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