Ivy lives in a world where there are only two goods, X and Y.Her utility function per period is given by:

U = 30x - 0.5x²+ 40y - y²

(a)Calculate Ivy's utility-maximizing amount of X and Y.

(b)Redo part (a) if the price of X is $10 and the price of Y is $5 and Ivy's income is $670.

(c)Show that at the point of utility-maximization in (b) the ratio of marginal utility of the two goods equals their price ratio.

(d)Provide a one-sentence interpretation for the value of the Lagrangian multiplier obtained in (b).

(e)At what income level is Ivy's utility of money maximized?

(f)Using the answers from earlier parts, calculate Ivy's income elasticity of demand for good X and Y.