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Mathematical Optimization for Economics
Nicolas Boccard
Departament d’EconomiaUniversitat de Girona, Spain
2011
N. Boccard (UdG) 2011 1 / 14
Mathematical Optimization Economic Problem
Economic Problem
• Economic choice x : quantity, price or quality
• Economic objective y: profit or utility
• Relationship between cause and effect: y = f (x)
• Economic restrictions over choice: x ≥ x1 and x ≤ x2
• Find optimal choice x∗ maximizing objective over domain
• Intuition: maximize mathematical function f (.)
• Solve first order condition (FOC) dfdx = 0
• Mathematical Solution x0 is only candidate economic solution
• Caution: several cases may appear, require detailed examination
N. Boccard (UdG) 2011 2 / 14
Mathematical Optimization Economic Problem
Interior Optimum
• Domain x1 ≤ x ≤ x2 connected (gray) segment
• FOC solution x0 interior ⇒ x∗ = x0
• Interior case: mathematical optimum x0 is economic optimum x∗
N. Boccard (UdG) 2011 3 / 14
Mathematical Optimization Economic Problem
Maximum Corner Solution
• Meaningful Economic Domain x1 ≤ x ≤ x2
• Mathematical optimum outside economic domain: x0 > x2
• Economic optimum: corner solution x∗ = x2
N. Boccard (UdG) 2011 4 / 14
Mathematical Optimization Economic Problem
Minimum Corner Solution
• Meaningful Economic Domain x1 ≤ x ≤ x2
• Mathematical optimum outside economic domain: x0 < x1
• Economic optimum: corner solution x∗ = x1
N. Boccard (UdG) 2011 5 / 14
Mathematical Optimization Economic Problem
Empty Economic Domain
• Incompatible (schizophrenic) conditions: x ≥ x1 and x ≤ x2
• Optimization Problem has NO Solution
• Economic model badly stated
N. Boccard (UdG) 2011 6 / 14
Mathematical Optimization Regimes Changes
Regimes Changes
• Decision-maker faces a variety of qualitatively different strategies
• Low choice of strategic variable x: aggressive strategy
• High choice of strategic variable x: accommodating strategy
• Behavior of decision-maker changes around some threshold x0
• Complex Objective f (x) ={
g(x) if x ≤ x0
h(x) if x ≥ x0
• Find overall optimum x∗ over both regimes
• Candidates xg and xh, unrestricted maximizers of g(.) and h(.)
N. Boccard (UdG) 2011 7 / 14
Mathematical Optimization Regimes Changes
Dominated High Choice
g h
y
g h
❺
• xh < x0 ⇒ leave area x ≥ x0 ⇒ optimum in area x ≤ x0
• In this case, xg < x0, thus it is the economic optimum
N. Boccard (UdG) 2011 8 / 14
Mathematical Optimization Regimes Changes
Dominated Low Choice
g h
y
g h
❻
• Symmetric case: xg > x0 ⇒ leave area x ≤ x0 ⇒ optimum in area x ≥ x0
• In this case, xh > x0, thus it is the economic optimum
N. Boccard (UdG) 2011 9 / 14
Mathematical Optimization Regimes Changes
Corner Solution
❼
g h
y
gh
• Corner Solution: tendency to leave both areas
N. Boccard (UdG) 2011 10 / 14
Mathematical Optimization Regimes Changes
g h
y
g h
❽
• Each regime has an interior maximum
• Multiple mathematical solutions
• Unique economic optimum found by comparing of g(xg ) and h(xh)
N. Boccard (UdG) 2011 11 / 14
Mathematical Optimization Comparative Statics
Comparative Statics
• Consider two ranked functions g < h, one parameter z
y
g h
g
h
z
N. Boccard (UdG) 2011 12 / 14
Mathematical Optimization Comparative Statics
• Solution xg to g = z, solution xh to h = z• We show that xg ≤ xh in 4 steps
y
g h
g
h
z
N. Boccard (UdG) 2011 13 / 14
Mathematical Optimization Comparative Statics
• Weighted averages of g and h• If α>βmeans f close to g, k close to h• Again xf < xk
y
g h
gf
kh α > β
f = αg + (1-α) h
k = βg + (1-β) h
f k
z
N. Boccard (UdG) 2011 14 / 14