Question

.Use separate graphs to sketch two indifference curves for people with each of the following utility functions:

U(x, y) = x + 2y.

U(x, y) = min{x,2y}.

What type of preferences are represented by a utility function of the form U(x, y) = square root of x+y? What about the utility function V(x,y) = 13x+13y?

Consider the utility function u(x,y) = ?^2 ?^3. What kind of preferences does it represent? Is the function v(x,y) = ?^4 ?^5 a monotonic transformation of u(x,y)? Calculate the marginal rate of substitution for u(x,y).

Answer 1

1).

Let’s assume the utility function is “U = X+2*Y”, where “X” and “Y” are the two goods. The MRS of the utility function is “1/2” which is constant, => the individual having perfect substitute type utility function between two goods. The following fig shows the utility function of the individual.

U3

Now, assume the utility function is “U = min(X, 2*Y)”, where “X” and “Y” are the two goods. Now, at the optimum “X=2Y”, => Y/X=1/2, => both goods “Y” and “X” are consumed in a fixed proportion ½, => the individual having perfect complement type utility function between two goods. The following fig shows the utility function of the individual.

2).

If the utility functions of the individual is “U = (X+Y)^0.5”, which is the positive monotonic transformation of the utility function “X+Y”, which is the perfect substitute type, => individual want to substitute “X” for “Y” in a constant rate “1”. Similarly, the utility functions of the individual is “V = 13*X+13*Y = 13*(X+Y)”, which is also the positive monotonic transformation of the utility function “X+Y”, which is the perfect substitute type, => individual want to substitute “X” for “Y” in a constant rate “1”. So, in both the cases the utility function is perfect substitute.