Please solve all parts.

B1 Suppose that a consumer has the utility function u(x1, x2) = 2122, where ci represents the quantity of good i consumed, i = 1, 2. The consumer has exogenous income m = 50 and faces prices p1 = 1 and p2 = 1 for goods 1 and 2, respectively. (a) Formulate the consumer's utility-maximisation problem, clearly indicating the decision vari- ables. What is the Lagrangian for this problem? Denote the Lagrange multiplier as A. Find bowallscens the first-order necessary conditions. Verify that the second-order sufficiency condition is satisfied. w yolabomani bind mol (b) Solve for x1 (P1, p2, m), x2(P1, p2, m), and A* (P1, p2, m). What is the maximised level of utility? By how much does the maximised level of utility increase for an infinitesimal increase in income? Explain your answer using the Lagrange multiplier. bidw is (c) Suppose that in the next year, prices increase to pi = 3 and p2 = 4. If m remains fixed at bru 150, then what are the consumer's new optimal choices and maximised utility level? dash ed(d) Given the new prices, p', = 3 and p2 = 4, how much extra income does the government wishyou need to give to the consumer in order to return them to their initial maximised level of utility from part (b)? ode to (e) What is the maximum amount that this consumer would be willing pay in order to prevent the price increases