Dynamic Models of Long Term Consequences of Disaster Relief Anton - PowerPoint PPT Presentation

Outline Motivation Effect of Disaster Aid Effect of Donations Dynamic Models of Long Term Consequences of Disaster Relief Anton Kleywegt 1 1 School of Industrial and Systems Engineering Georgia Institute of Technology Workshop on Humanitarian Logistics: Networks for Africa Bellagio, Italy 5–9 May 2008 Anton Kleywegt Dynamic Models of Long Term Consequences of Disaster Relief

Outline Motivation Effect of Disaster Aid Effect of Donations Motivation Effect of Disaster Aid on Real Estate Development Introduction Model Example Effect of Donations on Local Economy Introduction Model Example Anton Kleywegt Dynamic Models of Long Term Consequences of Disaster Relief

Outline Motivation Effect of Disaster Aid Effect of Donations CARE Refuses US Food Donations Anton Kleywegt Dynamic Models of Long Term Consequences of Disaster Relief

Outline Introduction Motivation Model Effect of Disaster Aid Example Effect of Donations Motivation Effect of Disaster Aid on Real Estate Development Introduction Model Example Effect of Donations on Local Economy Introduction Model Example Anton Kleywegt Dynamic Models of Long Term Consequences of Disaster Relief

Outline Introduction Motivation Model Effect of Disaster Aid Example Effect of Donations Real Estate Development Decisions ◮ Real estate development decisions influenced by benefits and costs ◮ Cost of risk/insurance part of cost of real estate ◮ Disaster aid affects cost of risk/insurance ◮ Therefore disaster aid affects real estate development decisions and resulting cost of risk/insurance and disaster aid Anton Kleywegt Dynamic Models of Long Term Consequences of Disaster Relief

Outline Introduction Motivation Model Effect of Disaster Aid Example Effect of Donations Real Estate Development Decisions Model: ◮ Piece of land A ⊂ R 2 ◮ Discrete time t = 0 , 1 , . . . ◮ For each x ∈ A , t = 0 , 1 , . . . , the land provides incremental value v ( x ) per unit area per unit time if developed ◮ For each x ∈ A , t = 0 , 1 , . . . , there is a random cost c ( x , t , ω ), ω ∈ Ω, per unit area per unit time ◮ Sequence { c ( x , t , ω ) : t = 0 , 1 , . . . } is i.i.d., with mean E [ c ( x , t , ω )] < ∞ Anton Kleywegt Dynamic Models of Long Term Consequences of Disaster Relief

Outline Introduction Motivation Model Effect of Disaster Aid Example Effect of Donations Real Estate Development Decisions Model: ◮ Decisions: For each piece of land x ∈ A , whether to develop the land or not ◮ Risk neutral — risk can be insured ◮ Economical to develop set of land A ∗ := { x ∈ A : v ( x ) ≥ E [ c ( x , t , ω )] } ◮ � Social surplus = A ∗ ( v ( x ) − E [ c ( x , t , ω )]) dx Anton Kleywegt Dynamic Models of Long Term Consequences of Disaster Relief

Outline Introduction Motivation Model Effect of Disaster Aid Example Effect of Donations Real Estate Development Decisions Model: ◮ Suppose disaster aid is provided ◮ Disaster aid policy: contribute φ ( c ) if cost is c (0 ≤ φ ( c ) ≤ c ) ◮ Economical to develop set of land A ′ := { x ∈ A : v ( x ) ≥ E [ c ( x , t , ω ) − φ ( c ( x , t , ω ))] } ◮ Note: A ∗ ⊂ A ′ ◮ � Social surplus = A ′ ( v ( x ) − E [ c ( x , t , ω )]) dx Anton Kleywegt Dynamic Models of Long Term Consequences of Disaster Relief

Outline Introduction Motivation Model Effect of Disaster Aid Example Effect of Donations Example ◮ Region A = (0 , 1] 2 ◮ Incremental value v ( x 1 , x 2 ) = x 1 ◮ Expected cost E [ c ( x , t , ω )] = x 2 Anton Kleywegt Dynamic Models of Long Term Consequences of Disaster Relief

Outline Introduction Motivation Model Effect of Disaster Aid Example Effect of Donations Example ◮ Economical to develop set of land A ∗ := { x ∈ A : v ( x ) ≥ E [ c ( x , t , ω )] } x ∈ (0 , 1] 2 : x 1 ≥ x 2 � � = � Social surplus = A ∗ ( v ( x ) − E [ c ( x , t , ω )]) dx � 1 � 1 1 = ( x 1 − x 2 ) dx 1 dx 2 = 6 0 x 2 Anton Kleywegt Dynamic Models of Long Term Consequences of Disaster Relief

Outline Introduction Motivation Model Effect of Disaster Aid Example Effect of Donations Example ◮ Government disaster relief policy φ ( c ) = 1 2 c ◮ Economic agents develop set of land � x ∈ (0 , 1] 2 : x 1 ≥ 1 � A ′ = 2 x 2 � 1 Social surplus = A ′ ( v ( x ) − E [ c ( x , t , ω )]) dx = 8 Anton Kleywegt Dynamic Models of Long Term Consequences of Disaster Relief

Outline Introduction Motivation Model Effect of Disaster Aid Example Effect of Donations Example 6 − 1 1 ◮ Relative loss in social surplus = = 1 8 1 4 6 1 � Expected optimal cost = A ∗ E [ c ( x , t , ω )] dx = 6 1 � Expected cost with disaster aid = A ′ E [ c ( x , t , ω )] dx = 3 Anton Kleywegt Dynamic Models of Long Term Consequences of Disaster Relief

Outline Introduction Motivation Model Effect of Disaster Aid Example Effect of Donations Motivation Effect of Disaster Aid on Real Estate Development Introduction Model Example Effect of Donations on Local Economy Introduction Model Example Anton Kleywegt Dynamic Models of Long Term Consequences of Disaster Relief

Outline Introduction Motivation Model Effect of Disaster Aid Example Effect of Donations Consumer and Producer Surplus Anton Kleywegt Dynamic Models of Long Term Consequences of Disaster Relief

Outline Introduction Motivation Model Effect of Disaster Aid Example Effect of Donations Surplus with Donation Anton Kleywegt Dynamic Models of Long Term Consequences of Disaster Relief

Outline Introduction Motivation Model Effect of Disaster Aid Example Effect of Donations Dynamics: Basic Cobweb Model Anton Kleywegt Dynamic Models of Long Term Consequences of Disaster Relief

Outline Introduction Motivation Model Effect of Disaster Aid Example Effect of Donations Dynamics: Basic Cobweb Model Anton Kleywegt Dynamic Models of Long Term Consequences of Disaster Relief

Outline Introduction Motivation Model Effect of Disaster Aid Example Effect of Donations Dynamics: Basic Cobweb Model Anton Kleywegt Dynamic Models of Long Term Consequences of Disaster Relief

Outline Introduction Motivation Model Effect of Disaster Aid Example Effect of Donations Effect of Donations on Local Economy Model: ◮ Long run local supply as a function of price s ( p ), strictly increasing, with inverse supply function s − 1 ( q ) ◮ Demand as a function of price d ( p ), strictly decreasing, with inverse demand function d − 1 ( q ) ◮ Equilibrium price p ∗ solution of d ( p ) = s ( p ) ◮ Equilibrium quantity q ∗ solution of d − 1 ( q ) = s − 1 ( q ) ◮ Supply does not adjust instantaneously to s ( p ′ ) when there is a new price p ′ Anton Kleywegt Dynamic Models of Long Term Consequences of Disaster Relief

Outline Introduction Motivation Model Effect of Disaster Aid Example Effect of Donations Effect of Donations on Local Economy Model: ◮ Discrete time t = 0 , 1 , . . . ◮ Local supply q t in time period t , q 0 given ◮ Donation ∆ t in time period t ◮ Market clearing price p t in time period t , d − 1 ( q t + ∆ t ) p t = assuming donated goods are sold ◮ If s ( p t ) ≥ q t , then q t +1 = q t + λ 1 ( s ( p t ) − q t ) where λ 1 ∈ [0 , 1] ◮ If s ( p t ) ≤ q t , then q t +1 = q t + λ 2 ( s ( p t ) − q t ) where λ 2 ∈ [0 , 1] Anton Kleywegt Dynamic Models of Long Term Consequences of Disaster Relief

Outline Introduction Motivation Model Effect of Disaster Aid Example Effect of Donations Effect of Donations on Local Economy Model: ◮ Consumer surplus U t in time period t , � q t +∆ t d − 1 ( q ) dq − p t ( q t + ∆ t ) = U t 0 or � q t +∆ t d − 1 ( q ) dq − p t q t = U t 0 ◮ Local producer surplus V t in time period t , � q t s − 1 ( q ) dq V t = p t q t − 0 Anton Kleywegt Dynamic Models of Long Term Consequences of Disaster Relief

Outline Introduction Motivation Model Effect of Disaster Aid Example Effect of Donations Effect of Donations on Local Economy Model: ◮ Local social surplus W t in time period t , � q t +∆ t � q t d − 1 ( q ) dq − s − 1 ( q ) dq − p t ∆ t W t = U t + V t = 0 0 or � q t +∆ t � q t d − 1 ( q ) dq − s − 1 ( q ) dq W t = U t + V t = 0 0 Anton Kleywegt Dynamic Models of Long Term Consequences of Disaster Relief

Outline Introduction Motivation Model Effect of Disaster Aid Example Effect of Donations Effect of Donations on Local Economy Dynamic Programming Model for Optimal Control: ◮ Discrete time t = 0 , 1 , . . . ◮ Total budget b for donations, � t ∆ t ≤ b ◮ Remaining budget b t at time t , b 0 = b ◮ State S t = ( q t , b t ) ◮ Decision ∆ t ∈ [0 , b t ] Anton Kleywegt Dynamic Models of Long Term Consequences of Disaster Relief