Microeconomics 1 Is Economics a Science? 2 Issues • Economics doesn’t predict well • All its assumptions are wrong • It isn’t even done in good faith 3 Economics doesn’t predict Three replies: • No one predicts • Economics actually does (sometimes) • Who needs to predict? 4 No one predicts • Large, complex systems are hard to predict • Chaos theory – even something as simple as π π₯ π₯ with π π₯ 1 4 π₯ 0.5 5 Chaos π π₯ 1 4 π₯ 0.5 0 0.01 0.02 0.03 0.04 0.05 1 0.0396 0.0784 0.1164 0.1536 0.19 2 0.152127 0.289014 0.411404 0.520028 0.6156 3 0.515939 0.821939 0.968603 0.998395 0.946547 5 0.00406 0.970813 0.427388 0.025467 0.6457 10 0.795154 0.503924 0.524371 0.836557 0.999652 15 0.393686 0.015682 0.494768 0.466403 0.316366 20 0.079928 0.590364 0.027768 0.774441 0.069137 6 Chaos • If a single variable with such a simple equation can generate chaos… • Just imagine what happens with the weather (the butterfly effect) • Or the entire global economy • Or the geopolitical system 7 Two additional complications In the social sciences: • We don’t even have the basic rules (no equivalents of flow equations in physics) • We’re dealing with self-reflective systems (a hurricane doesn’t change its mind if predicted correctly) 8 Economics actually does predict • Sometimes it doesn’t do too badly • Nice quantitative results on • • • Supply-and-demand in a single good market Auctions Matchings • Many qualitative insights 9 It is easier to predict when • The system is “small” and isolated • There are many “repetitions” – • similar examples that are causally independent • Experiments are possible • … Maybe we should learn when to expect a theory to predict 10 Who needs to predict? • Economics would surely like to be a predictive science • But it can be useful even if it isn’t • For instance, even if it can only critique reasoning • Compare with history 11 All assumptions are wrong • Well, yes • But think of • Robustness of findings? • Relevance to economic decisions? 12 The Ultimatum Game There is a sum of $100 to share between Players I and II Player I offers a way to divide the sum (say, integer values) Player II can say Yes or No Yes – they get the amounts offered No – they both get nothing What will happen? What does the theory say? 13 The Ultimatum Game Bernd Schwarze (b. 1944) Werner Güth (b. 1944) Güth, Schmittberger, Schwarze (1982) 14 Reference An Experimental Analysis of Ultimatum Bargaining Werner Güth, Rolf Schmittberger, Bernd Schwarze Journal of Economic Behavior and Organization, Vol. 3, No. 4 (Dec., 1982), pp. 367-388 Abstract There are many experimental studies of bargaining behavior, but suprisingly enough nearly no attempt has been made to investigate the so-called ultimatum bargaining behavior experimentally. The special property of ultimatum bargaining games is that on every stage of the bargaining process only one player has to decide and that before the last stage the set of outcomes is already restricted to only two results. To make the ultimatum aspect obvious we concentrated on situations with two players and two stages. In the ‘easy games’ a given amount c has to be distributed among the two players, whereas in the ‘complicated games’ the players have to allocate a bundle of black and white chips with different values for both players. We performed two main experiments for easy games as well as for complicated games. By a special experiment it was investigated how the demands of subjects as player 1 are related to their acceptance decisions as player 2. 15 What does the theory say? 100 π¦ππ 100,0 ππ 0,0 π¦ππ 99,1 99 ππ … 50 … 20 … π¦ππ … 0,0 80 50,50 … ππ 0,0 0 … … … 16 Well, We are tempted to predict: 100 π¦ππ 100,0 ππ 0,0 π¦ππ 99,1 99 ππ … 80 … 20 … π¦ππ … 0,0 50 50,50 … ππ 0,0 0 … … … The “Backward Induction” solution 17 Backward Induction In a finite game of perfect information we can go down to the leaves and work our way backwards to find the players’ choices Ernest Zermelo (1871-1953) 18 Backward Induction assumptions • Rationality • Common knowledge (or common belief) in rationality • To be precise, as many levels of belief as there are steps in the game 19 So in this case The backward induction seems to be 100 π¦ππ 100,0 ππ 0,0 π¦ππ 99,1 99 ππ … 50 … 20 … π¦ππ … 0,0 80 50,50 … ππ 0,0 0 … … … – But this assumes that the monetary sums are the “utilities” 20 Important • In a game as simple as the Ultimatum Game, it is impossible to test basic decision/game theoretic assumptions (such as transitivity) • We can only test them coupled with the assumption that only material payoffs matter 21 Emotional payoffs • Player II might be angry/insulted at a low offer • Player II as well as Player I might care for fairness • Player I might be altruistic • etc. • A way to tell some explanations apart: the Dictator Game 22 Is it rational to respond to emotions? In “Descartes’ Error” (1994) argued that it is wrong to think of emotions and Antonio Damasio (b. 1944) rationality as divorced; rather, rationality relies on emotions 23 Back to the Ultimatum Game • Having said all that, emotional payoffs should not be overstated • In the Ultimatum Game, if the payoffs were in millions of dollars rather than dollars, acceptance of low offers would likely to be higher • As well as when Player II has to wait before responding 24 The Ultimatum Game with delay Let Me Sleep on It: Delay Reduces Rejection Rates in Ultimatum Games Veronika Grimm, Friederike Mengel Economics Letters, Vol. 111, No. 2 (2011) pp. 113-115 Abstract Delaying acceptance decisions in the Ultimatum Game drastically increases acceptance of low offers. While in treatments without delay less than 20% of low offers are accepted, 60-80% are accepted as we delay the acceptance decision by around 10. min. 25 Not even in good faith • Can science be objective? • Aren’t we always affected by personal history, social class, our incentives? • If so, can we trust the “truths” that economists pretend to have “established”? • Should we check how many economists who believe in the free market also benefit from it (serve on boards of directors etc.) ??? 26 Can science be objective? Path-breaking studies on the history of madness, sexuality Michel Foucault (1926-1984) 27 Shouldn’t we be suspicious? • Well, yes • But – we can try to be (more) objective • Objectivity is a direction, not a place • Let’s remind ourselves of the distinction between Positive and Normative social science • And then ask the question about Postmodernism 28 Positive vs. Normative • Positive (~ descriptive) IS • Normative (~ prescriptive) OUGHT • Normative physics is called SciFi • But in the social sciences it makes sense 29 How do we judge theories • Positive – How close to reality it is • Normative – ??? • The king in “The Little Prince” 30 The Little Prince "It is contrary to etiquette to yawn in the presence of a king," the monarch said to him. "I forbid you to do so." "I can't help it. I can't stop myself," replied the little prince, thoroughly embarrassed. "I have come on a long journey, and I have had no sleep ..." "Ah, then," the king said. "I order you to yawn. It is years since I have seen anyone yawning. Yawns, to me, are objects of curiosity. Come, now! Yawn again! It is an order." "That frightens me ... I cannot, any more ..." murmured the little prince, now completely abashed. "Hum! Hum!" replied the king. "Then I—I order you sometimes to yawn and sometimes to—" He sputtered a little, and seemed vexed. For what the king fundamentally insisted upon was that his authority should be respected. He tolerated no disobedience. He was an absolute monarch. But, because he was a very good man, he made his orders reasonable. "If I ordered a general," he would say, by way of example, "if I ordered a general to change himself into a sea bird, and if the general did not obey me, that would not be the fault of the general. It would be my fault." Antoine de SaintExupery (1900-1944) 31 So what is a good normative theory? • I suggest: one that captures the kind of people/society we want to be • Normative as second-order positive • What type of a decision maker do I want to be? • What kind of a society/economy do I want to live in? 32 Be that as it may • Let’s not mix up positive and normative • There may never be eternal peace (positive) But this doesn’t mean we should start shooting each other (normative) • Our theories may never be perfectly objective (positive) But this doesn’t mean we shouldn’t try (normative) 33 Utility Maximization 34 Who maximizes utility ? Or rather, who behaves as if they did? Logical positivism and the emphasis on observables The revealed preferences paradigm 35 What’s observable ? Choices: between pairs or out of sets? Deterministic or stochastic? If sets – all sets? Only budget sets? These are all questions of modeling… 36 Binary relations π a set of alternatives π ⊂ π π is π – a binary relation reflexive if π₯π π₯ for all π₯ symmetric if π¦π π₯ whenever π₯π π¦ transitive if π₯π π§ whenever [π₯π π¦ πππ π¦π π§] complete if π₯π π¦ ππ π¦π π₯ (or both) for all π₯, π¦ 37 Equivalence relations π is a equivalence relation if it is reflexive, symmetric and transitive For example: equality is reflexive if π₯ π₯ for all π₯ symmetric if π¦ π₯ whenever π₯ π¦ π₯ π§ whenever [π₯ π¦ πππ π¦ transitive if π§] 38 Equivalence relations – examples π is a equivalence relation if it is reflexive, symmetric and transitive For example: π₯π π¦ iff π₯ and π¦ have the same (first) last name π₯π π¦ iff π₯ and π¦ have the same height π₯π π¦ iff π₯ and π¦ have the remainder after division by 3 39 Equivalence relations – examples More generally, define π₯π π¦ iff π π₯ π π¦ for some function π: π → π Then π is reflexive if π π₯ π π₯ for all π₯ symmetric if π π¦ π π₯ whenever π π₯ π π¦ transitive if π π₯ π π§ whenever [π π₯ π π¦ πππ π π¦ π π§ ] Are there others? 40 Equivalence classes An equivalence relation π divides the set π into equivalence classes: There is a partition of π, π΄ (finite or infinite) such that π₯π π¦ iff both π₯, π¦ belong to the same π΄ This also means that π is an equivalence relation iff there is some (set π and some) function π: π → π such that π₯π π¦ iff π π₯ π π¦ 41 Preference relations β½ is a preference relation if it is complete and transitive A fact: a complete relation is reflexive 42 Preference relations – weak and strict For a relation β½ define β» and ~ by π₯ β» π¦ ππ π₯ β½ π¦ ππ’π‘ πππ‘ π¦ β½ π₯ π₯ ~ π¦ ππ π₯ β½ π¦ πππ π¦ β½ π₯ (Define also βΌ and βΊ ) Facts: If β½ is transitive, then so is ~ If β½ is transitive, then so is β» 43 If Recall that is transitive, then so is π₯ ~ π¦ ππ π₯ β½ π¦ πππ π¦ β½ π₯ If we have π₯ ~ π¦ and π¦ ~ π§ then π₯ β½ π¦ πππ π¦ β½ π₯ π₯β½π§ So that we have π¦ β½ π§ πππ π§ β½ π¦ πππ π§β½π₯ π₯~π§ 44 If is transitive, then so is π₯ β» π¦ ππ π₯ β½ π¦ ππ’π‘ πππ‘ π¦ β½ π₯ Recall that We need to show that π₯ β» π¦ and π¦ β» π§ Implies π₯β»π§ Or: IF π₯ β½ π¦ ππ’π‘ πππ‘ π¦ β½ π₯ THEN and π¦ β½ π§ ππ’π‘ πππ‘ π§ β½ π¦ π₯ β½ π§ ππ’π‘ πππ‘ π§ β½ π₯ 45 We have π₯ β» π¦ and π¦ β» π§ that is, π₯ β½ π¦ ππ’π‘ πππ‘ π¦ β½ π₯ and π¦ β½ π§ ππ’π‘ πππ‘ π§ β½ π¦ then, by transitivity (of β½) π₯β½π§ 46 We have π₯ β» π¦ and π¦ β» π§ that is, π₯ β½ π¦ ππ’π‘ πππ‘ π¦ β½ π₯ and π¦ β½ π§ ππ’π‘ πππ‘ π§ β½ π¦ Could it be that π§ β½ π₯ also holds? No, because then we would have π§β½π¦ 47 Utility representation For a relation β½ on a set π and a function π’: π → ∞, ∞ we say that π’ represents β½ if, for all π₯, π¦, π₯ β½ π¦ πππ π’ π₯ π’ π¦ 48 Different notions of representation For a relation β½ on a set π and a function π’: π → ∞, ∞ (i) for all π₯, π¦ π₯ β½ π¦ πππ π’ π₯ π’ π¦ (ii) for all π₯, π¦ π₯ β» π¦ πππ π’ π₯ π’ π¦ (iii) for all π₯, π¦ π₯ ~ π¦ πππ π’ π₯ π’ π¦ Fact: If β½ is complete, (i) and (ii) are equivalent, and each implies (iii) (but not the other way around) 49 Equivalence of (i) and (ii) (i) For all π₯, π¦ π₯β½π¦βΊπ’ π₯ π’ π¦ (ii) For all π₯, π¦ π₯β»π¦βΊπ’ π₯ π’ π¦ Given completeness, this is just using the contrapositive: to say that π⇒π is equivalent to saying that π⇒ π 50 Contrapositives The (“material”) implication π⇒π is equivalent to π⇒ In fact, the material implication π ⇒ π is defined as And the truth value of π πβπ πβπ is Value of πβπ π π π ∨ – ∨ ∨ π – the same as the truth value of π⇒ π which is (again) defined as π β π πβπ 51 Back to the equivalence of (i) and (ii) (i) For all π₯, π¦ π₯β½π¦βΊπ’ π₯ π’ π¦ (ii) For all π₯, π¦ π₯β»π¦βΊπ’ π₯ π’ π¦ Given completeness: π₯β½π¦⇒π’ π₯ π’ π₯ π’ π¦ iff π’ π¦ ⇒ π₯ β½ π¦ iff π’ π¦ π’ π₯ ⇒π¦β»π₯ π¦β»π₯⇒π’ π¦ π’ π₯ Hence (i) is equivalent to For all π₯, π¦ π¦β»π₯βΊπ’ π¦ π’ π₯ – which is (ii) with π₯, π¦ reversed 52 And (i) [hence (ii)] implies (iii) (i) For all π₯, π¦ π₯β½π¦βΊπ’ π₯ π’ π¦ (iii) For all π₯, π¦ π₯~π¦ βΊπ’ π₯ π’ π¦ Because π₯~π¦ βΊ π₯β½π¦ βΊ π¦β½π₯ βΊ π’ π₯ π’ π¦ π’ π¦ π’ π₯ π₯~π¦ βΊπ’ π₯ π’ π¦ 53 Why doesn’t (iii) imply (i) ? (i) For all π₯, π¦ π₯β½π¦βΊπ’ π₯ π’ π¦ (iii) For all π₯, π¦ π₯~π¦ βΊπ’ π₯ π’ π¦ Because representing indifference doesn’t guarantee that preference is also “faithfully” represented For a given β½ and π’ that represents it, consider π£ π₯ π’ π₯ 54 Conditions for utility representation For a relation β½ on a finite set π, β½ is complete and transitive IFF There exists a function π’: π → ∞, ∞ that represents β½ 55 Interpretation I: meta-scientific For a relation β½ on a finite set π β½ is complete and transitive IFF There exists π’: π → ∞, ∞ that represents β½ • What is “utility” ? • The term derives its meaning from its usage (Ask not, “What?”, Ask, “How?”) • We’ll explain what it means to maximize utility in terms of observables 56 Logical Positivism • What is [good] “science”? • Theoretical terms should be defined by observations • Culminated in the “Received View” (Carnap, 1923) Rudolf Carnap (1891-1970) 57 Logical Positivism + Popper (1934) • A theory is meaningful only if it is refutable • It can never be verified, only refuted, or not-yetrefuted • Famous targets of critic: Marx’s historicism, Freud’s psychoanalysis Karl Popper (1902-1994) • (Evolution? Game theory?) 58 Logical Positivism and economics • Popper (1934) criticizes psychology • Samuelson (1938) pioneers “revealed preference theory” • Room for speculation… Paul Samuelson (1915-2009) 59 Economics and psychology Loewenstein (1988) suggested that, maybe, in the 1930s, economics didn’t think that psychology was such great company George Loewenstein (b. 1955) 60 Was it really Logical Positivism? Moscati is a serious historian who argues that I’m selling you fake history But the story is too good to kill Ivan Moscati (b. 1955) 61 Utility and marginal utility Adam Smith (as Plato) thought that there is no relationship between value and price Cf. water and diamonds Adam Smith (1723-1790) The Marginalist Revolution William Stanley Jevons 1835-1882 Carl Menger 1840-1921 Léon Walras 1834-1910 Alfred Marshall 1842-1924 Be that as it may A theorem such as: For a relation β½ on a finite set π β½ is complete and transitive IFF There exists π’: π → ∞, ∞ that represents β½ endows “utility” with meaning 64 Interpretation II: normative Suppose you ask me how to make a decision And I ask you, would you like your β½ to be complete ? • Many would say yes • Incompleteness is absence of decision (Kafka and his wedding engagements…) And then: would you like your β½ to be transitive ? • Again, many would say yes • An intransitive relation, let’s say a cyclical one (that’s more than just intransitive!) isn’t very useful 65 The normative interpretation – cont. And then I point out to you that For a relation β½ on a finite set π β½ is complete and transitive IFF There exists π’: π → ∞, ∞ that represents β½ And I can convince you that you would like to behave as if you were maximizing a utility function, or maybe just maximize one (consciously) 66 Interpretation III: descriptive Suppose I tell you that, when I analyze an economic problem, I assume that agents are utility maximizing. • Does it make sense? • Do you know many such agents? • What gives me the right to make predictions and give advice based on such a preposterous assumption? 67 The descriptive interpretation – cont. And then I point out to you that For a relation β½ on a finite set π β½ is complete and transitive IFF There exists π’: π → ∞, ∞ that represents β½ … and I may convince you that more agents might be described by my analysis than you would have imagined 68 Comments • Why do I need to convince you that this is how people behave? • Why not just test? • Indeed, if we test, it doesn’t matter which formulation we use • The whole point is that they’re equivalent • In fact, a characterization theorem is a sort of a framing effect • If economics were a successful science, it would not need axiomatizations • But it isn’t so successful. So it leaves room for rhetoric. 69 Compare with Social Choice For a relation β½ on a finite set π β½ is complete and transitive IFF There exists π’: π → ∞, ∞ that represents β½ • Pareto domination: transitive but not complete • Majority vote: complete but not transitive 70 Pareto Domination • Basically, unanimity (At least one has strict preference, the others – weak [or strict] ) • Transitivity seems obvious (A bit involved because of this “at least one strict” issue) • But completeness is utopian 71 Majority vote Condorcet showed that even if all individuals have complete and transitive preferences, the majority vote of them as a society might not be transitive Marie Jean Antoine Nicolas de Caritat, Marquis of Condorcet (1743-1794) 72 Condorcet’s Paradox Individuals 1,2,3 ; alternatives π₯, π¦, π§ Preferences are given by: 1 2 3 π₯ π¦ π§ π¦ π§ π₯ π§ π₯ π¦ 73 Condorcet’s Paradox – cont. Majority vote: π₯ 2,3 1,3 1 2 3 π₯ π¦ π§ π¦ π§ π₯ π§ π₯ π¦ π¦ π§ 1,2 74 The social choice perspective Shows that it isn’t trivial to assume that a relation β½ is complete and transitive complete Pareto domination Majority vote transitive + + 75 Indeed, it can happen in one’s mind • If we have different criteria for decision making, and we’re trying to aggregate them • Each can be thought of as an “individual” • Looking for unanimity we may not get completeness • Using majority – we may lose transitivity • The representation result suggests we should aggregate by a numerical trade-off 76 Is the utility unique? …There exists and a function π’: π → ∞, ∞ that represents β½ Can we say that we found the utility of the consumer? Well, all we asked is π₯ β½ π¦ πππ π’ π₯ So π£ π₯ π’ π¦ 10π’ π₯ could also work 77 How unique is the utility? OK, π£ π₯ 10π’ π₯ is just a change of the unit of measurement – we’re used to that And π£ π₯ 10π’ π₯ 15 Also involves “shifting” the zero; as in temperature 78 Cardinal utility If the data allow for any transformation π£ π₯ where π ππ’ π₯ π 0 , but only those, we say that π’ is cardinal As in πΉ π₯ 9 πΆ π₯ 5 32 πΆ π₯ 5 πΉ π₯ 9 160 9 79 But For π₯ β½ π¦ πππ π’ π₯ π’ π¦ to hold we can also use π£ π₯ π£ π₯ if π’ π₯ π’ π₯ πππ π’ π₯ 0 And many others. The function π’ π₯ is only ordinal. 80 Conditions for utility representation – beyond finite For a relation β½ on a countable set π, β½ is complete and transitive IFF There exists and a function π’: π → ∞, ∞ that represents β½ 81 Conditions for utility representation – beyond countable But: let π, π β½ π , π iff π or [π π π and π π′ ] It is complete and transitive 82 Lexicographic preferences Complete? Transitive? π ,π … but has no representation by any realπ, π valued function 83 Why is there no representation? If there were, we would need to have an entire (positive length) interval of utility values between π ,1 π’ π, 0 πππ π’ π , 1 For any π π, 0 π Which is a bit too much for the real line (as the π’ range) to carry 84 Continuity The relation β½ is continuous iff π₯ → π₯ implies that π₯ β½ π¦ whenever (π₯ β½ π¦ for all π) and π₯ βΌ π¦ whenever (π₯ βΌ π¦ for all π) 85 Continuity is very reasonable If π₯ → π₯ then π₯ β½ π¦ whenever (π₯ β½ π¦ for all π) and π₯ βΌ π¦ whenever (π₯ βΌ π¦ for all π) Almost everything we can think of, in terms of physical and physiological mechanisms, is continuous An exception: a vegetarian’s preferences for the amount of meat More generally, meaning may behave discontinuously 86 Lexicographic preferences aren’t continuous For any such π, π₯ β½ π¦ for all π But π¦ π, 1 π₯β½π¦ does not hold – we have π₯ π, 0 π₯ π 1 ,0 π π¦β»π₯ 87 Is continuity reasonable, then? • I would argue that the lexicographic preferences don’t typically appear in reality – apart from the case of endowing quantities with meaning • They do appear in speeches (“we will never risk human lives, but, given that, we will…”) • This may say more about the speeches than about real preferences • Anyway… 88 Continuous utility representation For a relation β½ on π , β½ is complete, transitive, and continuous IFF There exists and a continuous function π’: π → ∞, ∞ that represents β½ 89 Background: Countable and uncountable sets 90 A puzzle You run a hotel with infinitely many rooms β 1,2,3, … They’re all occupied, and a new person comes along and asks to be hosted Can you give them a room? 91 Well, you can: We start with β 1,2,3, … And, say, person 0 The set β∪ 0 has “as many elements” as β 0,1,2,3, … 1,2,3, … 92 “As many elements as’’ We can have a 1-1 mapping between β 1,2,3, … and β∪ 0 0,1,2,3, … β 1 2 3 4 5 … β∪ 0 0 1 2 3 4 … 93 What about all the integers? Again, we can have a 1-1 mapping between β 1,2,3, … and β€ 0, 1, 1, 2, 2, 3, 3, … β 1 2 β€ 0 1 3 1 4 2 5 2 … … 94 And the rationals? β π π π, π ∈ β ∪ 0 , π 0 π 1 3 4 5 … 1 2 1 3 1 4 1 5 … 2 5 … 3 5 … 4 5 … π 1 1 1 1 2 2 1 2 be “counted” too. 3 3 1 3 Consider π, π 4 4 1 4 5 5 1 5 And it turns out they can 0 2 … … 2 2 2 3 1 3 2 4 2 3 3 2 3 4 1 4 3 1 2 2 4 4 4 1 5 2 5 3 5 4 … … … 5 5 1 … … … 95 For any table… We can count the cells… 1 2 4 7 3 5 8 … 6 9 … 10 … … … … … … … … … … … … … 96 And thus β π 1 π, π ∈ β , π 0 can be “counted”, too 2 3 4 5 … 1 2 1 3 1 4 1 5 … 1 2 4 7 2 5 … 3 5 8 … 3 5 … 6 9 … 4 5 … 10 … π 1 1 1 1 2 2 1 2 3 3 1 3 4 4 1 4 5 5 1 5 … … 2 2 2 3 1 3 2 4 2 3 3 2 3 4 1 4 3 1 2 2 4 4 4 1 5 2 5 3 5 4 … … … 5 5 1 … … … … … … … … … … … … … … … 97 So it turns out that The naturals β 1,2,3, … the integers β€ 0, 1, 1, 2, 2, 3, 3, … and the rationals β π, π ∈ β€, π 0 all have “as many elements as” each other They are all countable 98 Admittedly, it’s weird… That a set β 1,2,3, … would have “as many elements as” supersets thereof β€ β 0, 1, 1, 2, 2, 3, 3, … π π π, π ∈ β€, π 0 But we simply don’t have a better definition of the (same) “number of elements” for infinite sets 99 And it can also happen with intervals That a set 0,1 has “as many elements as” its superset 0,2 In fact, π π₯ 0.5π₯ is a 1-1 mapping from 0,2 to 0,1 0,1 0,2 100 Any two intervals would clearly have the same “number of elements” (“cardinality”) In fact, π π₯ π π π π π₯ π π is a 1-1 mapping from π, π to π, π π, π π, π 101 Even infinite and finite And even the non-negative part of the line 0, ∞ has the same cardinality: π π₯ 1 π is a 1-1 mapping from 0, ∞ to 0,1 102 And since we have mappings from the entire line β into ∞, ∞ 1,1 , such as π π₯ ππππ‘π π₯ 103 All intervals of positive length have the same “number of elements” (“cardinality”) Because π π₯ 1 ππππ‘π π₯ 2 is a 1-1 mapping from β 1 ∞, ∞ into 0,1 104 So maybe all infinities are the same? Maybe the reals β ∞, ∞ are also countable? Is there a 1-1 mapping from the reals to the naturals β 1,2,3, … ? 105 Well, they aren’t Even 0,1 ⊂ β ∞, ∞ isn’t countable There is no 1-1 mapping from the reals to the naturals β 1,2,3, … Georg Cantor (1845-1918) 106 isn’t countable Assume it were Then we’d have 0,1 π₯ ,π₯ ,π₯ ,π₯ ,… For each π₯ ∈ 0,1 there is π 1 such that π₯ π₯ Each π₯ can be written in a decimal expansion (not always in a unique way): π₯ 0. π π π π π … Where π ∈ 0,1,2, , … , 9 107 If were countable We’d have π₯ 0. π π π π … π₯ 0. π π π π … π₯ 0. π π π π … π₯ 0. π π π π … We can construct π₯ with π π 0. π π π π … 2 πππ 10 so that π₯ π₯ – for all π 108 Why ? The decimal expansion isn’t unique so that fact that π 1 2 0. 500000 … 1 2 0. 499999 … π isn’t yet a proof that π₯ π₯ … but we get the point 109 Consumer theory 110 Consumer theory: Basic concepts and preview 111 A basic distinction What we can do and what we want to do are logically independent • Aesop’s fox: “sour grapes” • Groucho Marx: “I refuse to join any club that would have me for a member” • The fox is psychologically healthier • But they both commit the same “rationality sin” • And a converse one is “wishful thinking” 112 What can we choose π₯ π¦ – quantity of good 1 – quantity of good 2 π – price of good 1 π – price of good 2 feasible set – what are the possible values we πΌ – income may choose for these variables decision variables – what is up to us to control 113 The budget constraint π¦ πΌ π πΌ π π π¦ π π₯ π₯, π¦ 0 πΌ π₯ 114 The objective function π¦ Maximization of utility : πππ₯ π π₯, π¦ πΌ π We thus ask, which of the points in the feasible set has the highest π value? πΌ π π₯ 115 Indifference curves Even with two goods, it’s hard to visualize the utility : πππ₯ π π₯, π¦ In fact, because utility is only ordinal, it’s not clear we want to visualize it The information we’d miss is not very meaningful anyway… 116 Indifference curves π¦ Connect points with equal utility π π₯, π¦ π Each is an indifference class of β½ π₯ 117 Optimization π¦ πΌ π Match the indifference curves with the feasible set Try to find the highest indifference curve you can still be on πΌ π π₯ 118 Sneak preview Optimality is often found at the equality of slopes which will be identified by the marginality condition: π π π π π π π π or 119 The marginality condition π¦ A basic optimization tool: look for a point with πΌ π equal slopes Generally, neither necessary nor sufficient, but let’s see its logic first πΌ π π₯ 120 Why equate slopes? π¦ If the budget line is steeper than the πΌ π indifference curve… π₯ 121 Why equate slopes? π¦ If the indifference curve is steeper than the πΌ π budget line… π₯ 122 The slope of the budget constraint π π₯ π π₯ π π¦ π π π¦ π π π π πΌ π πΌ 0 π¦ π π π₯ 123 The slope of the indifference curve π π₯, π¦ π π₯ π π π π₯, π¦ π¦ π π π, π¦ π π π π π ≅0 π π π ≅ π π π₯ 124 … So, we want equality of slopes: π π π π or: π π π π 125 The economic meaning of the marginality condition $1 ≅ units of π₯ ≅ utility $1 ≅ units of π¦ ≅ utility 126 Thus… If: π π π π And vice versa if we’ll be better off moving $1 from π¦ to π₯ π π π π Unless… 127 Unless this is impossible, say π π π π and π¦ 0 … which can happen (“corner solution”) 128 Example π₯ # of six-pack of water bottles π¦ # of single water bottles π 15 π 3 price for a single 60 budget for water πΌ price for a six-pack 129 The budget constraint π¦ πΌ π 60 3 15π₯ 3π¦ 60 20 π₯, π¦ – a bundle πΌ π 60 15 4 π₯ 130 Example: linear utility For example π π₯, π¦ 6π₯ π¦ Which can happen: π₯ = six-packs of bottles π¦ = single bottles For such a function corner solutions will not be exceptional 131 Graphically π¦ π π₯, π¦ 6π₯ π¦ The six-packs are a better deal: π πΌ π 15 6∗3 6π Comparing the slopes: 20 π π π πΌ π 4 6, π 1 π π 15 3 5 6 π π 1 3 6 15 π π π₯ 132 What is exactly meant by A partial derivative of a function is the derivative relative to one variable while the others are held fixed The partial derivative of π π₯, π¦ relative to π₯: π₯, π¦ = , = π π₯, π¦ π₯, π¦ = , = π π₯, π¦ and relative to π¦: ? Partial derivatives graphically Examples of partial derivatives For π π₯, π¦ ππ₯ ππ¦ The partial derivatives are π π₯, π¦ π π₯, π¦ π π And for π π₯, π¦ ππ₯π¦ We get π π₯, π¦ π π₯, π¦ ππ¦ ππ₯ The economic meaning of π π π π $1 ≅ units of π₯ ≅ utility $1 ≅ units of π¦ ≅ utility 136 We could also have… π¦ π π₯, π¦ 6π₯ π¦ And if π πΌ π π π Can π πΌ π 6 6π π π 6π happen??? π₯ 137 Finally… π π₯, π¦ 6π₯ π¦ π¦ If π π π πΌ π 6π 6 π π No unique solution πΌ π π₯ (isn’t it a knife-edge case?) 138 Summing up The solution to the water consumption problem is: If If And if 6 6 π₯, π¦ πΌ ,0 π π₯, π¦ πΌ 0, π 6 – anywhere in πΌ πΌ , 0 , 0, π π Decreasing marginal utility • In the water bottles problems, the marginal utilities were constant • It might be more intuitive that the “extra utility” we get from a good decreases as we have more of it The utility from money Back in 1738, Daniel Bernoulli wrote, “And, because the marginal utility from money is in inverse proportion to the amount of money we have…” Daniel Bernoulli 1700-1782 141 “The marginal utility is inversely proportional…” π’′ π₯ π’ π₯ π∗ π ∗ πππ π₯ πππ π₯ π π 0 π 0 πππ π₯ 142 But what do you mean, Daniel? Rudolf Carnap (1891-1970) Karl Popper (1902-1994) • Theoretical terms should be defined by observations • How do you measure this “marginal utility”??? 143 Two possible answers: • Look, guys, you’re going to talk about this 200 years after me. Com’n. • In between us, there will be a psychologist who will show that more is observable than what these economists will choose to admit Daniel Bernoulli 1700-1782 144 Weber’s law in psychophysics Ernst Heinrich Weber 1795-1878 βπ π π π – stimulus βπ – increase in the stimulus (that can be discerned with a fixed probability, usually 75%) λ – a positive constant Weber’s law – cont. A person would notice, with probability 75% or more, that a change has occurred, namely that π βπ π Only if the physical change is large enough π βπ π 1 π Or: πππ π βπ πππ π πππ 1 π – a constant The function Hence, for physical quantities, the πππ function plays a special πππ π₯ role π₯ A comment re Unless otherwise stated, we’ll take the base of πππ to be π πππ π₯ πππ π₯ πππ π₯ Recall that any other base π 1 is a positive multiple thereof : πππ π₯ π₯ πππ π₯ πππ π πΎπππ π₯ for πΎ 0 Cobb-Douglas Preferences (After Charles Cobb, Paul Douglas) π π₯, π¦ We started with 6π₯ π¦ Or, more generally, a linear function: π π₯, π¦ ππ₯ ππ¦ π, π 0 We can now look at a simple function that allows for decreasing marginal utility: π π₯, π¦ ππππ π₯ ππππ π¦ π, π 0 Indifference curves for Cobb-Douglas π π₯,π¦ π¦ ππππ π₯ For example, for π π π₯, π¦ π π π π π₯ πππ π₯ ππππ π¦ π 1 πππ π¦ π π₯, π¦ π βΊ π πππ π₯π¦ βΊ π₯π¦ π πππ π₯π¦ Decreasing marginal utility π¦ Suppose 100 π π₯, π¦ πππ π₯ πππ π¦ Consider the indifference curve π π₯, π¦ 10 πππ π₯ π₯π¦ 10 100 π₯ πππ π¦ 3 1000 and two points on it 10,100 , 100,10 151 Indifference curves become less steep (going from left to right) π¦ Imagine that at 10,100 we reduce π¦ by 1. How much π₯ 100 should we add to compensate for this reduction? 10 10 100 π₯ 152 Let’s use the partial derivatives π π₯, π¦ πππ π₯ πππ π¦ And thus π π₯, π¦ π π₯, π¦ (For πΎ πππ′ πππ′ π₯ πΎ π₯ π¦ πΎ π¦ 1/πππ 10 ) 153 At The partial derivatives (marginal utilities) are π¦ As π¦ decreases from 100 to 99 the utility loss is 100 approximately and in order to compensate fot that we need extra π of product 1 (increase π₯ ) that satisfies, roughly, 10 π 10 100 π₯ π 154 By contrast, at The partial derivatives (marginal utilities) are π¦ As π¦ decreases from 10 to 9 the utility loss is approximately 100 and in order to compensate fot that we need extra π of product 1 (increase π₯ ) that satisfies, roughly, 10 π 10 100 π₯ π 10 155 A general point π¦ When the marginal utility of each product is decreasing, we will have indifference curves that are less steep as we go from upper left to lower right π₯ 156 Optimal solution for CD preferences π π₯, π¦ π¦ πΌ π ππππ π₯ π , π π₯ π π π πΌ π π₯ ππππ π¦ π π₯ π π¦ π π¦ ππ¦ ππ₯ 157 The slope of the indifference curves for CD π¦ When π₯ tends to 0 (and π¦ doesn’t) they become very steep When π¦ tends to 0 (and π₯ doesn’t) they become very flat The slope is the same along any ray from π₯ the origin (homothetic preferences) 158 Homothetic preferences π¦ Along every ray that starts at the origin 0,0 , the slope of all indifference curves is the same But it can change from one ray to another π₯ 159 Solving the CD problem – cont. Looking for π₯, π¦ where: π π₯ and or π π¦ π π π π ππ¦ ππ₯ π π πΌ 160 Solving the CD problem – cont. ππ¦ ππ₯ π π π₯ βΉ π π π¦ π π Let’s denote πΈ πΈ So that πΈ πΈ π π₯ π π¦ π π which has to hold for all π , π , πΌ ! 161 Solving the CD problem – nearly done… πΈ πΈ πΈ πΈ π π π π₯ Hence πΈ πΈ π π¦ π π π π π π πΌ πΌ πΌ 162 (Was that too quick?) πΈ πΈ π π βΉ π πΈ π πΈ Plug these in πΈ to get and thus π πΈ π πΈ π π₯ π π πΈ πΈ π π π π π and πΌ πΌ 1 πΈ πΈ πΈ π π¦ πΌ π πΌ πΌ π π πΈ π πΌ π π π πΌ 163 Solving the CD problem – wrapping up πΈ π π₯ π π π πΌ βΉπ₯ πΈ π π¦ π π π π 1 πΌ ππ π 1 πΌ ππ π πΌ βΉπ¦ π 164 Consumer theory: Lagrange Multipliers 165 Or, using Lagrange multipliers The “real” problem πππ₯ π π₯ π π₯, π¦ , π π¦ π₯, π¦ πΌ 0 166 Simplify our lives π¦ If π is monotone (the consumer πΌ π prefers more to less), it’s safe to assume the solution will be on π π₯ Can’t be optimal πΌ π π π¦ πΌ π₯ 167 Let’s simplify our lives even further Let’s ignore the non-negativity constraints π₯, π¦ 0 (Make a mental note not to forget these) And then there’s only one constraint, which is an equality: πππ₯ π . π‘. , π π₯ π π₯, π¦ π π¦ πΌ 168 Lagrange’s idea • We’ll build the constraint into the utility function • As if it could be violated, though at a cost • At the optimal solution it won’t be violated after all • But the trick will also have an economic meaning Joseph-Louis Lagrange 1736-1813 169 Solving using Lagrange multiplier πππ₯ π . π‘. , π π₯ π π₯, π¦ π π¦ πΌ Becomes πππ₯ β π₯, π¦, π , , π π₯, π¦ β π₯, π¦, π ππ π₯ π π¦ πΌ π – a new variable, the cost of violating the constraint 170 Lagrange’s method – cont. To find πππ₯ , , β π₯, π¦, π we take all partial derivatives and set them to zero πβ π₯, π¦, π ππ₯ 0 πβ π₯, π¦, π ππ¦ 0 πβ π₯, π¦, π ππ 0 171 The partial derivatives β π₯, π¦, π πβ ππ π π₯, π¦ ππ π₯ πβ ππ₯ π ππ πβ ππ¦ π ππ π π₯ π π¦ π π¦ πΌ πΌ 172 Setting them to zero πβ ππ πβ ππ₯ π ππ 0 πβ ππ¦ π ππ 0 π π₯ π π¦ πΌ 0 173 The resulting equations πβ ππ₯ πβ ππ¦ πβ ππ 0 0 0 π π₯ π π π π¦ ππ βΉ π π π ππ βΉ π π π πΌ βΉπ π₯ π π¦ πΌ 174 Conclusions πβ ππ 0βΉπ π₯ πβ ππ₯ πβ ππ¦ 0 βΉπ π π 0 βΉπ π π π π π π π π¦ πΌ π π π π βΉ π π π π 175 And in the CD problem We are again looking for π₯, π¦ where: π π₯ and or π π¦ π π π π ππ¦ ππ₯ π π πΌ 176 And the solution is, again πΈ π π₯ πΌ βΉπ₯ πΌ πΈ π π¦ πΌ βΉπ¦ πΌ Clearly, we didn’t need Lagrange here, but in more complex problem his method can really help… 177 Consumer theory: Ordinality 178 Ordinality Recall that the utility function is (only) ordinal • We won’t take the specific function too seriously • Any monotone transformation thereof is equally good • We should better verify that we only discuss properties that are common to all such transformations 179 “How unique” is ? We won’t be able to tell π¦ π π₯, π¦ log π₯ log π¦ from log π₯ log π¦ π₯π¦ π π₯, π¦ 5π₯π¦ or π π₯π¦ π π₯, π¦ or even π π₯ π π₯, π¦ 5π₯π¦ 10 180 Yet, the slope of the indifference curve is the same π ? π π ∗ πππ π₯ π¦ π π₯, π¦ π π₯ π π¦ π π π π₯, π¦ π₯ π π π¦ ππ₯ ππ₯ π¦ π ∗ πππ π¦ ππ¦ ππ₯ π₯ π¦ ππ¦ ππ₯ 181 More generally: If π is monotonically increasing π π₯, π¦ π π π₯, π¦ π π π ∗ π π π π ∗ π π π π′ ∗ π π′ ∗ π π π … a great relief 182 Hence the marginality condition π π π π Does not depend on the transformation because the slope is independent of π 183 But ... How about ? • The transformation π (π π₯, π¦ in the same way: multiply by π π π π₯, π¦ ) will modify both sides 0 • The values on both sides can change, but whether they’re equal or not – will not change • (Nor will the answer to the question, “which one is larger?”) 184 … Therefore… π π₯, π¦ π ∗ πππ π₯ π π₯, π¦ π π₯, π¦ πΌ log π₯ 1 π π₯, π¦ π ∗ πππ π¦ π₯ π¦ πΌ log π¦ , πΌ π π π π₯ π¦ … all describe the same preferences, and we can switch among them shamelessly 185 The normalized CD function π π₯, π¦ πΌ log π₯ π π₯ πΈ π π¦ πΈ πΌ log π¦ πΌπΌ 1 πΌ πΌ 1 πΌ πΌ π π₯ π¦ 1 1 1 πΌ πΌ π 186 Consumer theory: Monotonicity 187 Monotonicity of In bold strokes, “more is preferred to less” But: • What exactly is “more”? More π₯, more π¦, more both? • What exactly is “preferred to”? Strictly better? Just not worse? 188 Basic monotonicity π cannot decrease in any of the variables: If π₯ π₯ then π π₯, π¦ and π¦ π¦′ π π₯ ,π¦ Typically justified by free disposal 189 Free disposal If you don’t like it – throw it away • The quantities π₯, π¦ designate what’s legally yours, not necessarily what went into your stomach • Used to be less of an issue when I was a student • Less obvious when we think about the environment • And can even be an emotional problem (if we cherish values that are compromised by production/consumption of these goods) 190 Basic monotonicity – cont. Obviously allows π¦ π π₯, π¦ π₯ π¦ But also 20 π π₯, π¦ min π₯, 10 min π¦, 20 – the consumer can reach satiation 10 π₯ 191 Satiation • A type of nirvana • Luckily, doesn’t happen too naturally • Luckily? 192 A preview of the welfare theorem We will discuss general equilibrium And will find out that, under certain conditions, it is “nice” in the sense of Pareto The First Welfare Theorem: A general equilibrium yields Pareto optimal allocations 193 Pareto efficiency/optimality • An allocation is Pareto optimal/efficient if we can’t make some people better off without hurting others • Says nothing about justice or fairness Vilfredo Pareto 1848-1923 194 “Efficient” or “Optimal”? • “Efficient” sounds like we only try to produce as much as possible, and that’s not the case • "Optimal” sounds like it’s the “best”, at least as good as anything else – and it only means that there’s nothing better 195 The First Welfare Theorem • In any event, Pareto optimality/efficiency is a nice property to have • The First Welfare Theorem says that any allocation that is the result of a general equilibrium has this property • But consumer who reach satiation can destroy it • That’s why, as economists, we see something positive in the fact that people don’t reach satiation so easily… 196 Strict monotonicity π is strictly increasing in each variable: If π₯ π₯ and π¦ π¦′ or π₯ π₯ and π¦ π¦ then π π₯, π¦ π π₯ ,π¦ – necessarily satisfies basic monotonicity as well 197 Is strict monotonicity plausible? • How much water can you drink? • In many good we will reach satiation • Even if we still want something else (diamonds?) 198 Weak monotonicity Basic monotonicity + If π₯ π₯ and π¦ π¦′ then π π₯, π¦ π π₯ ,π¦ There may be satiation in some goods, but not in all 199 Example: weak but not strict monotonicity π¦ π π₯, π¦ 10 min π₯, 10 π¦ π₯ 200 Weak monotonicity suffices • For the consumer to be on the budget line π¦ • …and wish to sell any extras in the market • For the equilibria to be Pareto π₯ efficient/optimal 201 Consumer theory: Convexity 202 Problems • There are more complex feasible sets • The marginality condition doesn’t always help 203 More interesting “budget” sets Suppose you have to decide how many movies and how many theater shows to watch Good Price Minutes π₯ movie 40 120 π¦ theater 100 60 Budget 400 600 204 The budget(s) set π¦ 40π₯ 100π¦ 400 120π₯ 60π¦ 600 10 π₯, π¦ 4 0 3.75,2.5 5 10 π₯ 205 Let’s maximize utility π¦ Max π π₯, π¦ π₯ . π¦ . 10 4 3.75,2.5 5 40π₯ 100π¦ 400 120π₯ 60π¦ 600 π₯, π¦ 10 0 π₯ 206 Looking for a tangency point, say… π¦ Max π π₯, π¦ π₯ . π¦ . 10 πππ 120π₯ 4 3.75,2.5 5 ππππ πππ 60π¦ 600 π₯, π¦ 10 0 π₯ 207 Indeed, π¦ Max π π₯, π¦ 10 πππ π₯ 4 3,2.8 3.75,2.5 5 π¦ 10 π₯ π 1 πΌ π 1 π π₯ ππππ . π¦ . πππ 0.3 1 400 40 3 πΌ 0.7 400 2.8 The tangency with the first line is within the relevant range and we’re happy 208 But if preferences were different… π¦ Max π π₯, π¦ 10 πππ π₯ 4 3.75,2.5 π¦ π 1 πΌ π 1 π π₯ ππππ . π¦ . πππ 0.8 1 400 40 8 πΌ 0.2 400 0.8 8,0.8 5 10 π₯ The tangency point with this line is outside the range and we’re very unhappy 209 Looking for tangency with the other segment π¦ Max π π₯, π¦ 10 ππππ π₯ 4 3.75,2.5 4,2 5 π¦ 10 π₯ π 1 πΌ π 1 π . π₯ πππ . π¦ πππ 0.8 1 600 120 πΌ 0.2 4 600 2 And again there’s tangency with one line that’s in the relevant range (for this line) and we’re happy 210 Will this always work? π¦ Max π π₯, π¦ 10 πππ π₯ 4 3.75,2.5 5,2 5 π¦ 10 π₯ π 1 πΌ π 1 π π₯ ππππ . π¦ . πππ 0.5 1 400 40 5 πΌ 0.5 400 2 The tangency point with this line is again outside the range and we’re again unhappy 211 On the other hand… π¦ Max π π₯, π¦ 10 ππππ π₯ 2.5,5 4 3.75,2.5 5 π¦ 10 π₯ π 1 πΌ π 1 π . π₯ πππ . π¦ πππ 0.5 1 600 120 πΌ 0.5 2.5 600 5 The tangency point with the other line is also outside the relevant range and we’re very unhappy 212 So what’s going on? π¦ Max π π₯, π¦ 10 4 3.75,2.5 5 10 π₯ π₯ . π¦ 40π₯ 100π¦ 400 120π₯ 60π¦ 600 π₯ 3.75 π¦ 2.5 . Well, we can’t call it tangency, but we do have separation 213 More generally π¦ • When there are several linear inequality and they all have to be satisfied (“and”) we look for tangencies • If one of them is in the relevant range we’re happy • If not, we look at the extreme points • (We don’t need to look at all of them – if one of them is “in between” slope we’re done) • (And something like this works in higher dimensions, too) π₯ 214 Encouraged and cheered up, Let us now assume that π π 1, πΌ 200 But there are discounts for large quantities: above 100 the price of π₯ per unit goes down by 50% π if π₯ 100 215 Which bundles are feasible Distinguish between π¦ 100 π₯ 100 and 200 150 π₯ π 1 π 0.5 100 100 200 300 π₯ 216 The budget set In the range π¦ π₯ 100 the price is 200 π 1 π¦ 200 and the budget line: 150 π₯ 100 And, as usual, π₯, π¦ 100 200 300 0 π₯ 217 The budget set – cont. In the range π¦ π₯ 100 the price is π 200 0.5 and the constraint is 150 0.5 π₯ 100 100 π¦ 200 100 100 (Because we already spent 100 on the first 100 units) or 100 200 300 0.5π₯ π₯ (and π₯, π¦ π¦ 150 0) 218 The budget set, therefore: π¦ And we see we could also write π₯ π¦ 200 200 or 0.5π₯ 150 π¦ 150 and, as usual 100 π₯, π¦ 100 200 300 0 π₯ 219 Let’s maximize π¦ Max π π₯, π¦ π₯ 200 π₯ π¦ . π¦ . 200 or 150 0.5π₯ 100 100 200 300 π₯ π¦ π₯, π¦ 150 0 220 Looking for tangency π¦ Max π π₯, π¦ π₯ . π¦ . Let’s try tangency with 200 150 π π₯ 90,110 100 π¦ 100 200 300 π₯ π πππ 1 1 πΌ 0.45 200 90 π 1 1 π πΌ 0.55 200 π 110 We made it! 221 But there’s another tangency point π¦ Max π π₯, π¦ π₯ . π¦ . Tangency with 200 150 π. ππ π₯ 90,110 π¦ 135,82.5 100 100 200 300 π₯ π πππ 1 1 πΌ 0.45 150 135 π 0.5 1 π πΌ 0.55 150 82.5 π … and in this case 135 . 82.5 . 102.96 . . 90 110 100.50 222 Generally π¦ • If there are several linear inequalities and only one should be satisfied (“or”), we will look for all tangency points • We will need to compare them • And the intersection points • (We can save a bit: the intersection between two segments will not be better than both tangency points on them) π₯ 223 An “or” condition between inequalities π¦ • May appear when there are discounts • Or when we can buy in one of several markets, but not to mix between them • Or to order from one of several suppliers… π₯ 224 Convex sets For every two points in the set, the entire interval connecting them is also in the set 225 Non-convex sets There exists at least one pair of points in the set, such that some of the interval connecting them is outside the set 226 The interval connecting two points π¦ 5 For example, the interval connecting 1,5 1,5 and 3,1 3,1 1 1 3 π₯ 227 The interval formula – cont. π¦ 5 Consider, for example, the mid-point 1,5 What’s its π₯ value ? 1 2 3,1 1 1 2 3 3 2 π₯ 228 And, similarly π¦ 5 The midpoint’s π¦ value is 1,5 5 1 2 3 3 3,1 1 1 3 π₯ 229 In short, π¦ 5 The midpoint is 2,3 1,5 In other words, the average values: 2,3 3 1 1,5 2 3,1 1 3,1 2 2,3 1 1 2 3 π₯ 230 And what about other points? π¦ 5 Their coordinates are weighted average of 1,5 those of the extreme points – while always using the same weights π¦ π 1,5 3,1 1 π∗1 3 π₯ π 3,1 π ∗ 3, π ∗ 5 1 π 1 we get 1,5 and for π 0 we get 3,1 For 1 π₯ 1 1 π ∗1 231 Convex budget sets π¦ Obviously, the classic one: π π₯ π π¦ π₯, π¦ πΌ 0 π₯ 232 As well as… π¦ Anything we can get by intersection (“and”) of linear inequalities (or, more generally, the intersection of any convex sets) π₯ 233 But not… π¦ The union of convex sets need not be convex π₯ 234 A non-convex budget set π¦ Can be problematic: the marginality condition is no longer sufficient for optimality π₯ 235 Non-convex preferences π¦ π π₯, π¦ π₯ π¦ Increasing marginal utility (in each variable): π 2π₯ π 2π¦ π₯ 236 The marginality condition without convexity π¦ π π₯, π¦ π₯ π¦ The tangency point might be the worst point on the line π₯ 237 Examples of increasing marginal utility π π₯, π¦ π₯ π¦ • π₯, π¦ – minutes of watching a movie • π₯, π¦ – amount of heroin and cocaine • π₯, π¦ – practice time of two athletes for the Olympic games 238 Convex preferences π¦ The “better than” sets π π₯, π¦ π are convex, (for every π) π₯ 239 How can we tell if preferences are convex? π¦ • Drawing “better than” sets • Comparing the slope (marginal rate of substitution) of the curve along it • And another useful rule: π₯ 240 A sufficient condition for convex preferences If it so happens that π π₯, π¦ π π₯ π π¦ where each of π and π is concave (in its own variable) π π₯, π¦ π π₯, π¦ π π₯, π¦ π∗π₯ π ∗ πππ π₯ π∗π₯ π∗π¦ π ∗ πππ π¦ π ∗ πππ π¦ … then the preferences are convex 241 How come that concave imply convex preferences? π¦ • Assume they are concave • Decreasing marginal utility of each product • And we get a less steep indifference curve as we slide downwards (and to the right) π₯ 242 The importance of convexity • As we just argued, it simplifies our lives as economists who try to predict choices • But it also makes the optimization story more likely • How does the household “behave as if” it were maximizing a utility function? • Under convexity: small improvements would lead to an optimal solution 243 Consumer theory: Comparative statics / Sensitivity analysis 244 Consumer theory: Changes in income 245 Changing income • ICC—Income-Consumption Curve • Its habitat is the consumption bundles space • Income isn’t represented graphically • Ernst Engel 1821-1896 Engel Curve • Lives in the Income-Good (quantity) space • The quantities of the other goods are not represented graphically 246 The ICC for CD preferences π₯ π¦ π 1 πΌ π 1 π 1 πΌ π Recall that these preferences are homothetic: π¦ ICC The slope is constant along any ray that emanates from the origin 0,0 π₯ 247 The Engel Curve Consider the optimal solution π π₯ π¦ π π π° ππ 1 π 1 πΌ π Focus on the demand for one good and observe how it changes as a function of income πΌ 248 The Engel curve and income elasticity π π¦ π π π° ππ 1 1 πΌ π π π π₯ π = = 1 For CD preferences, income elasticity is 1 ≡ 1 if and only if the demand for π₯ is a linear function of income, that is π₯ πΌ ππΌ for some π 249 The general concept of elasticity • Given a function π§ π§ π€ we wonder how sensitive π§ is relative to changes in π€ • We have the (partial) derivative ππ§ ππ€ • But we want a “pure” measure, independent of measurement units: ππ§ ππ€ π , π§ π€ 250 Elasticity π , ππ§ ππ€ π§ π€ ππ§ π§ ππ€ π€ Constant elasticity: π§ ππ€ , π 0 βΊ π π , For instance: π§ π§ ππ€ βΊ π , π βΊ π , π€ 1 1 251 Constant Elasticity If π§ Then π , ππ§ ππ€ π§ π€ ππ€ πππ€ ππ€ /π€ πππ€ ππ€ π 252 Constant Elasticity – cont. π And if π , Then c· π log π§ for and log π§ π π π · log π€ π · π log π€ π log π€ π§ log π log ππ€ ππ€ 253 Responses to income changes π , 0 – π₯ increases, a “normal” good π , 0 – π₯ doesn’t change, a “neutral good” π , 0 – π₯ decreases, an “inferior” good in a certain range … why “in a certain range” ? 254 Further distinction among normal goods: π , 1 – a luxury good π , 1 – a proportional good 0 π , 1 – a basic/essential good 255 For CD preferences We got π , π , 1 And this makes sense: if there are only two goods, and one has income elasticity of 1, so should the other one: π π₯ π π¦ πΌ Suppose we increase income by 1% π 1.01 π₯ π 1 ? π¦ 1.01 πΌ 256 More generally Some weighted average of income elasticities (across all goods) is equal to 1 απ , 1 α π , 1 Hence it is impossible that all goods be luxury goods Or that all be basic goods 257 The ICC for linear preferences π π₯, π¦ ππ₯ ππ¦ Surely homothetic: π¦ The slope is constant not only along each ray (emanating from the origin), but also across rays π₯ 258 ICC for linear preferences – cont. If we have π π π¦ π π That is ICC π π π π The solution is π₯ π¦ π₯ 0 1 πΌ π 259 Engel Curve for linear preferences π π₯, π¦ ππ₯ ππ¦ π₯ The optimal solution is Say, π¦ 0 1 πΌ π π₯ π¦ πΌ πΌ 260 The ICC for linear preferences… And if π π π¦ π π That is π π π π The solution is ICC π₯ π₯ 1 πΌ π π¦ 0 261 Engel Curve for linear preferences π π₯, π¦ ππ₯ ππ¦ π₯ 1 πΌ π π¦ 0 The optimal solution is Say, π₯ π¦ πΌ πΌ 262 The ICC yet again Wait, but what if π π π¦ π π That is, π π ICC ? π π Any point on the budget line is a solution and the ICC becomes the entire orthant π₯ 263 Consumer theory: Changes in price 264 Changing price • • • PCC (Price Consumption Curve) • Resides in the bundles space • Price isn’t represented graphically Demand curve • Resides in the price-quantity space (for a given good) • The other quantities are not represented graphically Demand and cross demand curve 265 The PCC for CD preferences π₯ π¦ π 1 πΌ π 1 π 1 πΌ π π¦ πΌ π PCC 1 π₯ 266 Demand curve π₯ π₯ π 1 πΌ π = (π ≡ 1 = hyperbola ) π 267 Cross demand curve π¦ π¦ 1 π 1 π 1 πΌ π 1 πΌ π = π ≡0 constant) π 268 The slope of the demand curve π₯ We’d expect And this is indeed typical. Almost always true. Why almost? π 269 What happens when a price changes? Suppose π ↑ π′ πΌ π πΌ π′ π • The budget line is “tighter” • Its slope changes, too πΌ π 270 Income and substitution effects π′ π π₯ π₯ Indeed πΌ π Why? A B π₯ πΌ π′ π₯ • The consumer is “poorer” • The price ratio has changed πΌ π 271 Trying to tell these apart Budget line C goes through the old point, but with the πΌ π new slope C π₯ π₯ π₯ π₯ π₯ π₯ Overall change = income effect + substitution effect A B π₯ πΌ π′ π₯ πΌ π 272 For example π¦ π π₯, π¦ 60 π¦ πΌ log π₯ log π¦ 120, π 2, π π₯ πΌ · · 120 30 π¦ πΌ · · 120 30 2 A 30 π₯ 30 60 π₯ 273 Suppose the price has gone up π¦ π 60 3 2 π From budget line A π΄ 2,2,120 → 30,30 We switch to line B π¦ B 30 π₯ 20 π₯ A π΅ 3,2,120 → 20,30 30 60 π₯ 274 Introduce the third budget line πΆ 3,2, ? The price are the new ones 3,2 Which level of income would go through π΄ 30,30 ? πΌ 3 · 30 2 · 30 90 60 150 275 Hence we compare between… π΄ 2,2,120 → 30,30 60 π΅ 3,2,120 → 20,30 C πΆ 3,2,150 → 25,37.5 A 20 B 30 20 25 25 30 Overall change = income effect + substitution effect π₯ 20 π₯ π₯ 30 25 60 276 The substitution effect cannot be positive π₯ If the price of a good goes up, the substitution effect will not make us C want more of it A π 277 The substitution effect can be zero π₯ Say π π₯, π¦ π π₯, π¦ min π₯, π¦ π A,C π 278 Income effect Price increase real income has gone down The income effect is • negative for a normal good • zero for a neutral good • positive for an inferior good For a normal good the income and substitution effects are in the same direction 279 Giffen goods • For inferior goods, the income and substitution effects are in opposite directions • Typically, the substitution effect is stronger (that’s an empirical fact) • If this isn’t the case, the good is referred to as a Giffen good. Robert Giffen 1837-1910 π 0 280 Let us not confuse Giffen goods with • Uncertainty about quality (a $10 Rolex) • conspicuous consumption 281 Compensations: Slutsky and Hicks Back to the example 60 π π₯, π¦ πΌ C 37.5 π A 30 log π₯ B 2, π′ log π¦ 120 3, π 2 Suppose that the consumer is compensated so that she can consume bundle A π₯ 20 π₯ π₯ 30 25 60 282 What’s the impact on utility? π , π , πΌ → π₯, π¦ → π π₯, π¦ π π₯, π¦ π π₯, π¦ π₯π¦ log π₯ log π¦ π΄ 2,2,120 → 30,30 → 900 π΅ 3,2,120 → 20,30 → 600 πΆ 3,2,150 → 25,37.5 → 937.5 Evgeny Evgenievich Slutsky 1880-1948 150 – the compensated income, according to Slutsky Slutsky compensation: 150-120=30 283 Neither shocked nor upset π¦ • 60 It’s natural that rotating the budget line around a point (A) would change the optimal bundle C • A 60 And that’s perfectly fine with us… π₯ 284 However π¦ • Maybe the compensation shouldn’t be 60 that high? • Maybe it’s enough to go back to the old D utility level, rather than the old physical A quantities (which are no longer 30 optimal)? 30 60 π₯ 285 The compensation according to Hicks Consider the table again π ,π ,πΌ → π₯, π¦ π΄ 2,2,120 → 30,30 → 900 π΅ 3,2,120 → 20,30 → 600 πΆ 3,2,150 → 25,37.5 → 937.5 π· 3,2, ? → π₯, π¦ → π π₯, π¦ John Hicks 1904-1989 → 900 286 Hicks compensation – cont. For 3,2, πΌ we have 1 1 · ·πΌ 2 3 1 1 · ·πΌ 2 2 π₯ π¦ To get π π₯, π¦ π₯π¦ 1 πΌ 6 1 πΌ 4 900 we will require 1 1 πΌ· πΌ 4 6 πΌ 900 · 24 πΌ 900 21,600 146.96 287 The compensations of Slutsky and Hicks Slutzky: changes income to be able to be consume the original bundle Hicks: changes income to be able to be consume at the original utility level Hicks' sounds more fair, but requires knowledge (/estimation) of the utility function 288 The PCC for linear preferences π π₯, π¦ ππ₯ π¦ πΌ π ππ¦ If the solution is If the solution is 0, ,0 PCC 1 If π₯ it's any point in between 289 The demand "function" π₯ For π π it's π₯ For π π it's π₯ (And for π π 0 any value 0 π₯ ) The elasticity is π π π π π ≡ 1 ππ and not well in the range π defined outside it 290 The cross demand "function" π¦ For π π it's π¦ For π π it's π¦ (And for π π 0 any value 0 π¦ ) πΌ π The elasticity is π π π π π ≡0 ππ and not well in the range π defined outside it 291 An example of a Giffen good π π₯, π¦ min π₯ π¦, 100 0.5π₯ Both goods can satisfy hunger, but only good 1 has nutritional value π₯ – amount of nuts π¦ – amount of Styrofoam π₯ π¦ – total amount that fills the stomach 0.5π₯ – amount of nutritional food 100 0.5π₯ – the addition of 100 guarantees that the consumer starts seeking nutrition only after the hunger is somewhat satisfied 292 Describing the preferences π¦ π π₯, π¦ min π₯ π¦, 100 0.5π₯ Which need is the dominant one? Hunger or π π₯, π¦ 100 0.5π₯ nutrition? This depends on π₯ 100 π¦ β 100 0.5π₯ or π π₯, π¦ π₯ π¦ β 100 π¦ 200 0.5π₯ π₯ 293 Indifference curves π¦ π π₯, π¦ min π₯ π¦, 100 0.5π₯ We distinguish between the two regions: π π₯, π¦ 100 0.5π₯ π₯ 100 π¦ β 100 π¦ β 100 π π₯, π¦ π₯ π¦ 200 0.5π₯ 0.5π₯ π₯ 294 Let’s get the budget line into the picture π¦ π π₯, π¦ min π₯ π π₯ π π₯, π¦ 100 0.5π₯ π¦, 100 π π¦ 0.5π₯ πΌ And if π π 100 The consumer will only buy the first good π π₯, π¦ π₯ π¦ 200 π₯ 295 And this will be true for any income π¦ As long as π π π₯, π¦ 100 0.5π₯ π Only the first good is consumed 100 π π₯, π¦ π₯ π¦ 200 π₯ 296 The interesting case π¦ If π π π₯, π¦ 100 0.5π₯ 100 π – for low income the optimal bundle is on the π¦ axis – for high income the optimal bundle is on the π₯ axis – and in between on the dotted line π π₯, π¦ π₯ π¦ 200 π₯ 297 The ICC π¦ If π π – The ICC will first climb up the π¦ axis – then it will slide down the dotted line 100 – and finally – flatten onto the π₯ axis 200 π₯ – along the dotted line the second good is inferior 298 Next consider a change in price Let’s start with π and increase π π¦ π , πΌ 200π At low prices the optimal solution will be on the dotted line 100 When π is high enough, the optimal solution will be on the π¦ axis πΌ/π 200 π₯ And when π is higher (above π ) – on the π₯ axis 299 A possible PCC π¦ Importantly, there is a range in which it goes down In this range, an increase in the price of 100 good 2 (π ) results in an increase in the demanded quantity π¦ 200 π₯ 300 Endowments 301 Income as endowments • In a general equilibrium, there is no “income” • Rather, there is an initial endowment • A bundle of good that can be sold in the markets • Including leisure that will be sold in the labor market 302 The budget constraint π₯ – initial endowment of good 1 π¦ – initial endowment of good 2 π , π – prices π π₯ π π¦ π π₯ π π¦ π₯ π π¦ π¦ or π π₯ 303 Graphically π¦ π π₯ π¦ π₯ π π¦ π π₯ π π¦ π₯ 304 Optimization π¦ If, say, we pick π₯, π¦ so that β«ΧΧΧ§ΧΧ©β¬ π¦ π¦ π₯ π¦ β«ΧΧΧ¦Χ’β¬ π π₯ π₯ supply in market 1 π₯ π₯ π₯ π¦ π π¦ π¦ demand in market 2 π₯ 305 Will the consumer offer and demand or the other way around? π¦ Compare slopes: π π₯, π¦ π π₯, π¦ with π π π₯ 306 The effect of preferences π¦ For a given budget line with slope we can find preferences such that π₯ π¦ π₯ π¦ π¦ as well as preference for which the opposite holds π₯ π₯ 307 The effect of prices π¦ For given preferences, we can typically find prices such that π₯ π¦ π₯, π¦ π¦ as well as prices for which the opposite holds π₯ π₯ 308 Why “typically”? π¦ π π₯, π¦ π¦= 4 min π₯, π¦ π₯ 6 π¦ 4 Here the indifference curve has a slope of zero, and for any positive prices the consumer will wish to sell good 1 and buy good 2 π₯ 6 π₯ 309 Example π₯ – pears π¦ – apples π₯ π π₯, π¦ 10, π¦ 2log π₯ 10 log π¦ 310 A reminder – CD solution π π₯, π¦ π πlog π₯ π 1 π₯ π 1 π π₯ π π¦ π π₯ π π¦ ππΌ 1 π 1 π log π¦ = 1 π¦ π πΌ π π π₯ π π¦ πΌ π₯ π¦ π 1 1 πΌ π 1 π π 1 πΌ π 311 If pears are (relatively) expensive π¦ π πΌ π¦ 3 3 ∗ 10 1 ∗ 10 1 40 π₯ 2 1 πΌ 3π 2 1 · · 40 3 3 8.88 π¦ 1 1 πΌ 3π 1 1 · · 40 3 1 13.33 supply of pears π₯ π₯ demand for apples π¦ π₯ π 10 π¦ 8.88 13.33 1.11 10 3.33 π₯ 312 And if apples are (relatively) expensive π¦ π πΌ 2 2 ∗ 10 π₯ 2 1 πΌ 3π π¦ 1 1 πΌ 3π π¦ demand for pears π₯ supply of apples π¦ π₯ π 2 1 · · 70 3 2 1 1 · · 70 3 5 π₯ π¦ 5 ∗ 10 23.33 10 4.66 5 70 23.33 4.66 10 13.33 5.33 π₯ 313 The Labor Market 314 A simple model πΏ – leisure (in hours/day) π – an aggregate good π – wage per hour – the price of leisure π – the price of the aggregate good πΏ 24, π 0 the initial endowment 315 The budget constraint ππΏ π π π · 24 π ·0 24π or: π π πΏ 24 π 24 πΏ – consumed leisure πΏ – hours sold on the labor market 316 Graphically π ππΏ 24π π 24 π π 24π πΏ 317 Real wages Rather than π, π we can use only their ratio π€ π π Generally, if income is given by an initial endowment, only price ratios matter π€– how many units of π can be purchased for one hour of labor 318 The budget constraint in real terms Rather than ππΏ π π we have π€πΏ π Or, rather than π π π 24 πΏ π π€ 24 πΏ we get 24π 24π€ 319 π How much to work and how much to consume? π€πΏ 24π€ π 24π€ Say, π πΏ, π 1 πΌππππΏ 1 πΌ ππππ then πΌ 24π€ πΏ 24πΌ 24 πΏ π 1 1 πΌ 24π€ π€ 1 πΌ 24π€ 1 24πΌ 1 πΌ 24π€ 320 Supply of labor How much labor 24 πΏ will be offered as a function of π€? πΏ 24πΌ The labor supply is totally inelastic, independent of π€ Does it make sense? 321 Makes sense or not... π 24π It can't surprise us: ... we know by now what the PCC looks like for Cobb-Douglas 24πΌ 24 πΏ 322 Some weaknesses of the model • Assumes that labor itself doesn't affect well-being (one way or another) • Ignores psychological and social effects of work • Luckily, the world of economics has not been so naive 323 Some advantages of the model • Relates labor supply to leisure • Allows the analysis of a general equilibrium • Illustrates that the economic logic applies also to labor and leisure 324 A slightly more interesting story π π An initial endowment π 24π€ 0 on top of 24 hours of leisure The budget set is still convex So it makes sense to look for tangency, π and then verify it's in the relevant range 24 πΏ 325 Strict monotonicity could be helpful π π Weak monotonicity may not suffice 24π€ for the consumer to use all their resources Say, π π πΏ, π 24 πΏ min π, 0.5π πΏ 326 Solving the consumer problem π π π πΏ, π 24π€ πΌππππΏ πΏ πΌ π 1 πΏ π π Valid if 1 π 1 π π€ 1 24π€ πΌ π πΌ π π€ πΌ π 24πΌ πΌ ππππ 24π€ 24πΌ 1 πΌ 24π€ 24 and otherwise it is at 24, π 24 πΏ 327 Labor supply Demand for leisure is (as long as L πΏ π 24πΌ 24) Labor supply: 24 πΏ 24 πΌ π π€ 24πΌ 1 π€↑ ⇒ 24 πΏ ↑ πΌ 24 πΌ π π€ (as long as it's non-negative) upward sloping labor supply curve 328 There will be a corner solution π If π πΌ π π€ 24πΌ 24 1 πΌ 1 π πΌ 24 α 24π€ π Given π€, π : if πΌ is sufficiently close to 1 (That is, if the consumer "likes leisure 24 πΏ sufficiently" 329 Or... π Given πΌ, π : if π€ isn't sufficiently high πΌ 1 is equivalent to π€ π 24 24π€ π πΌ πΌ 1 π πΌ 24 πΏ 330 Alternatively π Given πΌ, π€: if π is high enough: π πΌ 1 is equivalent to π 24 πΌ 24π€ π 1 πΌ πΌ 24π€ πΏ 331 Producer Theory 332 A model of the producer π₯ – input, such as labor hours π¦ – output π – price of π₯ π – price of π¦ π – technology π¦ π π₯ 333 The feasible set π¦ π¦ π₯, π¦ π π π₯ 0 π₯ 334 Firm’s goal: maximize profit πππ₯ , π π₯, π¦ π π¦ π π₯ Or is it? • OK, it’s just a model • Organizational Behavior suggests many other theories • And yet, it is the stated purpose • And “profit” is a known and (roughly) measurable concept, more than is “utility” for the household 335 Profit maximization π¦ π π₯, π¦ πππ₯ π π¦ , π π₯ π π₯, π¦ Iso-profit line: π π₯, π¦ π¦ π π¦ π π₯ π π₯ π π π π π₯ 336 In search of an optimum π¦ We’re looking for πππ₯ , π π₯, π¦ π π¦ π π₯ Subject to π¦ π π₯, π¦ π π₯ 0 π₯ 337 Again, looking for tangency π¦ Between the highest iso-profit line π π₯, π¦ whose slope is π¦ π π π¦ π π₯ π π₯ π π π π and the technology constraint π¦ π π₯ with slope π¦′ π′ π₯ π₯ 338 Example π¦ π₯ π 10, π 100 Let’s take a derivative of the production function and equate slopes: 1 π π₯ 2 π₯ 1 10 10 100 π π we get π₯ π₯ 25 10 2 5 π¦ 5 and we can plug these into the profit function π 100 · 5 10 · 25 250 339 Example – cont. π¦ π₯ π 10, π 25 π¦ 100 Note that in the solution π₯ π 100 · 5 10 · 25 5 250 the marginal value/revenue of labor is 100 · 100 · 10 and the cost of labor (per hours) is π 10 340 Generally, π¦ π π₯ Tangency yields π π₯ π π or π π π₯ marginal revenue π marginal cost 341 Not too surprising In search of the maximum of π π₯, π¦ π π¦ π π₯ if we plug π¦ π π₯ we obtain a maximization problem in one variable: π π₯ π′ π₯ π π π₯ π π₯ π π′ π₯ π π′ π₯ π 0 π 342 Verbally, ππππππ‘ π ππ£πππ’π πΆππ π‘ Equating the derivative to zero yields ππππππππ π ππ£πππ’π ππ ππππππππ πΆππ π‘ ππΆ and, indeed, if ππ π π′ π₯ π ππΆ It would pay off to increase π₯ And a converse inequality suggests it would pay off to decrease π₯ 343 Convexity, again • As in consumer theory, we really like it, as economists • The marginality condition becomes sufficient for optimality (under convexity) • But, beyond that: it also provides a reasonable story about optimization • Again, small improvements can lead to a global optimum 344 Optimization dictates behavior in markets π π₯ π π Determines demand in the input market π₯ π and supply in the output market π¦ π¦ (say, demand in the labor market and supply in the apples market) π₯ 345 Changes in demand and supply Suppose that the price of π₯ goes up – or that the price of π¦ goes down: in both cases, the ratio goes up The firm’s demand for π₯ will go down π as will the supply of π¦ π¦ – which is what we’d expect as a downward sloping π¦′ demand curve (in the input market) and an upward π₯′ π₯ sloping supply curve (in the output market) 346 Another example π¦ π₯ π π₯ π₯ π 100 · 0.0025 π 2π₯ 10, π 100 10 100 0.05, π¦ 10 · 0.05 π π 0.0025 0.25 0.5 0.25 347 Just a sec… π¦ If π isn’t concave, tangency might not be such a great idea π π₯ π₯ π₯ 348 Tangency and convexity • π π₯ As in consumer theory, or in optimization without constraints, tangency isn’t a sufficient condition for 0 optimality • In a maximization problem, we also need concavity of the function • In consumer theory – convexity of two sets (the feasible and the desirable) 349 If both sets are convex If the feasible and all the desirable sets are π¦ convex, tangency is sufficient for optimality desirable This time we have the desirable set defined by a linear function (via the prices), and the feasible often “strictly” convex feasible In consumer theory this was the other way around, π₯ but the main idea is the same 350 The technology function Is it likely to be π¦ π¦ π π₯ π π₯ π₯ π₯ convex increasing returns to scale ? concave decreasing returns to scale 351 Wait – is it concave or convex? The question is whether the set that is above/below the graph is convex π¦ π¦ π π₯ π π₯ π₯ Convex function the set above the graph is convex π₯ Concave function the set below the graph is convex 352 The strings and the graph A string is never above the graph A string is never below the graph π¦ for a convex function π¦ π π₯ for a concave function π π₯ π₯ Convex function the set above the graph is convex π₯ Concave function the set below the graph is convex 353 Decreasing marginal productivity Makes sense in a variety of examples: • Climb a tree to pick apples • Study for an exam • Add workers to a limited space 354 Fixed and variable costs π¦ π π₯ π₯ π π₯ is neither convex nor concave 355 Several production factors π₯ – quantity of production factor 1 π₯ – quantity of production factor 2 π¦ – quantity of the output Technology is given by π¦ π π₯ ,π₯ 356 The profit function π π₯ ,π₯ ,π¦ π π¦ π π₯ π π₯ π – price of π₯ π – price of π₯ π – price of π¦ 357 Optimization Looking for π₯ ,π₯ , π¦ to maximize π π₯ ,π₯ ,π¦ π π¦ π π₯ π π₯ subject to π¦ π π₯ ,π₯ 358 Here Lagrange is really helpful β π₯ , π₯ , π¦, π π π¦ π π₯ π π₯ 0 πβ ππ₯ π ππ βΆ π π π 0 πβ ππ₯ π ππ βΆ π π π π πβΆπ 0 πβ ππ¦ 0 ππ¦ π π₯ ,π₯ π πβ ππ 359 And then π π π π π π π π Is this familiar? π π π π familiar 360 Cost minimization Suppose that the quantity π¦ is fixed for us Whatever it is, it’s better to produce it at lower cost πππ₯ π π₯ , π₯ , π¦ π π¦ π π₯ revenue π π₯ cost So we can think of πππ [ cost ] subject to π π₯ ,π₯ π¦ 361 That is… π₯ We will try to be on the lowest isocost line subject to the constraint of having enough π¦ … and then π π₯ ,π₯ π¦ doesn’t look too surprising… π π₯ π π₯ π π₯ 362 π₯ If the equality doesn’t hold Say, the iso-cost line is steeper than the iso-product line π π π π₯ ,π₯ π π₯ we can go down (to have lower cost) π¦ π π₯ π π without leaving the feasible set π π₯ 363 π₯ π π₯ π π₯ π In this situation The condition π π π π π π π π is equivalent to π π₯ ,π₯ π¦ which means that production factor 1 is (relatively) more expensive than 2 (and we’ll be better off using more of the second and less of the π₯ first) 364 And maybe someone will notice that A unit of π₯ yields, roughly, π units of π¦ hence 1 unit of π¦ ≅ units of π₯ ≅ dollars 1 unit of π¦ ≅ units of π₯ ≅ dollars and if, indeed, π π π π Then production factor 1 is (relative to marginal productivity) more expensive than is 2 365 A special case π₯ – input in plant 1 π₯ – input in plant 2 Overall output: π¦ and the condition π π₯ π π₯ or will define the right mix (again, under convexity assumptions, that is, concavity of both π , π ) 366 Example π π₯ π¦ πππ π₯ πππ π₯ 1 π₯ 1 π₯ πππ π₯ π₯ π₯ π π 367 Markets and Equilibria 368 Market πΆ – Quantity of good 1 π· – Quantity of good 2 No production Many consumers (many = 2) (But – they are price takers) 369 Consumer A’s preferences π¦ π· πΆ π₯ 370 Consumer B’s preferences π¦ π· πΆ π₯ 371 Consumer’s B preferences again π¦ π₯ πΆ π· πΆ π₯ π¦ 372 Edgeworth’s Box: putting them together π₯ π¦ π· π¦ π₯ πΆ 373 Preferences in Edgeworth’s box π₯ π¦ π· π¦ π₯ πΆ 374 What’s wrong with…? π₯ π¦ π¦ π₯ 375 Pareto domination An allocation , , Pareto dominates another allocation , , if π π₯ ,π¦ π π₯ ,π¦ And at least one of these π π₯ ,π¦ π π₯ ,π¦ is strict 376 Back to… π₯ π₯ π¦ π¦ π¦ π¦ π₯ , , Pareto dominates π₯ , , 377 Pareto Optimality An allocation , , is Pareto optimal (≡ Pareto Efficient) if there is no feasible allocation that Pareto dominates it 378 For example… π₯ π¦ π¦ π₯ Getting A to be better off within the feasible set will make B worse off (and vice versa). 379 There are many P.O. allocations π₯ π¦ π¦ π₯ The Contract Curve connects all such allocations 380 Words of warning “Pareto Efficient” sounds too weak • It is not only about producing a lot • It is also about producing the right products • And allocating them efficiently 381 More warnings… “Pareto Optimal” sounds too strong • “Optimal” sounds like “best” • But we may prefer an allocation that is not Pareto optimal to one that is. 382 The utility frontier π Each allocation defines a point. ππ The frontier corresponds to Pareto Optimal allocations π 383 Pareto optimality π π§ is Pareto optimal π§ π‘ isn’t π‘ π 384 But… π π§, π€ are Pareto optimal, π‘ is not. π§ Pareto dominates π‘, but that’s all π§ π‘ π‘, π€ , π§, π€ are incomparable π€ π 385 Pareto domination - incomplete • Not every two allocations can be compared • It is basically a unanimity relation • In this case, “optimal” says much less than “optimum” 386 The contract curve & the pareto frontier π’ π· πΆ π’ 387 Competitive equilibrium Equilibrium – a list of prices, such that, when everyone responds optimally to them, markets clear (No excess demand, and excess supply only if the price is zero) 388 “Everyone responds optimally“ • In general, there are also firms • Which have demand in input markets (such as labor) and supply in output markets • Their profits are divided among households according to ownership • But let’s focus on equilibrium without production 389 Agent Endowment π₯ , π¦ Given prices π , π Has a budget constraint: π π₯ π π¦ π π₯ π π¦ 390 Graphically π¦ π¦ π₯ π π₯ π π¦ π π₯ π₯ π π₯ π π¦ π π¦ π¦ π₯ 391 Will agent buy good 1 and sell 2? π¦ As above, it depends on the slope of A’s indifference curve at π₯ , π¦ π¦ π₯ π₯ 392 Conversely, π¦ Given the preferences, there will typically be prices for which A will want to sell good 1 and buy good 2, π¦ and prices for which the opposite is true. π₯ π₯ 393 Equilibrium Looking for prices where demand = supply, as in: π₯ π¦ π· π¦ πΆ π₯ 394 Existence Theorem Kenneth J. Arrow (1921-2017) Gérard Debreu (1921-2004) Under certain assumptions… there exists (at least one) competitive equilibrium 395 Existence in the 2x2 case (Misleadingly simple) If 0, both want to sell π¦ and buy π₯ If ∞ – vice versa As the line tilts, equilibrium is reached 396 Properties an equilibrium might have • Existence • Uniqueness • Dynamic stability (relative to perturbations) • Robustness (to initial conditions) 397 Existence Is there a point that is a “steady state” ? 398 Uniqueness Is the steady state unique? 399 Dynamic stability Will a small perturbation result in a big change? Will dynamic forces push us back to the equilibrium? Far away? 400 Different notions of dynamic stability Will dynamic forces push us back to the equilibrium? Far away? Neither? 401 Robustness If we missed the initial conditions by a bit, will the prediction change dramatically? 402 Partial equilibrium was so nice… π π π· • Existence • Uniqueness • Dynamic stability (relative to perturbations) π • π Robustness (to initial conditions) π 403 Existence of partial equilibrium If demand is a continuous decreasing curve, π π π· and supply is a continuous increasing curve, mild conditions suffice to have an intersection π π π 404 Uniqueness of partial equilibrium If demand is a continuous decreasing curve, π π π· and supply is a continuous increasing curve, the intersection has to be unique π π π 405 Dynamic stability of partial equilibrium We could tell plausible stories about the way π π π· market forces would push the price towards the equilibrium price in case of excess demand or excess supply π π π 406 Robustness of partial equilibrium If the demand or the supply curve was not quite π π π· what we had in mind, it’s not the end of the world For “nearby” initial conditions there will be a “nearby” equilibrium π π π 407 Unfortunately For general equilibrium we only have existence (in general) • Existence + • Uniqueness – • Dynamic stability – • Robustness – 408 Pareto optimality of equilibrium (Simple, but not misleading) At equilibrium both indifference curves are tangent to the same line, HENCE TO EACH OTHER 409 The First Welfare Theorem All competitive equilibria define Pareto optimal allocations Intuition: prices carry information They serve as a communication device among the agents, allowing decentralization 410 Decentralization • Just imagine how hard it is for a central planner to solve the “problem of the economy” • How much misinformation can be involved • Decentralize: no risk of deception; using everyone to solve their bit of the puzzle 411 Why are equilibria allocations efficient? Suppose π , π is an equilibrium (“price list” or “price vector”) and that , , is an allocation corresponding to it Could another feasible allocation , , Pareto dominate , , ? 412 Equilibrium assumes optimality , , is the allocation of an equilibrium (price list) π , π The bundle π₯ , π¦ is optimal for π΄ given π , π The bundle π₯ , π¦ is optimal for π΅ given π , π 413 Optimality for The bundle π₯ , π¦ π¦ is optimal for π΄ given π , π Hence if π π₯ π₯ ,π¦ π π¦ π π₯ π π¦ then π π₯ π₯ ,π¦ (wherever π₯ , π¦ π π₯ ,π¦ is in the feasible set) π₯′ , π¦′ 414 Optimality for plus monotonicity Moreover, if π¦ π π₯ π π₯ π π¦ then π₯ ,π¦ π₯′ , π¦′ π π¦ π π₯ ,π¦ π π₯ ,π¦ π₯ 415 Why? π¦ π₯ ,π¦ We claim that π π₯ π π¦ π π₯ π π¦ Implies π π₯ ,π¦ π π₯ ,π¦ Otherwise, i.e., π π₯ , π¦ π π₯ ,π¦ There would be another point π₯′′ , π¦′′ π₯′′ , π¦′′ and is (strictly) preferred to both π π₯ ,π¦ π₯′ , π¦′ that is feasible π₯ So π₯ , π¦ π π₯ ,π¦ π π₯ ,π¦ couldn’t have been optimal Note: weak monotonicity suffices here (but not basic) 416 Hence, with monotonicity, optimality for π΄ implies π π₯ π π¦ π π₯ π π¦ βΉ π π₯ ,π¦ π π₯ ,π¦ π π¦ π π₯ π π¦ βΉ π π₯ ,π¦ π π₯ ,π¦ and π π₯ 417 Contrapositives π π₯ π π¦ π π₯ π π¦ βΉ π π₯ ,π¦ π π₯ ,π¦ IFF π π₯ ,π¦ π π₯ ,π¦ βΉ π π₯ π π¦ π π₯ π π¦ and π π₯ π π¦ π π₯ π π¦ βΉ π π₯ ,π¦ π π₯ ,π¦ IFF π π₯ ,π¦ π π₯ ,π¦ βΉ π π₯ π π¦ π π₯ π π¦ 418 Using these, optimality for π΄ (with weak monotonicity) implies π π₯ ,π¦ π π₯ ,π¦ βΉ π π₯ π π¦ π π₯ π π¦ π π₯ π π¦ π π₯ π π¦ and π π₯ ,π¦ π π₯ ,π¦ βΉ 419 For the two agents π π₯ ,π¦ π π₯ ,π¦ βΉ π π₯ π π¦ π π₯ π π¦ π₯ ,π¦ π π₯ ,π¦ βΉ π π₯ π π¦ π π₯ π π¦ π and π π₯ ,π¦ π π₯ ,π¦ βΉ π π₯ π π¦ π π₯ π π¦ π π₯ ,π¦ π π₯ ,π¦ βΉ π π₯ π π¦ π π₯ π π¦ 420 If we didn’t have Pareto optimality , , Pareto dominates , , means that π π₯ ,π¦ π π₯ ,π¦ π π and π₯ ,π¦ π₯ ,π¦ with at least one of these inequalities being strict 421 Putting these together π π₯ ,π¦ π π₯ ,π¦ π π π₯ ,π¦ π₯ ,π¦ with at least one strict, given π π₯ ,π¦ π π₯ ,π¦ βΉ π π₯ π π¦ π π₯ π π¦ π₯ ,π¦ π π₯ ,π¦ βΉ π π₯ π π¦ π π₯ π π¦ π π π₯ ,π¦ π π₯ ,π¦ βΉ π π₯ π π¦ π π₯ π π¦ π π₯ ,π¦ π π₯ ,π¦ βΉ π π₯ π π¦ π π₯ π π¦ 422 We get π π₯ π π¦ π π₯ π π¦ π π₯ π π¦ π π₯ π π¦ Important: these are the same prices!!! with at least one strict Adding them up π π₯ π₯ π π¦ π¦ π π₯ π₯ π π¦ π¦ 423 But this contradicts feasibility: Feasibility of , , means that π₯ π₯ π₯ π₯ π¦ π¦ π¦ π¦ 424 Multiplying by prices π₯ π₯ π₯ π₯ π¦ π¦ π¦ π¦ yield π π₯ π₯ π π₯ π₯ π π¦ π¦ π π¦ π¦ π π¦ π¦ π π₯ π₯ π π¦ π¦ π π₯ π₯ π π¦ π¦ ) and, adding up, π π₯ π₯ (while Pareto domination implied π π₯ π₯ π π¦ π¦ 425 The proof • Works the same way with any number of goods and consumers • Requires some extra work if we allow production • Still, simple enough to be convincing • While not entirely trivial – basically, changing the order of summation 426 The theorem relies on Explicitly: • Optimization • Weak monotonicity • Same prices Implicitly: • Price taking behavior • No explicit uncertainty – in fact, no asymmetry of information • No externalities 427 Correspondingly The conceptual message of the first welfare theorem may not be valid if one: • Agents are not “rational” • Some have market power • There are asymmetries of information • There are externalities (as in the case of public goods) 428 And the conceptual message is limited π’ Only guarantees Pareto optimality. … says nothing about equality, fairness… π’ 429 How do we deal with inequality? To competitive equilibrium answer: use lump-sum transfers What’s that? 430 Lump-Sum Transfers • Transfers of goods between agents before we let them start the economic activity • Changing the initial endowments once and not touching equilibrium allocations that result • The point: we don’t want to meddle with economic incentives 431 The Second Welfare Theorem (Under some conditions) for any π· Pareto Optimal allocation, there π¦ exists lump-sum transfers for π₯ πΆ which the allocation is an equilibrium allocation (posttransfers) 432 The message and its limitations • If you’re concerned about inequality, try to solve it without messing up the price mechanism • Don’t kill the coordination device • The problem: we don’t really know any examples lumpsum transfers • Most reasonable ideas won’t be “lump-sum” if used more than once 433 The assumptions • The second welfare theorem needs to assume more than the first (basically, also convexity) • But less than the existence proof (which isn’t constructive and doesn’t come with a convergence result) 434 Preferences over Consumption Streams 435 We haven’t dealt with • Saving for the future • Uncertainty • In fact, these are two things that financial markets do for us: • Smoothing consumption over time (saving and borrowing) • Smoothing consumption over states of nature (insurance) 436 Multiple periods π¦ – income in period 0 π¦ – income in period 1 π – consumption in period 0 π – consumption in period 1 π – interest rate 437 Preferences The classical model π π ,π ,π … 0 πΏ π’ π πΏπ’ π πΏ π’ π … 1 – the discount factor with two periods: π π , π π’ π πΏπ’ π 438 The budget set Without financial markets: only (π¦ , π¦ ) With saving/borrowing at π: every $1 today can be converted to $ 1 every $1 tomorrow can be converted to $ π tomorrow today 439 The budget set π 1 π π¦ 1 π π¦ 1 π π π¦ 1 1 π π¦ or: 1 π¦ π¦ π¦ 1 1 π π π 1 π π¦ π¦ π π π¦ 440 Optimal choice π π ,π π π¦ log π π 1 π π πΏ π π π π¦ πΏ log π π πΏπ π 441 We might have π π π¦ π¦ saving loan π¦ π π¦ π 442 Will I save or borrow? Depends on: • The endowment π¦ , π¦ • The time preferences • The interest rate 1 π 443 Effect of endowment: π π π¦ loan π¦ saving π¦ π π¦ π 444 Effect of time preferences π slope of indifference curve: π high πΏ π¦ π¦ low πΏ π¦ π π¦ π 445 Effect of interest rate slope: 1 π π π high π π¦ π¦ low π indifference slope: π¦ π π¦ π 446 A special (but important) case: Fixed income: π¦ π 1 π π¦ 1 πΏ π 1 π¦ π¦ π¦ π π¦ π 1 πΏ π 447 The above is a bit more general… π π ,π π’ π πΏπ’ π π π’ π π πΏπ’ π πΏ But at π¦ π¦ we are left with πΏ 448 Back to Cobb-Douglas π π ,π log π πΏlog π Budget constraint: 1 For π¦ π¦ π 2 π π¦ π¦ π * π * π π 2 2 π π¦ π π¦ π * π¦ ⇔ πΏ π * π¦ ⇔ πΏ 449 Equilibrium with time preferences Agent A: π log π πΏlog π π π 2π¦ Agent B: π log π πlog π π π 2π¦ Equilibrium should be at π such that: 1 π 1 πΏ π or: 1 πΏ 1 π 1 π 1 450 Indeed… π * y π¦ π¦ The sum of demands should be zero + π * 0 y π¦ π¦ 2 and we can divide by π¦ The equilibrium interest rate is sort of an average of those corresponding to the agents’ time preferences 451 Overlapping generations Identical preferences: π π log π πΏ log π But different endowments: π΄: π¦ ,π¦ π΅: π¦ ,π¦ π¦ π¦ 452 Edgeworth box for young & old π¦ π΅ π¦ π¦ π΄ π¦ 453 For A: π π ,π 1 π * π * π¦ π π¦ 1 π¦ π¦ π¦ π π log π π πΏ log π 1 π π¦ π¦ Demand in period 0 For B (reverse π¦ and π¦ ): π * π¦ π¦ π¦ Equilibrium: π * π¦ π * π¦ π¦ π¦ πΏ Sum of demands should be zero 0 π¦ π¦ 454 Conclusions • Financial markets allow agents to smooth consumption over time • There might be trade between agents with different time preferences • Or with different endowments • Or both. Or… 455 Preferences under Risk and Uncertainty 456 Uncertainty Has been with us since days of yore: 6 ”And the messengers returned to Jacob, saying, We came to thy brother Esau, and also he cometh to meet thee, and four hundred men with him. 7 Then Jacob was greatly afraid and distressed: and he divided the people that was with him, and the flocks, and herds, and the camels, into two bands; 8 And said, If Esau come to the one company, and smite it, then the other company which is left shall escape.” King James version, Genesis chapter 32, 6-8. 457 Preferences The most common model: Expected Utility maximization Implicit in Pascal’s Wager (1670) Explicit in Bernoulli’s resolution to the St. Petersburg Paradox (1738) Became dominant with von-Neumann and Morgenstern (1947) 458 Expected utility Maximize the expectation, not of the variable, but of its utility: πππ πΈ π₯ π΅ππ πΈπ π₯ π π₯ π π₯ π π’ π₯ β― π π₯ π π’ π₯ β― π π’ π₯ 459 Betting Starting with wealth level π You’re offfered a bet: π It’s a fair bet if πΈ π 0 or πΈ π 100 .5 100 .5 π π Expected value maximization βΉ indefference 460 But… π’ π₯ π’ π 0.5π’ π + 0.5π’ π 100 100 π’ concave βΉ πΈ π’ π π 100 π π 100 π π’ π π₯ 461 And vice versa for convex : π’ π₯ 0.5π’ π + 0.5π’ π π’ convex βΉ πΈ π’ π 100 π π’ π 100 π’ π π 100 π π 100 π₯ 462 Risk aversion Defined as: For any fair bet π πΈ π 0 and any wealth level, π is at least as good as π π Risk loving: the other way around 463 Under EU maximization Risk aversion βΊ π’ concave Risk loving βΊ π’ convex (Both can be defined in the strict sense) 464 Insurance problem Car worth 100,000 might be stolen with probability 1% Insurance premium 2,000 Expected value maximization: don’t insure 0.99 100,000 0.01 0 99,000 98,000 465 Insurance problem - cont. π’ π₯ π’ 98,000 0.99π’ 100,000 + 0.01π’ 0 However, a risk averse agent might have: 0.99 98 100 π’ 100,000 0.01 π’ 0 π’ 98,000 π₯ 466 How do we know… that the premium is greater than expected loss? • Because insurance companies make money • At least those who are around • What’s the difference between the insurance company and the agent? 467 Law of large numbers (LLN) If π are i.i.d. (identically and independently distributed) and π Then: π → π ∑ π where π πΈ π 468 Law of large numbers – cont. Important: • Identicality (or something close) • Independence (or something close) • Large π 469 Gambling and state lottery The state lottery offers us a gamble with negative expectation Say, pay $1 to get $1M with probability 0.000,000,5 How do we know that? 470 State lotteries π’ π₯ A risk averse agent will not buy it 1 1 π’ π 2π 1 π’ π 2π Not even risk neutral 1 But… 1π π’ π π 1 π π 1π 471 Back to the consumer’s problem: States of nature: π₯, π¦ – (wealth in state 1, wealth in state 2) π¦ π₯ 472 The betting example Tail Bet on Tail π 100 π π 100 Bet on Head π 100 π π 100 Head 473 Preferences π π₯, π¦ π π·π’ π₯ 1 π ·π’ π¦ For example: 100 π π₯, π¦ π · log π₯ 1 π · log π¦ π π 100 Risk aversion: concave π’ π 100 π π 100 474 The budget constraint Say, a fair bet on any amount: π₯ π¦ π π A risk averse agent will choose to remain at π, π π 475 The insurance problem π¦ π – wealth apart from car car stolen 1 π 0.01 π car not stolen π 0.99 π 100,000 π₯ 476 The insurance problem – cont. π¦ Preferences: π π₯, π¦ π 0.99π’ π₯ 0.01π’ π¦ 98 Budget constrains: π Say, any amount up to 100,000 can be insured with premium of 2% π 98 π 100 π₯ 477 The insurance problem – cont. Insure at π, 0 π 100,000 Pay 0.02π for sure βΉ π 100 0.02π, π π π 0 π π 0.02π 100, π 98, π π 100 98 The segment connecting them is 0.98π₯ 0.02π¦ W 98 478 Insurance problem– solution π π₯, π¦ 0.99 log π₯ Budget line: 0.98π₯ π₯* 0.99 π¦* 0.01 . . 0.01 log π¦ 0.02π¦ W 98 π 98 . . π 98 π 98 π 98 . . π 98 π 98 The agent will not be fully insured 479 Why? At full insurance, π 98, π 98 the slope of the indifference curve is: . π . ,π . . β π₯ π¦ 99 while the slope of the budget constraint is: π 0.98, π 0.02 . . 49 480 Thus… the indifference curve is steeper: π¦ certainty line π 98 π π 98 π 100 π₯ 481 And this is quite general: With Expected Utility: π π₯, π¦ π·π’ π₯ 1 π π’ π¦ So that: β π₯ π¦ The slopes of all indifference curves along the certainty line are the same, and independent of π’ 482 Financial markets 1,0 – An asset that pays 1 in state 1 and 0 in state 2 costs π 0 π 1 0,1 – An asset that pays 1 in state 2 and 0 in state 1 costs π 0 π 1 π π 1 It stands to reason that Why? 483 No Arbitrage Imagine that 1,0 costs π 0.4 0,1 costs π 0.5 and Then I could buy both, and at a total cost of 0.9 have 1 for sure, making a sure gain of 0.1 (ignoring transaction costs) 484 No Arbitrage – part II And now imagine that 1,0 costs π 0.6 0,1 costs π 0.5 And, say, Then I could sell both, and this means I will get 1.1 but will have to pay only 1 – whatever the state: I’ll be making a sure gain of 0.1 (again, ignoring transaction costs) 485 No Arbitrage – conclusion 1,0 – An asset that pays 1in state 1 and 0 in state 2 costs π 0,1 – An asset that pays 1in state 2 and 0 in state 1 costs π If π π 1 doesn’t hold, market forces will push the prices in this direction. So, for some 0 π 1, 1,0 costs π 0,1 costs 1 π 486 The market odds 1,0 costs π 0,1 costs 1 Hence 1 π, π 1 π π ,π each has a net worth of 0 And so does π 1 π , ππ For any π (positive or negative) 487 The no arbitrage theorem In a perfect market (no transaction costs, buying and selling of any amount is allowed with linear pricing) there are no arbitrage opportunities IFF there is a probability such that each asset is valued by its Stephen A. Ross (1944-2017) expectation (relative to that probability) 488 The geometry of No Arbitrage π¦ (state 2) Suppose 0,1 0.6,0.6 1,0 costs π 0.6 0,1 costs π 0.5 0.5,0.5 1,0 π₯ (state 1) 489 In that case… π¦ (state 2) 0,1 1,0 costs π 0.6 0,1 costs π 0.5 0.5 1,0 0.5 0,1 0.5,0.5 which is strictly lower than 0.6,0.6 0.5,0.5 0.5 0.6,0.6 0.5 0.5,0.5 0.55,0.55 1,0 π₯ (state 1) so selling both assets pays off 490 Similarly Similar sure-gain trades can be π¦ (state 2) generated whenever the two line segments (each connecting an 0,1 uncertain asset to its certainty equivalent on the diagonal) are not parallel 1,0 π₯ (state 1) 491 But If they are parallel then, by Thales’s π¦ (state 2) Theorem we get two similar triangles and the average of the 0,1 certainty equivalent equals the certainty equivalent of the average 1,0 π₯ (state 1) 492 And the same holds if the two assets we π¦ (state 2) start out with are not the unit vectors, or not symmetric around the diagonal π₯ (state 1) 493 Or on the same side of the diagonal π¦ (state 2) π₯ (state 1) 494 The budget constraint An agent with endowment πΆ, π· can purchase π π and add π 1 0 ππ π 0 units of 1 π, π π , ππ to her endowment The agent’s budget constraint consists of points π₯, π¦ such that for some π : πΆ π 1 π ,π· ππ π₯, π¦ Or: we can think of the budget line as all the points π₯, π¦ that have the same cost as πΆ, π· at prices π, 1 π : ππ₯ 1 π π¦ ππΆ 1 π π· 495 Agent’s demand & supply π log π₯ Max Budget: π₯* π¦* π ππ₯ ππΆ 1 π 1 1 1 π π¦ π log π¦ ππΆ 1 π π· π π· ππΆ 1 π π· 496 Will the agent buy the asset? That is, will the agent want more in state 1, π₯* π ππΆ ππΆ 1 π π· In case πΆ π·: only if In case π π: only if π· π π· π· πΆ πΆ πΆ that is, if πΆ πΆ? π π 497 Two main reasons for agents to trade: • Consumption smoothing across states: the agent agrees with the market, π π· • π, but experiences uncertainty πΆ Thinking they know better: the agent faces no uncertainty, π· πΆ, but thinks a state is more likely than evaluated by the market π π … And we often have combinations of the above 498 Equilibrium in financial markets π· Suppose the agents agree on probabilities: πΆ π π₯, π¦ π log π₯ 1 π log π¦ π π₯, π¦ π log π₯ 1 π log π¦ But have uncertain endowments πΆ, π· , π·, πΆ π· πΆ 499 What will be the equilibrium price? Agent’s A demand (in state 1): π₯* πΆ π ππΆ 1 π π· πΆ π log π₯ 1 π log π¦ And recall that, for CD… 1 π πΆ π π· Agent’s B demand πΆ β· π· : 1 Sum to zero implies π 1 π πΆ π π· π₯ π¦ π π 1 πΌ π 1 π 1 πΌ π πΆ π: π· π πΆ π· 500 Thus With identical preferences: π log π₯ 1 π log π¦ And endowments: πΆ, π· , π·, πΆ • No aggregate uncertainty: πΆ π·, πΆ π· • The equilibrium price is the probability π π, and the equilibrium allocations are “full insurance” 501 But the agents can disagree: π π₯, π¦ π log π₯ 1 π log π¦ π π₯, π¦ π log π₯ 1 π log π¦ Suppose that πΆ, πΆ is the endowment of each… 502 πΆ πΆ πΆ πΆ Say, π π Agent A thinks state 1 is more likely than does agent B. 503 Equilibrium with disagreement: Sum π₯∗ πΆ 1 π πΆ π π₯∗ πΆ 1 π πΆ π π π 1 π π 1 π π πΆ πΆ 0βΉ 2 π ∗ ∗ for π π∗ π will be the “average” belief. 504 Puzzles and Anomalies 505 Puzzles : Time preferences • Do you prefer $100 today, or $110 in a week? • Do you prefer $100 in 50 weeks to $110 in 51 weeks? 506 Dynamic consistency What we plan today to do tomorrow is also what we indeed wish to do tomorrow. Violated in the previous example… 507 Puzzles : EU Do you prefer $3000 for sure to $4000 with probability 80%? And how about $3000 with π 0.25 to: $4000 with π 0.20 508 Puzzles - cont. You get $1000. You get $2000. Do you prefer: Do you prefer: a. $500 for sure b. $1000 with π 0.5 a. $500 for sure b. $1000 with π 0.5 509 Prospect theory Kahneman and Tversky: • Small probabilities loom large • Reference points matter 510 Other anomalies: • Decoy and Compromise Effects • Mental Accounting • Disposition Effect 511
