Part 2

PREFERENCES AND UTILITY 1

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 2

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 3

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 !!!!

4

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 5

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 6

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 7

 

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) 8



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?

9



Utility

Game…   Clearly, the numbers are arbitrary The only consistent thing is the ranking that we obtain 10



Utility

Utility could be represented by a Table Bundles Example Utility P Q R S 1 0 0 2 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 11



Utility

Notice that several tables of utility can represent the same ranking Bundles Another example Example Utility P 1 1 Q R S 0 0 4 0 0 2 •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) 12

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….

13

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 14

Utility

   Clearly, for an economist it is the same to use U=X*Y than to use U=(X*Y) good are preferred to others 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 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 15

Economic Goods

 In the utility function, the assumed to be “goods”  more is preferred to less x and y are Quantity of y Preferred to x*, y*

?

Worse than x*, y*

y* x* ?

Quantity of x 16

Indifference Curves

 An indifference curve shows a set of consumption bundles among which the individual is indifferent Quantity of y Combinations (x 1 , y 1 ) and (x 2 , y 2 ) provide the same level of utility

y 1 y 2

U 1 Quantity of x

x 1 x 2

17

Indifference Curve Map

 Each point must have an indifference curve through it Quantity of y Increasing utility U 1 U 2 U 3 Quantity of x

U

1 <

U

2 <

U

3 18

Transitivity

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

Quantity of y 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 C A B But B is preferred to A because B contains more

x

and

y

than A U 2 U 1 Quantity of x 19

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 20

Convexity

 Formally, If the indifference curve is convex, then the combination ( x 1 + x 2 )/2, ( y 1 + y 2 )/2 will be preferred to either ( x 1 , y 1 ) or ( x 2 , y 2 ) Quantity of y 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.

(y 1 + y 2 )/2 y 1 y 2

U 1 Quantity of x

x 1 (x 1 + x 2 )/2 x 2

21

Marginal Rate of Substitution

    Important concept !!

MRS YX 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 !!!

MRS YX is the negative of the slope of the indiference curve (where Y is in the ordinates axis) 22

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

MRS yx

 

dy dx

 1

y 1 y 2 U 1

Quantity of x

x 1 x 2

23

Marginal Rate of Substitution

   Notice that if indifference curves are strictly convex, then the MRS is decreasing (as x increases, the MRS yx 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 MRS yx ) decreases If the assumption that “balanced bundles” are preferred to “extreme bundles” (convexity of preferences assumption” holds then the MRS yx is decreasing!! 24

Marginal Rate of Substitution

 MRS changes as  x and it is decreasing and y change Quantity of y

y 1 y 2 x 1

At (

x

1 ,

y

1 ), 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

U 1 At (

x

2 ,

y

2 ), 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

Quantity of x

x 2

25

Utility and the MRS

 Suppose an individual’s preferences for hamburgers ( y ) and soft drinks ( be represented by x ) can utility  10 

x



y

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/

x

2 26

Utility and the MRS

  MRS yx = dy / dx = 100/ x 2 Note that as x rises, MRS falls   when x when x = 5, MRS yx = 20, MRS yx = 4 = 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.

27

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

dU

 

U



x dx

 

U



y dy

Along any indifference curve, utility is constant (

dU

= 0) dU/dy and dU/dx are the marginal utility of y and x respectively 28

Another way of computing the MRS

 Therefore, we get:

MRS yx

 

dy dx

 

U



x



U



y MRS

is the ratio of the marginal utility of

x

to the marginal utility of

y Marginal utilities are generally positive (goods)

29

Example of MRS

 Suppose that the utility function is utility 

x



y

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

30

Deriving the MRS

 Thus,

MRS yx

 

U

* 

x



U

* 

y

 0.5

x

0.5

y



y x

Notice that the MRS is decreasing in x: The MRS falls when x increases 31

Examples of Utility Functions

 Cobb-Douglas Utility utility = U ( x,y ) = x  y  where  and  are positive constants   The relative sizes of  and  importance of the goods indicate the relative The algebra can usually be simplified by taking ln(). Let’s do it in the blackboard….

32

Examples of Utility Functions

 Perfect Substitutes U ( x , y ) =U=  x +  y Y= -(  /  ) x + ( 1/  ) U , indifference curve for level U Quantity of y The indifference curves will be linear.

The MRS = (  /  ) is constant along the indifference curve.

U 1 U 2 U 3 Quantity of x 33

Examples of Utility Functions

 Perfect Substitutes U ( x , y ) =U=  x +  y d U=  d x +  d y   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 34

Examples of Utility Functions

 Perfect Complements Quantity of y 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 U 3 U 2 U 1 Quantity of x 35

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   =  36

Examples of Utility Functions

 CES Utility (Constant elasticity of substitution)  The elasticity of substitution (  ) is equal to 1/(1  )  Perfect substitutes   =   Fixed proportions   = 0 37