When Does Variety Increase with Quality?

Abstract

In this paper we provide a novel explanation of the relationship between product quality and variety. We develop a simple theoretical model with nearly rational consumers, who randomize among all alternatives that yield the same utility for them but could make a mistake of choosing a slightly worse alternative. We show that it is optimal for a firm to offer more varieties of its higher quality, more profitable, products, in order to price discriminate more effectively. When the market is thin at the very top quality level, the number of varieties first increases and then decreases, which results in the overall hump-shaped relationship between the number of varieties and quality. We find empirical support for our qualitative predictions in the Australian car market.

Appendix A

1.1 Appendix A1: Background on the Dataset

Price, characteristics and sales data for cars are compiled for the 17 makes that offer 30 or more varieties in the Australian market. These makes range from the low-price Daewoo and Daihatsu to the premium-priced BMW and Mercedes-Benz. We exclude all varieties with prices of A$200,000 or more, because we believe that such cars are not generally sold as off-the-shelf varieties and, thus, their numbers of varieties are not informative.

The issue with the data that we have is that while price and characteristics are available for each variety of a model, the sales data are available only for the whole model and, in a few cases, only for the groups of models. For example, in the price and characteristics data, a distinction is made between the GMH Commodore and what could be considered as two higher-quality models of the Commodore: the Berlina and Calais; but in the sales data all three models are aggregated as the Commodore.

As a result, in order to perform the tests in Sect. 6.3 we calculate an estimate of sales for each variety, by assuming that all varieties of a model have equal shares of sales. This may underestimate (and is unlikely systematically to overestimate) sales for cheaper varieties within a model and, therefore, will bias against finding support for our model.

1.2 Appendix A2: Proofs for the Two Propositions

Proof of Proposition 1

Using (19), it is straightforward to verify that as $\varepsilon _{i}\rightarrow +0$ for all i (as marginal cost $c\rightarrow +0$ ), the number of varieties is given by

$$\begin{aligned} n_{i}=a_{i}\varepsilon _{N}^{-\frac{i}{N+1}}+O\left( \varepsilon _{N}^{\frac{1}{N+1 }}\right) , \end{aligned}$$

(25)

where coefficients $a_{i}$ satisfy the second-order difference equation

$$\begin{aligned} \frac{a_{i+1}a_{i-1}}{a_{i}^{2}}=k_{i}, \end{aligned}$$

(26)

subject to the following initial and terminal conditions:

$$\begin{aligned} a_{0}=1,\text { }\frac{a_{N-1}}{a_{N}^{2}}=1. \end{aligned}$$

(27)

Therefore, the ratio of numbers of varieties in the limit goes to zero:

$$\begin{aligned} \lim _{c\rightarrow +0}\frac{n_{i}}{n_{i+1}}=0. \end{aligned}$$

(28)

$\square$

Proof of Proposition 2

For $i>j$, for small marginal cost and small $p_i$, we have $\varepsilon _{i}\sim 1$. (Note that if marginal cost is so small such that $\varepsilon _{i}$ is much smaller than $k_i$, then Proposition 1 applies.) Given that $k_i<<1$ and $\varepsilon _{i}\sim 1$, we can drop the second term on the left-hand side of Eq. (19) and easily verify that $n_{i+1}<n_{i}$ as long as $4n_{i}\varepsilon _{i+1}>1$. Since $n_{i}\ge 1,$ a sufficient condition is $\varepsilon _{i+1}>1/4$: If the cost of proliferation is not too small²⁵ and there are very few high-type consumers, the number of varieties will eventually fall.

For $i \le j$, the same type of argument as in Proposition 1 applies; thus, the number of varieties is increasing in quality.

Therefore, this case results in an overall hump-shaped set of varieties. $\square$

1.3 Appendix A3: Data Transformation for Estimating the Beta Density Function

To generate the number of varieties at 50 points suitable for estimation of the beta density function, we first construct a kernel estimate of the density for each make using the actual data. Second, we select values of the density function at the Stata default option of 50 points. To ease estimation, we convert these values to integers by multiplying them by 10 million and rounding. This yields numbers of varieties at the 50 points. Descriptive statistics of the new densities reveal they are similar to the descriptive statistics for the original prices.