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Intermediate Microeconomics W3211 Lecture 3

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Intermediate Microeconomics W3211 Lecture 3 3/1/2016 1 Intermediate Microeconomics W3211 Lecture 3: Introduction Preferences and Choice Columbia University, Spring 2016 Mark Dean: [email protected] 2 3 4 The Story So Far…. Today’s Aims • So far, we have described consumer choice as 1. Use our tools to describe various different types of preferences constrained optimization: ‘Standard’ preferences Monotonic 1. CHOOSE a consumption bundle Convex Introduce some other handy classes of preferences 2. IN ORDER TO MAXIMIZE preferences Perfect substitutes 3. SUBJECT TO the budget constraint Perfect compliments Cobb Douglas • And spent some time talking about how one can represent Varian Ch. 3 & 4, Feldman and Serrano Ch 2 preferences • Indifference curves • MRS • Utility function 5 6 A Short Diversion: Proofs A Short Diversion: Proofs In your homework, you are asked to prove something A proof is basically just a sequence of statements which follow from each other In class I claimed that two utility functions u and v represent the If X then Y same preferences if and only if there is a strictly increasing If Y then Z function f such that u(x)=f(v(x)) for all x. I would like you to prove half of this statement: if there is such a function f, then u and v These statements don’t necessarily need to be ‘mathematical’, represent the same preferences but maths can be an easy way to make your argument There are many types of proof, but two will be particularly useful Proving things often gets people confused. for the course While they are not central to the course, I will expect you to be Direct: If you are trying to prove that X implies Y prove a able to complete simple proofs sequence of steps X implies A This is not because I am mean spirited, but because this type of A implies B thinking is important … Both in economics and beyond H implies Y If you are worried about this, make use of office hours By Contradiction: If you are trying to prove that X implies Y, Talk to the TAs (or me) sooner rather than later show that assuming X and not Y leads to a contradiction 1 3/1/2016 7 8 A Short Diversion: Proofs A Short Diversion: Proofs In most cases, the proofs that you will be required to do just An example (from last lecture): A non transitive preference require manipulating the definitions. relation cannot have a utility representation. Start by remembering the definitions: Transitivity: ≿, ≿implies ≿ Utility representation: there is a function u such that ≿ if and only if for all x and y Proof by contradiction: assume that preferences are not transitive, but there IS a utility representation 1. If preferences are non transitive, then there is some x,y and z such that ≿, ≿but NOT ≿ 2. If there is a utility representation then ≿implies ≿implies This implies that But if this is a utility representation, this implies ≿ Contradiction 10 Different Types of Preference We now have two tools to represent preferences Utility functions Indifference curves (and MRS) We are going to use them to think about various different types Different Types of of preferences Preferences To start with we are going to think about what we will call Defining Standard Preferences ‘standard’ preference This adds two assumptions to the ones that we have already made: that preferences are 1. monotonic 9 2. convex. 11 12 Monotonic Preferences Monotonic Preferences Monotonic preferences are ones in which having more is always What do the indifference curves of monotonic preferences look better like? Two types of monotonicity Weak monotonicity: if bundle x has more of every good than bundle y then x is strictly preferred to y for every i implies ≻ Strict monotonicity: if bundle x has more of at least one good and no less of any good than bundle y then x is strictly preferred to y for every I and for some i implies ≻ Why might monotonicity fail? Bads Satiation 2 3/1/2016 13 14 Monotonic Indifference Curves Non Monotonic Indifference Curves Good 2 Good 2 One good and one Two goods bad a positively sloped a negatively sloped indifference curve. indifference curve. Good 1 Bad 1 Some amount of good 2 can compensate for Bad 1 15 16 Satiation Indifference Curves Exhibiting Satiation x Another example of non-monotonic preferences 2 For some goods there is a perfect optimal amount Satiation Examples (bliss) Salt in a dish point Anchovies on pizza Classes of Intermediate Microeconomics Sometimes there is a bundle that is the optimal one Called satiation/bliss point x1 17 18 Indifference Curves Exhibiting Indifference Curves Exhibiting Satiation Satiation x2 x2 Satiation Satiation (bliss) (bliss) point point Better Better x1 x1 3 3/1/2016 19 20 Monotonic Preferences Convexity Convexity: Roughly speaking - mixtures of bundles are preferred to the bundles themselves. What do the indifference curves of monotonic preferences look like? E.g., say that the consumer is indifferent between x and y Downward sloping Weakly monotonic – can be horizontal or vertical the 50-50 mixture of the bundles x and y is Strictly monotonic – strictly downward sloping What does the MRS of monotonic preferences look like z = (0.5)x + (0.5)y Positive (as it is the negative of the slope of the indifference curve) Weakly monotonic – can be 0 or infinity Then: convexity says that z is at least as preferred as x or y Strictly monotonic – strictly positive real number What does a mixture mean? What do monotonic utility functions look like? Marginal utility is always positive: The ‘average’ of the number of each good given by the two 0for every i: strictly monotonic bundles 1 0for every i and 0some i : weakly monotonic 1 ⋮ 1 21 22 Convexity Interpretation of Convexity Again, two flavors of convexity: Why are we assuming convexity? Weakly convex: If x is indifferent to y, then any mixture of x and y is weakly preferred to either Does it make sense? ∼implies 1 ≿ Well, think about constructing a meal Say you were indifferent between 2lb of mash potato (and no steaks) or 2 steaks (and no mash potato) It is likely that you would prefer 1lb of mash potato and 1 steak to either bundle Strictly convex: If x is indifferent to y, then any mixture of x and y is strictly preferred to either Can think of many similar examples Holidays and cars ∼implies 1 ≻ Visits to the dentist and visits to the theatre Economics classes and novels What about ice cream and crab paste? Arguably still yes Depends on whether we think that these things are consumed at exactly the same time 23 24 Convexity Graphically: Convexity What do the indifference curves of convex preferences look like? x Convexity x2 implies that the 0.5x+0.5y set of weakly x2+y2 preferred bundles is a 2 convex set y y2 Indifference curve is x1 x1+y1 y1 convex 2 4 3/1/2016 25 26 Strict Convexity Weak Convexity. Preferences are x’ Preferences are strictly convex weakly convex x x z’ 2 when all if at least one z mixtures z are mixture z is x strictly z equally preferred to y preferred to a y original bundles component y y’ 2 x and y bundle x1 y1 27 28 Non-Convex Preferences More Non-Convex Preferences x2 x2 The mixture z The mixture z z is less preferred z is less preferred than x or y. than x or y. y2 y2 x1 y1 x1 y1 29 30 Convexity MRS & Ind. Curve Properties What do the indifference curves of convex preferences look Good 2 MRS decreases with x1 if like? and only if preferences They are convex MRS = 5 are strictly convex Weakly convex preferences – weakly convex indifference curves Strictly convex preferences – strictly convex indifference curves As x1 becomes large MRS becomes small What does the MRS of convex preference look like? Intuition? You have a lot of good 1 and you really want some good 2! MRS = 0.5 Good 1 Obvious from calculus: convex function has an INCREASING derivative 5 3/1/2016 31 32 MRS and Convexity MRS & Ind. Curve Properties Convexity is equivalent to saying that MRS decreases x MRS increases with good 1 2 MRS = 0.5 as x1 increases Meaning: the amount of good 2 that the person nonconvex needs to compensate them for the loss of good 1 preferences decreases the more of good 1 they have This is very reasonable: the more you have of one good, the more you’re willing to exchange for some MRS = 5 of the other good x1 From this point of view, convexity is very natural Obvious from calculus: concave function has an DECREASING derivative 33 34 MRS & Ind. Curve Properties Convexity What do the indifference curves of convex preferences look x2 like? They are convex Weakly convex preferences – weakly convex indifference curves MRS = 1 Strictly convex preferences – strictly convex indifference curves MRS What does the MRS of convex preference look like? = 0.5 Decreasing as increases MRS = 2 Weakly convex – weakly decreasing Strictly convex – strictly decreasing What do the utility functions of convex preferences look like? This is only for the math fetishists x1 Convex preferences have utility functions which are quasi-concave 1 min ,) 36 Other Types of Preferences It is now going to be useful to think about some other types of preferences These will be useful as we illustrate different ideas going forward Perfect Substitutes Perfect Compliments Different Types of Cobb Douglas Preferences Preferences Some other beasts in the menagerie of preferences 35 6 3/1/2016 37 38 Perfect Substitutes Graphically: Perfect Substitutes x2 If a consumer always regards units of commodities 1 and 2 as Slopes are constant at - 1 equivalent, then the commodities are perfect substitutes and Bundles in I all have a total of 15 units and only the total amount of the two commodities in bundles 15 I2 2 are strictly preferred to all bundles in I , which determines their preference rank-order 1 have a total of only 8 units in them.
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