Distortion on Top

In this subsection, I investigate the property of no distortion on top in detail. Almost all existing works have reported that a quality distortion does not happen on a con-sumer of a highest taste. Remember ¯q defined in the Chapter II, where ¯q is the socially efficient quality level for the consumer of the taste ¯θ. Both in the discrete/discrete and the continuous/continuous model, the monopolist sells ¯q to the consumer of the type ¯θ.

In our finite/continuous model, neither the monopolist nor the social planner provides ¯q to the consumer of the type ¯θ. They are both serving consumer interval, including ¯θ at the end of interval, not only to ¯θ. Since they should consider other consumers besides ¯θ, the product quality may be distorted downward from ¯q. Thus, for more meaningful interpretation, we need to check whether the monopolist’s choice is equal to the social best for the highest group of consumers, rather than compare to ¯q.

Simulation Result III.3. (Distortion on Top) Generally, q^∗_n 6= q^m_n. That is, the highest quality provided by the monopolist may be distorted from the social best selection.

As we saw in the previous section, the distortion on top does not happen in the case that n = ∞. When we have a uniform consumer distribution and a quadratic marginal cost function, we can analytically verify there is no distortion on top. Unfor-tunately, these are just special examples. In usual environments, quality for a highest interval tends to be distorted either downward or upward. The Figure 1 shows the instances of downward distortion on top.

From the Propositions II.3 and II.6, in order that q_n^∗ is equal to q_n^m,

(3.2) C⁰(q^m_n) = θ^m_n = E θ|θ ∈ [θn^∗, ¯θ]
= C⁰(q^∗_n).

The equation (3.2) implies that θ^m_n should be equal to a conditional mean over the interval [θ^∗_n, ¯θ]. It usually does not happen. If θ_n^m is greater than the conditional mean, q_n^m > q_n^∗, the highest quality product can be distorted upward.

The Table 2 reports simulation results using six different truncated normal dis-tributions. Each distribution has a right-side tail of different thickness and length.

There, the monopolist and the social planner choose only three different products. In

all distributions, q_n^m < q_n^∗, confirming the Simulation Result III.3.

Next, I define a degree of quality distortion on top as a ratio of qualities provided to a highest consumer group by the monopolist and by the social planner. That is, q_n^m/q_n^∗. If the degree of quality distortion on top is greater than 1, it means upward distortion. Note that as the ratio recedes from 1 either upward or downward, the quality is heavily distorted.

The first task is to find a relationship between the degree of quality distortion and a center of distribution.

Simulation Result III.4. (Degree of Quality Distortion on Top and Center of Distribution)

• Generally, the degree of quality distortion on top is larger when the distribution is left-centered, rather than right-centered.

• In the concentrated distribution, the degree of quality distortion on top becomes larger as the center of distribution goes to both extremes.

• In the dispersed distribution, the degree of quality distortion on top becomes smaller as the center of distribution goes to right.

The Simulation Result III.4 is nothing but a summary for the Figure 5. The key element to explain the Simulation Result III.4 is for whom the monopolist cares. In a left-centered distribution, its main concern may be consumers of low tastes. Since the low quality product market is large, the monopolist wants to reduce consumers who will buy nothing. Thus, θ₁^m will be quite small. In a right-centered distribution, it cares for consumers of high tastes. It does not need to have a very small θ₁^m. Let θ^m_1,l be θ^m₁ in the left-centered distribution and θ^m_1,r be in the right-centered distribution.

Obviously, θ^m_1,l < θ^m_1,r. The product quality increases as the consumer type increases.

Since ¯θ − θ^m_1,l > ¯θ − θ_1,r^m, the quality can travel a longer span in the left-centered distribution, and so it may arrive at a higher level.

The above scenario can be applied to the second and third argument in the Sim-ulation Result III.4, except the right-centered concentrated distribution. In the case, consumer population of highest tastes is very large. Now, the right-side population is extremely important for the monopolist’s profit. To fully exploit their tastes, the monopolist may offer a high quality product with a high price.

Next, I relate the degree of quality distortion with the degree of concentration.

Simulation Result III.5. (Degree of Quality Distortion on Top and Degree of Concentration)

• The degree of quality distortion on top approaches 1 as the distribution becomes more dispersed (flatter).

• If the distribution is not too dispersed, the degree of quality distortion on top becomes smaller as the distribution becomes more concentrated.

• In the left-centered distribution and under the proper degree of concentration, the degree of quality distortion on top may be bigger than 1 (upward distortion).

The first argument is related to a uniform distribution property. As σ goes to infinity, a distribution becomes a uniform distribution. As previously mentioned, q_n^∗ = q_n^m = ¯q when adopting the uniform distribution. Then, the degree of quality distortion on top becomes 1. For the second argument, we should consider a proportion of consumers of high tastes. The more concentrated the distribution is, the more thin the right-side tail. Then, the consumers of high tastes are ignorable by the monopolist.

The highest quality product will be more distorted downward. The third argument is observed in the Figures 5 and 6. This upward distortion will be explained in detail in the next subsection.