Rachel’s preferences for goods x and y are represented by the following utility function: U(X,Y ) = X1/2.Y 1/2.

(a) Find the marginal rate of substitution. Comment if it is increasing, constant or decreasing in x.

(b) Graph the indi?erence curves corresponding to U = 2 and to U = 4.

(c) Assume that Y = 1. Graph U(X,1) (X in the x-axis and U(.) in the y-axis).

(d) Assume that Y = 1. Graph MUx(X,1) (X in the x-axis and MUx(.) in the y-axis)

(e) Now assume that the price of good x is px = 4 and the price of good y is py = 1. We also know that Rachel’s income is I = 24.

(e.i) Write the constrained optimization problem that Rachel faces.

(e.ii) Find the optimal bundle (X∗,Y∗).

(e.iii) Find Rachel’s utility at the optimal point, and ?nd another bundle that gives her the same level of satisfaction.

(e.iv) Draw a graph that includes the following curves: a) Rachel’s budget constraint (dont forget to label the axis and the intercept, and to indicate the slope of the budget line). b) Rachel’s optimal bundle. c) The indi?erence curve associated with Rachel’s optimal bundle.