**Money in the utility function (MIUF)**

Money in the utility function (MIUF) presents the axiomatic basis for including money in the utility function. Individuals differ in their tastes or preferences over goods and in their income or wealth. An additional axiom is sometimes added for analytical convenience. This is that the individual never reaches satiation for any good. That is, he continues to prefer more of each good to less of it. In view of the definition of goods, this axiom implies that a good never ceases to be a good for the individual, no matter how much or how little he possesses of it.

**Axioms Of Utility Theory**

Microeconomic theory defines the “rational” individual as one whose preferences are consistent and transitive. The definitions of these terms are specified by the following axioms of utility theory:

**Axiom 1: Consistent preferences** : If the individual prefers a bundle of goods A to another bundle B, then he will always choose A over B.

**Axiom 2: Transitive preferences** : If the individual prefers A to B and B to a third bundle of goods C, then he prefers A to C.

To these two axioms in the theory of the demand for commodities, monetary theory usually adds the following one:

**Axiom 3: Real balances as a good**

In the case of financial goods that are not “used directly in consumption or production” but are held for exchange for other goods in the present or the future, the individual is concerned with the former’s exchange value into commodities. That is, their real purchasing power over commodities and not with their nominal quantity. For example, 100 bank notes each with a face value of $1 have a nominal quantity or value of $100. Assume that the individual wishes to hold a certain amount of real purchasing power in money balances. And this demand of his equals $100 at a certain set of prices. If prices of commodities were to double, the individual would no longer demand $100 but $200 of money balances in order to keep his demand constant in terms of real purchasing

power.

**The Axioms Of Consistency And Transitivity**

The axioms of consistency and transitivity ensure that the individual’s preferences among goods can be ordered monotonically and represented by a utility or preference function. Axiom 3 ensures that financial assets, when considered as goods in such a utility function, should be measured in terms of their purchasing power and not their nominal quantity. The inclusion of money (and other financial assets) directly into the utility function can be justified on the grounds that the utility function expresses preferences. And that, since more of financial assets is demanded rather than less, they should be included in the utility function just like other goods.

**Utility Function Example**

Given these axioms, let the individual’s period utility function be specified as: U(.) = U(x1, . . ., xK,n,mh)

where:

*xk =* quantity of the kth commodity,

*k =* 1, …, K

*n =* labor supplied, in hours

*mh =* average amount of real balances held by the individual or household for their liquidity services.

Note that : has K+2 goods, consisting of K commodities, labor and real balances. Axioms 1 to 3 only specify U(.), an ordinal utility function.

*Note: A utility function that gives a consistent and transitive ranking of preferences, without any other characteristics of measurability, is said to be ordinal or unique up to an increasing monotonic transformation. That is, if U(x1,…, xs) is the individual’s utility function, then F[U(x1, …, xs)], where ∂F/∂U is greater than 0, is also an admissible utility function with identical demand functions for xi, i = 1, …, s.*

Uk = ∂U/∂xk is greater than 0 for all k, Un = ∂U/∂n is smaller than 0, Um = ∂U/∂mh is greater than 0. All second-order partial derivatives of U(.) are assumed to be negative. That is, each of the commodities and real balances yield positive marginal utility and hours worked have negative marginal utility.

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