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MBA 299 – Section Notes

MBA 299 – Section Notes. 4/18/03 Haas School of Business, UC Berkeley Rawley. AGENDA. Administrative CSG concepts Discussion of demand estimation Cournot equilibrium Multiple players Different costs

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MBA 299 – Section Notes

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  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@haas.berkeley.edu • Office hours Room F535 • Monday 1-2pm • Friday 2-3pm

  5. GENERAL STRATEGY FOR CSG • Using regression coefficients to find Pm* and Qm* • You know how to do this • Cournot with >2 players • Cournot with different cost structures • Mathematical model and intuition • 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

  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 • Price • 157 • 208 • 168 • 299 • 331 • 89 • 441 • 300 • 203 • 213 • 244 • 359 • 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 • 7.764721 • 6.693324 • 6.184149 • 8.23695 • 7.701652 • 7.455877 • 6.999422 • 6.753438 • 8.329899 • 8.740977 Set-up and run regression 1 • Quantity • 4624 • 3187 • 4306 • 1430 • 1043 • 6927 • 311 • 1414 • 3361 • 3108 • 2356 • 807 • 485 • 3778 • 2212 • 1730 • 1096 • 857 • 4146 • 6254 P(Q) = a + b*lnQ => 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 => 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 • 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 • 8.236421 • 8.983314 • 7.815611 • 8.377931 • 8.723231 • 8.735686 • 8.61214 • 8.195058 • 8.062748 • 8.81863 • 8.504918 • 8.427925 • 8.383433 • 8.182559 • 8.765927 • 9.008347 Set-up and run regression 1 • Quantity • 6864 • 6038 • 6505 • 4609 • 3776 • 7969 • 2479 • 4350 • 6144 • 6221 • 5498 • 3623 • 3174 • 6759 • 4939 • 4573 • 4374 • 3578 • 6412 • 8171 P(Q) = a + b*lnQ => 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 => 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 • 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 • 7.143618 • 7.219642 • 7.301148 • 7.358831 • 7.366445 • 7.406103 • 7.427739 • 7.562162 • 7.945201 • 8.126223 • 8.153062 • 8.251142 • 8.273847 • 8.380686 • 8.535426 • 8.692826 Set-up and run regression 1 • Quantity • 594 • 860 • 864 • 880 • 1266 • 1366 • 1482 • 1570 • 1582 • 1646 • 1682 • 1924 • 2822 • 3382 • 3474 • 3832 • 3920 • 4362 • 5092 • 5960 P(Q) = a + b*lnQ => 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 => 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 • Price • 459 • 615 • 540 • 888 • 1001 • 333 • 1305 • 933 • 590 • 687 • 911 • 1106 • 1198 • 489 • 802 • 951 • 987 • 1180 • 671 • 1267 • ln (Q) • 8.354204 • 8.170186 • 8.278936 • 7.839526 • 7.693937 • 8.53287 • 7.198931 • 7.79111 • 8.256607 • 8.093462 • 7.788626 • 7.499977 • 7.345365 • 8.346879 • 7.917536 • 7.73587 • 7.652071 • 7.409742 • 8.116417 • 7.249215 Set-up and run regression 1 • Quantity • 4248 • 3534 • 3940 • 2539 • 2195 • 5079 • 1338 • 2419 • 3853 • 3273 • 2413 • 1808 • 1549 • 4217 • 2745 • 2289 • 2105 • 1652 • 3349 • 1407 P(Q) = a + b*lnQ => 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 => 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 N>2Homogeneous Consumers and Firms • Solution • Profit 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-c • qi*=1/2[(a-c)/b + qj] • Solving the n equations • q1=q2= . . .=qn=(a-c)/[(n+1)b] • Note that qj < a – c as we assumed • 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 ji

  14. COURNOT DUOPOLY N=2Homogeneous Consumers, Firms Have Different Costs • Solution • Profit 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-c • qi*=1/2(a-qj*-ci) • Solving the pair of equations • qi=2/3a - 2/3ci + 1/3cj • qj=2/3a - 2/3cj + 1/3ci • Note that qj < a – c as we assumed • 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

  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 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 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)1/2*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

  18. 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)

  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)/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

  20. FIRM SPECIFIC DEMAND WITH DIFFERENTTIATED PRODUCTSStep 1 • Let’s 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 >= 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

  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 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]/2t Dz(pz,px) = N[1 - (t - px + pz)/2t]

  22. EQUILIBRIUM: MUTUAL BEST RESPONSE • Firm i’s best response to a price pj is • max (pi - c)*Di(pi,pj) • Observe that: • marginal benefit(MB)= dp*Di(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)/2t]*dpi = (pi - c)*(-dDi/dpi*dpi) • Solving for pi • pi *= (t+ pj* + c)/2 => px*=pz*=c + 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 • => 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

  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)/4*[1/(1-.9)]  D*(v-c) + 0 since 10/4  1 this is true, hence cooperation/collusion is sustainable

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