In the vertical differentiation model, the consumers have utilityU=spU=spif they purchase and 0 otherwise.is distributed uniformly over interval[0,1][0,1], sis the quality and p is the price. The size of the total population is normalized to 1.

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

b. The cost function isc(q,s)=q(cs2)c(q,s)=q(cs2), whereccis a constant. Find out the optimalqqandssfor both the firm's profit maximization problem and the social welfare maximization.

Please provide a brief explanation.