A monopoly produces and sells widgets to a continuous, unit measure population of consumers (think of each consumer as infinitesimally small), according to the demand function Q = 1 P if 0 P 1, and Q = 0 if P > 1. In this equation, Q >= 0 represents the quantity demanded when the price chosen by the monopoly is P >= 0. The monopoly profit is pi= ((1 t)P c)Q, where 0 <= c < 1 is the unit cost (i.e., the constant marginal cost) of production and 0 <= t <=1 is the tax rate that the monopoly pays to the government. There is no fixed cost (i.e., the fixed cost is equal to zero). The monopoly chooses its price P to maximize its profit, given the demand, the tax rate, and its technology. This is the singleprice monopoly model with perfect information. Let P*, Q*, and pi* denote, respectively, the equilibrium price, quantity, and profit.

(i) Find the equilibrium welfare, W* . Remember that you should add the monopolyprofit, the consumer surplus, and the tax revenue, all variables in equilibrium. Calculate the derivative dW*/dt . Is this derivative positive, zero, or negative?