topic of investment timing in multi-stage public-private partnership (PPP) projects has not been received much attention so far. This study investigates optimal expansion timing decisions in multi-stage PPP projects under an uncertain demand and where the first-stage greenfield project involving a dedicated asset is immediately implemented as the PPP contract is closed, whereas the timing of the later expansion is flexibly decided according to the demand. In this setting, the optimal multiple stopping timing theory is adopted to model the expansion framework. Furthermore, we integrate a government subsidy, including an investment subsidy and revenue subsidy, into the expansion timing decisions. Through a hypothetical three-stage investment plan for a sanitary sewerage project, the optimal expansion strategies and the value of the multi-stage project before and after the subsidy are provided using a least squares Monte Carlo simulation. Also, the influences of a dedicated asset on the expansion strategies and project value are illustrated. In addition, we compare the incremental value before and after the subsidy and earlier expansion derived from two types of subsidies. The comparisons show that there is more incremental value for the revenue subsidy, and that the investment subsidy brings an earlier expansion.

Citation: Jinghuan Li, Yu Li, Shuhua Zhang. Optimal expansion timing decisions in multi-stage PPP projects involving dedicated asset and government subsidies. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2065-2086. doi: 10.3934/jimo.2019043
##### References:

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##### References:
The schematic diagram of a three-stage PPP project
The project value for different demand levels
The optimal exercise boundaries for the i-th (i = 1, 2) expansions
The influences of the dedicated asset ratio
The project value and subsidy amount under different demands
The influences of the investment subsidy proportion
Revenue subsidy at different demand levels
The influences of the revenue subsidy price
The comparison of the subsidy amount
The comparison of the incremental value
The comparison of the exercise boundary under the same subsidy amount
Default parameters used in the calculations
 Constant Symbol Value Unit Concession Period $T_{c}$ 30 Year Investment period $T$ 10 Year Planned investment times $N$ 3 time Construction period $\nu$ 1 Year Refraction time $\delta$ 2 Year Capacity of i-th stage $m_{i}$ 40,000 $m^3$/day Unit price $p$ 1.8 CNY/$m^3$ Unit operational cost $c$ 0.8 CNY/$m^3$ Construction cost parameter $b$ 2917.8 Construction cost parameter $\gamma$ 0.9427 Drift $\alpha$ 6% Volatility rate $\sigma$ 15% Discount rate $\rho$ 8% Dedicated asset ratio $\eta$ 10%
 Constant Symbol Value Unit Concession Period $T_{c}$ 30 Year Investment period $T$ 10 Year Planned investment times $N$ 3 time Construction period $\nu$ 1 Year Refraction time $\delta$ 2 Year Capacity of i-th stage $m_{i}$ 40,000 $m^3$/day Unit price $p$ 1.8 CNY/$m^3$ Unit operational cost $c$ 0.8 CNY/$m^3$ Construction cost parameter $b$ 2917.8 Construction cost parameter $\gamma$ 0.9427 Drift $\alpha$ 6% Volatility rate $\sigma$ 15% Discount rate $\rho$ 8% Dedicated asset ratio $\eta$ 10%
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