Chapter 2: Product Selection and Quality Relax A1. The goods - PDF document

Chapter 2: Product Selection and Quality Relax A1. The goods produced by the monopolist are not given. A2. No price discrimination. • Choice of quality and pricing behavior by a monopolist. • Consumers can learn the quality before purchasing (search goods), – too low of too high quality relative to social optimum? – too much or too little product variety? • Consumers can learn the quality after purchasing (experience goods), – what are the incentives of the monopolist to supply quality? – How to provide information to consumers? 1

1 Product Space • Goods are always differentiated by some charac- teristics. • There are always interactions between groups of goods. • What is the differentiation between the goods within an industry? Hotelling (1929), Chamberlin (1952, 1962) • A good is a bundle of characteristics: quality, location, availability, consumer’s information about its existence, its quality..... 2 types of differentiation: 1. vertically differentiated product space = quality; 2. horizontally differentiated product space = location. 2

1.1 Vertical Differentiation • All consumers agree over the most preferred mix of characteristics. • Higher quality is preferred to lower quality (BMW is preferred to Subaru). Model of vertical differentiation • single quality/good monopolist. • Each consumer consumes 0 or 1 unit of a good. • N consumers. • s : quality index (service), s > 0 . • A consumer has the following preferences ( θs − p if he buys the good of quality s at price p u = 0 if he does not buy where θ > 0 is a taste parameter. • θ , density f ( θ ) , a cumulative distribution F ( θ ) ∈ [0 , 1] and F (0) = 0 and F ( ∞ ) = 1 . • F ( θ ) : fraction of consumers with a taste parameter < θ . 3

What are the demand functions? 1. If 1 quality s at price p , the demand is D ( p ) = N [1 − F ( p s )] 2. If 2 qualities: s 1 < s 2 and that prices are p 1 < p 2 , different cases: • if s 2 p 2 ≥ s 1 p 1 , then demands are D 2 ( p 1 , p 2 ) = N [1 − F ( p 2 )] s 2 D 1 ( p 1 , p 2 ) = 0 • If low quality good is not dominated. e θ = p 2 − p 1 s 2 − s 1 : indifferent consumer. – Consumers with θ ≥ e θ buy quality 2. 1 − F ( p 2 − p 1 s 2 − s 1 ) : proportion of consumers who buy quality 2 . – Consumers with θ < e θ and θ ≥ p 1 s 1 buy quality 1. Thus F ( p 2 − p 1 s 2 − s 1 ) − F ( p 1 s 1 ) : proportion of consumers who buy quality 1. – if θ < p 1 s 1 no purchase. D 2 ( p 1 , p 2 ) = N [1 − F ( p 2 − p 1 )] s 2 − s 1 D 1 ( p 1 , p 2 ) = N [ F ( p 2 − p 1 ) − F ( p 1 )] s 2 − s 1 s 1 4

1.2 Horizontal Differentiation • Spatial differentiation: location. Model of Hotelling (1929) • Linear city of length 1. • N consumers are distributed uniformly along the city. • 2 shops are located at the 2 ends of the city, shop 1 is at x = 0 and of shop 2 is at x = 1 . • Both sell the same physical good. • Consumers have transportation costs t per unit of length. • They consume either 0 or 1 unit of the good. • p 1 and p 2 are the prices charged by the 2 shops. • Price of going to shop 1 for a consumer at x is p 1 + tx . • Price of going to shop 2 for a consumer at x p 2 + t (1 − x ) . • The utility of a consumer located at x is 5

⎧ ⎪ ⎪ s − p 1 − tx if he buys from shop 1 ⎨ U = s − p 2 − t (1 − x ) if he buys from shop 2 ⎪ ⎪ ⎩ 0 otherwise What are the demands? 1. If p 2 − p 1 < t , and prices are not too high – There exists an indifferent consumer located at x ( p 1 , p 2 ) = p 2 − p 1 + t e . Thus demands are 2 t D 1 ( p 1 , p 2 ) = N e x ( p 1 , p 2 ) D 2 ( p 1 , p 2 ) = N (1 − e x ( p 1 , p 2 )) – See graph 2. If p 2 − p 1 > t , shop 2 has no demand. D 1 ( p 1 , p 2 ) = N if p 1 < s − t D 1 ( p 1 , p 2 ) = N s − p 1 if p 1 > s − t t – See graph 3. If p 1 and p 2 belong to [ s − t, s ] , – 2 local monopolies 6

– the indifferent consumer does not consume D 1 ( p 1 , p 2 ) = N s − p 1 t D 2 ( p 1 , p 2 ) = N s − p 2 t – See graph • Residual demand for good 1, as price of good is given has a kink (see graph). Remark 1 Endogenous transportation costs: Dos San- tos Ferreira and Thisse (1996), IJIO. 7

2 Product Selection • Spence (1975, 1976) 2.1 Single-product Monopolist • Monopolist chooses p and s . • p = P ( q, s ) : inverse demand function, ∂P ( . ) ∂s > 0 , • C ( q, s ) : cost function, ∂C ( . ) ∂s > 0 . 1. Social planner maximizes the gross consumers surplus minus cost, W ( q, s ) Z q Max P ( x, s ) dx − C ( q, s ) q,s 0 FOC q P ( q, s ) = ∂C ( q,s ) (price=MC) ∂q R q ∂P ( x,s ) dx = ∂C ( q,s ) FOC s 0 ∂s ∂s R q ∂P ( x,s ) dx = ∂C ( q,s ) ⇒ q 0 ∂s q ∂s R q ∂P ( x,s ) dx 0 ∂s average marginal valuation of quality. q 8

2. Monopolist maximizes his profit function Π m ( q, s ) qP ( q, s ) − C ( q, s ) Max q,s FOC q q ∂P ( q,s ) + P ( q, s ) = ∂C ( q,s ) (MR=MC) ∂q ∂q FOC s q ∂P ( q,s ) = ∂C ( q,s ) (MR=MC) ∂s ∂s 3. Comparison : • Quantities and prices: usual inefficiencies, p m > MC . • Quality: – the social planner is concerned about the average marginal value of quality, – the monopolist is concerned with the marginal (consumer) marginal ( quality) valuation. • Why? – The social planner looks at the effect of an increase in quality on all consumers . – The monopolist considers the effect of an increase of quality on the marginal consumer . 9

Summary The incentive to provide quality is related to the marginal willingness to pay for quality for the marginal consumer in the case of the monopolist, and for the average consumer in the case of the social planner. Result 1 For a given output level, the monopolist undersupplies quality relative to the optimum when R q ∂P ( x,s ) dx > ∂P ( q,s ) 0 ∂s q ∂s and conversely. 2.1.1 Example of underprovision of quality • Vertical differentiation model • Utility is ( θs − p if he buys the good of quality s at price p u = 0 if he does not buy • θ , f ( θ ) , F ( θ ) ∈ [0 , 1] and F (0) = 0 and F ( ∞ ) = 1 . • Normalization: N = 1 . • If only one quality s at price p , the demand is D ( p ) = [1 − F ( p s )] and the inverse demand function 10

is p = P ( q, s ) = sF − 1 (1 − q ) F − 1 ( . ) (increasing) inverse function of F . • Average marginal valuation for quality ( social planner ) R q R q ∂P ( x,s ) 1 dx = 1 0 F − 1 (1 − x ) dx q 0 ∂s q • Marginal marginal valuation for quality ( monopolist ) ∂P ( q,s ) = F − 1 (1 − q ) ∂s • As x ≤ q , ⇒ F − 1 (1 − x ) > F − 1 (1 − q ) and thus R q 0 F − 1 (1 − x ) dx > qF − 1 (1 − q ) ⇒ Average valuation for quality > marginal valuation. • For a given q , the monopolist undersupplies quality. • But it is not clear that the quality of the monopoly is lower, because quantities and prices are different! • See exercise 22 page 104: θ ∼ U [0 , 1] and the cost function is C ( q, s ) = cs 2 2 q 11

2.2 Multi-product Monopolist • Too many or too few products? 2.2.1 Single-product monopolist and underprovi- sion • f fixed cost of introducing (or producing) the good. • Monopolist introduces the good only if Π m > f . • Social planner introduces the good if W > f . • As W > Π m , there exit values of f such that W > f > Π m and thus – the monopolist does not introduces the good – the social planner would do it. Result 2 With only one potential product, a monopoly situation may imply too few products. Why? • In general a monopolist cannot appropriate all the social surplus. • Except if he can price-discriminate (Chapter 3). 12

2.2.2 Multi-product monopolist and overprovision • 2 goods are substitutes, ∂D j ∂p i > 0 , i 6 = j and i = 1 , 2 . • The monopolist sets p 1 > MC 1 , thus demand for good 2 increases. • It is thus profitable to produce good 2 which would not be profitable if p 1 = MC 1 . Exercise 2.3 • Horizontal differentiation model • linear city of length 1; consumers are uniformly distributed. N = 1 . • transportation cost: t . • Unit demands. • Product diversity: monopolist can sell at: x = 0 and x = 1 . • f fixed set-up cost of establishing one store. • MC = 0 . • What is the benefit to open a second shop? – If the monopolist opens 1 shop, he sets the price p = s − t , and profit is s − t − f 13

– If he opens 2 shops, the price increases to s − 1 2 t , and profit is s − 1 2 t − 2 f – Thus the increase in profit of adding one shop is ∆Π = 1 2 t − f – Social planner: minimizes costs (transportation costs and set-up costs) – If 1 shop, the cost is R 1 0 txdx + f R 1 – If 2 shops, the cost is 2 0 txdx + 2 f 2 – Thus the gain of adding one shop is ∆ W = 1 4 t − f • Comparison: – for 1 4 t < f < 1 2 t the monopolist opens the second shop whereas it is not socially optimal to do so. 2.3 Product Selection and Discrimina- tion • Monopolist can discriminate: sell at a higher price to consumers with high valuation. • Cons. have no incentive to reveal high valuation. 14