Quick summaryOne-dimensional vertical (quality) differentiation model is extended to two dimensionsUse to analyze product and price competitionTwo stage game

1. Firms first compete on product position2. Then they compete on price

Firms do not tend towards maximal differentiationWhen firms have equal opportunities, they end up with max-min differentiation, just like in Irmen and Thisse

Choice of product featuresNeed to consider consumer preferences

In quality differentiation, all consumers agree that more of quality is better.

The vary in their willingness to pay for qualityNeed to consider strategic responses of competitorsTwo forces determine the locational equilibrium (both vertical and horizontal differentiation models)

A demand force that draws firms close togetherA strategic force that causes them to differentiate

Early research on vertical differentiation maximal differentiation shows maximal quality differentiation when demand is inelastic, but when one option is to not buy, quality differs exist, but not maximal

Model: market and firmsGoods have more than one type of quality

product qualityservice quality

Two firms, indexed by 1 and 2choose one product to marketthe product has two characteristics, x and y, defined as a point, (xi,yi) where these define the two types of quality, and are both bounded by (allowably different) maximum and minimum valuesproduction has constant marginal cost, set equal to zero regardless of product position (does not depend on quality characteristics), and no fixed cost

Model: consumersConsumers prefer more of each quality to less, and prefer a low price to a higher priceConsumer reservation price, R, is sufficiently high that every consumer in the market buys one unit from either firm 1 or firm 2Consumer valuation is defined by

so consumers care about absolute rather than relative pricesThe parameters 1 and 2 are uniformly distributed over the population. We restrict the range to [0,1], where the scale is different for (x,y) if one characteristic is more important than the other.

1 2 for 1, 2iU R p i

Characteristic competitionAsymmetric characteristics competition – each firm has a relative advantage in one quality dimension (figure a), for example “ease of use” and “power” (Apple versus Windows)Dominated characteristics competition – one firm has a relative advantage on both characteristics (Windows 10 vs Windows 7)For both types of competition, the relative positions of the products are described by the ratioThis ratio equals the tangent of the angle between the horizontal axis and a line from the origin perpendicular to a line joiningThis angle is called the angle of competition

1 2 1 2( ) / ( )x x y y

1 2 1 2, to , x x y y

Indifference spacePanel a shows the angle of competition with asymmetric characteristics (firm 1 dominates in x, firm 2 dominates in y)Consumers buy the good that maximizes their utility – there is a set of consumers indifferent between the two goods where

as shown in panel b when prices are equal, in which case the indifference line lies along the angle of competitionDifferences in prices shifts the indifference line up or down. More skewed dominance in increases or decreases the angle of competition

2 1 1 22 1 1

2 1 2 1

ˆ ( ) p p x xy y y y

or x y

Indifference spacePanel a shows the angle of competition with dominant characteristics competition (firm 2 has the superior product)When prices are equal the indifference line is outside the characteristic space – everyone buys from firm 2. Hence, for the indifference equation to be in the product space we need and we graph as before

Relatively greater dominance by firm 2 in y increases the angle of competition. Relatively greater dominance in x decreases the angle of competition

2 1 1 22 1 1

2 1 2 1

ˆ ( ) p p x xy y y y

2 1p p

Price EquilibriumSequential game: firms first choose quality, then choose price, with subgame perfect criterionAn equilibrium consists of product choice for firms 1 and 2 such that neither would choose a different product unilaterally, for whatever prices equilibrium that follows, using backward inductionProfit function for firm i, i=(1,2) isA noncooperative Nash equilibrium is a pair of prices such that

1 2 1 2( , ) ( , )i i ip p p D p p

* *1 2( , )p p

* * *( , ) ( , ) 0 , (1,2)i i j i i j ip p p p p i j

Asymmetric characteristics equilibriumThe indifference line is positively sloped with angle

With product positions fixed the indifference line is shifted up or down by changes in Look at the problem from firm 1’s perspective (so p2 is fixed).We see in the next figure there are four boundaries

1 1 2

2 1

tan x xy y

2 1.p p

is the lowest price at which no consumers buy from firm 1 is the lowest price at which no consumers buy from firm 1These are the upper and lower bounds of the price the firm will charge givenThe two remaining key prices are those where the indifference line passes through (0,0), at price , and when it passes through (1,1), at price

1up

1lp

2p̂

1np

1mp

Price equationsReplacing in the boundary equations with the boundary point coordinates gives

where, for example, in (3) we set These are all increasing in , and, if it is in the equation, increasing in

The greater firm 1’s relative advantage over 2, the higher the price it can charge to generate a similar demand

1 2 and

1 21, 0.

2p̂1 2 2 1.x x y y and decreasing in

Characteristic i dominance, i=x,yIf α45o we have and which is characteristics x dominance, while if then 45o α we have characteristic y dominance and

1 2 2 1x x y y 1 1 1 1l n m up p p p

1 2 2 1x x y y 1 1 1 1l m n up p p p

𝑝1𝑢

𝑝1𝑢

𝑝1𝑛

𝑝1𝑛

𝑝1𝑚 𝑝1

𝑚𝑝1𝑙

𝑝1𝑙

α45o characteristic x dominance45o α, characteristic y dominance

11

22

Characteristic x dominance

Three regionsIn demand for firm 1 increases at an increasing rateIn demand for firm 1 increases at an constant rateIn demand for firm 1 increases at an increasing rate

1xR

2xR

3xR

To see the rate of changeSuppose we are in . As price falls from to the area of demand grows at an increasing rate We can easily see that z1 is given by

and demand by

1xR

1up 1

mp

Total demand under characteristic x dominance

It is easy to see that

Note D2=1-D1 and we have three regions that correspond to the regions in the above graph. In region 1 D1 is convex, in region 2 it is linear and in region 3 it is convex. Firm 2’s demand follows the opposite pattern

1xR2

xR3xR

Equilibrium pricesStart in region 2. Total profit is price times quantity, so Maximizing this with respect to px gives a unique solution

if in region 2 so but that is only satisfied if (the condition for characteristic x dominance which we started with). A similar condition holds that puts firm 2 in the same region.A similar analysis under characteristic y dominance results in so whichever characteristics is dominant in the asymmetric dominance situation we end up in the middle region

21 .xp D

1 2 2 1( ) ( )x x y y *1 1 1n mp p p

*1 1 1m np p p

Dominated characteristic competitionIn this case one firm (2) has higher quality in both characteristics. With characteristic x dominance we have 1 1 1 1

m u l np p p p

1np

1lp 1

up1mp

With the same type of tortured logic they show that there are price equilibrium in all three regions. This is different from the asymmetric characteristic competition.

There is a similar picture if y is dominant, but the ordering of prices becomes because the blue lines are flatter.

R2R1

R3

1 1 1 1m l u np p p p

1

2

Product equilibrium strategyProduct positions are dependent on the equilibrium pricesProcess1. Establish which regions need to be considered2. Calculate the firms profit function within each relevant region3. Find the FOC for profit along with the demand restrictions that

define the region indicate the maximum profit location in a region4. Compare the maximum profit in each region to find the optimal

location choice

Location decision under asymmetric dominanceNeed only look in region 2 for each firm. From the demand for each firm, and the equilibrium prices we find

Now look at table 1

Implications about produce choiceMaximum quality in both directions yields the highest profit, therefore both firms want to choose that locationBoth can’t be there, and it is not clear which will, so we can arbitrarily pick oneGiven that one firm is in the supreme position of highest quality in both characteristics, it is the strategy of the lower quality firm that determines how much differentiation there isOne solution is max-min (panel a on next page), in which both firms have maximum quality in one characteristic, one has maximum quality in the first characteristic, and the other has minimal quality in the second characteristicOther solutions are max-max, and max partial

Max-min

Max-max

Max-partial

What is interesting about these three pictures?If the range in quality in one direction is approximately equal to that of the other, the result is max-minAs the ratio of the ranges grows, we move to max-maxBut as it gets very large, we get max-partial. Moreover, as the range of x quality relative to the range of y quality gets bigger and bigger, the partial differentiation gets further away from max-max (but not necessarily closer to max-minSo why? (they do a lousy job of explaining why these results make sense)

Both firms want to have maximum quality. When quality dimensions are approximately equal, but because of strategic force to avoid price competition, only one locates there. The other firm, to maintain demand as much as possible keeps quality high in one dimension, but differentiates in the other dimension to reduces price competition.When quality are very different maintaining high quality in the lesser dimension does not bring enough demand to counter the price competition effect. So the second firm relinquishes demand (by lowering the quality on the second dimension) but gains in price.But the strategic tradeoff doesn’t always dominate. If the ranges of quality get sufficiently large, relative quality would get too low (they would lose too much demand for any diminishment of the price competition to compensate, so chooses an intermediate level in the second quality dimension.

So why is this important?Depending on the range of quality differences, we go, as the range of x relative to the range of y increases, from max-min to max-max to max-partial in differentiation along the two quality dimensionsThis is different from everything we’ve seen before, because within one model we are seeing all possibilitiesAlso shows that the way quality differentiates and how consumers value different quality components are important in determining the equilibrium (different consumer values would show up as different scales, increasing or decreasing the ranges of x and y)Helps explain the wide variety of quality we see among differentiated products