In this article we will discuss about the change in utility function of consumer through monotonic transformation.
The indifference curve theory is based on the ordinal measurement of utility. That is why the numbers the utility function assigns to the alternative commodity combinations do not have any cardinal significance, they only have ordinal significance, i.e., they only indicate whether the utility level derived from a particular combination of the goods is higher or lower than that obtained from another combination according as the number assigned to the former is higher or lower than the latter.
But the difference between any two ordinal utility numbers is meaningless, for they cannot tell how high is the utility level in one case than that in the other.
Now, if a particular set of utility numbers associated with various combinations of Q1 and Q2 represents a utility function with all its preferences and indifferences, then any positive monotonic transformation of it is also a utility function with the same preferences and indifferences between the combinations.
A function F (U) is a positive monotonic transformation of U = f(q1, q2) if F(U1) > F(U0) whenever U, > U0. For example, the transformations W = aU + b, (a > 0), and W = U2 are positive monotonic, provided all utility numbers are non-negative.
Assume that the utility function, to begin with, is U = f(q1, q2). Now form a new utility function W = F(U) = F[f(q1, q2)] by applying a positive monotonic transformation to the original utility function. By definition, the function F(U) is an increasing function of U. It can be shown that maximising W subject to the budget constraint is equivalent to maximising U subject to the budget constraint.
Imagine that (q01, q02) is the combination that uniquely maximises U = f(q1, q2) subject to the budget constraint. Let (q(1)1, q(1)2) be any other combination which also satisfies the budget constraint. Then by the theory, U0 = f(q01,q02) >U1 = f(q(1)1, q(1)2).
But by the definition of monotonicity, U0 > U, => F(U0) > F(U1) => W(q01, q02) > W(q(1)1, q(1)2), which proves that the utility function W(q1, q2) is also maximised at the combination (q01, q02).
What is obtained here is that if W = F(U) is a utility function of the consumer which is a positive monotonic transformation of U = f(q1, q2), then W is maximum at the same point or combination, here (q01, q02), on the budget line as that at which U is maximum.
This gives the preference-indifference rankings or the indifference map of the consumer would be the same, and therefore, the constrained equilibrium point and the demand functions for the goods would be the same for any two utility functions of which one is a positive monotonic transformation of the other.
Note, however, that if there is a change in the utility function of the consumer through a positive monotonic transformation, then the whole set of utility numbers for the different commodity combinations would change.
For example, the utility number at each point on an IC may change from, say, 3 to 300. But, the utility numbers in the indifference curve theory have no cardinal significance. What matters here is the indifference-preference rankings of the consumer.