Consider a Hotelling duopoly in which firms are located at the extreme points of the unit interval and consumers of mass 1 are uniformly distributed on the unit interval. The price of the two products is given and equal to 1, p₁ = p₂ = 1 (e.g. because the price is fixed upstream); production costs are zero. The quality of product i is denoted by si ∈ [2, 3]. The quality of each product is drawn independently from the uniform distribution on this interval [2, 3]. Both firms observe the two qualities of the product, consumers do not observe the qualities. A consumer located at x 2 [0, 1] derives utility Es₁- x-p₁ from product 1 and Es₂ -(1-x)- p₂ from product 2, where Es_i is the expected quality of product i given the information available to consumers.

1. Suppose that firms can simultaneously disclose their own quality s_i at zero cost. Consumers then decide which product to buy. Characterize the equilibrium of this game. Prove that it is the unique equilibrium.

2. Suppose now that disclosure is costly, i.e. a firm has to spend a given advertising cost a ∈ (0, 1/4] to disclose its own quality. Suppose that firm i conditions its action on s_i only. Characterize the equilibrium of the game in which firms first decide whether to advertise their own quality truthfully or not to disclose any information and then consumers make their choices. Note that firms know the cost a and make their disclosure decisions simultaneously.

3. Suppose that instead of advertising their own quality, firms can costly advertise only the quality difference s₁ - s₂, i.e. firms can only engage in comparative advertising. Characterize the equilibrium in which both firms simultaneously decide whether to disclose the quality difference at cost a ∈ (0, 1/4].

4. Discuss verbally the welfare properties of the equilibria determined in (2) and (3).

5. Consider now a model in which firms can choose not to advertise, to use non-comparative advertising or to use comparative advertising. In the last two cases the same advertising cost a applies. Provide verbally an intuition about the properties of the equilibrium of this game. To simplify the argument, consider an alternative setting in which there are only two discrete types s_i 2 {2, 3}.