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Intermediate Microeconomics W3211 Lecture 2

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Intermediate Microeconomics W3211 Lecture 2 3/1/2016 1 Intermediate Microeconomics W3211 Lecture 2: Introduction Indifference Curves and Utility What do your preferences look like? Columbia University, Spring 2016 Mark Dean: [email protected] 2 3 4 The Story So Far…. Today’s Aims • In Monday’s lecture we described how we would model 1. Describe how to represent preferences on our handy graphs consumer choice as constrained optimization: Indifference Curves 1. CHOOSE a consumption bundle 2. Describe how to represent preferences using maths Utility functions 2. IN ORDER TO MAXIMIZE preferences 3. Describe ‘standard’ preferences 3. SUBJECT TO the budget constraint Monotonic Convex • Consumption bundles and budget constrains were dealt with fairly thoroughly 4. Introduce some other handy classes of preferences • Preferences may still seem a bit mysterious Perfect substitutes Perfect compliments • Today, we will attempt to de-mystify Cobb Douglas Reminder: Varian Ch. 3 & 4, Feldman and Serrano Ch 2 6 Going from Three Dimensions to Two • Think back to our nice, simple example with two goods • Made it nice and easy to graph the consumption bundles and budget constraints Indifference Curves Or: how to see your preferences 5 1 3/1/2016 7 8 Going from Three Dimensions to Two Going from Three Dimensions to Two • Think back to our nice, simple example with two goods • Unfortunately, we now have three pieces of information associated with each bundle • Made it nice and easy to graph the consumption bundles • Amount of good 1 and budget constraints • Amount of good 2 Budget constraint is • Whether this bundle is preferred or not to others m /p2 • How can we represent this information? p1x1 + p2x2 = m. • One way would be to use three dimensions: • Bundles that are more preferred are placed higher on dimention 3 than those which are less preferred m /p1 x1 9 10 Going from Three Dimensions to Two Going from Three Dimensions to Two • While this would work, graphing in three dimensions is a pain • As we will see, we can get a lot more intuition if we can work in two dimensions • So how can we go from three dimensions to two? • Luckily, cartographers have the answer 11 12 Going from Three Dimensions to Two Going from Three Dimensions to Two On a map, contour lines link areas of equal height We will use indifference curves which link areas of equal preference 2 3/1/2016 13 14 Going from Three Dimensions to Two Going from Three Dimensions to Two 12 10 8 2 2 6 4 Good 4 6 8 2 9 0 11.522.533.544.555.566.577.588.599.510 Good 1 15 16 Indifference Curves Graphically # of oranges x’ x” x”’ A simple example x’ 2 goods: Apples Oranges x” Suppose our agent likes both Take some bundle x’ x”’ An indifference curve links all bundles of goods which are indifferent to x’ # of apples For example, say that ′∼′′∼′′′ 17 18 Indifference Curves Indifference Curves I 3 curves # of Suppose # of 1 oranges oranges All bundles in I are x zxy x 1 strictly preferred to all in I2 All bundles in I are strictly z z 2 preferred to all in I3. I2 y y I3 # of apples # of apples 3 3/1/2016 19 20 Indifference Curves Indifference Curves # of # of oranges oranges Set of Set of x bundles weakly x bundles strictly preferred to x. preferred to x I(x) I(x’) I(x) # of apples # of apples 21 22 Can indifference curves Can indifference curves intersect? intersect? Take 2 curves I and I Take 2 curves I and I # of I 1 2 # of I 1 2 oranges 2 oranges 2 I1 Suppose z y so these I1 Suppose z y so these are different curves are different curves Can they intersect? Can they intersect? x x No! From I1, x y. From I2, x z. y y From transitivity y z. z z Contradiction!! # of apples # of apples 23 24 The Marginal Rate of Substitution The Marginal Rate of Substitution One crucial thing we will want to know about preferences: Why is this of interest? “If I take away one apple, how many oranges would I have to At this stage you can take my word for it… give you to keep you indifferent” ….or you can think about the following Or more accurately, at what rate do I have to change oranges for apples to keep you indifferent? “If the rate at which I am willing to trade off apples to oranges is higher than the relative price of apples and oranges, am I This is the marginal rate-of-substitution (MRS) between apples and oranges maximizing my preferences”? Mathematically WARNING: Sometimes MRS is defined without the minus sign ∆ There is no consensus about which is correct , lim ∆→ ∆ I will accept either definition from you, as long as you are consistent! Such that , ∼ ∆, ∆ 4 3/1/2016 25 26 The Marginal Rate of Substitution Marginal Rate of Substitution Claim: It is very easy to see the marginal rate of substitution from Good 2 the indifference curve. Bundle (x1, x2) x2 The MRS at a particular point is the negative of slope of the indifference curve at that point x2 Bundle (y1, y2) y2 x1 x1 y1 Good 1 27 Marginal Rate of Substitution Good 2 Slope of indifference Bundle (x1, x2) curve at (x , x ) is the x 1 2 2 MRS at this point Preferences and Utility Or: Your preferences in numbers x1 Good 1 28 29 30 Where are the Numbers? Where are the Numbers? So far we have represented the objectives in our consumer’s Okay, so we are going to work with utility functions problem with preferences and weird symbols. But first I want you to understand how utility functions are used I bet you are crying out to get away from these symbols and in economics use some proper, god fearing numbers to represent what the consumer wants to maximize This is slightly counterintuitive… This would allow us to do lots of cool things …but useful for you to understand what is going on For example take derivatives For many people this is the most confusing bit of the course, so Moreover, you have probably heard of the concept of a ‘utility hold on! function’ Reports how ‘happy’ a particular bundle of goods makes someone So why can’t we work with utility functions? 5 3/1/2016 31 32 In the beginning…. In the beginning…. Back in the days of Marshall (1890’s) utility was considered to be Three problems with this approach a real, measurable, cardinal scale 1. What evidence did we have that there was utility lurking in Lurking in people’s brains was the ‘happiness’, or utility, the brain associated with different bundles of goods No way to directly measure it People made choices in order to maximize this happiness 2. All attempts to derive utility from first principles (and so explain choice) failed The nature of utility could be derived from first principles Calories 3. What did it mean for the utility of bundle x to be twice that of Water bundle y? Warmth Etc. 33 34 20th Century: Preferences! Utility from Preferences This lead to a change of approach in 20th century Despite this realization, it is still useful to be able to work with utility functions Preferences were taken as the primitive thing that people For one thing, we can take derivatives! maximized So the question became whether we can build a utility Why? What is the big advantage of preferences over utility? function from preferences We can plausibly measure preferences in a way we cannot Let’s start with a set of preferences ≿ on different bundles of measure utility goods How? Can we build a utility function u that represents (or contains the same information as) ≿ One good way would be through choice x is preferred to y if x is chosen over y i.e. we can find a way of assigning utility numbers to bundles such that, for any two bundles x and y Side note: Neuoscientists and neuro-economists are trying to take us back to the days of Marshall ≿if and only if 35 36 Utility from Preferences When do we have a utility function? If we can do this, then we can ‘pretend’ that the consumer is maximizing utility So, when does a preference relation allow a utility Maximizing preferences: choosing x such that ≿for all representation? available y Maximizing utility: choosing x such that for available y Answer: as long as it is well behaved! Complete These are the same thing if the utility function represents the Transitive preferences: i.e. Reflexive ≿if and only if (also, if the set of available options is not countable, we need continuity, but don’t worry so much about this) 6 3/1/2016 37 38 When do we have a utility function? What do Utility Numbers Mean? Lets say we have If preferences are not well behaved, will there be a utility representation? Three alternatives x,y and z Preferences ≿ over x,y and z A utility function u that represents these preferences No! 1 2 For example: failure of transitivity 3 ≿, ≿but NOT ≿ Does it mean anything that the utility of z is three times the utility of Say u represents these preferences y? ≿implies Not really: Consider instead the utility numbers ≿implies 1 By the power of maths, this implies 1.5 But if u represents the preferences this would imply ≿ 2 Contradiction These represent the same preferences Would lead to the same choices A preference relation allows a utility representation if and only if it is well behaved But now the utility of z is twice that of y 39 40 What do Utility Numbers Mean? What do Utility Numbers Mean? Utility numbers only have ordinal meaning Theorem: Take two utility functions u and v.
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