Show That I Might Fail to Hold if the Underlying Utility Function Ux1x2 is Not Continuous
Here is an elaborated discussion on the utility function, highlighting:- 1. Meaning of the Utility Function 2. Constructing a Utility Function 3. Some Very Common Utility Functions.
Meaning of the Utility Function:
A utility function is a way of assigning a number to each possible consumption bundle such that larger numbers are assigned to more-preferred bundles than less-preferred ones and the same number is assigned to equally preferred bundles. This simply means that a bundle (x1, x2) is preferred to a bundle (x'1, x'2) if and only if the utility of (x1, x2) exceeds that of (x'1, x'2).
(x1, x2) > (x'1, x'2) iff u(x1, x2) > u(x'1, x'2)
and u(x1, x2) = u(x'1, x'2) iff (x1, x2) ~ (x'1, x'2)
The only restriction on this transformation rule is that when u1 increases u2 must increase.
Ordering the Bundles of Goods:
The sole important property of utility assignment is related to the ordering the bundles of goods. The magnitude of the utility function is important only because it ranks the different consumption bundles; the magnitude of the utility difference between any two consumption bundles is not a decision variable.
Due to the emphasis on ordering of goods, rather than measuring the magnitude of utility, this kind of utility is known as ordinal utility.
In Table 5.1 we illustrate different ways of assigning utilities to three bundles of goods, all of which order the bundles in the same way. In this case the consumer prefers X to Y and Y to Z because X is associated with a higher number than Y, which, in its turn, is associated with a higher number than Z.
So only the ranking of the consumption bundles matters. There is no unique way to assign utilities to bundles of goods, i.e., the utility function is not unique. One way of assigning utility numbers to bundles of goods is through a monotonic transformation.
If, for example, m(x1, x2) represents a way to assign utility numbers to the bundle (x1, x2), then multiplying u(x1, x2) by 2 (or any positive number) is an example of a monotonic transformation. A monotonic transformation is a way of transforming one set of numbers into another in such a way that the order of the numbers is not disturbed.
A monotonic transformation of a function f(u) occurs when each number u is transformed into some other number f(u) such that u1 > u2 which implies that f(u1) > f(u2). Mathematically speaking, there is no fundamental difference between a monotonic transformation and a monotonic function.
The graph of a monotonic function will always have a positive slope, as shown in Fig 5.1 (a). A monotonic function is one that is always increasing. Part (b) illustrates a function that is not monotonic, since it sometimes increases and sometimes decreases.
In short, a monotonic transformation of a utility function is one that represents the same preferences as the original utility function. Since utility is a way to label indifference curves a monotonic transformation is just a relabeling of them.
A related point may also be noted in this context. Since a larger label is put on indifference curves containing more-preferred bundles than those containing less-preferred bundles, the labelling does not alter the preferences. In other words, the labelling will represent the same preferences.
Constructing a Utility Function :
It may be noted at the outset that intransitive preferences cannot be represented by a utility function. An example of this is X > Y > Z > A.
A utility function, is a way to label the indifference curves such that large numbers are assigned to higher indifference curves. This point becomes clear from the indifference map shown in Fig. 5.2. The distance of each indifference curve from the origin is measured along the diagonal line OR drawn through the origin.
The line is a utility function for the simple reason that if preferences are monotonic then the line through the origin must intersect every indifference curves only once. Since every bundle is getting a label, and those bundles on higher indifference curves are getting larger labels, the diagonal line is indeed a utility function, as suggested by Hal Varian.
By drawing a diagonal line from the origin, it is possible to label indifference curves, i.e., to label to each indifference curve to show how far it is from the origin, measured along the diagonal line, so long as preferences are monotonic. Almost any type 'reasonable' preference (consistent choice) can be represented by a utility function.
Some Very Common Utility Functions :
Various types of preference and the associated indifferences curves may be represented by utility functions.
From a chosen utility function such as u(x1, x2) it is fairly easy to draw a set of indifference curves we have just to plot all the points (x1, x2) such that' u(x1, x2) is a constant, i.e., all the points on an indifference curve represent the same level of utility. The set of all (x1, x2) such that u(x1, x2) equals a constant is called a level set. For each different value of the constant such as u1, u2, u3, etc. we get a different indifference curve.
Here;
u3(x1,x2) > u2(x1, x2) > u1(x1, x2)
Various types of indifference curves from different types of utility functions:
We can draw different types of indifference curves from different types of utility functions.
Let us suppose the utility function is of the following form:
u(x1, x2) = x1x2
We have already noted that an indifference curve is just the set of all x1 and x2 such that u = x1x2 for some constant k. If we solve for x2 as a function of x1, the equation of the indifference;
x2 = u/x1,
which is the formula for hyperbola.
This curve is shown in Fig. 5.3 for these values of u.
Example :
Suppose the utility function is of the following type v(x1, x2) = x1 2x2 2. In this case also we get indifference curves of the type shown in Fig. 5.3.
This point is proved thus:
This utility function may be expressed as v(x1x2) = x1 2x2 2 = (x1x2)2 = u(x1, x2)2. In this case the utility function v(x1, x2) is just a square of the original one m(x1x2). Since u(x1, x2) is always non-negative, v(x1, x2) is a monotonic transformation of the original function.
So this utility function v(x1, x2) = x1 2x2 2 will have exactly the same shape as those shown in Fig. 5.3. It is because v(x1, x2) describes exactly the same preferences as u(x1, x2) since it orders all of the bundles in the same way. Hence all indifference curves look alike. Only the labeling of the curves will be different.
The labels that were 1, 2, 3 will now be 1, 4, 9. But the set of bundles that has v(x1, x2) = 9 is exactly the same as the set having u(x1, x2) = 3. Thus v(x1, x2) describes the same preferences as u(x1, x2).
Finding Utility Function from Indifference Curves :
It is also possible to move in the opposite direction — to find a utility function that represents some indifference curves. This is done by following a simple intuitive approach. Given a description of the preferences, we have to think about what the consumer is trying to maximise, i.e., what combination of the goods describes our representative consumer's choice behaviour. A few examples will make the point clear.
Perfect Substitutes :
If apples and oranges are perfect substitutes the combination of the two is not important. What is important is their total number. Therefore we pick up the utility function u(x1, x2) = x1 + x2. Alternatively we could take the square of the number of apples (or, oranges) and use the utility function v(x1, x2) = (x1 + x2)2 = x1 2 + 2x1x2 + x2 2 because it is also a monotonic transformation of u(x1, x2).
If 3 units of x1 is as valuable as one unit of x2, the MRS will be —1/3. It is because the consumer will require three units of x2 to compensate him for giving up one unit of x1. Since the desired rate of commodity substitution is different from one-to-one, the utility function takes the form w(x1, x2) = 3x1 + x2. In this case since x2 is thrice as valuable to the consumer as x1, this utility function yields indifference curves with a slope of -1/3.
The general form of the utility function representing preferences for perfect substitutes is;
u(x1, x2) = k1x1 + k2x2
where the two positive numbers (k1, k2 > 0) measure the 'value' of x1 and x2 to the consumer. In this case the slope of a typical indifference curves is – k1/k2.
Perfect Complements:
In case of perfect complements such as left shoe and right shoe, since the consumer cares about the number of pairs of shoes in his possession, he (she) has to decide how many pairs of shoes to choose. This choice forms his utility function. The number of complete pairs of shoes possessed by the consumer is the minimum number of right shoes, x1 and the number of a left shoes x2 in his stock.
Thus the utility function for perfect complements is expressed as u(x1, x2) = min (x1, x2).
Suppose a consumer is currently choosing a bundle such as (5, 5). If he acquires an extra unit of x1, he gets (6, 5) which will keep him on the same indifference curve. Since the 6th unit of x1 is useless without the 6th unit of x2, min {5, 5} = min {6, 5} = 5. Any monotonic transformation would be quiet appropriate in case of this type of utility function.
Even if the consumer consumes the two goods in different proportions, not necessarily one-to-one, the utility function will be the same. Let us suppose 2 biscuits (x1) are consumed with every cup of tea (x2). Then the optimal (correct) consumption bundle will be min {x1, 1/2x2}.
Since any monotonic transformation of this utility function will describe the same preferences, we can express the utility function as u(x1, x2) = min{2x1, x2) by multiplying x1 and x2 by 2 to be able to express this as integer numbers rather than as fractions.
The general form of the utility function in case of perfect complements is:
u(x1, x2) = min {k1x1, k2x2}
where k1 and k2 are positive numbers indicating the proportions in which x1 and x2 are consumed, i.e., k1 = p1x1/m, k2 = p2x2/m and k1 + k2= 1.
Quasi-Linear Preferences:
In case of quasi-linear preference a consumers indifference curves are vertically shifted replica of one another, as shown in Fig. 5.4. In this case all the indifference curves are just vertically shifted versions of the origin indifference curve (the first one). The equation of an indifference curve in this case is x2 = a – v (x1) where a is a constant.
The value of a is different for different indifference curves. The height of each indifference curve is some function of x1 – v(x1) plus a constant a. Here, a3 > a2 > a1 implying that higher values of a give higher indifference curves. The utility function representing such preferences is u(x1, x2) = a = v(x1) + x2.
This is obtained by solving the original equation for a and setting it equal to u.
In this case since the utility function is linear in x2, but (possibly) non-linear in x1 it is known as quasi-linear (partly linear) utility.
It is very easy to work with quasi-linear utility function. But these are very abstract, not at all a reflection of reality.
Cobb-Douglas Preferences :
A commonly used utility function is the Cobb-Douglas utility function.
u(x1, x2) = x1 b1x2 b2
where b1 and b2 are positive numbers describing the preferences of the consumer. Different values of the parameters b1 and b2 lead to different shapes of the indifference curves. Two examples of such indifference curves are given in Fig. 5.5.
Part (a) shows the case where b1 = 1/2 and b2 = 1/2 and part (b) shows the case where b1 = 1/5 and b2 = 4/5.
Cobb-Douglas preferences are of interest to us because they generate well-behaved preferences (indifferent curves). These are examples of nice convex monotonic indifference curves.
Monotonic Transformation :
Cobb-Douglas preferences are treated as standard examples of well-behaved indifference curves because a monotonic transformation of the Cobb-Douglas utility function will represent exactly the same preferences.
1. Since logarithm is a monotonic transformation we have, by taking the natural log of utilities,
v(x1, x2) = 1n (x1 b1x2 b2) = b11nx1 + b21nx2.
2. If we raise utility to the 1/(b1 + b2) power,
we have b1/x1 b1+b2/b2/x2 b1+b2
Let us now define a new number α = b1/b1 + b2
We can now express the utility function as v(x1, x2) – x1 α x2 β
This simply means that we can always take a monotonic transformation of the Cobb-Douglas utility function that makes the sum of the exponents exactly add up to 1.
Two Predictions of the Cobb-Douglas Utility Function:
1. In case of Cobb-Douglas utility function the expenditure on each commodity is a constant fraction of wealth irrespective of prices. The Cobb-Douglas utility function in case of two commodities is given by u(x1, x2) = Ax1 αx2 β for 0 < α, β < 1 and A > 0. It is increasing for all (x1, x2) > 0 and this is homogeneous of degree one because it is a logical deduction of the Cobb-Douglas production function. Here we use the increasing transformation of a log x1 + β log x2, a strictly concave function, as our utility function.
With this specification the constrained utility maximisation problem can be expressed as:
Max. U = α log x1 + β log x2 ……..(1)
subject to p1x1 + p2x2 = m
Here U is increasing. So the budget constraint will hold with strict equality at any solution.
Since, log 0 = – α, the optimal choice of x1 and x2 is strictly positive and must satisfy the first order conditions. Writing the consumption levels simply as x1 and x2, we get
Thus with the Cobb-Douglas utility function, the expenditure on each commodity is a constant fraction of wealth for all prices. A share a goes to x1 and a share β goes to x2.
2. For Cobb-Douglas utility function budget shares of all goods are independent of prices and income.
To prove the given statement we shall first derive the optimum quantity of two goods x1 and x2 in terms of p1, p2 and m. Then we shall put the value of x and y in the expression for budget share.
Source: https://www.microeconomicsnotes.com/utility-function/utility-function-meaning-and-construction-microeconomics/13639
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