A consumer wants to maximise his utility function: U(I. y) = 1 subject to a budget constraint; M = pxx + 9in 1+3! where M, xand y are the consumer income. the respective two goods the consumer wishes to consume. a) Find out the demand function for good X (the expression of the utility-maximising quantity of X that this person will wish to buy) as a function of the parameters Pg, Py. and M. (20 marks) b) Suppose that the budget constraint is given by x 90 = 4y. write out the Lagrange function for the constrained maximisation problem and derive the rst-order conditions for the utility function maxim isation. (20 marks) c) Find the optimal values of Lagrange multiplier, x and ythat maximise the utility function. Also. calculate maximal (optimal) value of the utility function. (20 marks) d) Show that at the optimal value of the utility function. the ratio of the marginal utility of the two goods equals the ratio of output prices. (20 marks} e} Prove that the law of diminishing marginal utility holds for each of the two goods. (20 marks)