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 Kleywegt1

1School 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 Motivation Effect of Disaster Aid Effect of Donations Introduction Model Example

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 Introduction Model Example

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 Motivation Effect of Disaster Aid Effect of Donations Introduction Model Example

Real Estate Development Decisions

Model:

◮ Piece of land A ⊂ R2 ◮ 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 Motivation Effect of Disaster Aid Effect of Donations Introduction Model Example

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 Motivation Effect of Disaster Aid Effect of Donations Introduction Model Example

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 Motivation Effect of Disaster Aid Effect of Donations Introduction Model Example

Example

◮ Region A = (0, 1]2 ◮ Incremental value v(x1, x2) = x1 ◮ Expected cost E[c(x, t, ω)] = x2

Anton Kleywegt Dynamic Models of Long Term Consequences of Disaster Relief

Outline Motivation Effect of Disaster Aid Effect of Donations Introduction Model Example

Example

◮ Economical to develop set of land

A∗ := {x ∈ A : v(x) ≥ E[c(x, t, ω)]} =

• x ∈ (0, 1]2 : x1 ≥ x2
• Social surplus

=

• A∗ (v(x) − E[c(x, t, ω)]) dx

= 1 1

x2

(x1 − x2) dx1dx2 = 1 6

Anton Kleywegt Dynamic Models of Long Term Consequences of Disaster Relief

Outline Motivation Effect of Disaster Aid Effect of Donations Introduction Model Example

Example

◮ Government disaster relief policy φ(c) = 1 2c ◮ Economic agents develop set of land

A′ =

• x ∈ (0, 1]2 : x1 ≥ 1

2x2

• Social surplus

=

• A′ (v(x) − E[c(x, t, ω)]) dx

= 1 8

Anton Kleywegt Dynamic Models of Long Term Consequences of Disaster Relief

Outline Motivation Effect of Disaster Aid Effect of Donations Introduction Model Example

Example

◮ Relative loss in social surplus =

1 6 − 1 8 1 6

= 1

4

Expected optimal cost =

• A∗ E[c(x, t, ω)] dx

= 1 6 Expected cost with disaster aid =

• A′ E[c(x, t, ω)] dx

= 1 3

Anton Kleywegt Dynamic Models of Long Term Consequences of Disaster Relief

Outline Motivation Effect of Disaster Aid Effect of Donations Introduction Model Example

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 Introduction Model Example

Consumer and Producer Surplus

Anton Kleywegt Dynamic Models of Long Term Consequences of Disaster Relief

Outline Motivation Effect of Disaster Aid Effect of Donations Introduction Model Example

Surplus with Donation

Anton Kleywegt Dynamic Models of Long Term Consequences of Disaster Relief

Outline Motivation Effect of Disaster Aid Effect of Donations Introduction Model Example

Dynamics: Basic Cobweb Model

Anton Kleywegt Dynamic Models of Long Term Consequences of Disaster Relief

Outline Motivation Effect of Disaster Aid Effect of Donations Introduction Model Example

Dynamics: Basic Cobweb Model

Anton Kleywegt Dynamic Models of Long Term Consequences of Disaster Relief

Outline Motivation Effect of Disaster Aid Effect of Donations Introduction Model Example

Dynamics: Basic Cobweb Model

Anton Kleywegt Dynamic Models of Long Term Consequences of Disaster Relief

Outline Motivation Effect of Disaster Aid Effect of Donations Introduction Model Example

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 Motivation Effect of Disaster Aid Effect of Donations Introduction Model Example

Effect of Donations on Local Economy

Model:

◮ Discrete time t = 0, 1, . . . ◮ Local supply qt in time period t, q0 given ◮ Donation ∆t in time period t ◮ Market clearing price pt in time period t,

pt = d−1(qt + ∆t) assuming donated goods are sold

◮ If s(pt) ≥ qt, then

qt+1 = qt + λ1(s(pt) − qt) where λ1 ∈ [0, 1]

◮ If s(pt) ≤ qt, then

qt+1 = qt + λ2(s(pt) − qt) where λ2 ∈ [0, 1]

Anton Kleywegt Dynamic Models of Long Term Consequences of Disaster Relief

Outline Motivation Effect of Disaster Aid Effect of Donations Introduction Model Example

Effect of Donations on Local Economy

Model:

◮ Consumer surplus Ut in time period t,

Ut = qt+∆t d−1(q) dq − pt(qt + ∆t)

• r

Ut = qt+∆t d−1(q) dq − ptqt

◮ Local producer surplus Vt in time period t,

Vt = ptqt − qt s−1(q) dq

Anton Kleywegt Dynamic Models of Long Term Consequences of Disaster Relief

Outline Motivation Effect of Disaster Aid Effect of Donations Introduction Model Example

Effect of Donations on Local Economy

Model:

◮ Local social surplus Wt in time period t,

Wt = Ut+Vt = qt+∆t d−1(q) dq− qt s−1(q) dq−pt∆t

• r

Wt = Ut + Vt = qt+∆t d−1(q) dq − qt s−1(q) dq

Anton Kleywegt Dynamic Models of Long Term Consequences of Disaster Relief

Outline Motivation Effect of Disaster Aid Effect of Donations Introduction Model Example

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 bt at time t, b0 = b ◮ State St = (qt, bt) ◮ Decision ∆t ∈ [0, bt]

Anton Kleywegt Dynamic Models of Long Term Consequences of Disaster Relief

Outline Motivation Effect of Disaster Aid Effect of Donations Introduction Model Example

Effect of Donations on Local Economy

Dynamic Programming Model for Optimal Control:

◮ Market clearing price pt in time period t,

pt = d−1(qt + ∆t)

◮ Transitions

qt+1 = qt + λ1(s(pt) − qt) if s(pt) ≥ qt qt + λ2(s(pt) − qt) if s(pt) ≤ qt bt+1 = bt − ∆t

◮ Single stage reward

r(qt, bt, ∆t) = Wt = qt+∆t d−1(q) dq − qt s−1(q) dq

◮ Total Objective

max

∆0,∆1,...

• t

αtWt where α ∈ (0, 1) is a discount factor

Anton Kleywegt Dynamic Models of Long Term Consequences of Disaster Relief

Outline Motivation Effect of Disaster Aid Effect of Donations Introduction Model Example

Example

◮ Long run local supply s(p) = max

• 0, p−b

2a

• , where a, b > 0,

inverse supply function s−1(q) = 2aq + b for q ≥ 0

◮ Demand d(p) = max {0, α − βp}, where α, β > 0, inverse

demand function d−1(q) = max

• 0, α−q

β

• for q ≥ 0

◮ Equilibrium price p∗ = 2aα+b 1+2aβ ◮ Equilibrium quantity q∗ = α−bβ 1+2aβ

Anton Kleywegt Dynamic Models of Long Term Consequences of Disaster Relief

Outline Motivation Effect of Disaster Aid Effect of Donations Introduction Model Example

Example

◮ Market clearing price pt in time period t,

pt = max

• 0, α − (qt + ∆t)

β

• ◮ Transitions

qt+1 =            qt + λ1

• max
• 0, pt−b

2a

• − qt
• if ∆t ≤ −(1 + 2aβ)qt + α − bβ

qt + λ2(max

• 0, pt−b

2a

• − qt)

if ∆t ≥ −(1 + 2aβ)qt + α − bβ bt+1 = bt − ∆t

◮ Single stage reward

Wt = α min{α, qt + ∆t} β − min{α, qt + ∆t}2 2β − aq2

t − bqt

Anton Kleywegt Dynamic Models of Long Term Consequences of Disaster Relief

Outline Motivation Effect of Disaster Aid Effect of Donations Introduction Model Example

Example

◮ a = 0.25 ◮ b = 20 ◮ α = 100 ◮ β = 1 ◮ Equilibrium price p∗ = 47 ◮ Equilibrium quantity q∗ = 53

Anton Kleywegt Dynamic Models of Long Term Consequences of Disaster Relief

Outline Motivation Effect of Disaster Aid Effect of Donations Introduction Model Example

Dynamics with No Donations

Anton Kleywegt Dynamic Models of Long Term Consequences of Disaster Relief

Outline Motivation Effect of Disaster Aid Effect of Donations Introduction Model Example

Dynamics with Immediate Donation

Anton Kleywegt Dynamic Models of Long Term Consequences of Disaster Relief

Outline Motivation Effect of Disaster Aid Effect of Donations Introduction Model Example

Dynamics with Optimal Control

Anton Kleywegt Dynamic Models of Long Term Consequences of Disaster Relief

Outline Motivation Effect of Disaster Aid Effect of Donations Introduction Model Example

Dynamics: Compare Controls

Anton Kleywegt Dynamic Models of Long Term Consequences of Disaster Relief

Outline Motivation Effect of Disaster Aid Effect of Donations Introduction Model Example

Dynamics: Compare Controls

◮ Total Discounted Surplus under Null Control = 13500 ◮ Total Discounted Surplus under Immediate Control = 12000 ◮ Total Discounted Surplus under Optimal Control = 18500

Anton Kleywegt Dynamic Models of Long Term Consequences of Disaster Relief