Question

2. Consider a world in which a risk-neutral monopolist offers a product for sale. The product costs c = $60. In each period, the product can either be fully functional or totally defective. It is totally defective with probability 0.2 and is thus fully operative with probability p = 0.8. These events are independent across periods. Consumers, who are all risk-neutral, have valuation of V = $120 for a fully-functional product and zero for a totally defective product. In any transaction, the equilibrium price is equal to a consumer's expected valuation. There is no discounting Suppose the monopolist provides a one-time replacement warranty. That is, the monopolist replaces the product if it is defective. However, if the replacement is also defective, the monopolist does not replace it. (1) Determine the monopolist's expected profit, n, under this one-time replacement warranty policy. (ii) Without choosing numbers for p, c, and V, prove that * > 0. That is, the higher is the quality of the monopolist's product, the higher is her expected profit.'

Answer 1

From the given information, we can compute the following table regarding the consumer valuation of the product:

Product	Functional	Defective
Valuation (V)	$120	0
Probability	ρ = 0.8	(1 - ρ) = 0.2

Cost, C = $60

Now, Expected Valuation (EV) = ρ.V_F + (1 – ρ).V_D

                                       = 0.8(120) + (0.2)0

                                       = 96

1. Profit, ∏ = EV – C – (1 – ρ).V [the extra (1 – ρ).V at the end is because of the one-time replacement warranty]

= 96 – 60 – (0.2)(120)

   = 36 – 24

   = 12

               ii. Profit, ∏ (V, C, ρ) = ρ.V_F + (1 – ρ).V_D – C – (1 – ρ).V

                                                = ρ.V – C – (1 – ρ).V [since, V_D = 0]

                                                = 2ρ.V – C – V

            Now, ∂∏/∂ρ = 2V >0