Q UANTITY P REMIA AND W ITHIN -S TORE S EARCH C OSTS In Kyung Kim Indiana University August 30, 2013 1 / 42
P RICES E XPECTATION WITHIN A S TORE ◮ When consumers are shopping in a store, they usually do not expect to pay more per unit for a larger package than they do for a smaller package. ◮ Same unit price makes sense under the law of one price. ◮ Quantity discount also makes sense under the law of diminishing marginal utility. ◮ It is hard to find any reason for consumers to purchase a larger package with a higher unit price. 2 / 42
A N E XAMPLE OF U NIT P RICE D ISPERSION AND Q UANTITY S URCHARGE 3 / 42
L ITERATURE R EVIEW ◮ Costly search ◮ Stigler (1961), Rothschild (1973), Reinganum (1979), Burdett and Judd (1983) ◮ Information clearinghouse ◮ Rosenthal (1980), Varian (1980), Baye and Morgan (2001) ◮ Empirical work ◮ Hypothesis test ◮ Sorensen (2000), Brown and Goolsbee (2002), Baye, Morgan, and Scholten (2004) ◮ Structural estimation ◮ Hong and Shum (2006), Moraga-González and Wildenbeest (2008) 4 / 42
C ONTRIBUTION OF T HIS P APER Explore within-store unit price dispersion and search costs. ◮ Provide an explanation of quantity premium based on model with costly price information. ◮ Recover determinants of comparison costs. ◮ Investigate how location on shelves and aisles’ impact on quantity discount, premia and unit price dispersion. 5 / 42
T HE M ODEL A SSUMPTION AND N OTATION ◮ Three players: retailer, heavy users, and light users ◮ A product is sold by the piece and a K pack of the same product is also available. ◮ Light users’ demand for the product is � 1 if P L ≤ V L , D L = 0 otherwise . ◮ Heavy user’s demand is � K if P ∗ H ≤ V H , D H = 0 otherwise . 6 / 42
T HE M ODEL A SSUMPTION AND N OTATION ◮ Two types of light users: ◮ Type 1 light users have a higher reservation value than heavy users. ( V 1 L > V H ) ◮ Type 2 light users have a lower reservation value than heavy users. ( V 2 L < V H ) ◮ The proportion of the type 1, type 2 light users, and heavy users: γ 1 , γ 2 , γ 3 (= 1 − γ 1 − γ 2 ) . ◮ Neither type of light user has any incentive to purchase a multi-pack: V 1 L ≤ KV H . 7 / 42
T HE M ODEL A SSUMPTION AND N OTATION ◮ Heavy users’ strategy set is { Compare Unit Prices , Do Not Compare }. ◮ It incurs a constant positive comparison cost, C ∗ > 0 for heavy users to check the single unit price. ◮ Take product assortment and shelf space allocation as given. ◮ Focus on the decision of single unit price and multi-pack unit price. 8 / 42
T HE M ODEL A SSUMPTION AND N OTATION ◮ The retailer’s profit per unit is equal to the unit price it charges minus the wholesale price per unit. π = P − W. ◮ The profit from the sale of one individual unit is either π 1 L = V 1 L − W 1 L or π 2 L = V 2 L − W 2 L . ◮ The profit from the sale of one multi-pack is Kπ H = K ( V H − W H ) . ◮ Assume that when the multi-pack unit price is higher than the single unit price ( P H = V H , P L = V 2 L ), the profit per unit from the sale of one multi-pack is larger than the profit from the sale of one individual unit: π H > π 2 L . (1) 9 / 42
T HE M ODEL A SSUMPTION AND N OTATION ◮ If heavy users never compare the two unit prices, � V 1 if B > A, L P H = V H and P L = V 2 B < A, if L where A ≡ ( γ 1 + γ 2 ) π 2 L + Kγ 3 π H and B ≡ γ 1 π 1 L + Kγ 3 π H . ◮ If heavy users always compare the two unit prices, � V 1 if B > D, L P H = V H and P L = V 2 B < D, if L where B ≡ γ 1 π 1 L + Kγ 3 π H and D ≡ ( γ 1 + γ 2 + Kγ 3 ) π 2 L . ◮ Therefore, the retailer’s strategy set is { Quantity Discount , Quantity Premium }. 10 / 42
T HE M ODEL S TRATEGIES ◮ The retailer’s strategy is ◮ Quantity Discount with probability α . ◮ Quantity Premium with probability (1 − α ) . ◮ The heavy user’s strategy is ◮ Compare Unit Prices with probability β . ◮ Do Not Compare with probability (1 − β ) . 11 / 42
T HE M ODEL P AYOFF M ATRIX Heavy User Compare Do Not Compare Quantity Discount ( B, − C ∗ ) ( B, 0) Store ( D, K ( V H − V 2 Quantity Premium L ) − C ∗ ) ( A, 0) ◮ K ( V H − V 2 L ) can be defined as a potential search benefit. 12 / 42
T HE M ODEL E QUILIBRIA ◮ Under the assumption (1), D < A . ◮ Three separate cases, D < B < A , D < A < B , and B < D < A , lead to different equilibria. ◮ For instance, when D < B < A and C ∗ < K ( V H − V 2 L ) , we have a mixed strategy equilibrium with and β = ( γ 1 + γ 2 ) π 2 L − γ 1 π 1 C ∗ L α = 1 − . (2) L ) , K ( V H − V 2 Kγ 3 ( π H − π 2 L ) ◮ The higher the comparison cost or the smaller the potential benefit of search, the higher the probability of quantity premium. 13 / 42
T HE M ODEL A N E XAMPLE OF M IXED S TRATEGY E QUILIBRIUM Heavy User Compare Do Not Compare Quantity Discount (10 , − 1) (10 , 0) Store Quantity Premium ( 5 , 2) (15 , 0) ◮ The retailer charges a premium on multi-packs one out of three times. ( α = 2 / 3) ◮ Heavy users compare unit prices one out of two times. ( β = 1 / 2) 14 / 42
T HE M ODEL A N E XAMPLE OF P URE S TRATEGY E QUILIBRIUM Heavy User Compare Do Not Compare Quantity Discount (15 , − 2) (15 , 0) Store Quantity Premium ( 5 , − 1) (10 , 0) ◮ Quantity Discount with probability 1 is the retailer’s dominant strategy. ◮ Do Not Compare with probability 1 is the heavy user’s dominant strategy. 15 / 42
T ABLE : Equilibria Condition Cost of Comparison Equilibrium V 1 L > V H > V 2 L C ∗ < K ( V H − V 2 L ) Mixed Strategy Equilibrium (QP/QD, Check/No Check) D < B < A C ∗ > K ( V H − V 2 L ) Pure Strategy Equilibrium (QP , No Check) C ∗ < K ( V H − V 2 L ) D < A < B Pure Strategy Equilibrium (QD, No Check) C ∗ > K ( V H − V 2 L ) C ∗ < K ( V H − V 2 L ) Pure Strategy Equilibrium (QP , Check) B < D < A C ∗ > K ( V H − V 2 L ) Pure Strategy Equilibrium (QP , No Check) V 1 L = V H > V 2 L C ∗ < K ( V H − V 2 L ) Mixed Strategy Equilibrium (QP/ND, Check/No Check) D < B < A C ∗ > K ( V H − V 2 L ) Pure Strategy Equilibrium (QP , No Check) C ∗ < K ( V H − V 2 L ) D < A < B Pure Strategy Equilibrium (ND, No Check) C ∗ > K ( V H − V 2 L ) C ∗ < K ( V H − V 2 L ) Pure Strategy Equilibrium (QP , Check) B < D < A C ∗ > K ( V H − V 2 L ) Pure Strategy Equilibrium (QP , No Check) V 1 L > V H = V 2 L C ∗ < K ( V H − V 2 L ) B < A Pure Strategy Equilibrium (ND, No Check) C ∗ > K ( V H − V 2 L ) C ∗ < K ( V H − V 2 L ) A < B Pure Strategy Equilibrium (QD, No Check) C ∗ > K ( V H − V 2 L ) 16 / 42
T ABLE : Equilibria Condition Cost of Comparison Equilibrium V 1 L > V H > V 2 L C ∗ < K ( V H − V 2 L ) Mixed Strategy Equilibrium (QP/QD, Check/No Check) D < B < A C ∗ > K ( V H − V 2 L ) Pure Strategy Equilibrium (QP , No Check) C ∗ < K ( V H − V 2 L ) D < A < B Pure Strategy Equilibrium (QD, No Check) C ∗ > K ( V H − V 2 L ) C ∗ < K ( V H − V 2 L ) Pure Strategy Equilibrium (QP , Check) B < D < A C ∗ > K ( V H − V 2 L ) Pure Strategy Equilibrium (QP , No Check) V 1 L = V H > V 2 L C ∗ < K ( V H − V 2 L ) Mixed Strategy Equilibrium (QP/ND, Check/No Check) D < B < A C ∗ > K ( V H − V 2 L ) Pure Strategy Equilibrium (QP , No Check) C ∗ < K ( V H − V 2 L ) D < A < B Pure Strategy Equilibrium (ND, No Check) C ∗ > K ( V H − V 2 L ) C ∗ < K ( V H − V 2 L ) Pure Strategy Equilibrium (QP , Check) B < D < A C ∗ > K ( V H − V 2 L ) Pure Strategy Equilibrium (QP , No Check) V 1 L > V H = V 2 L C ∗ < K ( V H − V 2 L ) B < A Pure Strategy Equilibrium (ND, No Check) C ∗ > K ( V H − V 2 L ) C ∗ < K ( V H − V 2 L ) A < B Pure Strategy Equilibrium (QD, No Check) C ∗ > K ( V H − V 2 L ) 16 / 42
T ABLE : Equilibria Condition Cost of Comparison Equilibrium V 1 L > V H > V 2 L C ∗ < K ( V H − V 2 L ) Mixed Strategy Equilibrium (QP/QD, Check/No Check) D < B < A C ∗ > K ( V H − V 2 L ) Pure Strategy Equilibrium (QP , No Check) C ∗ < K ( V H − V 2 L ) D < A < B Pure Strategy Equilibrium (QD, No Check) C ∗ > K ( V H − V 2 L ) B < D < A C ∗ > K ( V H − V 2 L ) Pure Strategy Equilibrium (QP , No Check) V 1 L = V H > V 2 L C ∗ < K ( V H − V 2 L ) Mixed Strategy Equilibrium (QP/ND, Check/No Check) D < B < A C ∗ > K ( V H − V 2 L ) Pure Strategy Equilibrium (QP , No Check) C ∗ < K ( V H − V 2 L ) D < A < B Pure Strategy Equilibrium (ND, No Check) C ∗ > K ( V H − V 2 L ) B < D < A C ∗ > K ( V H − V 2 L ) Pure Strategy Equilibrium (QP , No Check) V 1 L > V H = V 2 L C ∗ < K ( V H − V 2 L ) B < A Pure Strategy Equilibrium (ND, No Check) C ∗ > K ( V H − V 2 L ) C ∗ < K ( V H − V 2 L ) A < B Pure Strategy Equilibrium (QD, No Check) C ∗ > K ( V H − V 2 L ) 17 / 42
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