0 mba 299 – section notes 4/18/03 haas school of business, uc berkeley rawley
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MBA 299 – Section Notes
4/18/03
Haas School of Business, UC Berkeley
Rawley
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AGENDA
Administrative
CSG concepts– Discussion of demand estimation– Cournot equilibrium
• Multiple players• Different costs
– Firm-specific demand for differentiated products and how that can look very different than the demand faced by a monopolist
Problem set on Cournot, Tacit Collusion and Entry Deterrence
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AGENDA
Administrative
CSG concepts– Discussion of demand estimation– Cournot equilibrium
• Multiple players• Different costs
– Firm-specific demand for differentiated products and how that can look very different than the demand faced by a monopolist
Problem set on Cournot, Tacit Collusion and Entry Deterrence
![Page 4: 0 MBA 299 – Section Notes 4/18/03 Haas School of Business, UC Berkeley Rawley](https://reader035.vdocument.in/reader035/viewer/2022062314/56649d805503460f94a64e50/html5/thumbnails/4.jpg)
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ADMINISTRATIVE
In response to your feedback– Slides in section and on the web– More math– More coverage of CSG concepts
CSG entries due Tuesday and Friday at midnight each week
Contact info:– [email protected]
Office hours Room F535– Monday 1-2pm– Friday 2-3pm
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GENERAL STRATEGY FOR CSG
Estimate monopoly price Pm* and
quantity Qm*
– Your price should never be above Pm
*
Estimate perfect competition price Pc* and quantity Qc*
– Your price should always be above Pc*
Use Cournot equilibrium to estimate reasonable oligopoly outcomes
Use firm-specific demand with differentiated products to find another sensible set of outcomes
Using regression coefficients to find Pm*
and Qm*
You know how to do this
Cournot with >2 playersCournot with different cost structures
Mathematical model and intuition
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AGENDA
Administrative
CSG concepts– Discussion of demand estimation– Cournot equilibrium
• Multiple players• Different costs
– Firm-specific demand for differentiated products and how that can look very different than the demand faced by a monopolist
Problem set on Cournot, Tacit Collusion and Entry Deterrence
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ESTIMATING THE OPTIMAL PRICE AND QUANTITYMARKET A – Monopoly Scenario
Quantity462431874306143010436927
3111414336131082356
807485
3778221217301096
85741466254
Price157208168299331
89441300203213244359399188255281335350175111
ln (Q)8.4390158.0668358.367765
7.265436.9498568.8431825.7397937.2541788.1199948.0417357.7647216.6933246.184149
8.236957.7016527.4558776.9994226.7534388.3298998.740977
P(Q) = a + b*lnQ =>
Adj R2 = 0.986t-stat = - 36.0a = 1104b = - 111.7
Q(p) = e(p-a)/b
MR = MC => P(Q) + (dP/dQ)Q = MC = 50
P* = c - b = $162Q* = 4,605 units
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2
Set-up and run regression
Set MR = MC and solve
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ESTIMATING THE OPTIMAL PRICE AND QUANTITYMARKET B – Monopoly Scenario
Quantity68646038650546093776796924794350614462215498362331746759493945734374357864128171
Price159205177288345100459311197190230366399156268298310369171
88
ln (Q)8.8340468.7058288.7803268.4357668.2364218.9833147.8156118.3779318.7232318.735686
8.612148.1950588.062748
8.818638.5049188.4279258.3834338.1825598.7659279.008347
P(Q) = a + b*lnQ =>
Adj R2 = 0.993t-stat = - 50.8a = 2970b = - 318.3
Q(p) = e(p-a)/b
MR = MC => P(Q) + (dP/dQ)Q = MC = 220
P* = c - b = $538Q* = 2,074 units
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2
Set-up and run regression
Set MR = MC and solve
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ESTIMATING THE OPTIMAL PRICE AND QUANTITYMARKET C – Monopoly Scenario
Quantity594860864880
1266136614821570158216461682192428223382347438323920436250925960
Price450401387388338320309307311297287269300185179161153139111
89
ln (Q)6.3868796.7569326.7615736.7799227.1436187.2196427.3011487.3588317.3664457.4061037.4277397.5621627.9452018.1262238.1530628.2511428.2738478.3806868.5354268.692826
P(Q) = a + b*lnQ =>
Adj R2 = 0.958t-stat = - 21.0a = 1434b = - 153.5
Q(p) = e(p-a)/b
MR = MC => P(Q) + (dP/dQ)Q = MC = 20
P* = c - b = $174Q* = 3,692 units
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2
Set-up and run regression
Set MR = MC and solve
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ESTIMATING THE OPTIMAL PRICE AND QUANTITYMARKET D – Monopoly Scenario
Quantity42483534394025392195507913382419385332732413180815494217274522892105165233491407
Price459615540888
1001333
1305933590687911
11061198
489802951987
1180671
1267
ln (Q)8.3542048.1701868.2789367.8395267.693937
8.532877.198931
7.791118.2566078.0934627.7886267.4999777.3453658.3468797.917536
7.735877.6520717.4097428.1164177.249215
P(Q) = a + b*lnQ =>
Adj R2 = 0.995t-stat = - 60.0a = 6530b = - 722.9
Q(p) = e(p-a)/b
MR = MC => P(Q) + (dP/dQ)Q = MC = 200
P* = c - b = $923Q* = 2,337 units
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2
Set-up and run regression
Set MR = MC and solve
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MONOPOLY PRICES ARE THE CEILING, MARGINAL COST IS THE FLOOR
Use monopoly prices as the ceiling on reasonable prices to charge
Use marginal cost as the floor on reasonable prices to charge
Remember, the goal is not to sell all of your capacity, the goal is to maximize profit!
Problem: The range from MC to PM* is hugeProblem: The range from MC to PM* is huge
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AGENDA
Administrative
CSG concepts– Discussion of demand estimation– Cournot equilibrium
• Multiple players• Different costs
– Firm-specific demand for differentiated products and how that can look very different than the demand faced by a monopolist
Problem set on Cournot, Tacit Collusion and Entry Deterrence
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COURNOT EQUILIBRIUM WITH N>2Homogeneous Consumers and Firms
Set-up
P(Q) = a – bQ (inverse demand)
Q = q1 + q2 + . . . + qn
Ci(qi) = cqi (no fixed costs)
Assume c < a
Firms choose their q simultaneously
SolutionProfit i (q1,q2 . . . qn)
= qi[P(Q)-c]=qi[a-bQ-c]
Recall NE => max profit for i given all other players’ best play
So F.O.C. for qi, assuming qj<a-cqi
*=1/2[(a-c)/b + qj]
Solving the n equationsq1=q2= . . .=qn=(a-c)/[(n+1)b]
Note that qj < a – c as we assumed
ji
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COURNOT DUOPOLY N=2Homogeneous Consumers, Firms Have Different Costs
Set-up
P(Q) = a – Q (inverse demand)
Q = q1 + q2
Ci(qi) = ciqi (no fixed costs)
Assume c < a
Firms choose their q simultaneously
SolutionProfit i (q1,q2) = qi[P(qi+qj)-ci]
=qi[a-(qi+qj)-ci]
Recall NE => max profit for i given j’s best play
So F.O.C. for qi, assuming qj<a-cqi
*=1/2(a-qj*-ci)
Solving the pair of equationsqi=2/3a - 2/3ci + 1/3cj
qj=2/3a - 2/3cj + 1/3ci
Note that qj < a – c as we assumed
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AGENDA
Administrative
CSG concepts– Discussion of demand estimation– Cournot equilibrium
• Multiple players• Different costs
– Firm-specific demand for differentiated products and how that can look very different than the demand faced by a monopolist
Problem set on Cournot, Tacit Collusion and Entry Deterrence
![Page 16: 0 MBA 299 – Section Notes 4/18/03 Haas School of Business, UC Berkeley Rawley](https://reader035.vdocument.in/reader035/viewer/2022062314/56649d805503460f94a64e50/html5/thumbnails/16.jpg)
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MODELING HETEROGENEOUS DEMANDN Consumers
Spectrum of preferences [0,1]– Analogy to location in the product space
Consumer preferences (for each consumer)
BL(y) = V - t*(L - y)2
L = this consumer’s most-preferred “location”t = a measure of disutility from consuming non-L(L - y)2 = a measure of “distance” from the optimal
consumption pointNote that different consumers have different values of L
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STOCHASTIC HETEROGENEOUS DEMAND
L is drawn at random from some distribution (e.g.,)– Normal: f(x) = [1/(2)1/2*exp(-(x-)2/2)]– Uniform: f(x) = 1/(b-a), where x is in [a,b]– Exponential (etc.)
Here we will assume L ~ U[0,1]
Also assume – V > c +5/4t (so all consumers want to buy in
equilibrium)– All other assumptions of the Bertrand model hold
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MARKET DEMAND WITH UNDIFFERENTIATED PRODUCTSStep 1
Let’s say firms X and Z both locate their products at 0 yx = yz = 0
Consumers are rational so they will only buy if consumer surplus is at least zero
BL(0) - p = V - tL2 - p >= 0
To derive market demand we need to find the consumer who is exactly indifferent between buying and not buying (the marginal consumer)
LM = [(V - p)/t]1/2
If Li < LM the consumer buys, if Li > LM she doesn’t buy (since consumer surplus is decreasing in L)
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MARKET DEMAND WITH UNDIFFERENTIATED PRODUCTSStep 2
Use the value of LM and the distribution of L to find the number of consumers who want to buy at price p
– Here it is NLM = N[(V - p)/t]1/2
– How many consumers would want to buy at price p if L is distributed Normally with mean = 0 and variance = 1?
• N*f(L) = N*[1/(2*pie)1/2*exp(-L2/2)]
Observe that we’ve expressed demand as a function of price
– Market demand = D(p) = N[(V - p)/t]1/2
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FIRM SPECIFIC DEMAND WITH DIFFERENTTIATED PRODUCTS
Step 1
Let’s say firm X locates at 0 and Z locates at 1yx = 0, yz = 1– In the CSG game you are randomly assigned your product location,
this example shows how the maximum difference between you and your competitors in the brand location space impacts optimal pricing
Consumers are rational so they will only buy if consumer surplus is at least zero . . .
BL(y) - p = V - t*(L - y)2 - p >= 0
. . . and they will only buy good X if it delivers more surplus than good Z (and vice versa)
BL(yx) = V - tL2 -px > BL(yz) = V - t*(L - 1)2 - pz
BL(yz) = V - t*(L - 1)2 -pz > BL(yx) = V - tL2- px
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FIRM SPECIFIC DEMAND WITH DIFFERENTTIATED PRODUCTS
Step 2
In this example the marginal consumer is the one who is indifferent between consuming good X and good Z
V - tL2 -px = V - t*(L - 1)2 -pz
Solving we find: LM = (t - px + pz)/2t
Observe– If Li<LM the consumer will buy X
– If Li>LM the consumer will buy Z
Therefore, firm-specific demand is:Dx(px,pz)=N[(t - px + pz)/2t] = ½N – [px-pz]/2tDz(pz,px) = N[1 - (t - px + pz)/2t]
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EQUILIBRIUM: MUTUAL BEST RESPONSE
Firm i’s best response to a price pj ismax (pi - c)*Di(pi,pj)
Observe that:marginal benefit(MB)= dp*Di(pi,pj) andmarginal cost (MC) = (pi -c)*-dDdD = dD/dp * dpdD/dp = -N/2t
Setting MB = MC to maximize profit N[(t - pi + pj)/2t]*dpi = (pi - c)*(-dDi/dpi*dpi)
Solving for pi
pi *= (t+ pj
* + c)/2 => px*=pz
*=c + t
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HOW DOES THIS RELATE TO CSG?
t is a measure of brand loyalty you can roughly approximate values of t from “information on brand substitution”
– For example in market D it appears that t is small (less than 1)– Products in market C have the highest brand loyalty so t is
large
While it is not easy to calculate t directly you can use the information on the market profiles and the data generated by the game to get a rough sense of its value. Use your estimates of t along with a Cournot equilibrium model to find optimal prices.
– For a detailed explanation of how to estimate t more precisely (well beyond the scope of this class) see Besanko, Perry and Spady “The Logit Model of Monopolistic Competition – Brand Diversity,” Journal of Economics, June 1990
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AGENDA
Administrative
CSG concepts– Discussion of demand estimation– Cournot equilibrium
• Multiple players• Different costs
– Firm-specific demand for differentiated products and how that can look very different than the demand faced by a monopolist
Problem set on Cournot, Tacit Collusion and Entry Deterrence
![Page 25: 0 MBA 299 – Section Notes 4/18/03 Haas School of Business, UC Berkeley Rawley](https://reader035.vdocument.in/reader035/viewer/2022062314/56649d805503460f94a64e50/html5/thumbnails/25.jpg)
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QUESTION 1: COURNOT EQUILIBRIUM
Q(p) = 2,000,000 - 50,000p
MC1 = MC2 = 10
a.) P(Q) = 40 - Q/50,000
=> q1 = q2 = (40-10)/3*50,000 = 500,000
b.) i = (p-c)*qi = (40-1,000,000/50,000-10)*500,000 = $5M
c.) Setting MR = MC => a – 2bQ = c
Q* = (a-c)/2b
=> m = {a – b[(a-c)/2b]}*(a-c)/2b = (a-c)2/4b
=> m = (40-10)/4*50,000 =$11.25M
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QUESTION 2: REPEATED GAMES AND TACIT COLLUSION
Bertrand model set-up with four firms and = 0.9
(cooperate) = D*(v – c)/4(defect) = D*(v – c)(punishment) = 0
Colluding is superior iff{D*(v-c)/4* t} = D*(v-c)/4*[1/(1-.9)] D*(v-c) + 0since 10/4 1 this is true, hence cooperation/collusion is sustainable