MBA 299

1
MBA 299 Section Notes

• 4/18/03
• Haas School of Business, UC Berkeley
• Rawley

2
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

3
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

4
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
• rawley_at_haas.berkeley.edu
• Office hours Room F535
• Monday 1-2pm
• Friday 2-3pm

5
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
gt2 players Cournot with different cost
structures Mathematical model and intuition
6
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

7
ESTIMATING THE OPTIMAL PRICE AND QUANTITYMARKET
A Monopoly Scenario

• Quantity
• 4624
• 3187
• 4306
• 1430
• 1043
• 6927
• 311
• 1414
• 3361
• 3108
• 2356
• 807
• 485
• 3778
• 2212
• 1730
• 1096
• 857

Price 157 208 168 299 331 89 441 300 203 213 244 3
59 399 188 255 281 335 350 175 111
ln (Q) 8.439015 8.066835 8.367765 7.26543 6.949856
8.843182 5.739793 7.254178 8.119994 8.041735 7.76
4721 6.693324 6.184149 8.23695 7.701652 7.455877 6
.999422 6.753438 8.329899 8.740977
Set-up and run regression
1
P(Q) a blnQ gt
Q(p) e(p-a)/b
Adj R2 0.986 t-stat - 36.0 a
1104 b - 111.7
2
Set MR MC and solve
MR MC gt P(Q) (dP/dQ)Q MC 50 P c - b
162 Q 4,605 units
8
ESTIMATING THE OPTIMAL PRICE AND QUANTITYMARKET
B Monopoly Scenario

• Quantity
• 6864
• 6038
• 6505
• 4609
• 3776
• 7969
• 2479
• 4350
• 6144
• 6221
• 5498
• 3623
• 3174
• 6759
• 4939
• 4573
• 4374
• 3578

Price 159 205 177 288 345 100 459 311 197 190 230
366 399 156 268 298 310 369 171 88
ln (Q) 8.834046 8.705828 8.780326 8.435766 8.23642
1 8.983314 7.815611 8.377931 8.723231 8.735686 8.6
1214 8.195058 8.062748 8.81863 8.504918 8.427925 8
.383433 8.182559 8.765927 9.008347
Set-up and run regression
1
P(Q) a blnQ gt
Q(p) e(p-a)/b
Adj R2 0.993 t-stat - 50.8 a
2970 b - 318.3
2
Set MR MC and solve
MR MC gt P(Q) (dP/dQ)Q MC 220 P c -
b 538 Q 2,074 units
9
ESTIMATING THE OPTIMAL PRICE AND QUANTITYMARKET
C Monopoly Scenario

• Quantity
• 594
• 860
• 864
• 880
• 1266
• 1366
• 1482
• 1570
• 1582
• 1646
• 1682
• 1924
• 2822
• 3382
• 3474
• 3832
• 3920
• 4362

Price 450 401 387 388 338 320 309 307 311 297 287
269 300 185 179 161 153 139 111 89
ln (Q) 6.386879 6.756932 6.761573 6.779922 7.14361
8 7.219642 7.301148 7.358831 7.366445 7.406103 7.4
27739 7.562162 7.945201 8.126223 8.153062 8.251142
8.273847 8.380686 8.535426 8.692826
Set-up and run regression
1
P(Q) a blnQ gt
Q(p) e(p-a)/b
Adj R2 0.958 t-stat - 21.0 a
1434 b - 153.5
2
Set MR MC and solve
MR MC gt P(Q) (dP/dQ)Q MC 20 P c - b
174 Q 3,692 units
10
ESTIMATING THE OPTIMAL PRICE AND QUANTITYMARKET
D Monopoly Scenario

• Quantity
• 4248
• 3534
• 3940
• 2539
• 2195
• 5079
• 1338
• 2419
• 3853
• 3273
• 2413
• 1808
• 1549
• 4217
• 2745
• 2289
• 2105
• 1652

Price 459 615 540 888 1001 333 1305 933 590 687 91
1 1106 1198 489 802 951 987 1180 671 1267
ln (Q) 8.354204 8.170186 8.278936 7.839526 7.69393
7 8.53287 7.198931 7.79111 8.256607 8.093462 7.788
626 7.499977 7.345365 8.346879 7.917536 7.73587 7.
652071 7.409742 8.116417 7.249215
Set-up and run regression
1
P(Q) a blnQ gt
Q(p) e(p-a)/b
Adj R2 0.995 t-stat - 60.0 a
6530 b - 722.9
2
Set MR MC and solve
MR MC gt P(Q) (dP/dQ)Q MC 200 P c -
b 923 Q 2,337 units
11
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 huge
12
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

13
COURNOT EQUILIBRIUM WITH Ngt2Homogeneous
Consumers and Firms

• Set-up
• P(Q) a bQ (inverse demand)
• Q q1 q2 . . . qn
• Ci(qi) cqi (no fixed costs)
• Assume c lt a
• Firms choose their q simultaneously

Solution Profit i (q1,q2 . . . qn)
qiP(Q)-c qia-bQ-c Recall NE gt max profit
for i given all other players best play So
F.O.C. for qi, assuming qjlta-c qi1/2(a-c)/b
?qj Solving the n equations q1q2 . .
.qn(a-c)/(n1)b Note that qj lt a c as we
assumed
j?i
14
COURNOT DUOPOLY N2Homogeneous Consumers, Firms
Have Different Costs

• Set-up
• P(Q) a Q (inverse demand)
• Q q1 q2
• Ci(qi) ciqi (no fixed costs)
• Assume c lt a
• Firms choose their q simultaneously

Solution Profit i (q1,q2) qiP(qiqj)-ci q
ia-(qiqj)-ci Recall NE gt max profit for i
given js best play So F.O.C. for qi, assuming
qjlta-c qi1/2(a-qj-ci) Solving the pair of
equations qi2/3a - 2/3ci 1/3cj qj2/3a - 2/3cj
1/3ci Note that qj lt a c as we assumed
15
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

16
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 consumers most-preferred location
• t a measure of disutility from consuming non-L
• (L - y)2 a measure of distance from the
  optimal consumption point
• Note that different consumers have different
  values of L

17
STOCHASTIC HETEROGENEOUS DEMAND

• L is drawn at random from some distribution
  (e.g.,)
• Normal f(x) 1/(2??)1/2exp(-(x-?)2/2?)
• Uniform f(x) 1/(b-a), where x is in a,b
• Exponential (etc.)
• Here we will assume L U0,1
• Also assume
• V gt c 5/4t (so all consumers want to buy in
  equilibrium)
• All other assumptions of the Bertrand model hold

18
MARKET DEMAND WITH UNDIFFERENTIATED PRODUCTSStep
1

• Lets 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 gt 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)/t1/2
• If Li lt LM the consumer buys, if Li gt LM she
  doesnt buy (since consumer surplus is decreasing
  in L)

19
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)/t1/2
• How many consumers would want to buy at price p
  if L is distributed Normally with mean 0 and
  variance 1?
• Nf(L) N1/(2pie)1/2exp(-L2/2)
• Observe that weve expressed demand as a function
  of price
• Market demand D(p) N(V - p)/t1/2

20
FIRM SPECIFIC DEMAND WITH DIFFERENTTIATED
PRODUCTSStep 1

• Lets say firm X locates at 0 and Z locates at 1
• yx 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 gt 0
• . . . and they will only buy good X if it
  delivers more surplus than good Z (and vice
  versa)
• BL(yx) V - tL2 -px gt BL(yz) V - t(L - 1)2
  - pz
• BL(yz) V - t(L - 1)2 -pz gt BL(yx) V - tL2-
  px

21
FIRM SPECIFIC DEMAND WITH DIFFERENTTIATED
PRODUCTSStep 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 LiltLM the consumer will buy X
• If LigtLM the consumer will buy Z
• Therefore, firm-specific demand is
• Dx(px,pz)N(t - px pz)/2t ½N px-pz/2t
• Dz(pz,px) N1 - (t - px pz)/2t

22
EQUILIBRIUM MUTUAL BEST RESPONSE

• Firm is best response to a price pj is
• max (pi - c)Di(pi,pj)
• Observe that
• marginal benefit(MB) dpDi(pi,pj) and
• marginal cost (MC) (pi -c)-dD
• dD dD/dp dp
• dD/dp -N/2t
• Setting MB MC to maximize profit
• N(t - pi pj)/2tdpi (pi -
  c)(-dDi/dpidpi)
• Solving for pi
• pi (t pj c)/2 gt pxpzc t

23
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

24
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

25
QUESTION 1 COURNOT EQUILIBRIUM

• Q(p) 2,000,000 - 50,000p
• MC1 MC2 10
• a.) P(Q) 40 - Q/50,000
• gt q1 q2 (40-10)/350,000 500,000
• b.) ?i (p-c)qi (40-1,000,000/50,000-10)500,0
  00 5M
• c.) Setting MR MC gt a 2bQ c
• Q (a-c)/2b
• gt ?m a b(a-c)/2b(a-c)/2b (a-c)2/4b
• gt ?m (40-10)/450,000 11.25M

26
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)/41/(1-.9) ? D(v-c)
  0
• since 10/4 ? 1 this is true, hence
  cooperation/collusion is sustainable