1.1 Preference and Utility Flashcards
What will Utliity depend on in our models?
depends on the quantities consumed of all goods
What are the 5 modern assumptions of preferences over bundles of goods?
1) Completeness
2) Transitivity
3) Continuity
4) Non-satiation
5) Convexity
What is Completeness?
A consumer, when confronted with any two bundles of goods ( apples and oranges), can tell us which one she prefers, or whether she is indifferent between them.
Show completeness Mathematically?
How can you show completeness on 3 diagrams with indifference curves?
What is the key implication of completeness and whats wrong with this?
Key implication: the consumer can make choices over the entire range of possibilities.
• Consumers may find it impossible to rank some options
Also behavioural economics will disagree.
What is the assumption of transitivity?
Preferences are such that if bundle x is preferred to bundle y, and bundle y is preferred to bundle z, then x is preferred to z.
Show Transitivity mathematically showing 3 bundles A B and C, you can show it as utility or as just bundles.
What does Transitivity imply you can get from Utlitiy?
You can get a number out of each bundle, telling us the ultitiy you get from the bundle.
Is height transitive and is Winning at Tennis transitive, use Emily, Emmanuella and Jason as examples, what is usually transititive?
If Jason is taller than Emily but Emily is taller than Emmanuella, it must mean that Jason is taller than Emmanuella.
If Jason beats emily at Tennis and Emily beats Emmanuella at Tennis, it doesn’t mean Jason will beat Emmanuella at Tennis.
So any numbers must be transitiative.
What is the assumption of Continuity and how can we show on an indifference curve diagram?
If bundle A is preferred to bundle B and bundle C is close to bundle B, then bundle A is preferred to bundle C.
Or If A is preferred to B and C is close to A, then C is preferred to B.
Give a literally example of Continuity?
If i have 2 shopping baskets and A is preferred to B. if I add one tomato to the basket B, you still prefer basket A to B. So small variations shouldn’t lead to big reservals in preferences.
What are the implications of Completeness, Transtivitiy and Continuity imply?
Preferences can be represented by a utility function and indifference curves.
( By writing a utility function preferences must satisfy all the 3 things)
What else is Utility from the implications of the first 3 assumptions?
Utility is ordinal
What does it mean if Utlitiy is ordinal?
Puts a rank on any number of consumption bundles.
With ordinal utility, goods are ranked only in terms of more or less preferred, there is no attempt to determine how much more one good is preferred to another. ( e.g. 2 bundles apples(A) and organes (B) Bundle A has utlitiy of 300 and bundle B has a ulitiy of 200, hence A is preferred to B, but the exact numbers have no meaning, it just a device to order preferences.
What is a key implication of Utility being ordinal?
different utility functions can reflect the same
preferences, if ordering is preserved
What does different utility functions can reflect the same preferences, if ordering is preserved mean, by answering these questions?
Example 1) We squared the utlitiy function, it doesn’t change preference ordering
Example 2) Yes they do as there is a change in preference ordering.
If indifference curves intersected, what would it mean?
It would mean that our standard assumptions would not hold so far ( Completeness, Transitvitiy and Continuity)
Show ordinal Utility using cobb douglas function u(x1,x2) = x1^3/4, x2^1/4 where X1>0 and x2>0 and U>0 with u(1,1) u(1,4) and u(1,9)
Then consider the function v(x1,x2) = [u(x1,x2]^2 = x1^3/2 x2 with utility values v(1,1) v(1,4) v(1,9)
Show that Utility is ordinal?
What does this example represent?
A montonic transformation is a way of transforming a set of numbers into another set that preserves the order of the original set
What are 2 ways to check whether 2 utility functions reflect the same preferences?
Method 1: eyeballing…sometimes you can see one utility function is a positive monotonic transformation of the other
Method 2 : Checking the MRS (The same preferences imply the same indifference curves, which imply the same MRS)
Does these functions represent a monotonic transformation of each other?
Yes they do as logs are opposite to expotentials
What does Marginal rate of subsitution mean and how do you calculate it
the marginal rate of substitution is the rate at which a consumer can give up some amount of one good in exchange for another good while maintaining the same level of utility. It is diminishing too.
You partially differentiate the utlitiy functions with respect to X1 and X2.
By calculating the MRS find out whether these preferences represent a monotonic transformation of each other?
How do you test for Non-satiation?
The First order partial deratives need to be positive for both goods. ( increasing x1 and or x2 increases ultiity)
Check whether Non-saitation holds
It always does for Cobb douglas. For all positive values of x1 and x2.
If non- saiation is satisfied for both goods, where is the preferred set?
indifference curve slope downwards and the preferred set is above the indifference curve
Does Non-saitation satisfied for this consumer?
Not everywhere as some places are upward sloping.
Does consumers prefer extreme or averages, when function is convex?
They prefer averages to extremes. ( all the points on the line, represent a superior level of utility than just A or B.
What bundle gives the highest Utility?
C
Give an example of a non convex preference, what does it imply and draw it?
Tea and coffee in the same cup, you wouldn’t want to mix tea and coffee together, you prefer extremes here, so function is concave to the origin. The combination of the 2 don’t lie in the preferred set ( thus convexity fails.
Show that this cobb douglas function is convex?
What types of preferences will be look at throughout the studies ?
Cobb douglas ( and in logarithmic form e.g. alogx1 which is the same thing) ( always downward sloping indifference curves. Quasi linear (e.g. x2^1/2 is still quasi linear, as it is still a function of x1) Perfect complements ( cannot solve for non-saitation but you assume so and convexity, there indifference curves are right angles
Lets say we have a cobb douglas function u(x1,x2) = x1^1/2 x2^1/2. If x1 and x2 = 4, what must utility be and if x1 = 1 and Utliity is 4, what is the value of x2, show on diagram? Show the iC in red of u=0
a) 4
b) 16 ( remember were ever you are on indifference curve, you must get the same utlitiy, bundle must give you a utility of 4.
c) Any bundle on x1 axis and any bundle on x2 axis gives you 0 utility.
Lets look at an example perfect subistutes, what do we know about the IC’s? Also what do we know about their MRS’s, SKETCH THE DIAGRAM AND DOESN NON-SATIATION HOLD AND CONVEXITY.
Where is the preferred set?
They are downward sloping linear lines, there MRS’s are constant.
We know with perfect substitutes, convexity cannot hold as the person prefers extremes to averages, we can plug numbers. Non satiation is satisfied as the Partial first order derative is positive for both goods.
Lets say you have perfect subisutes but you hate them? where is the preferred set and where would you maxmise utlitiy?
This implies that the First order partial derivative for both goods is negative ( MU is negative for both goods).
You would maxmise utlitiy at U(0,0)
Can we differentiate perfect complements and what does MIN mean
Nope and min means utlitiy which is the lowest in the bracket
What is the min[2,10], min[2,12], min [2,1] min [3,3] and what does this show? show on diagram?
1) 2
2) 2 ( by adding good 2 it doesn’t help)
3)1
4)3
and individual can get better off but increasing in both goods e.g. min [2,2] and min [3,3] utlitiy does rise from 2 to 3, but you do one of the goods it doesn’t change.
Increasing good 1 and not good 2 has no change on utlitiy.
Is it possible that preferences are not transitive but can be represented by a utility function?
Assume that non-satiation and transitivity hold. Can indifference curves cross? Explain using a diagram.
For ii) the diagram is a non convex shape
How does a non -convex but non-saitation diagram look like?
