Strictly convex preferences imply that indifference curves are C-shaped. By C-shaped, we really just mean the downward sloping part of the letter C. To determine the shape of an indifference curve, we may analyze the marginal rate of substitution (the slope along an indifference curve) that we calculate, and see what happens as we increase X. But note that to stay along the same indifference curve, we must also decrease Y. If the absolute value of the marginal rate of substitution is diminishing as we increase X and decrease Y, then indeed the indifference curve has the C-shape. If |MRS(X, Y)| is constant as we increase X and decrease Y, then the indifference curve is \-shaped. If |MRS(X, Y)| is increasing as we increase X and decrease Y, then the indifference curve is )-shaped.
For each of the following utility functions, (i) calculate the absolute value of the marginal rate of substitution |MRS(X, Y)| and simplify; (ii) determine whether |MRS(X, Y)| is diminishing, constant, or increasing as we move down an indifference curve (increasing X↑ and decreasing Y↓); and (iii) indicate whether the associated indifference curves are C-shaped, \-shaped, or )–shaped. I will do (a) to show you what to do.
a) U(X, Y) = X3/5Y2/5
b) U(X, Y) = X1/3Y2/3 .
c) U(X, Y) = [X − 5][Y – 10] for (X, Y) such that X > 5 and Y > 10 d)U(X,Y)=X2 +Y
e) U(X, Y) = X1/2 + Y1/2
f)U(X,Y)=2X+ Y.