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

  • * Corresponding author: Jinghuan Li

    * Corresponding author: Jinghuan Li 

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)

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  • 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.

    Mathematics Subject Classification: Primary: 49J20, 65C05, 65M06.


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  • Figure 1.  The schematic diagram of a three-stage PPP project

    Figure 2.  The project value for different demand levels

    Figure 3.  The optimal exercise boundaries for the i-th (i = 1, 2) expansions

    Figure 4.  The influences of the dedicated asset ratio

    Figure 5.  The project value and subsidy amount under different demands

    Figure 6.  The influences of the investment subsidy proportion

    Figure 7.  Revenue subsidy at different demand levels

    Figure 8.  The influences of the revenue subsidy price

    Figure 9.  The comparison of the subsidy amount

    Figure 10.  The comparison of the incremental value

    Figure 11.  The comparison of the exercise boundary under the same subsidy amount

    Table 1.  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%
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