Reference no: EM133426052

1. Consider some utility function U(x,y), where x and y are two goods consumed by a consumer, as follows: U(x,y)=0.2xy5 +0.8y12

(a) Make use of the Implicit Function Theorem to:
i. first identify whether y can be defined as a function of x at the point (1, 2); and

ii. then calculate the marginal rate of substitution (MRS) for the consumer at this point 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) to approximate the new level of utility of the consumer at the point (1, 2 21 ) 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) set out in (a) above. Now show that the gradient vector ∇U(x,y) will be perpendicular to the indifference curve of U at (1,2). Show all calculations.

(d) Set up the Hessian matrix of U(x,y) and evaluate it at the point (1,2).