# Vertical Product Differentiation Model

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- Additional Contributors: None
Categories: CategoryModel CategoryMicro CategoryIndustrialOrganisation CategoryImperfectCompetition

Related Models: Horizontal_Product_Differentiation_Model

Created: 2008-08-20

Suggested Citation: See license page.

## Introduction

The defining feature of "Vertical" models of product differentiation is that if any two of the goods in competition were offered at the **same** price then **all** consumers would choose the same one -- i.e. ignoring prices all products can be *consistently* ranked/ordered by **all** consumers. This differs from Horizontal_Product_Differentiation_Model where the defining characteristic is that the population of consumers is heterogeneous in their preferences for the various products even without the introduction of price.

This feature has a substantial effect on the nature of the results obtained. Most significantly in other types of model as the fixed costs of entry decline (relative to the size of the market) the outcome converges to the competitive equilibrium in which firms charge marginal costs and there are a very large number (infinitely many) of firms each identical and with no market power. By contrast in these vertical differentiated models it is possible that a very limited oligopoly with a firms charging a markup over marginal costs even as (relative) entry costs tend towards zero. As Sutton (sutton_1986) explains (p. 394):

"Whether or not it does [i.e. converge to competitive outcome or stay with oligopoly] depends on the nature of technology and tastes. Specifically, what matters is the relationship between consumers' willingness to pay for quality improvements, and the increase in *unit variable cost* associated with such improvements."

## Model

This is based on that presented in shaked_ea_1983 and sutton_1986.

Suppose a consumer with income $$Y$$ who purchases *one unit* of product with quality $$u$$ and price $$p$$ thereby derives utility (if she buys nothing utility is $$u_{0} Y$$):

$$ \[ V = u (Y - p) \] $$

**Remark:** So here consumers are differentiated by income with wealthier consumers obtaining a greater value ($$vY$$ from the same good -- but all consumers still ranking goods in the same order.

Let there be some number of firms $$1,...,n$$, each able to produce product of quality $$u$$ at (marginal) at per (marginal) unit cost $$c(u)$$. Let product qualities and prices as chosen by the $$n$$ firms be $$u_{1}, ..., u_{n}$$ and $$p_{1}, ..., p_{n}$$ and without loss of generality assume qualities are ordered (from low to high). Furthermore define $$c_{k}=c(u_{k})$$.

Assume consumers are uniformly distributed by income. (NB: This assumption plus the form of the utility function are not crucial to the results).

## Solving

Define:

$$ \[ r_{k-1,k} = \frac{u_{k}}{u_{k}-u_{k-1}} \] $$

A consumer is indifferent between good $$k$$ and $$k-1$$ if:

$$\[ u_{k-1} (t_{k} - p_{k-1}) = u_{k} (t_{k} - p_{k}) \]$$

Rearranging one obtains (for $$k=1$$ set $$p_{0}=0$$):

$$ \[ t_{k} = p_{k-1} + (p_{k}-p_{k-1})r_{k-1,k} \] $$

And consumers with income above $$t_{k}$$ prefer $$k$$ to $$k-1$$.

Then the profit of firm $$k$$ with price $$p_{k}$$ is zero if $$p_{k} \lt c_{k}$$ and otherwise is:

$$ \[ \pi_{k} = (p_{k} - c_{k})(t_{k+1} - t_{k}) \] $$

Now seek a Nash Equilibrium in prices. To do this sufficient to prove that for any set of qualities and given any set of other firm prices a firm's profit function is a single-peaked function of its own price $$p_{k}$$ (since then each firm's profit function is concave ...). Doing this involves some tedious algebra and is omitted here but can be found in Lemma 1 (p. 1475) of [shaked_ea_1983].

## Comments

**Remark:** (Informally) if either of the following conditions hold one has a situation analogous to horizontal differentiation case and one can have infinite entry:

- Marginal costs rise steeply
- There is a consumer whose derivative of utility with respect to quality is zero (here that would be a consumer with income zero)

If these do not hold (and that is the more interesting case) then only a finite number of firms survive in equilibrium.