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Part 2. PREFERENCES AND UTILITY. Objectives of the chapter. Study a way to represent consumer’s preferences about bundles of goods What are bundles of goods? =combinations of goods. For instance: X=slices of pizza Y=glasses of juice Bundles: P: X=1, Y=1 Q: X=3, Y=0 R: Y=3, X=0

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### Part 2

PREFERENCES AND UTILITY

Objectives of the chapter

- Study a way to represent consumer’s preferences about bundles of goods
- What are bundles of goods? =combinations of goods. For instance:
- X=slices of pizza
- Y=glasses of juice

- Bundles:
- P: X=1, Y=1
- Q: X=3, Y=0
- R: Y=3, X=0
- S: X=2, Y=1

Objectives of the chapter

- John’s preferences are such that:
- P is preferred to both Q and R
- S is preferred to P

- This way of representing preferences would be very messy if we have many bundles
- In this chapter we study a simple way of representing preferences over bundles of goods
- This is useful because in reality there are many bundles of goods

Axioms of rational choice

- Before describing this simple method to represent preferences over bundles, we will study what requirements must the preferences satisfy in order for the method to work
- These requirements are the axioms of rational choice
- Without these requirements, it would be very difficult to come up with a simple method to represent preferences over many bundles of goods
- It is easy to read a tube map, but not so much to read a tube-bus-and rail map !!!!

Axioms of Rational Choice

- Completeness
- if A and B are any two bundles, an individual can always specify exactly one of these possibilities:
- A is preferred to B
- B is preferred to A
- A and B are equally attractive

- In other words, preferences must exist in order to be able to describe them through a simple method

- if A and B are any two bundles, an individual can always specify exactly one of these possibilities:

Axioms of Rational Choice

- Transitivity
- if A is preferred to B, and B is preferred to C, then A is preferred to C
- assumes that the individual’s choices are internally consistent
- If transitivity does not hold, we would need a very complicated method to describe preferences over many bundles of goods

Axioms of Rational Choice

- Continuity
- if A is preferred to B, then bundles suitably “close to” A must also be preferred to B
- If this does not hold, we would need a very complicated method to describe individual’s preferences

Utility

- Given these assumptions, it is possible to show that people are able to rank all possible bundles from least desirable to most
- Economists call this ranking utility
- if A is preferred to B, then the utility assigned to A exceeds the utility assigned to B
U(A) > U(B)

- if A is preferred to B, then the utility assigned to A exceeds the utility assigned to B

Utility

- Game…
- Someone state the preferences using numbers from 1 to 10
- Can someone use different numbers from 1 to 10 but state the same ordering?
- Can someone use numbers 1 to 100 and state the same preferences?

Utility

- Game…
- Clearly, the numbers are arbitrary
- The only consistent thing is the ranking that we obtain

Utility

- Utility could be represented by a Table

U(P)=1>U(Q)=0 because we said that P was preferred to Q

U(B)=U(C) because Q and R are equally preferred

Utility

- Notice that several tables of utility can represent the same ranking

- We can think that the rankings are real. They are in anyone’s mind. However, utility numbers are an economist’s invention
- The difference (2-1, 4-1…) in the utility numbers is meaningless. The only important thing about the numbers is that they can be used to represent rankings (orderings)

Utility

- Utility rankings are ordinal in nature
- they record the relative desirability of commodity bundles

- Because utility measures are not unique, it makes no sense to consider how much more utility is gained from A than from B. This gain in utility will depend on the scale which is arbitrary
- It is also impossible to compare utilities between people. They might be using different scales….

Utility

- If we have many bundles of goods, a Table is not a convenient way to represent an ordering. The table would have to be too long.
- Economist prefer to use a mathematical function to assign numbers to consumption bundles
- This is called a utility function
utility = U(X,Y)

- Check that the previous example of the three columns table is obtained with the following utility functions:
- U=X*Y,
- and U=(X*Y)2

Utility

- Clearly, for an economist it is the same to use U=X*Y than to use U=(X*Y)2 because both represent the same ranking (see the table), so both functions will give us the same answer in terms of which bundles of good are preferred to others
- Any transformation that preserves the ordering (multiply by a positive number, take it at a power of a positive number, take “ln”) will give us the same ordering and hence the same answer
- We can use this property to simplify some mathematical computations that we will see in the future

Economic Goods

Preferred to x*, y*

?

?

Worse

than

x*, y*

- In the utility function, the x and y are assumed to be “goods”
- more is preferred to less

Quantity of y

y*

Quantity of x

x*

Indifference Curves

- An indifference curve shows a set of consumption bundles among which the individual is indifferent

Quantity of y

Combinations (x1, y1) and (x2, y2)

provide the same level of utility

y1

y2

U1

Quantity of x

x1

x2

Indifference Curve Map

Increasing utility

U3

U2

U1

- Each point must have an indifference curve through it

Quantity of y

U1 < U2 < U3

Quantity of x

Transitivity

- Can any two of an individual’s indifference curves intersect?

The individual is indifferent between A and C.

The individual is indifferent between B and C.

Transitivity suggests that the individual

should be indifferent between A and B

Quantity of y

But B is preferred to A

because B contains more

x and y than A

C

B

U2

A

U1

Quantity of x

Convexity

- Economist “believe” that:
- “Balanced bundles of goods are preferred to extreme bundles”
- This assumption is formally known as the assumption of convexity of preferences
- Using a graph, shows that if this assumption holds, then the indifference curves cannot be strictly concave, they must be strictly convex

Convexity

- Formally, If the indifference curve is convex, then the combination (x1 + x2)/2, (y1 + y2)/2 will be preferred to either (x1,y1) or (x2,y2)

This means that “well-balanced” bundles are preferred

to bundles that are heavily weighted toward one

Commodity (“extreme bundles”).

The middle points are better than the

Extremes, so the middle is at a higher indifference

Curve.

Quantity of y

y1

(y1 + y2)/2

y2

U1

Quantity of x

x1

(x1 + x2)/2

x2

Marginal Rate of Substitution

- Important concept !!
- MRSYX is the number of units of good Y that a consumer is willing to give up in return for getting one more unit of X in order to keep her utility unchanged
- Let’s do a graph in the whiteboard !!!
- MRSYX is the negative of the slope of the indiference curve (where Y is in the ordinates axis)

Marginal Rate of Substitution

- The negative of the slope of the indifference curve at any point is called the marginal rate of substitution (MRS)

Quantity of y

y1

y2

U1

Quantity of x

x1

x2

Marginal Rate of Substitution

- Notice that if indifference curves are strictly convex, then the MRS is decreasing (as x increases, the MRSyx decreases)
- See it in a graph: As “x” increases, the amount of “y” that the consumer is gives up to stay in the same indifference curve (that is MRSyx) decreases
- If the assumption that “balanced bundles” are preferred to “extreme bundles” (convexity of preferences assumption” holds then the MRSyx is decreasing!!

Marginal Rate of Substitution

At (x1, y1), the indifference curve is steeper.

At this point, the person has a lot of y,

So, he would be willing to give up more y

to gain additional units of x

At (x2, y2), the indifference curve

is flatter. At this point, the person

does not have so much y,

so he would be willing to give up

less y to gain

additional units of x

- MRS changes as x and y change
- and it is decreasing

Quantity of y

y1

y2

U1

Quantity of x

x1

x2

Utility and the MRS

- Suppose an individual’s preferences for hamburgers (y) and soft drinks (x) can be represented by

Solving for y, we get the indifference curve for level 10:

y = 100/x

- Taking derivatives, we get the MRS = -dy/dx:
- MRS = -dy/dx = 100/x2

Utility and the MRS

MRSyx = -dy/dx = 100/x2

- Note that as x rises, MRS falls
- when x = 5, MRSyx = 4
- when x = 20, MRSyx = 0.25

- When x=20, then the individual does not value much an additional unit of x. He is only willing to give 0.25 units of y to get an additional unit of x.

Another way of computing the MRS

- Suppose that an individual has a utility function of the form
utility = U(x,y)

- The total differential of U is

Along any indifference curve, utility is constant (dU = 0)

dU/dy and dU/dx are the marginal utility of y and x respectively

Another way of computing the MRS

- Therefore, we get:

MRS is the ratio of the marginal utility of x to the marginal utility of y

Marginal utilities are generally positive (goods)

Example of MRS

- Suppose that the utility function is

We can simplify the algebra by taking the logarithm of this function (we have explained before that taking the logarithm does not change the result because it preserves the ordering, though it can make algebra easier)

U*(x,y) = ln[U(x,y)] = 0.5 ln x + 0.5 ln y

Examples of Utility Functions

- Cobb-Douglas Utility
utility = U(x,y) = xy

where and are positive constants

- The relative sizes of and indicate the relative importance of the goods
- The algebra can usually be simplified by taking ln(). Let’s do it in the blackboard….

Examples of Utility Functions

U3

U2

U1

- Perfect Substitutes
U(x,y) =U= x + y

Y= -(/ )x + (1/ )U , indifference curve for level U

The indifference curves will be linear.

The MRS =(/ ) is constant along the

indifference curve.

Quantity of y

Quantity of x

Examples of Utility Functions

- Perfect Substitutes
U(x,y) =U= x + y

dU= dx + dy

- Notice that the change in utility will be the same if (dx= and dy= 0) or if (dx= 0 and dy= ). So x and y are exchanged at a fixed rate independently of how much x and y the consumer is consumed
- It is as if x and y were substitutes. That is why we call them like that

Examples of Utility Functions

U3

U2

U1

- Perfect Complements
utility = U(x,y) = min (x, y)

The indifference curves will be

L-shaped. It is called complements

because if we

Are in the kink then utility does not

increase by we increase the quantity

of only one good. The quantity of both

Goods must increase in order to increase

utility

Quantity of y

Quantity of x

Examples of Utility Functions

- CES Utility (Constant elasticity of substitution)
utility = U(x,y) = x/ + y/

when 0 and

utility = U(x,y) = ln x + ln y

when = 0

- Perfect substitutes = 1
- Cobb-Douglas = 0
- Perfect complements = -

Examples of Utility Functions

- CES Utility (Constant elasticity of substitution)
- The elasticity of substitution () is equal to 1/(1 - )
- Perfect substitutes =
- Fixed proportions = 0

- The elasticity of substitution () is equal to 1/(1 - )

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