A monopolist produces a single good :1: and sells it to two consumer types, A and B, whose preferences are given by the utility functions 3 UA(.1',y) = 24:12 x2 + y and UB(.1:,y) = 22:: ix? + y, where :1: is the quantity consumed of the monopolist's good and y is the consumer's disposable income. There is an equal number (say N) of both types of consumer within the population and the monopolist has a constant marginal cost equal to 4. (3) Suppose that the monopoly can accurately identify consumer type and wishes to design specic \"take it or leave it" contracts to perfectly price discriminate and maximise prots. These contracts take the form of a specic quantity qr sold for an overall price of r,- for each consumer type i = A, B. (i) Write down the prot-maximisation problem and the two constraints faced by the monopolist. (ii) Determine the specic quantity and overall price for each contract, 1.8., ((391,731) and (QBJB)- Suppose now that the monopoly cannot accurately identify consumer type and so the contracts must be tailored in such a way that the consumers of each type have the incentive to self select a given contract. (i) Write down the wlingnesstopay and self-selection con- straints for the monopolist's prot-maximisation problem and highlight which constraints are slack and which constraints bind. (ii) Graphically or otherwise determine the revised contracts (qA,rA) and (gang) for this second-degree price discrimination problem that maximise the monopolist's prot. How would your answer to part (b) change if the number of type B consumers was ve times the number of type A consumers? Explain why the answer is different to the original one in (b) where the populations are of equal size