Esami corretti Economia industriale

4/9/2014

2 imprese stesso prodotto competizione sulle quantità Costo investimento CF=150 Cournot gioco simultaneo converne o meno?

1. P=120-2Q=120-2q₁-2q₂

   Π₁= (120-2q₁-2q₂-12) q₁ = 120q₁-2q₁q₂-2q₁²-12q₁

   ∂Π₁/∂q₁ = 120-4q₁-2q₂-12=0

   q₁ = 27 - q₂/2

   ∂Π₂/∂q₂ = 120-2q₁-4q₂-12 = 0

   q₂ = 27 - q₁/2

   q₁ = 27 - 1/2 (27 - q₁/2)

   P = 48

   Π₁ = (48 - 12) 18 = 648 = Π₂

2. C₁ = 6 e C₂ = 12

   Π₁ = (114 -2q₁-2q₂) q₁

   ∂Π₁/∂q₁ = 114-4q₁-2q₂ = 0

   q₁ = 28,5 - q₂/2

   Π₂ = (108 - 2q₁ -2q₂) q₂

   ∂Π₂/∂q₂ = 108-2q₁-4q₂ = 0

   q₂ = 27 - q₁/2

   q₁ = 28,5 - 1/2(27 - q₁/2)

   P = 110 - 2(20 +17) = 46

   Π₁ = 650 Π₂ = 578

Unità 1: Hotelling

N = 100 t = 1 c = 6 c(2 = 3

P1 + t Xn = P2 + t (1 - Xn)

Xn = (P2 + t - P1) / 2t

D1 = N (P2 + t - P1) / 2t = (50P2 / t) + 50 - (50P1 / t)

π1a = (P1 - C1) Q1a = (P1 - 6) (50P2 / t + 50 - 50P1 / t)

∂π1/∂P1 = 50P2 / t + 50 - 100P1 / t + 300 / t = 0

100P1 / t = 50P2 / t + 50 + 300 / t

P1 = P2 / 2 + t / 2 + 3

D2 = 100 (P1 + t - P2) / 2t = 50P1 / t + 50 - 50P2 / t

π2 = (P2 - 3) (50P1 / t + 50 - 50P2 / t)

∂π2/∂P2 = 50P1 / t + 50 - 100P2 / t + 150 / t = 0

100P2 / t = 50P1 / t + 50 + 150 / t

D2 = P1 / 2 + t / 2 + 1,5

Pn = 1 / 2 (P2 / 2 + t / 2 + 1,5) + t / 2 + 3

DP1 = P1 / 4 + t / 4 + 0,75 + t / 2

3/4 P1 = 3,75 + 3t / 4

P1 = t + 5

P2 = t / 2 + 2,5 + t / 2 + 1,5

P2 = t + 4

8/9/15

Prodotti Differenziati

duopolio

C1 = C2 = 2

QA = 10 - 2PA + PB

QB = 10 - 2PB + PA

A -> impresa leader

B -> follower

Stackelberg

Determinare equilibrio

Π'B = (10 - 2PB + PA) (PB - 2) = 10PB - 2PB² + PAPA - PB - 20 + 4PB - 2PA

ΠB = 10 - 4PB + PA + 4 = 0 -> PB = 14 + PA / 4

Π'A = (10 - 2PA + 14 + PA / 4) (PA - 2) = (13.5 - 1.75PA) (PA - 2)

= 13.5PA - 1.75PA² - 27 + 3.5PA = 17PA - 1.75PA² - 27

Π'A = 17 - 3.5PA = 0 -> PA = 4.85 e quindi PB = 4.71

Q = QA + QB = 10 - 2RA + PB + 10 - 2PB + PA = 20 - PA - PB

ΠB = (20 - PA - PB) (PB - 2) = 20PB - PAPB - PB² + 40 + 2PA + 2PB

= 22PB - PAPB - PB² - 40 + 2PA

Π'B = 22 - PA - 2PB = 0

PB = 22 - PA / 2

Π'A = (20 - PA - 11 + PA / 2) (PA - 2) = (9 - PA / 2) (PA - 27)

= 9PA - 18 - PA / 2 + PA

Π'A = 10 - PA = 0 -> PA = 10 quindi PB = 22 - 10 / 2 = 6

ΠB = 16

ΠA = 32

Allora QA = 5 e QB = 5.43

Dunque ΠA = 14.15 e ΠB = 14.71

π_M = (60-9₁)q₁ - 40q₂

π_M = 60 - 2q₁ - 40 - 0 - q_M

q_M = 10

se q₁⁰ > q₁^DET = 10 > 20 - 2√F -> F > 25 -> entrata bloccata

se q₁⁰ = 10 -> q₁⁰ + 20 - 10⁰₂ = 5

se q₁^DET π_M = π_ACC

(60 - 9⁰₁)q⁰₁ - 40q⁰₁ > 50

q₁⁰ - 20q₁ + 50 < 0

^{20 + √400}_-200

17,07

2,93

produco sempre non meno di q₁^M quindi

10 < q₁⁰ < 17,07

17,07 > 20 - 2√F -> F > 2,14

↓ N = ↓ compl. = π

3/5/'96

1) Duopolio Hotelling

C₁=1 C₂=4 t=1

● P e Π di equilibrio

P₁ + X_n = ^{P₂ - P₁ + 1}/₂

X_n = ^{P₂ - P₁ + 1}/₂

D₁ = ¹⁰⁰/₁ ( ^{P₂ - P₁ + 1}/₂) = 50P₂ - 50P₁ + 50

D₂ = ¹⁰⁰/₁ ( ^{P₁ - P₂ + 1}/₂) = 50P₁ - 50P₂ + 50

Π'₁ = (P₂ - 1) (50P₂ - 50P₁ + 50) = 50P₂P₂ - 50P₂P₁ + 50P₁ - 50P₂ + 50P₂ - 50P₁ - 5

Π₁ = 50P₂ - 100P₁ + 100 = 0

P₁ = ^{50P₂ + 100}/100 = ^P₂/2 + 1

Π'₂ = (P₂ - 4) (50P₁ - 50P₂ + 50) = 50P₁P₂ - 50P₁ + 50P₁ - 50P₂ + 200P₁ + 200P₂ - 20

Π₂ = 50P₁ - 100P₂ + 250 = 0

P₂ = ^{50P₁ + 250}/100 = ^P₁/2 + 2,5

P₂ = ¹/₂ ( ^P₂/2 + 1) + 2,5 → ³/₄ P₂ = 3 → P₂ = 4 e P₁ = 3

Π₁ = 200

P₂ = C₂, quindi Π₂ = 0

D₁ = 100 e D₂ = 0

V = 4 P₂ = 4

NON CAMBI UN CAZZO

Non cooperano

Π₁ = 180p₁ - 2p₁² + p₁p₂

Π₂ = 180p₂ - 2p₂² + p₁p₂

Π'₁ = 180 - 4p₁ + p₂ = 0 ➔ p₁ = ^{180 + p₂}/₄

Π'₂ = 180 - 4p₂ + p₁ = 0 ➔ p₂ = ^{180 + p₁}/₄

p₁ = 45 + ¹/₄(45 + ^p₁/₄) = 45 + ⁴⁵/₄ + ^p₁/₁₆ ➔ [p₁ = 60] e [p₂ = 60]

Π₁ = (180 - 120 + 60) · 60 = 7200 = Π₂

Deviare

1 cooperа, 2 devіa ➔ p₁ = 90

p₂ = ^{180 + p₁}/₄ = 67,5 quindi Π = 6075 e Π₂ = 9112,5

1. V/F
2. VERO
3. VERO guerra sui prezzi
4. VERO
5. FALSO

21/3/13

(1) Q = 50 - P₂/2

N = 3

C₁¹ = C₂¹ = C₃¹ = 10

SIMMETRICHE

P = 100 - 2Q

π₁ = (100 - 2q₁ - 2q₂ - c)q₁

π₁ = 100 - 4q₁ - 2q₂ - c = 0

q₁^* = (100 - 2q₂ - c)/4

π₂ = (100 - 2q₁ - 2q₂ - 10)q₂

π₂ = 90 - 2q₁ - 4q₂ = 0

q₂^* = (90 - 2q₁)/4

q₁ = 25 - 1/2(45/2 - q₁/2) - c/4

(3/4)q₁ = 25 - 45/4 - c/4

3q₁ = 55 - c

q₁ = (55 - c)/3

q₂ = 45/2 - 1/2(55 - c)/3

= 45/2 - 55/6 + c/6

= 80/6 + c/6

π₁ = (100 - 2(55 - c)/3 - 8(80 + c)/8₃ - c)(55 - c)/3

= (100 - 110/3 + 2c/3 - 80/3 - c/3 - c)((55 - c)/3)

=(300 - 110 + 2c - 80 - c - 3c)/3

((55 - c)/3)

=(110/3) (55 - c)/3 = 2(55 - c)²/3