Optimal Pricing Strategy for Second Degree Price Discrimination

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1 Optimal Pricing Strategy for Second Degree Price Discrimination Alex O Brien May 5, 2005 Abstract Second Degree price discrimination is a coupon strategy tat allows all consumers access to te coupon. Purcases are made by consumer self selection based on utility, a function of self price, coupon face value, and assle cost. Tis work builds on a previous study and develops a grapical analysis and model based on similar assumptions. We find an optimal self price and coupon value to segment a population based on maximizing a firm s profit.

3 social value. Trougout, utility is used to refer to consumer utility, and profit is used to refer to firm utility. Our model of utility is described in detail later. We investigate optimal strategies for setting price and coupon value in order to maximize profit. Tis increases overall social welfare, as bot te firm increases its profits and most consumers pay less money. Te goal of tis paper is to lay a foundation of common assumptions and provide a single uniform model tat is able to describe complex situations wit a grapical representation. Tis elps us understand te relationsips between coupon face value, self price, and assle cost. 2 Model Assumptions Te model considers a single firm selling to a unit mass of customers of type θ, were θ is uniformly distributed on te interval [0,]. Te firm sells only one type of product at price p 0 wit a marginal cost of w 0. Utility is modeled by a piecewise linear function of, were 0 is data driven and depicts te valuation increase across te θ interval. Tat is, is te rate at wic utility increases over te population. Furtermore, 0 is te minimum value, or intrinsic value, te product as to society. Hassle cost, H(θ) 0, is positive for only a fraction β of consumers of type θ X, were X delineates were consumers begin to ave a assle cost tat effects utility. Hassle cost is discussed later in detail. Te value of X is determined by te data of te specific firm, product, and most importantly consumers. Te utility function, U(θ), for eac consumer of type θ is determined by weater or not tey consider using a coupon wit a face value c 0. It is important to note tat we assume consumers ave certain caracteristics and purcasing patterns in te market, and tese caracteristics are wat determines, β, X,, w, and. Tat is, tese parameters are data driven. Te consumer utility function is { UC (θ), 0 θ ( + β)x U(θ) = U NC (θ), ( + β)x < θ, were and U C (θ) = U NC (θ) H + c () U NC (θ) = + θ p. (2) Te utility functions, U C (θ) and U NC (θ), are restricted to teir domains, as sown in Figure (). We call U NC (θ) utility wit no consideration of a coupon and U C (θ) utility wit consideration of a coupon. Consumers of type θ are not allowed to coose wic function tey measure teir utility wit, but at te same time, it is teir consumption patterns upon wic utility is based. For simplification, we assume tat income increases as θ increases. Tat is, consumers of type θ close to zero ave a low income and consumers of type θ close to ave a ig income, and it is in tis way tat te utility domains are cosen. In oter words, te wealtier consumers do not consider using a coupon wile te poorer consumers do consider using a coupon. Te middle class consumers ave a positive assle cost and must determine if te coupon value outweigs teir assle cost. 3

4 U C Figure : Utility Function Domains U NC If U(θ) 0, ten we say tat consumer θ purcases te product. purcasing if teir utility is negative. Notice tat A consumer foregoes and U NC (θ) 0 θ U C (θ) 0 θ [ (p ) (p ), (3) (c H) ] = p c + H. (4) Hassle cost is te consumer s disutility from using a coupon and is a time consideration. For example, te assle of keeping track of te coupon, organization costs, or a variety of oter costs tat influence a consumer s desire to use a coupon. Hassle cost is defined by { 0, 0 θ X H(θ) =, X θ ( + βx), were > 0. Again, notice tat assle cost is only a factor in utility for a fraction β of consumers of type θ X. Note tat consumers of type θ ((+βx), ] do not consider te use of coupons because tey are too wealty, tus, tey ave no applicable assle cost. Hassle cost is depicted in te figures as te dased piecewise constant function. Since utility is te determining factor of purcases, we are concerned wit were U(θ) 0. Specifically, we determine exactly were U(θ) = 0, say at ˆθ, and consumers of type θ ˆθ purcase for all θ witin te respective utility domain. Altoug U(θ) is piecewise linear, it is not continuous, and tus utility can alternate between positive and negative along te θ axis, as seen in Figure (2). Note tat U(θ) = 0 can ave 3 solutions. 4

5 Figure 2: Grapical representation for U(θ) 3 A Model of Profit, Segmentation, and Utility wit a Grapical Representation and Analysis We now develop a model tat determines an optimal price and coupon value tat segments te population to maximize profit. Profit, π, is obtained were utility is non-negative, or were consumers purcase te product. Remember tat profit is calculated by price, p, less cost, w, less any losses from a coupon, c, multiplied by te uantity sold. Te model calculates uantity sold as te number of consumers wit a non-negative utility. Tat is, uantity sold is te sum of te distance of all θ [0, ] were utility is non-negative, or were U(θ) 0. Let V = (p, c, a 2, s 2, s 3, s 4 ). Te model is as follows: max π(v ) = a 2 (p c w)s 2 + (p c w)s 3 + (p w)[ (( + β)x + s 4 )] Subject to: p c = X + a s a 2 s 2, (5) a, a 2 {0, }, (6) a i =, (7) i X p c + + s 3 ( + β)x (8) p s 4 = ( + β)x (9) 0 s βx, (0) 5

6 0 s 2 X, () 0 s 3 βx, (2) 0 s 4 ( ( + β)x), (3) p, c 0. (4) To understand te model, we first explain te constraints. Consider constraints (5) troug (7). From te domain of U(θ) and H(θ), we know tat U(θ) = 0 wen θ = (p c )/ for some θ [0, ( + β)x]. Note tat H(θ) = 0 for θ [, X) and remember tat X is te point determined by te data of te population were assle cost is positive for all θ [X, ( + β)x]. Tus we set (p c )/ = X + a s a 2 s 2, were a and a 2 are binary variables. Notice tat s is te distance from X to (p c )/ wen (p c )/ > X, and s 2 is te distance from (p c )/ to X wen (p c )/ < X. Constraint (7) allows only one of te a i s to be so tat eiter s or s 2 is removed from te profit calculation. We let 0 s βx and 0 s 2 X wic guarantees 0 (p c )/ ( + β)x, and tus utility becomes non-negative witin our population and is restricted to te specified utility domain. If (p c )/ < X, ten profit is increased by (p c w)s 2. If (p c )/ = X, ten s = s 2 = 0 and tere is no increase in profit. If (p c )/ > X, ten profit is not effected by s. Altoug we allow for X < (p c )/ ( + β)x, if te utility becomes nonnegative for some θ (X, ( + β)x], ten we use s 3 for profit calculation witin tis domain and s is simply used as a reference for utility. Tat is, if X < (p c )/, ten assle cost drops te utility function down and allows contstraint (8) to coose an s 3. Figures (3) and (4) depict a grap of U(θ) tat sows te existence of s and s 3. However, U(θ) < 0 over [0, X), so tere is no s and terefore no additional profit is made by te firm from tis segment. Figure (3) represents an infeasible grap of U(θ), but we see a corresponding feasible grap of U(θ) in Figure (4). Tat is, our model does now allow Figure (3) because U(θ) 0 for ( + β)x θ. Tus, te model forces p to decrease. We ten obtain an intersection depicted by te feasible grap of U(θ) in Figure (4). Tis is clarified wit te understanding of te remaining constraints. 0 X (+β) X 0 X (+β) X S S 3 Figure 3: Infeasible grap of U(θ) Figure 4: Corresponding feasible grap of U(θ), wit s, s 3 > 0 Again, constraint (8) is used to determine utility on te middle segment. On tis interval, 6

7 H(θ) = and s 3 is te distance between (p c + )/ and ( + β)x, or were U(θ) 0 for θ [X, ( + β)x]. We see tat te objective function drives s 3 to be as large as possible and tat profit for tis segment is increased by (p c w)s 3. Figure (5) depicts anoter infeasible grap of U(θ) wile Figure (6) depicts te corresponding feasible grap wit positive s 2 and s 3. Again notice ow te model forces te utility function to cange in Figure (6). Constraint (2) bounds s 3 so tat we keep (p c + )/ in te correct domain. S 2 S 3 Figure 5: Infeasible grap of U(θ) Figure 6: Corresponding feasible grap of U(θ), wit s 2, s 3 > 0 Due to te domain divisions of U(θ) and constraint (9), we see tat (p )/ = 0 for some θ [( + β)x, ]. We let s 4 be te distance between ( + β)x and (p )/, or were U(θ) < 0. Because of tis, we see in te objective function tat te distance between (+β)x and were U(θ) 0 is [ (( + β)x + s 4 )]. On tis domain coupons are not considered, and tus profit is not effected by c. Terefore profit increases by (p w)[ (( + β)x + s 4 )]. Note tat s 4 is bounded by constraint (3). Figure (7) represents yet anoter infeasible grap of U(θ) and Figure (8) sows te corresponding feasible grap wit s 4 > 0. S 4 Figure 7: Infeasible grap of U(θ) Figure 8: Corresponding feasible grap of U(θ), wit s 4 > 0 Profit is iger wen s 2 and s 3 are larger, and wen s 4 is smaller. However, te oter 7

8 variables in te model greatly effect weater te objective will attempt to minimize or maximize any of te s i s. 4 Bounds on p and c We now make some observations on p and c and see tat our model constraints provide bounds for bot p and c. Teorem 4. Te model constraints provide a lower bound for c and bot an upper and a lower bound for p. In particular, Proof: wic implies c, and ( + β)x + p +. We first sow c. From constraints (8) and (9) we ave tat X p c + Since s 4 0, we ave tat s 4 0. Terefore, Since s 3 0, we conclude tat + s 3 p s 4, (5) X p + c + + s 3 s 4. (6) c + + s 3 0. (7) c. (8) We now sow te bounds on p. From constraint (4), p 0. However, we can find a greater lower bound by using constraint (9). We see tat Since s 4 0, we ave tat p s 4 = ( + β)x p = ( + β)x + + s 4. (9) p ( + β)x + > 0. (20) Tus, we ave sown a lower bound on p. We use euation (9) and notice from constraint (3) tat s 4 ( ( + β)x). Tus, we substitute into euation (9) to obtain p ( + β)x + ( ( + β)x) + = +. Terefore, we ave tat ( + β)x + p +. 8

9 Parameter or Caracteristic Assumed Value for Example X /2 β /3 8 2 w 2 Table : Parameters for Example p c π \$9 \$0 \$9/3 \$,444 \$7 \$,250 \$6 \$,333 \$9 2 \$0 \$9/3 2 \$,638 \$7 2 \$,250 \$6 2 \$,833 \$9 3 \$0 \$9/3 3 \$,889 \$7 3 \$,583 \$6 3 \$,96 Table 2: Values for p, c, and π 5 A Numerical Example Te following example contains different scenarios for values of p and c and demonstrates te difficulty in determining an optimal p and c. We first make educated assumptions and pick values for our population parameters. Table () summarizes tese parameters. Selecting values for p and c, we see teir effect on profit, π, as sown in Table (2). Note tat te bounds on p give us an upper bound of \$9 and a lower bound of \$9/3, or \$6.33. From te table, we ave tat wen p is its upper bound, te firm gains zero profit. Tus, it is not optimal for p to be its upper bound. We also notice tat te lower bound on p may be optimal. Note tat te first four cases wit c = are not allowed by our model. Furtermore, te tree cases were p = \$6 are also not allowed by our model, as te lower bound on p is \$6.33. Tese cases are simply used as a reference to see te effects of different p and c values on profit. For profit calculations we assume tat θ is representative of,000 people in our population. Tus we multiply profit by,000 to gain a more realistic dollar value. From tis example we see tat te igest profit obtained is wen p = \$6 and c = 3. 9

10 Literature Model Our Model utility function fixed domains domains dependent for utility on p and c 2 cases one model p unbounded p bounded forces c ensures c c = is optimal claim c > is optimal Table 3: Comparison of models However, tis p value is not attainable in our model. Te igest profit allowed by our model is wen p = \$9/3, te lower bound on p, and wen c = 3 >. Tis leads us to Conjecture (5.). Conjecture 5. If (c*,p*) is optimal, ten < c* < (p*-w). Also, tere exists an optimal (c*,p*) suc tat p*= ( + β)x +. In comparing our model wit te one in an article from te Journal of Marketing Researc [], we discover some interesting differences. Table (3) summarizes our comparison. Our model restricts te domain of te utility function wile te literature allows te utility of consumer θ to cange depending on te firms desicions. Tat is, te literature s model allows consumers to measure teir utility differently depending on θ, p, and c. Because of tis, te literature is forced to divide te model into two cases, X < (p )/() and X (p )/(). We develop a uniform model tat accounts for all possiblilities troug te use of binary variables and a new utility function. Because of te fixed domains of utility and te constraints tat force U(θ) 0 for eac domain of U(θ), we find tat our model places bounds on p. At first glance tis seems to restrict te amount of possible profit, but upon furter investigation, we see tat te model also insures c. Since c is allowed to increase beyond, it essentially reduces te price consumers pay and tus increases teir utility so tat we obtain an euivalent or greater number of consumers wit a non negative utility tan te literature. Te literature does not place bounds on p, but does owever add an explicit constraint in teir model tat forces c. From a marketing standpoint tere is always some kind of relationsip between assle cost and coupon value. Most marketing managers would argue tat allowing coupon value to exceed assle cost is detrimental to profit as it would not accurately segment te population. Te firm is essentially allowing te market a discounted price and not gaining sales troug price sensitivity segmentation. However, we ave used similar assumptions in developing our model and still find an optimal p and c tat maximizes profit given tese assumptions. We assume tat is independent of c, but te literature assumes is dependent on c. Te literature also forces c to agree wit te insigt of te marketing assumptions. We find tat tis constraint is not needed. In fact, using similar assumptions to develop our model, we find tat te opposite is true. 0

11 In our example, c > provides a iger profit tan c =. We find tis to be true because increasing c essentially reduces p to be below te lower bound for all consumers tat consider a coupon. Tus again we are lead to believe tat conjecture (5.) is true. Wile we claim c > is optimal for our model, te literature finds tat c = is optimal for teir model. Note tat tis is teir upper bound on c. In tis paper we combine two cases in [] to develop an alternative model based on similar assumptions tat bound p and c. We find it interesting tat our model ensures c wile te literature forces c. However, tis is not suprising. Because we ensure c, we still ave a feasible optimal solution for maximizing profit. We claim tat c > and te lower bound on p is optimal. However, because we assume a piecewise utility function wit fixed domains U(θ), we are over simplifying te population. Terefore, it may not be realistic to apply our model to an empiricle coupon offering. 6 Conclusions Tis paper is beneficial to bot firms and consumers. For consumers, te understanding of tis paper is most beneficial in understanding teir purcasing beaviors. For firms, wen offering a coupon in a Second Degree price discrimination situation, te firm can find an optimal price and coupon value tat maximizes profit. Furtermore, te firm gains an optimal segmentation of te population. For tese reasons, a wole field of study in marketing is devoted to consumer beavior, wic includes te study of assle cost and purcasing patterns. In conclusion, tis newfound knowledge can be applied to better understand firm and consumer beavior. References [] Eric T. Anderson and Inseong Song. Coordingating price reductions and coupon events. Journal of Marketing Researc, 4:4 422, [2] Greg Saffer and Z. Jon Zang. Competitive coupon targeting. Marketing Science, 4: , 995.

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