Utility (Micro Economics)
Micro Economics
- Chapter 4. Utility
Theme: The Consumer theory and Utility functions in economics. This time we reviews utility functions as an alternative way to represent consumer preferences that is easier to work with. Several common types of utility functions are discussed along with their properties, indifference curves, and marginal utilities. The analysis shows how marginal rate of substitution can be derived from utility functions. Action items involve practicing calculations with different utility function forms to gain economic intuition and technical proficiency.
Overview
1. Review of Consumer Theory Foundations
2. Utility Functions to Represent Preferences
Utility functions assign real numbers to bundles enabling an alternative way to order them. Key properties are outlined for utility functions to represent particular preference relations. Continuous preferences can be represented by continuous utility functions per an important theorem.
3. Indifference Curves and Maps from Utility Functions
4. Properties of Different Utility Function Forms
Several common functional forms are analyzed including: perfect substitutes, perfect complements, quasi-linear, and Cobb-Douglas. Key characteristics are discussed along with resulting indifference curve shapes.
5. Marginal Utility and Marginal Rate of Substitution
Review of previous page:
We started consumer theory with the budget constraint. And as the first blocks of our consumer theory, we introduced these alternatives for the consumer. So they were basically consumption bundles. They are the vectors of commodities. We have infinitely possible consumption bundles for the consumer, but given the constraints such as the budget constraint, well, not all of those consumption bundles were affordable. And given the prices and the income to spend on some particular commodities, consumers are basically restricted in terms of what they can choose due to their constraints. So basically, these budget constraint, and then hence budget set, we were able to find all those bundles that are affordable to the consumer.
Last time we started talking about preferences, and that is basically ordering of all consumption bundles. Okay, so that is a preference. And given that there are infinitely many bundles, well, it's natural to think that there are infinitely many different preference orderings of those consumption bundles. In order to build that part, we basically use what is known as binary relations.
Completeness: For any two bundles x and y it is always possible to state
x ≻" y
Reflexivity: Any bundle x is always at least as preferred as itself; i.e.
x ≻" x.
Transitivity: For any x, y, and z; If
x is at least as preferred as y, and
y is at least as preferred as z, then
x is at least as preferred as z; i.e.
= x ≻" y and y ≻" z → x ≻" z.
So always relate to consumption bundles and we can basically say whether we like one bundle weekly over the other strictly or we are indifferent between them by doing that we are able to build our preference ordering basically from the more preferred bundles to least allowing also in differences to exist.
But, some issues happen with these preference ordering, there issue will be solved mainly by the utility function so these utility functions will be a way to represent preferences and they're gonna be very convenient to work with comparing to all these preference ordinates of infinite domain consumption bundles.
Ex)
(x, x') → 100
(y, y') → 7
(z, z') → (?) < 7
: Utility = ordinal concept
A utility function is a way of assigning a number to every possible consumption bundle such that more-preferred bundles get assigned larger numbers than less-preferred bundles. That is, a bundle (x1, x2) is preferred to a bundle (y1,y2) if and only if the utility of (x1,x2) is larger than the utility of (y1, y2): in symbols, (x1, x2) ≻ (y1, y2) if and only if u(x1, x2) > u(y1, y2).
The only property of a utility assignment that is important is how it orders the bundles of goods. The magnitude of the utility function is only important insofar as it ranks the different consumption bundles; the size of the utility difference between any two consumption bundles doesn’t matter. Because of this emphasis on ordering bundles of goods, this kind of utility is referred to as ordinal utility.
4.1 Cardinal Utility
There are some theories of utility that attach a significance to the magnitude of utility. These are known as cardinal utility theories. In a theory of cardinal utility, the size of the utility difference between two bundles of goods is supposed to have some sort of significance.
A preference relation that is complete, reflexive, transitive and continuous can be represented by a continuous utility function. Continuity means that small changes to a consumption bundle cause only small changes to the preference level.
A utility function U(x) represents a preference relation.
e.g. if U(A)=6, and U(B)=2,
then bundle x is strictly preferred to bundle y
= then U(A) is more strictly preferred than U(B).
***BUT NOT, than 3 Times.
(But x is not preferred three times as much as is y)
4.2 Constructing a Utility Function
So let me remind you that these preferences are orderings of all consumption bundles. Now these utility functions are another way to represent them by simply taking these vectors of these consumption bundles and turn them into some real number.
Now, any difference curve, which we have seen last time, it contains equally preferred bundles, the bundles that are all equally preferred by the consumers, meaning that, all of those bundles that are on the same indifference curve will have exactly the same utility level.
preference ordering ex)
e.g. U(2,3) = 6 > U(4,1) = U(2,2) = 4.
Call these numbers (= 6 or 4) as utility levels.
and, U(2,3)&U(4,1)&U(2,2) as Utility function.
Suppose that we are given an indifference map as in Figure 4.2. We know that a utility function is a way to label the indifference curves such that higher indifference curves get larger numbers. How can we do this?
One easy way is to draw the diagonal line illustrated and label each "indifference curve" with its distance from the origin measured along the line.
How do we know that this is a utility function? It is not hard to see that if preferences are monotonic then the line through the origin must intersect every indifference curve exactly once. Thus every bundle is getting a label, and those bundles on higher indifference curves are getting larger labels and that’s all it takes to be a utility function.
 |
| Figure 4.2 |
4.2++ Utility Functions & Indiff. Curves
Well, if we have this "three-dimensional plot" of consumption and utility levels, so here we have this, in a sense, indifference curve diagram with these quantities or commodities of 1 and 2, and additionally we have this third dimension that captures utility, well, it's going to look like this, and in fact it will look a little bit better as we introduce these indifference curves on them.
Comparing more bundles will create a larger collection of all indifference curves and a better description of the consumer’s preferences.
And, comparing all possible consumption bundles gives the complete collection of the consumer’s indifference curves, each with its assigned utility level.
This complete collection of indifference curves completely represents the consumer’s preferences.
The collection of all indifference curves for a given preference relation is an indifference map.
An indifference map is equivalent to a utility function in terms of representing preferences.
4.3 Some Examples of Utility Functions
There is no unique utility function representation of a preference relation.
◆① Suppose U(x1,x2) = x1x2 represents a preference relation.
Consider the bundles (4,1), (2,3) and (2,2).
U(x1,x2) = x1x2, so
U(2,3) = 6 > U(4,1) = U(2,2) = 4;
that is, (2,3) > (4,1) ∼ (2,2).
◆② Define V = U²
Then V(x1,x2) = x1²x2² and
V(2,3) = 36 > V(4,1) = V(2,2) = 16 so again
(2,3) > (4,1) ∼ (2,2).
: V preserves the same order as U and so represents the same preferences(not only for these 3 bundles, but all)
◆③ Define W = 2U + 10.
Then W(x1,x2) = 2x1x2+10 so
W(2,3) = 22 > W(4,1) = W(2,2) = 18. Again,
(2,3) > (4,1) ∼ (2,2).
: W preserves the same order as U and V and so represents the same preferences.
Conclusion:
If, U is a utility function that represents a preference relation "≻~ "and F is a strictly increasing function,
then V = f(U) is also a utility function representing "≻~".
F is also called monotonic transformation.
4.3++ Goods, Bads and Neutrals
◆ A good is a commodity unit which increases utility (gives a more preferred bundle).
◆ A bad is a commodity unit which decreases utility (gives a less preferred bundle).
◆ A neutral is a commodity unit which does not change utility (gives an equally preferred bundle).
Case 1.
Is this utility function constant along the indifference curves? Does it assign a higher label to more-preferred bundles? The answer to both questions is yes, so we have a utility function.
 |
| ex. red pencil and blue pencil example |
Case 3.
Instead of U(x1,x2) = x1x2, consider W(x1,x2) = min{ax1,bx2}.
What do the indifference curves for this “perfect complementarity” utility function look like? (e.g. a = b = 1)
 |
| ex. left shoe–right shoe case |
Case 4.
A utility function of the form U(x1,x2) = f(x1) + x2 is linear in just x2 and is called quasi-linear.
E.g. U(x1,x2) = (2x₁)½ + x₂
Each curve is a vertically shifted copy of the others
Any point of X1', all slopes are same.
Cobb-Douglas Indifference Curves
Any utility function of the form U(x₁,x₂) = x₁ª x₂ᵇ
with a > 0 and b > 0 is called a Cobb- Douglas utility function.
E.g. U(x₁,x₂) = x₁½ x₂½ (a = b = 1/2), V(x₁,x₂) = x₁ x₂³ (a = 1, b = 3)
(but, must have more than 0, meaning a≠0 b≠0)
 |
| ## All curves are hyperbolic, asymptoting to, but never touching any axis. |
Cobb-Douglas indifference curves look just like the nice convex mono- tonic indifference curves that we referred to as “well-behaved indifference curves” in Chapter 3. Cobb-Douglas preferences are the standard exam- ple of indifference curves that look well-behaved, and in fact the formula describing them is about the simplest algebraic expression that generates well-behaved preferences. We’ll find Cobb-Douglas preferences quite useful to present algebraic examples of the economic ideas we’ll study later.
Of course a monotonic transformation of the Cobb-Douglas utility function will represent exactly the same preferences, and it is useful to see a couple of examples of these transformations.
First, if we take the natural log of utility, the product of the terms will become a sum so that we have
v(x₁,x₂) = ln(xª₁xᵈ₂)=a lnx₁ +d lnx₂
The indifference curves for this utility function will look just like the ones for the first Cobb-Douglas function, since the logarithm is a monotonic transformation.
4.4 Marginal Utility
Consider a consumer who is consuming some bundle of goods, (x₁,x₂). How does this consumer’s utility change as we give him or her a little more of good 1? This rate of change is called the marginal utility with respect to good 1. We write it as MU₁ and think of it as being a ratio,
MU₁ = ΔU / Δx₁ = u(x₁ +Δx₁,x₂)−u(x₁,x₂) / Δx₁
that measures the rate of change in utility (ΔU) associated with a small change in the amount of good 1 (Δx1).
+Practical info :
Marginal means “incremental”.
The marginal utility of commodity i is the rate-of-change of total utility as the quantity of commodity i consumed changes; i.e.
E.g.
if U(x₁,x₂) = x₁½x₂² then
MU₁ = ∂U / ∂x₁ = 1/2 (x₁)-½ (x₂)²
E.g.
if U(x₁,x₂) = x₁½x₂² then
MU₂ = ∂U / ∂x₂ = 2 (x₁)½ (x₂)
: MU’s depend on x₁ and x₂.
4.5 Marginal Utility and MRS (: Marginal Rates-of-Substitution)
A utility function u(x₁,x₂) can be used to measure the marginal rate of substitution (MRS) defined in Chapter 3. Recall that the MRS measures the slope of the indifference curve at a given bundle of goods; it can be interpreted as the rate at which a consumer is just willing to substitute a small amount of good 2 for good 1.
This interpretation gives us a simple way to calculate the MRS. Con- sider a change in the consumption of each good, (Δx₁,Δx₂), that keeps utility constant—that is, a change in consumption that moves us along the indifference curve. Then we must have
MU₁ Δx₁ + MU₂ Δx₂ = ΔU = 0
Solving for the slope of the indifference curve we have
MRS = Δx₂ / Δx₁ = - MU₁ / MU₂
+Practical info :
The general equation for an indifference curve is: U(x1,x2) ≡ k, a constant.
dx₂ / dx₁ = - (∂U / ∂x₁) / (∂U / ∂x₂)
(-1) * (U를 x₁으로 미분한 것) / (U를 x₂으로 미분한 것)
This is the MRS.
Example (문제풀이)
:
Suppose U(x1,x2) = x1x2. Then
∂U / ∂x1 = (1)(x2) = x2
∂U / ∂x2 = (x1)(1) = x1
so MRS = dx2 / dx1 = - (∂U / ∂x1) / (∂U/∂x2) = -x1/x2 = - x₂ / x₁
Means, Utility of (1,8) is 8, Utility of (6.6) is 36
Summary
1. A utility function is simply a way to represent or summarize a prefer- ence ordering. The numerical magnitudes of utility levels have no intrinsic meaning.
2. Thus, given any one utility function, any monotonic transformation of it will represent the same preferences.
3. The marginal rate of substitution, MRS, can be calculated from the
utility function via the formula MRS = Δx₂/Δx₁ = −MU₁/MU₂.
[Reference]
[1] Hal R. Varian - Intermediate Microeconomics_ A Modern Approach, 8th Edition -W.W. Norton & Co. (2010)
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