MBA 299 – Section Notes

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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





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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

mailto:[email protected]

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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





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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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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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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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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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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





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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





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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





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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

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MBA 299 – Section Notes