A firm has either a high quality or a low quality product (the firm cannot choose the quality of its product). The firm faces a continuum of consumers with a total mass one. Each consumer wishes to buy at most one unit. There are 2 types of consumers: 2/5 of the consumers are of type 1 and their utility is 10 - p if they buy a high quality product and 5 - p if they buy a low quality product; 3/5 of the consumers are of type 2 and their utility is 6 - p if they buy a high quality product and 3 - p if they buy a low quality product. Both types of consumers obtain a utility of 0 if they do not buy. The per unit cost of production is 2 if the firm has a high quality product and 0 if it has a low quality product.

1. Suppose that consumers can tell the quality of the product before they buy. Determine the prices that each type of firm would charge. (Note that if quality is high, the firm can sell only to type 1 consumers if p > 5 but to all consumers if p ≤ 5; likewise, if quality is low, the firm can sell only to type 1 consumers if p > 3 but to all consumers if p ≤ 3.)

2. Suppose now that consumers cannot tell the quality of the product before they buy and can only infer it from the price that the firm charges. Show that there exists a separating equilibrium in which high- and low-quality firms behave differently and therefore consumers can infer the quality of the product from the price that the firm charges. In this equilibrium, a low-quality firm behaves as in (1). What is the price that the high-quality firm needs to charge in order to separate itself from the low-quality firm? Show that charging this price is profitable for the high quality firm. Does the high-quality firm signal its quality by charging a high price or a low price? Provide an intuition.