# Numan Morgan Utility Function | Consumer | Microeconomics

Consumer has different choices for clothes, toothpastes, soaps, biscuits, air-tickets, bonds and debentures in their day today life. Sometime such choices are made under uncertainty. Microeconomics is interested in such kind of choice behaviors of individual. For understanding the theorem, we are using the example of lotteries. This is because the choice of lotteries is made under uncertainly. Consumers do not know whether such choices will give a prize or not.

Lotteries:

State lotteries can be evaluated either of two perspectives familiar to economists as a consumer commodity or as a source of public revenue. As a commodity, the lottery is notable for its broad market penetration and rapid growth. Alternatively, the lotteries choice is imagined because it tells us how the consumer makes the choice. The individual selects the lottery to win the price. The choice of lottery leads to wining the price or not winning the price.

In simple method, a lottery is denoted as:

The above notation shows that a consumer receives a price x with probability p and price y with probability 1-p. The above notation is simply a probability function with a chance of winning and not winning lottery. The price for wining any lottery differs. Wining of lottery may give further lottery ticket, goods or money. In the modern world, the prices for winning lottery are offered in the form of money only. Modern microeconomics studies the risky behaviors of consumers under different circumstances.

There are number of risk behaviors such as drinking alcohol, smoking, drinking and driving. Such behaviors do not always win a price. The cost of accidents or injury is much higher for the risk adverse consumer. Now, we put the consumer behavior in terms of lotteries. The consumer behavior for lottery is put under risk and into number of frameworks.

There are number of assumptions made to understand the consumer’s perception for lottery:

Above notation shows that getting a price with probability one is same with getting the price for certainty.

It means that consumer does not care about the order in which the lottery is described.

This notation describes the perception of a lottery depends on the net probabilities of receiving the various prices.

Above notations are different from each other. The first and second assumptions are harmless and do yield satisfactory result. Assumption 3 called as reduction of compound lotteries. The consumer treats compound lotteries different than one short lottery. The third assumption is not important to understand lottery framework.

In lottery framework, we can define that ℓ is a space of lotteries available to consumer. Consumer prefers the lottery space depend on choices. There are number of lotteries are available to consumer at different points and places. Suppose consumer is offered two lotteries at particular place then he will choose best one in it. From the previous discussion, we assume that the revealed preference and choice rule.

It is assumed that they are complete, reflexive and transitive. The consumer can choose one lottery but the prices are not restrictive. Consumer may get further lottery, car and money as a prize. Suppose we assume that there are three prices as x, y, z. The probability of getting each price is one third.

The reduction of compound lotteries, we get the lottery which is equivalent to the lottery as follows:

According to third assumption, consumer only cares about the net probabilities involved in it. This is equivalent to the notation of original lottery.

Utility Expected:

There exists a continuous utility function. We denote it is as u. Such utility function describes the consumer’s preferences.

It is explained as follows:

Above utility function is not unique. Any monotonic transformation would do as well. Under assumptions, we can find monotonic transformation of the utility function. It has a very convenient property.

The expected utility is as follows:

The expected utility explains that the utility of a lottery is the expectation of the utility from its prizes. The utility of any lottery can be computed as each outcome and summing over the outcome. From the above equation we can say that probability of winning a lottery from x or y adds the utility to consumer. The utility is a linear over probability and it is separable over the outcomes. The utility function and existence of utility function is not an issue. A well behaved preference ordering can be represented by a utility function. But we can still prove it.

We require additional axioms and they are as follows:

Above equation is closed set for all x, y and z in ⎯ ℓ

Assumption 1 explains the continuity which is innocuous. Assumption 2 explains that lotteries with indifferent prices are indifferent. From the above equation (55), suppose the lottery is given as p0x⊕ (1-p) 0 z and we know that x ∼ y, then we substitute y for x to construct a lottery and it is p0y⊕(1-p)0z. The consumer regards as being equivalent to the original lottery. It is only possible assumption.

For technical things, we can make further assumptions in the following paragraphs:

For our understanding, we assume that there is worst lottery w and best lottery b. the consumer never wins the prize for worst lottery. For any x in space of lottery (⎯ ℓ), ≥ x ≥ w

A lottery p0b ⊕ (1-p) 0w is preferred to q0b ⊕ (1-q) 0 w if and only if p > q. It means probability of winning price for best lottery is higher. Consumer always thinks positive and prefer best lottery. Consumer has either knowledge of it or he is asking to seller, friend and relative. Consumer’s choice is bounded to get the best price for best lottery.

Above 3 assumptions are purely for convenience. Assumption 4 can be derived from other axioms. A consumer prefers a best lottery from worst lottery. Sometimes they use their experience and knowledge for it. Best lottery has high probability to get best price to consumer. The expected prize for best lottery is high and prize is announced before consumer chooses the lottery. Similarly, consumer choose best lottery because probability of getting the best prize from the best lottery is comparatively higher.

Expected Utility Theorem:

If (⎯ ℓ, ≥) satisfy the above axioms then there is a utility function u is defined on ℓ. It satisfies the expected utility property

The proof of above theorem is given as follows:

Define u (b) = 1 and u (w) = 0. It means that the best lottery gives utility one and the worst lottery gives the utility zero. Now we need to find the utility of an arbitrary lottery z. The utility of set u (z) = Pz where Pz is defined by

The consumer is indifferent between z and a gamble between the best and the worst outcome of lottery. It gives probability Pz of the best outcome. Sometimes it is defined as the price for the z lottery. It is normally observed for all lottery frameworks. Every lottery is subjected to declared prize.

We have two things which are important for price of z lottery.

They are explained as follows:

1. Pz is existing in each form of selected lottery. There are two sets {p in [0, 1]: p 0 b ⊕ (1- p) 0 w ≥ z} and {p in [0, 1]: z ≥ p 0 b ⊕ (1-p) 0 w} are closed and non-empty. Every point in [0, 1] is in one or the other of the two sets. It is probability to win a prize for best lottery and not winning a prize. Since the unit interval is connected, there must be some p in both. It is because each probability will give the prize or not give prize. But every lottery has the desired outcome therefore it is written as Pz.

2. Pz is unique in the lottery framework. If we assume that Pz and P’z are two distinct numbers then each number satisfies lotteries in the framework. The fourth assumption is that the lottery gives a bigger probability of getting the best price. But at the same time, other lottery cannot be indifferent to one which gives a smaller probability. Therefore Pz is unique and utility is well defined. Consumer cannot define the difference of probability between the two lotteries.

There is need to check the utility (u) from the lottery to each consumer. Each lottery has the expected utility property. It follows from the some simple substitutions.

From the above equations, one uses utility framework and the definition of Px and Py. Substitution 2 uses lottery framework 3, it explains only the net probability of obtaining best lottery or worst lottery. It is matter to us because substitution 3 uses the construction of the utility function.

It follows from the construction of the utility function that:

Again we are using the simple probability between the x and y lottery. Now we need to verify that u is utility function for consumer. Suppose that probability of wing the lottery x > y then the following equations are possible.

From the axiom U 4 it must give us the u (x) > u (y). It means that utility from x is greater than the utility of y.

Uniqueness of the Expected Utility Function:

Now we have understood that the expected utility function is u: ℓ R. Any monotonic transformation of u will also be a utility function. It describes the consumer’s choice behavior.

Suppose u (.) is an expected utility function then V (.) = au (.) + c where a > 0. It means any final transformation of an expected utility function is also an expected utility function. It is linear function of expected utility function.

It is clearer when we transform the entire utility function in terms of linear expected utility function.

Above equation is much similar with the above equation. We have added the vector in the above function.

From the above equation, it is not much hard to see the converse that any monotonic transform of u that has the expected utility property. It must be final transformation and it is stated another way.

Uniqueness of Expected Utility Function:

An expected utility function is unique up to the line of transformation.

The proof of such transformation is given as follows:

In this explanation, the monotonic transformation preserves the expected utility property. But the condition is that it must be an affine transformation. Suppose we assume that f: R’ = R which is a monotonic transform of u. Such monotonic transformation has the expected utility property.

Then:

Or

We have understood that above two equations are same and they are equivalent to the definition of an affine transformation. 