1. Consider the following utility function for consumer i, who consumes two goods, namely x and y : U:(x, y) = 0.2xy + 0.8y2 (a) Make use of the Implicit Function Theorem to calculate the marginal rate of substitu- tion (MRS) for consumer i at the point (x, y) = (1,2) in order to illustrate how much less of good y the consumer would need to consume to compensate for gaining 1 unit of good x while remaining on the same indifference curve. (b) Now use your calculations in (a) above to approximate the change in the level of utility of the consumer from the original location (x, y) = (1, 2) to the point (x, y) = (1,2;) (without using direct substitution). (c) What are the condition(s) for a point to be a regular point on a function? Show that these condition(s) hold in the case of point (1,2) and function U;(x, y). Now show that the gradient vector VU;(x,y) will be perpendicular to the indifference curve of U; at (1, 2). Show all calculations.