Question

1. (18 points) For each of the following situations, i) Find the Marginal Rate of Substitution at the given bundle, and ii) use a graph to indicate the given bundle, and accurately draw the in that goes through that bundle. Be sure to label you graph carefully and accurately. In all cases put the amount of good X on the horizontal axis, and the amount of good Y on the vertical axis. a) The consumers utility function is given by U(XY)-X12*Y12, and the given bundle is X = 16 and Y- 4 MRS ii) Draw your graph in this space:

Answer 1

U (X, Y) = X^1/2 middot Y^1/2 (i) MRS = MU_X/ MU_Y: MU_x = partial differential V/ partial differential X = 1/2 X^1/2 - 1 Y^1/2 = 1/2 X^-1/2 Y^1/2 MU_Y = partial differential U/ partial differential Y = x^1/2 (1/2) Y^1/2 - 1 = 1/2 X^1/2 Y^-1/2 So, MRS = (1/2) X^-1/2 Y^-1/2/ (1/2) X^1/2 Y^-1/2 = Y^1/2 + 1/2/ X^1/2 + 1/2 = Y/X At X = 16 and Y = 4: MRS = 4/16 = 1/4 = 0.25 implies MRS = 1/4 = 0.25 (IC is convex to the origin) U(X, Y) = Min (X, 4Y) (i) MRS = 0 or infinity [for complementary goods, MRS is either 0 or infinity] (X = 5 and Y = 1) (IC Is L shaped) U(X, Y) = 4 X + 2 Y (i) MRS = MU_X/ MU_Y = partial differential U/ partial differential X/ partial differential U/ partial differential Y = 4/52 = 2 implies MRS = 2 (IC is straight line with constant slope = 2)