5. In the vertical differentiation model, the consumers have utility U = s - p if they purchase and 0 otherwise. is distributed uniformly over interval [0,1]. s is the quality and p is the price. The size of total population is normalised to 1.

a) Derive the demand function and the social surplus function.

b) If cost function is c(q, s) = q( cs^2 ), where c is a constant. Find out the optimal q and s for both the firm's profit maximisation problem and the social welfare maximisation.