Answer to Question #240301 in Microeconomics for Sam

Question

a) Define what is meant by a monotonic transformation of some consumer's utility function: u (21,22). b) Suppose u (01,02) = 3x1 + 2x2. Sketch indifference curves for this consumer corresponding to utility levels 10, 20 and 30. c) Suppose u (C1, 12) = 11 + 12. Sketch indifference curves for this consumers corresponding to utility levels 10, 20 and 30. d) Can these two utility functions be said to describe the same preferences: Is the function in b) a monotonic transformation of the function in c). Explain.

Answer 1

a) Montonic Transformation: Monotonic transformation is the way to transform a utility function into another utility function such that the marginal utility is preserved.

If the consumer's utility function is U(x₁,x₂),

we represent a monotonic transformation by V= f[U(x₁,x₂)].

Now to have a monotonic transformation V > U . It can Be U² or U³, or U+10 to preserve the order, V must be a strictly increasing function of U.

b) u(x₁, x₂) = 3x₁ + 2x₂

IC1 = is for Utility level 10

IC2 = is for Utility level 20

IC3 = is for Utility level 30

c) u(x₁, x₂) = x₁ + x₂

IC1 = is for Utility level 10

IC2 = is for Utility level 20

IC3 = is for Utility level 30

d) From the indifference curves above, it is seen that the two utility functions u(x₁, x₂) = 3x₁ + 2x₂ and u(x₁, x₂) = x₁ + x₂ have the same kind of preferences. For reference, see the below table:

for every bundle of values (1,1),(1,2) etc...the preference has same kind of arithmetic progression.

So it can be said that the function u(x₁, x₂) = 3x₁ + 2x₂ is a monotonic transformation of function u(x₁, x₂) = x₁ + x₂