1. Beach Nourishment with Strategic Interaction between Adjacent Communities: A Game Theoretic Model Camp Resources XV August 7-8, 2008 Sathya Gopalakrishnan Martin Smith Nicholas School of the Environment & Earth Sciences Duke University

2. 2 MOTIVATION Dynamic Coastal Environment Sea Level Rise Changing Storm Patterns Gradual Landward movement of shoreline 90 % Sandy beaches in the US face erosion Source: www.noaanews.noaa.gov/stories2006/s2622.htm (Leatherman, 1993)

3. 3 MOTIVATION Growing development in coastal towns that thrive on revenue from tourism Policy intervention to protect property and infrastructure Beach Nourishment: Process of artificially re-building the beach by replacing lost sand Source: www.darylboyette.comwww.darylboyette.com Source: www.ncdot.org/doh/preconstruct/highway/photo/...www.ncdot.org/doh/preconstruct/highway/photo/

4. 4 earthtrends.wri.org Photo credit: Program of the Study of Developed Shorelines, Western Carolina University Factors influencing Coastal Change Sea Level Rise (~ x100) Alongshore Sediment Transport Max transport (~ 45 degrees) Human Manipulation

5. 5 Simplified Dynamics of Beach Erosion and Nourishment From Smith, Slott, and Murray, 2007 (in review) Beach Nourishment: Process of artificially re-building the beach by replacing lost sand periodically with sand from off-shore locations

6. 6 Beach Nourishment analogous to Rotational Harvest in Forestry Benefits from maintaining beach width: Recreational flow & Storm Protection High fixed cost of nourishment Coastal dynamics influencing erosion rate Rotational/Periodic Nourishment Faustmann problem applied in reverse

7. 7 THE MODEL: Single Community = For each nourishment interval T Discount rate Width Fixed cost Variable cost Community chooses a series of nourishment intervals Infinite horizon problem (assuming stationarity) Community chooses Optimal T* Net Benefits NB(T) Benefits B(T) Costs C(T) -

8. 8 If there are two communities How does strategic interaction between adjacent coastal communities affect volume and frequency of re-nourishment? What are the policy implication of physical and economic interdependency? Non-cooperative vs Cooperative solutions? Can we estimate welfare effects? Does North Carolina data support the model?

9. 9 Spatial interaction between adjacent communities Adapted (modified) from Smith, Slott, and Murray, 2007 (in review)

10. 10 State Equations for Beach Width Uniform Retreat Exponentia l Retreat Diffusion Parameter Sediment Transfer from Beach B Narrow Beach Portion experiencing exponential retreat In the absence of Beach A Slower erosion due to Beach A Sediment Transfer to Beach A Wide Beach

11. 11 Beach Width with No Spatial Interaction (K = 0) 10-year Nourishment Interval Initial Width Beach A = 100 Beach B = 200 Baseline erosion = 2 ft/yr Exponential retreat rate for nourished beach = 0.10 Portion of Beach A facing exponential retreat = 0.35 Portion of Beach B facing exponential retreat = 0.675

12. 12 Beach Width with Spatial Interaction (K = 0.5) 10-year Nourishment Interval Increase in K (0.5) Slower in erosion at both A and B Change in curvature Concave Function for A Time (Years) Beach Width (Feet)

13. 13 Beach Width with Higher Diffusion (K = 1) Nourishment Interval at Wider Beach = 10 years Beach B nourishes every 10 years Beach A experiences accretion No Nourishment at A Positive externality Time(Years) Beach A Beach B Similar relation between rich and poor communities? Beach Width (Feet)

14. 14 Strategic interaction in a 2-player game PLAYERS: Community A and Community B (Coastal Managers) ACTION: Nourishment Interval T ∈ [0, Tmax] PAYOFFS: Present Value Net Benefits NB(T) = f(x(T)) INFORMATION: Full Information Game Net Benefits for i Beach Width at i Tj*Tj* Ti*Ti*

15. 15 To solve for the optimal nourishment interval Beach A: Find Optimal T A * given T B Beach B: Find Optimal T B * given T A T A * = f(T B ) B’s Reaction / Best Response Function NASH EQUILIBRIUM Mutual Best Responses / Intersection of reaction functions T B * = f(T A ) A’s Reaction / Best Response Function

16. 16 Preliminary Results: To solve for Optimal Rotation Kinky Reaction Functions Not Unique Nash Equilibrium Non-autonomous solution path?

17. 17 Preliminary Results Non-cooperative vs Cooperative solutions Maximize of joint net benefits Optimal Rotation Length Increases at the Narrow Beach Decreases at the Wider Beach Joint Net Benefits under Cooperative solution greater than sum of Net Benefits under Competitive Solution

18. 18 Next Steps Non-autonomous solution path 1.Optimal Control Problem:  Differential Game: Optimal nourishment  Accounting for Fixed Costs?  No fixed costs - ‘Bang-Bang Solution’ - Nourish every time period 2.Discrete Choice Dynamic Programming Problem  Binary Choice Variable at each time period: Nourish / Don’t Two-stage model with endogenous initial width Choice variables {T*, x0}

19. 19 Questions & Suggestions

20. 20 ADDITIONAL SLIDES

21. 21 Reference Case Simple Case - Starting with a Straight Shoreline Choosing Initial Width [x0(A) and x0(B)] and mu(A) determines the portion of beach B facing exponential decay

22. 22 State Equations for Beach Width 50-year horizon; No Interaction50-year horizon; With Interaction Slower Erosion in both beaches

23. 23 Preliminary Analysis: Sensitivity of Diffusion Parameter Beach i assumes that Beach j will never nourish ( Tj ≥ 1000 ) Beach A Beach B K ⇒ TA*TA* K ⇒ TB*TB* Large K (> 0.75) Narrow beach gains from sediment transfer Delays Nourishment Optimal Nourishment Interval (T*) varying K Diffusion Parameter (K) Optimal Nourishment Interval (Years) High K creates incentive for wider beach to delay nourishment and reduce gradient differential?

24. 24 Sensitivity to Diffusion Parameter No Nash Equilibrium K = 0.25

25. 25 State Equations with Nourishment if Width of beach i at time t depends on the nourishment interval at beach j

26. 26 The Resource Problem Housing markets directly influenced by physical coastal processes Coastal property values affected by erosion Beach as a dynamic natural resource that generates value Storm protection Recreational flow Value of beach reflected in property values Environmental Economics Optimal Beach Management Natural Resource Economics

27. 27 Beach as a Dynamic Natural Resource Different from traditional resource economics problems Economic Value of the resource derived from maintaining resource base (preventing erosion) rather than harvesting the resource Analogous problem can be solved using tools from resource economics Wide Beach Storm Protection Recreational Flow Value

28. 28 State Equations for Beach Width Portion experiencing exponential retreat In the absence of Beach A Transition Function Sediment Transfer to Beach A Wide Beach Slower erosion due to Beach A