A firm’s has quality x ∈ {0,…,100}, each of which is equally likely. Regardless of x, with probability 1/10 , the firm can only send message m = ∅ while with probability 9/10 it can send one of two messages m = ∅ or m = x. Consumer observes the firm’s message and forms a belief about the firm’s expected quality b = E [x|m]. The firm wants to maximize b, i.e., it wants to maximize the consumer’s expectation of its quality.

QUESTION

A firm’s has quality x ∈ {0,…,100}, each of which is equally likely. Regardless of x, with probability 1/10 , the firm can only send message m = ∅ while with probability 9/10 it can send one of two messages m = ∅ or m = x. Consumer observes the firm’s message and forms a belief about the firm’s expected quality b = E [x|m]. The firm wants to maximize b, i.e., it wants to maximize the consumer’s expectation of its quality.

Note the distinction between 0, which is a potential value that x can take and ∅, which is not a potential value for x but rather a message that is “silent” about x.

Is there an equilibrium with a cutoff type x∗ such that: (i) every firm with x < x∗ sends a message m=∅, (ii) every firm with x≥x∗ sends a message m=x if it can? If so, what is x∗?

ANSWER

Equilibrium Analysis: Maximizing Consumer Expectation of Quality in Firm Communication

Introduction

In this essay, we will explore the concept of equilibrium in a firm’s communication strategy with consumers, with the objective of maximizing the consumer’s expectation of the firm’s quality. We will analyze the scenario where a firm’s quality is denoted by x, which can take values between 0 and 100. The firm has two possible messages to convey to the consumer: an empty message (m = ∅) and a message containing the firm’s quality (m = x). We will determine if there exists an equilibrium with a cutoff type x∗, where firms with quality below x∗ send an empty message, while firms with quality above or equal to x∗ send a quality-revealing message.

Equilibrium Analysis

To analyze the equilibrium, let’s consider the perspective of the consumer. The consumer receives a message from the firm and forms a belief about the expected quality of the firm, denoted by b = E[x|m]. The firm aims to maximize b, i.e., the consumer’s expectation of its quality, by strategically selecting the appropriate message.

Firms with quality x < x∗ sending m = ∅:

For firms with quality below x∗, it is optimal to send a silent message (m = ∅). This strategy is due to the fact that no matter what message is sent, the consumer cannot distinguish between different qualities below x∗. Hence, the firm’s choice is irrelevant, and it is best to remain silent.

Firms with quality x ≥ x∗ sending m = x if possible:

For firms with quality above or equal to x∗, they have a strategic advantage of being able to convey their quality through the message m = x. As x increases, the consumer’s belief in the firm’s quality, b, will also increase. Therefore, it is in the firm’s best interest to reveal its quality by sending the message m = x.

Determining the cutoff type x∗

To find the equilibrium cutoff type x∗, we need to consider the point at which the firm is indifferent between sending a silent message (m = ∅) and revealing its quality (m = x). This occurs when the consumer’s belief in the firm’s quality, b, is the same regardless of the message received.

For firms sending m = ∅, b = E[x|m = ∅] = E[x], which represents the average quality of the firm in the absence of any information.

For firms sending m = x, b = E[x|m = x], which is equal to the firm’s actual quality x.

To find the equilibrium cutoff x∗, we set b = E[x|m = ∅] equal to b = E[x|m = x] and solve for x.

Since the quality x is equally likely between 0 and 100, the average quality E[x] is 50.

Thus, we have:

E[x|m = ∅] = E[x] = 50

E[x|m = x] = x

Equating the two expressions, we obtain:

50 = x

Therefore, the equilibrium cutoff type x∗ is 50. Firms with quality x < 50 will send a silent message (m = ∅), while firms with quality x ≥ 50 will send a quality-revealing message (m = x) if possible.

Conclusion

In conclusion, there exists an equilibrium in which firms with quality below the cutoff type x∗ = 50 send a silent message (m = ∅), while firms with quality above or equal to x∗ send a quality-revealing message (m = x) if possible. This equilibrium is optimal for firms seeking to maximize the consumer’s expectation of their quality. By strategically selecting the appropriate message, firms can effectively communicate their quality and enhance consumer beliefs, ultimately leading to a more favorable perception of their products or services.