PM/RM Concept in Bertrand Model

Algorithm – Project 1Kavi PandyaRoll- 131020

SEM-5

04-09-2015

Understanding Bertrand

Cournot Model

• Firms compete in quantities

• Ex:

Bertrand Model

• Firms compete in prices

• Ex:

Quantity: 1.2 lakh 1.1 lakh Price: Rs. 110 Rs. 92

Features Of Bertrand Model :

1. There are only two firms in the market.2. They are producing differentiated goods.3. Both the firms have different production cost.4. Firms have control over their product's price and cost.

PM/RM in Bertrand

Equations: Production Level(xi) for two firms are:1st firm : x1 = a - b1*p1 + b2*p22nd firm : x2 = a - b1*p2 + b2*p1

Utility(Ui) of firms:PM : Ui = (pi - ci)*xiRM : Ui = pi*xi

pi = Price,ci = Production Costbi = +ve constants

LEMMA’s :

4. If both the firms are in PM, thenIf the Higher-price firm switches to RM then the lower-price firm should also switch to RM.

1. If a firm switches from RM to PM then its price increases

2. If a firm switches from PM to RM then it is in capacity to reduce its price to certain extent and the minimum price is given by:

p1rm = [(a + b2*p2)/(2*b1)] - [(( ((a + b2*p2)/b1)^2 - 4*(p1pm - c1pm)(a -b1*p1pm + b2*p2)/b1 )^1/2)/2]

3. If both the firms are in RM, then(i). If lower-price firm switches to PM then the higher-price firm should stick to RM(ii). If higher-price firm switches to PM then the lower-price firm should also switch to PM.

Cost Delegation Game in Bertrand

STEP-1 : Agents determine price:Firm 1: dU1/dp1 = 0, gives: p1 = (a + b2*p2 + c1*b1)/(2*b1)Firm 2: dU2/dp2 = 0, gives: p2 = (a + b2*p1 + c2*b1)/(2*b1)

STEP-2 : Owners determine cost:Firm 1: dU1/dc1 = 0, gives: c1 = (a + b2*p2)/(b1)Firm 2: dU2/dc2 = 0, gives: c2 = (a + b2*p1)/(b1)

Equations: Utility(Ui) of firms:Ui = (pi - ci)*xi

Production Level(xi) for two firms are:1st firm : x1 = a - b1*p1 + b2*p22nd firm : x2 = a - b1*p2 + b2*p1

Utility(Ui) of firms:U1 = (p1 - c1)*(a - b1*p1 + b2*p2)U2 = (p2 - c2)*(a - b1*p2 + b2*p1)

Overview : Project-1

1. Understanding Bertrand Model

2. PM/RM Delegation Game in Bertrand Model + 4 Lemmas

3. Generalized Cost Delegation Game in Bertrand Model

THANKS