M. Shum Economics 106: Problem Set #3Problem Set: Discrete-Choice Demand ModelsAttached is a table of market shares, prices, and characteristics on th e top-selling br ands of cerealin 1992. The data are aggregated from household-level scanner data (collected at supermarketcheckout counters).The market shares below are shares of total cereal purchases observed in the dataset. For thepurposes of this problem set, assume that all households p urchased some cereal during 1992 (sothat non-purchase is not an option).∗Assume that brand #51, the composite basket of “all otherbrands”, is the outside goo d.Two sets of prices are given in the table. Shelf prices are those listed on su permarket shelves, anddo not include coupon discounts. Transactions prices are prices actually paid by consumers, net ofcoupon discounts. Estimate using the transactions prices. Note that you should subtract the priceof brand #51, the “outside good”, fr om the prices of the top fifty brands.Assume a utility specification for uij, household i’s utility from brand j:uij= Xjβ − αpj+ ξj+ νijwhere Xjare characteristics of brand j, ξjis an unobserved (to the econometrician) quality param-eter for brand j, and νijis a d isturbance term which is identicially and independently distributed(i.i.d.) over households i and brand s j. Denote the mean utility level from brand j asδj≡ Xjβ − αpj+ ξj.1. Assumin g that the νij’s are distributed i.i.d. type I extreme value, derive the resulting expressionsfor the market shares of each brand j, j = 1, . . . , 51.Next we implement the BLP two-step estimator.2. Invert the resulting system of demand functions to get estimates of the mean utility levels δjasa function of the s hares sj.3. Estimate the second stage regression of δjon Xjand pjin different ways:(a) OLS(b) 2SLS: using average characteristics for all other brands pro duced by the same manufacturer asbrand j as instruments for pj(c) 2SLS: usin g average characteristics for all other brands produced by rivals to the manufactureras brand j as instruments for pj(d) 2SLS: using average characteristics for all other brands as instruments for pj∗This is not far from the truth; from an alternative data source (the IRI Marketing FactBook), one finds out thatin 1992, 97.1% of American households purchased at least some cereal during the year.1M. Shum Economics 106: Problem Set #3How do your results differ?4. From the aggregate demand functions derived in question 1, derive the formulas for the deriva-tives∂sj∂pj′and the elasticities ǫij≡∂sj∂pj′pj′sj, where j and j′are any two pairs of brands.What is the difference between ǫikand ǫjk. Explain the implication of this.5. Ass uming that the m anufacturers of th e top fifty brands compete in Bertrand fashion, derive thefifty first-order conditions which define prices in this market, assuming constant marginal costs ofproduction for each brand (and ignoring advertising costs). In other words, assume that th e totalcost f unction for b rand j Cj(qj) = cjqj.These FOCs are a system of linear equations in the u nknowns c1, . . . , c50. Using the expressionderived in question 4 above, rewrite these FOCs completely in terms of the known prices, s hares,and parameters (in particular, α).Solve for the marginal costs from this system of equations. Recall that linear equations of the formAx = b can be solved by x = A−1b.After deriving these costs, solve for the markuppj− cjpjassociated with each brand.2Table 1: Brand characteristicsName Avg trans- Avg Avg In-sample Sgmnt Cals Fat Sugaraction Shelf Ad MarketPrice Price Expn Share($/lb) ($/lb)1 KGaCorn Flakes 1.81 1.95 7.109b5.67cFam 100 0 22 GM Cheerios 3.16 3.47 7.287 4.38 Fam 110 2 13 KG Rice Krispies 2. 96 3.20 6.034 4.04 Fam 120 0 34 KG Frosted Flakes 2.52 2.68 7.867 3.82 Fam 120 0 135 KG Raisin Bran 2.34 2.50 5.591 2.73 Fam 200 1.5 186 GM Total 3.61 4.04 3.926 2.36 Adult 110 1 57 GM HoneyNut Cheerios 3.14 3.41 4.030 2.26 Fam 120 1.5 118 KG Special K 3.48 3.78 3.531 2.16 Adult 110 0 39 PT Grape N uts 2.14 2.29 6.740 2.12 Adult 200 1 710 NB SpoonSize ShdWt 2.81 3.05 0.025 2.08 Adult 170 0.5 011 QK 100% Natural 2.24 2.55 1.612 1.96 Adult 220 8 13.512 KG Frosted Mini Wheats 2. 62 2.75 6.106 1.84 Adult 170 1 1013 KG NutriGrain 2.87 3.10 2.508 1.55 Adult 100 1 014 KG Mueslix 3.31 3.58 1.975 1.53 Adult 200 4 1315 GM Wheaties 2.55 2.86 2.257 1.52 Fam 110 1 416 PT Raisin Bran 2.23 2.57 4.361 1.46 Fam 190 1 2017 RL Muesli 3.34 3.93 0.215 1.26 Adult 210 2.7 1418 KG Corn Pops 3.51 3.69 3.198 1.46 Fam 120 0 1419 GM Raisin Nut Bran 2.98 3.22 1.659 1.35 Adult 210 4.5 1620 GM Basic 4 3.27 3.63 2.510 1.31 Adult 210 3 1221 GM Cocoa Puffs 3.46 3.67 2.097 1.28 Kids 120 1 1422 GM Golden Grahams 3.24 3.54 2.953 1.24 Kids 120 1 1123 GM Cinn. Toast Crunch 3.36 3.56 2.963 1.23 Kids 130 3 1024 KG Froot Loops 3.53 3.76 3.110 1.20 Kids 120 1 1525 KG Low Fat Granola 2.68 3.10 2.327 1.17 Adult 190 3 1226 GM Trix 3.96 4.22 3.236 1.13 Kids 120 1.5 1327 GM Triples 2.33 2.80 3.036 1.12 Adult 120 1 628 KG Crispix 3.28 3.49 3.225 1.12 Adult 110 0 329 GM Kix 3.67 3.93 3.801 1.08 Kids 120 0.8 630 GM Lucky Charms 3.45 3.72 3.079 1.08 Kids 120 1 1331 GM AppleCinn. Cheerios 3.02 3.35 3.120 1.06 Fam 120 2 1332 KG Cracklin Oat Bran 3.19 3.51 2.279 1.06 Adult 190 6 1533 NB Big Biscuit ShdWt 2.79 3.05 0.000 0.99 Adult 156 1.2 9.634 PT Honey Bunches of Oats 2.85 3.18 3.749 0.95 Adult 125 2.2 635 PT Great Graines 2.90 3.43 2.648 0.89 Adult 215 5.5 10.536 GM Otml Raisin Crisp 2.71 3.04 1.641 0.97 Adult 210 2.5 1937 QK Oat Squares 2.43 2.71 1.472 0.94 Adult 220 3 938 RL Rice Chex 3.40 3.53 0.875 0.89 Adult 120 0 239 GM Total Raisin Bran 3.00 3.50 1.874 0.89 Adult 180 1 1940 KG Product 19 3.38 3.70 1.408 0.89 Adult 100 0 441 KG Apple Jacks 3.64 3.91 1.465 0.84 Kids 120 1 1642 QK Capt Crunch 2.55 2.86 1.714 0.83 Kids 105 1.5 11.543 NB Shredded Wheat 2.82 3.00 2.925 0.80 Adult 160 0.5 044 PT Fruity Pebbles 3.21 3.48 1.710 0.83 Kids 110 1 1245 GM Clusters 3.14 3.52 1.425 0.78 Fam 210 3.5 1446 KG Cinnamon MiniBuns 2.75 3.14 0.002 0.76 Fam 120 0.5 1247 KG Double Dip Crunch 3.01 3.52 1.454 0.73 Adult 110 0 1148 GM MultiGrain Cheerios 3.34 3.74 2.520 0.75 Fam 110 1 1549 PT Honeycomb 3.40 3.67 2.567 0.74 Kids 110 0 1150 QK Popeye 1.77 1.77 0.000 0.67 Kids 120 1 13.351 basket of all other brands 2.68 0.645d24.29aKG: Kelloggs GM: General Mills PT: Post (Phillip Morris) RL: Ralston QK: Quaker Oatsbquarterly expns, $million. Source: Leading