Microeconomics 2 - Fall 2023 Lecture Notes PDF

Microeconomics 2 Igor Letina Department of Economics University of Bern Fall 2023 Last updated: September 18, 2023 1 Chapter 1: Preliminaries 2 Ch 1.1: Mathematical Statements I A statement is a...

Microeconomics 2 Igor Letina Department of Economics University of Bern Fall 2023 Last updated: September 18, 2023 1 Chapter 1: Preliminaries 2 Ch 1.1: Mathematical Statements I A statement is a “sentence” that can either be true or false (but never both). I Examples: I We are in Bern. I Two is an odd number. I What about: I “Micro is cool.” I “This statement is false.” 3 Ch 1.1: Mathematical Statements I Let A be a statement. Then ¬A (“not A”) is also a statement. I If A is true then ¬A is false. I If A is false then ¬A is true. I Example: I A =“Every student loves micro.” Hence ¬A = “No student loves micro.” Correct? 4 Ch 1.1: Mathematical Statements I Suppose A and B are statements. Then we can define a compound statement A ) B (implication) by setting SoW A B A)B 1 T T T 2 F T T 3 F F T 4 T F F I A ) B is just another statement. I If A ) B is true, then A is sufficient for (the truth of) B and B is necessary for (the truth of ) A. 5 Ch 1.1: Mathematical Statements I Converse of A ) B is B ) A. I If A ) B and B ) A are both true then A , B is true. I We say: I A is equivalent to B. I A holds if and only if B holds. I A is necessary and sufficient for B. 6 Ch 1.2: How to prove A ) B I Direct proof I Find a sequence of accepted statements Si for i = 1,... , n, such that A = S1 ) S2 ) S3 · · · ) Sn = B 7 Ch 1.2: How to prove A ) B I Proof by contradiction I Assume that A is true but B is false. Show that A and ¬B together imply C, but it is already known that C is false. Hence ¬B must be false if A true. Thus A implies B. I Proof by contraposition I It is a special case of the proof by contradiction. I Prove ¬B ) ¬A. SoW A B A)B ¬B ¬A ¬B ) ¬A 1 T T T F F T 2 F T T F T T 3 F F T T T T 4 T F F T F F 8 Ch 1.2: How to prove A ) B I Proof by mathematical induction I Only useful for statements that can be indexed by natural numbers, but can be used also if there is an infinite number of statements. I How to prove that statement S(n) holds for any n? I Need to prove: 1. S(1) is true; 2. whenever S(k) is true for some k, then S(k + 1) is also true. I First step is called the basis step, second step is called the inductive step and S(k) is called the inductive hypothesis. 9 Chapter 2: Pure exchange economy Varian, Chapter 17 10 Ch 2.1: Introduction to General Equilibrium I Basic supply and demand model I Key assumption: one market! 11 Ch 2.1: Introduction to General Equilibrium I Known as partial equilibrium analysis. I Useful simplification. I But leaves some questions open: I where do resources of consumers come from? I what happens with firm profits? I how do di↵erent markets interact? I In general equilibrium analysis we model all markets in an economy. 12 Ch 2.1: Introduction to General Equilibrium I Main questions: I how does a market economy work when considered as a whole? I are the outcomes desirable? I how can we a↵ect the outcomes of the market interaction? I Initially, we will ignore firms and focus on a pure exchange economy. 13 Ch 2.2: Model Nothing is less real than realism. Details are confusing. It is only by selection, by elimination, by empha- sis that we get at the real meaning of things. Georgia O’Kee↵e 14 Ch 2.2: Model I There is a finite number of agents (consumers) n and a finite number of commodities k. I The set of all agents is N = {1, 2,... , n} and the set of all commodities is K = {1, 2,... , k}. I We will speak of a agent i for some i 2 N and commodity j for some j 2 K. 17 Ch 2.2: Model I A commodity is a good or a service which is completely specified physically, temporally and spatially. I For example: I good: eggs. I service: haircut. I physically: big eggs, small eggs. I temporally: eggs today, eggs tomorrow. I spatially: eggs in Bern, eggs in Basel. 18 Ch 2.2: Model I Each agent is described by I a utility function ui : Rk ! R I an initial endowment of each commodity !i = (!i1 , !i2 ,... , !ik ) where !ij 0 for all i and all j. I Agents can trade all commodities. I After trading, agents have consumption bundles xi = (x1i , x2i ,... , xki ), where x1i is the quantity of commodity 1 that agent i holds after the trade (and so on). 19 Ch 2.2: Model I An allocation x is a collection of all consumption bundles: x = (x1 , x2 ,... , xn ). I A feasible allocation is an allocation which is physically possible. That is, x is feasible if for every commodity j 2 K it holds n X n X xji = !ij. i=1 i=1 I Sometimes, it is useful to define a feasible allocation with free disposal as any allocation such that n X n X xji  !ij. i=1 i=1 20 Ch 2.2: Model I When there are two commodities and two agents, we can represent the exchange economy with an Edgeworth box. Francis Ysidro Edgeworth (1845-1926) 21 Ch 2.2: Model I Suppose that the agent 1 is endowed with !11 and !12. The agent 2 is endowed with !21 and !22. Sketch the Edgeworth box. 22 Ch 2.3: Walrasian Equilibrium Léon Walras (1834-1910) Professor of Political Economy University of Lausanne 23 Ch 2.3: Walrasian Equilibrium I Also known as a competitive equilibrium. I Assumption: I there are market prices p = (p1 ,... , pk ) I one for each good I pj 0 for all j I agents take them as given (compare to e.g. Cournot equilibrium) I Agents can trade (exchange) their endowments at the market prices. I Each agent maximizes her utility, given the own endowment and the market prices. 24 Ch 2.3: Walrasian Equilibrium I Formally, each agent i 2 N solves max ui (x1i ,... , xki ) x1i ,...,xki k X k X such that pj xji = pj !ij j=1 j=1 I The solution to this problem (assumed unique in this class), for each price and endowment vector, is the consumer’s demand function xi (p, p!i ) : Rk+1 ! Rk. 25 Ch 2.3: Walrasian Equilibrium Definition 2.1 (Walrasian equilibrium) A Walrasian equilibrium is given by a vector of prices p⇤ = (p⇤1 ,... , p⇤k ) and an allocation x⇤ = (x⇤1 ,... , x⇤n ) , such that, for each agent i 2 N , it holds x⇤i = xi (p⇤ , p⇤ !i ) (that is, P Pn x⇤i maximizes agent’s utility) and ni=1 x⇤,j i  j i=1 !i for all j 2 K (that is, there is no excess demand). 26 Ch 2.4: Relative prices Lemma 2.2 For all consumers, price vectors and endowments, xi (p, p!i ) = xi (↵p, ↵p!i ) for all ↵ > 0. 27 Ch 2.4: Relative prices I Thus, in this model, nominal prices are irrelevant, only relative prices are important. I This is a similar idea as the “neutrality of money” in macroeconomics. I It is a consequence of the fact that the agents know all prices in the economy (instantaneously!) and are able to perfectly calculate the optimal bundle. 28 Ch 2.5: Walras’ law I The excess demand function for some good j, given a vector of endowments !i is n h X i j z (p) = xji (p, p!i ) !ij. i=1 I Denote the vector of all excess demand functions as z(p). I Note that z(p) depends only on relative prices. I z(p) depends on the endowments !. I if xji (p, p!i ) is continuous for all i and j, then z(p) is also continuous. 29 Ch 2.5: Walras’ law Proposition 2.3 (Walras’ law) For any price vector p, we have pz(p) ⌘ 0, i.e., the value of the excess demand is identically zero. 30 Ch 2.6: Some properties Proposition 2.4 (Market clearing) If demand equals supply in k 1 markets, and pk > 0, then demand must equal supply in the k-th market. Proposition 2.5 (Free goods) If (p⇤ , x⇤ ), is a Walrasian equilibrium and z j (p⇤ ) < 0, then p⇤,j = 0. That is, if some good is in excess supply at a Walrasian equilibrium it must be a free good. 31 Ch 2.6: Some properties I Under which conditions can we be sure that there is no excess supply in equilibrium? Definition 2.6 (Strongly increasing utility functions) We say that a utility function is strongly increasing if for any y x and y 6= x, it holds u(y) > u(x). Proposition 2.7 (Equality of demand and supply) If all utility functions are strongly increasing and p⇤ is a Walrasian equilibrium, then z(p⇤ ) = 0. 32 Ch 2.7: Existence of an equilibrium I When can we be sure that a Walrasian equilibrium exists? I A question of fundamental importance. I Before proving the existence of equilibrium, we will introduce mathematical concepts that will be used in the proof. 33 Ch 2.7: Existence of an equilibrium I We have shown before that only relative prices matter. I That is, we can multiply the entire prize vector p with any ↵ > 0 and the outcome of the model will not change. I This is called normalizing the price by some constant k (think of expressing a price in CHF and in EUR). I There are many (actually infinitely many) ways to do this. 34 Ch 2.7: Existence of an equilibrium I For example, we can normalize for a “numeraire” good (that is money). I Start with a vector of prices p̂. I Suppose you want good 1 to be your numeraire good. I Then normalize the price vector with ↵ = 1 p̂1 : p̂j pj = for all j = (1,... , k). p̂1 I Then, p1 = 1 and for all other goods pj captures how many units of good 1 you have to trade for 1 unit of good j 35 Ch 2.7: Existence of an equilibrium I We will use a di↵erent normalization. I Let ↵ = Pk 1 and j=1 p̂j p̂` p` = P k for all ` = (1,... , k). j j=1 p̂ I this normalization capture the fraction of the bundle of all goods you would have to give up in exchange for 1 unit of good `. 36 Ch 2.7: Existence of an equilibrium I Furthermore, prices normalized in this way sum up to 1: k k " # X X p̂` ` p = Pk j `=1 `=1 j=1 p̂ Pk p̂` = Pk`=1 j j=1 p̂ =1 I This will be useful in the existence proof because it implies that the prices are in the unit simplex. 37 Ch 2.7: Existence of an equilibrium Definition 2.8 (Simplex) A (k 1)-dimensional simplex is a set S k 1 such that 8 9 < X k = S k 1 = p 2 Rk+ : pj = 1 : ; j=1 I Draw a one-dimensional and two-dimensional simplex. 38 Ch 2.7: Existence of an equilibrium I A mathematical result: Proposition 2.9 (Brouwer fixed-point theorem) If f : S k 1 ! S k 1 is a continuous function from the unit simplex to itself, there is some x in S k 1 such that x = f (x). I Proof is omitted, but it is easy to obtain intuition for k = 2. I We are now (finally!) in a position to prove the main existence result. 39 Ch 2.7: Existence of an equilibrium Proposition 2.10 (Existence of Walrasian equilibria) If z : S k 1 ! Rk is a continuous function that satisfies Walras’ law, pz(p) ⌘ 0, then there exists some p⇤ such that z(p⇤ )  0. 40 Ch 2.8: Welfare I Now that we know that an equilibrium allocation of resources exists, we can analyze if it is “good.” I But what does “good” mean? I Efficient? I Fair? I Equal? I What if we care about all of the above? I When is it justified to decrease the welfare of one person to increase the welfare of another? I This is a difficult question that reasonable people can disagree about. 41 Ch 2.8: Welfare I In the next section we will focus on a (precisely defined) concept of efficiency. I However, this does not imply that only efficiency matters! I As economists, we should focus on understanding mechanisms that can best achieve the goals of the society, and not attempt to prescribe those goals! 49 Ch 2.9: Pareto efficiency Vilfredo Pareto (1848-1923) Professor of Political Economy University of Lausanne 50 Ch 2.9: Pareto efficiency Definition 2.11 (Weak Pareto efficiency) A feasible allocation x is weakly Pareto efficient if there is no feasible allocation x0 such that all agents strictly prefer x0 to x. Definition 2.12 (Strong Pareto efficiency) A feasible allocation x is strongly Pareto efficient if there is no feasible allocation x0 such that all agents weakly prefer x0 to x, and some agent strictly prefers x0 to x 51 Ch 2.9: Pareto efficiency I It should be easy to see that an allocation which is strongly Pareto efficient is also weakly Pareto efficient. I The reverse is in general not true. I However, under weak conditions we can guarantee that the two conditions are equivalent. 52 Ch 2.9: Pareto efficiency I We will say that a utility function is continuous if for every allocation x, limy!x u(y) = u(x). I For more see Varian, Chapter 7. 53 Ch 2.9: Pareto efficiency Proposition 2.13 (Equivalence of WPE and SPE) Suppose that the utility functions of all agents are continuous and strongly increasing. Then an allocation is weakly Pareto efficient if and only if it is strongly Pareto efficient. 54 Ch 2.9: Pareto efficiency I From now on, when we say Pareto efficient we will mean WPE. I Note that Pareto efficient allocations can be very unequal. For example, with strongly increasing utility functions, one person having everything is Pareto efficient. I We can depict the set of Pareto efficient allocations in an Edgeworth box. 55 Ch 2.10: Fundamental welfare theorems Proposition 2.14 (First Theorem of Welfare Economics) If (x, p) is a Walrasian equilibrium and the utility functions of all agents are strongly increasing, then x is Pareto efficient. 56 Ch 2.10: Fundamental welfare theorems Proposition 2.15 (Second Theorem of Welfare Economics) Suppose x⇤ is a Pareto efficient allocation and that preferences are represented by continuous and strongly increasing utility functions. Suppose further that a competitive equilibrium exists from the initial endowments !i = x⇤i and let it be given by (p0 , x0 ). Then, in fact, (p0 , x⇤ ) is a competitive equilibrium. 57 Ch 2.11: Welfare maximization I Now we turn to the question of social welfare. I Suppose that there exists some social welfare function W : Rn ! R. I W (u1 ,... , un ) gives us the “social utility.” I An important assumption: W is increasing in each of its arguments. 58 Ch 2.11: Welfare maximization I An allocation is socially optimal if it solves: max W (u(x1 ),... , u(xn )) x n X n X s.t. xji = !ij for all j = 1,... , k. i=1 i=1 59 Ch 2.11: Welfare maximization Proposition 2.16 (Welfare maximization implies Pareto efficiency) If x⇤ maximizes a social welfare function, then x⇤ is Pareto efficient. 60 Ch 2.11: Welfare maximization I We will say that a utility function is concave if given any x, y and any ↵ 2 (0, 1) it holds u(↵x + (1 ↵)y) ↵u(x) + (1 ↵)u(y). If the inequality is strict, then we say that the utility function is strictly concave. 61 62 63 64 65 66 Ch 2.11: Welfare maximization Proposition 2.17 (Pareto efficiency maximizes some welfare function) Let x⇤ be a Pareto efficient allocation with x⇤,j i > 0 for all i and all j. Let the utility functions be concave, continuous and strongly increasing. Then,P there is some choice of weights a⇤i such that x⇤ maximizes a⇤i ui (xi ) subject to resource constraints. 67 Ch 2.12: Overview I We have established that under quite general conditions a Walrasian equilibrium will exist in an exchange economy. I We proved that any Walrasian equilibrium is Pareto efficient. I We proved that (essentially) any Pareto efficient allocation can be reached in a Walrasian equilibrium. I In addition, the only tool that we need in order to achieve di↵erent Pareto efficient allocations is redistribution of initial endowments. I We showed that if an allocation is socially optimal, then it is Pareto efficient, and if it is Pareto efficient then it maximizes a particular welfare functions — a weighted sum of utilities of agents. 68  Chapter 3: Production Economy Varian, Chapter 18 Jehle and Reny, Advanced Microeconomic Theory, Chapter 5.3 69 Ch 3.1: Plan I We will formally state the model with production. I We will state the main results without proof. I We will more carefully examine special examples of production economies. 70 Ch 3.2: Firms I We now expand the model by allowing for production. I Production occurs when firms transfer inputs into outputs. I Suppose there is a set of firms F = {1,... , f }. I We will denote a particular production plan (or an input-output vector) of some firm ` 2 F as y` 2 Rk. I For any specific production plan y` and a commodity j 2 K, if: I y`j < 0 we will say that the commodity j is an input; I and if y`j > 0 then we will call the commodity j an output; I For example y` = ( 2, 1) means that the firm ` can produce 1 unit of good 2 with the input of 2 units of good 1. 71 Ch 3.2: Firms I Given a price vector p, the profit of firm ` is simply k X py` = pj y`j. j=1 I For example if p = (1, 3) and y` = ( 2, 1) then py` = 1 · ( 2) + 3 · 1 = 1. I Denote with Y` the set of all feasible production plans of firm `. 72 Ch 3.2: Firms I Suppose we have one production plan for each firm: (y1 ,... , yf ). P I Then, y = f`=1 y` is an aggregate production plan. I It is a very convenient way to think about the aggregate production: it summarizes all inputs and outputs of the entire economy, even when some firms produce goods that are used as inputs by other firms. I The set of all aggregate feasible production plans is given by the Minkowski sum: f X Y = Y` `=1 73 Ch 3.2: Firms Hermann Minkowski (1864-1909) ETH Zurich (1896-1902) 74 Ch 3.2: Firms I Firms are assumed to be maximizing profits. I Formally: max py`. y` 2Y` I Assume a solution to this problem exists and denote it y` ⇤. I We are interested in a point at which all firms are maximizing their profits. 75 Ch 3.2: Firms I The following proposition gives us a simple way to find such points. Proposition 3.1 (Aggregate profit maximization) An aggregate production plan y maximizes aggregate profit if and only if each firm’s production plan y` maximizes its individual profit. 76 Ch 3.2: Firms I Difficulties in analyzing economies with production is that behavior of aggregate supply might be strange. The following assumption rules out these difficulties. Assumption 3.2 (Production technology) 1. 0 2 Y`. 2. Y` is closed and bounded. 3. Y` is strongly convex. That is, for all distinct y` and y`0 , and all ↵ 2 (0, 1), there exists ȳ` 2 Y` such that ȳ` ↵y` + (1 ↵)y`0 and the equality does not hold. 77 Ch 3.2: Firms Proposition 3.3 (Properties of Y ) If each Y` satisfies Assumption 3.2, then the aggregate production possibility set Y also satisfies Assumption 3.2. Without proof. Proposition 3.4 (Properties of supply) If Y` satisfies Assumption 3.2, then for every p 0 the solution to the firm’s problem is unique and the firm’s supply function y` (p) is well defined and continuous on Rk++. Without proof. Since we are maximizing a continuous function py` over a compact set Y` , a maximum exists. It is unique because Y` is assumed strongly convex. 78 Ch 3.3: Consumers I What happens to firm profits? I In a private ownership economy all firms are owned by (some) consumers. I However, a single consumer does not have to own an entire firm. I Furthermore, a single consumer can own multiple (shares of) firms. I Here we show how this can be easily incorporated into the existing model. 79 Ch 3.3: Consumers I Let ✓i,` be the share of firm ` that consumer i owns. I Shares have to satisfy some (trivial) regularity properties. I Nobody can own more than the entire firm, or less than no shares in the firm. That is, 0  ✓i,`  1, for all i 2 N and all ` 2 F. I Each firm has to be fully owned. That is n X ✓i,` = 1, for all ` 2 F. i=1 80 Ch 3.3: Consumers I We have to incorporate firm profits into consumer’s budget constrain. I Let mi (p) be the budget of consumer i. I Then: f X mi (p) = p!i + ✓i,` py` (p) `=1 81 Ch 3.3: Consumers I Formally, each agent i 2 N solves max ui (x1i ,... , xki ) x1i ,...,xki k X such that pj xji = mi (p) j=1 I If each Y` satisfies the Assumption 3.2 and if ui is continuous, strongly increasing and strictly quasiconcave, then the solution to the consumer’s problem will exist and be unique. I The solution gives the demand function xi (p, mi (p)) : Rk+1 ! Rk. 82 Ch 3.3: Quasiconcave functions I We say that a utility function is quasiconcave if given any x, y it follows that u(↵x + (1 ↵)y) min{u(x), u(y)} for all ↵ 2 (0, 1). If the inequality is strict, then we say that the utility function is strictly quasiconcave. I Note that concavity is a stronger assumption than quasiconcavity. I Any concave function is quasiconcave. I A quasiconcave function does not have to be concave. 83 Ch 3.4: Main results Definition 3.5 (Walrasian equilibrium with production) A Walrasian equilibrium is given by a vector of prices p⇤ = (p⇤1 ,... , p⇤k ), an allocation x⇤ = (x⇤1 ,... , x⇤n ) and a production plan y⇤ = (y1⇤ ,... , yf⇤ ), such that: 1. for each agent i 2 N , it holds x⇤i = xi (p⇤ , mi (p⇤ )) (that is, x⇤i maximizes agent’s utility); 2. for each firm ` 2 F , it holds y`⇤ = y` (p⇤ ) (that is, y`⇤ maximizes firm’s profits); P Pn Pf 3. and ni=1 x⇤,j i  j i=1 !i + ⇤,j `=1 y` for all j 2 K (that is, there is no excess demand). 84 Ch 3.4: Main results Proposition 3.6 (Existence of an equilibrium with production) If 1. each Y` satisfies the Assumption 3.2, 2. each ui is continuous, strongly increasing and strictly quasiconcave, P 3. and y + ni=1 !i 0 for some y 2 Y , then a Walrasian equilibrium with production exists. Without proof (based on Theorem 5.13 in Jehle and Reny). 85 Ch 3.4: Main results I A feasible allocation (x, y) is (weakly) Pareto efficient if there is no other feasible allocation (x0 , y0 ) such that all agents strictly prefer x0 to x. I The equivalence result (Proposition 2.13) extends to the production economy. I As before, we assume all ui are continuous and strongly increasing so that the equivalence holds. Proposition 3.7 (FTWE with production) If (x⇤ , y⇤ , p⇤ ) is a Walrasian equilibrium, then (x⇤ , y⇤ ) is Pareto efficient. Without proof. 86 Ch 3.4: Main results Proposition 3.8 (STWE with production) Suppose (x⇤ , y⇤ ) is a Pareto efficient allocation in which x⇤ 0, and each ui is continuous, strongly increasing and strictly quasiconcave. Further, suppose each Y` satisfies the Assumption 3.2. Then, there exists a price vector p⇤ such that: 1. if ui (xi ) > ui (x⇤i ), then p⇤ xi > p⇤ x⇤i for all i 2 N ; 2. if y` 2 Y` , then p⇤ y`  p⇤ y`⇤ for all ` 2 F. Without proof. 87 Ch 3.5: Robinson Crusoe economy 88 Ch 3.5: Robinson Crusoe economy I In a Robinson Crusoe economy there is only one consumer (i.e. Robinson Crusoe). I There is only one producer: Robinson Crusoe! I There are two goods: I h is time (time to work, or time to relax), I y is coconuts. I Each good will have a price: I w is the price for time (think of it as the wage), I p is the price of coconuts. 89 Ch 3.5: Robinson Crusoe economy I “Robinson Crusoe the producer” uses time as input and produces coconuts. I “Robinson Crusoe the producer” maximizes profits at prices (w, p). I “Robinson Crusoe the producer” is owned entirely by “Robinson Crusoe the consumer”. I Hence, all profits generated are transferred to “Robinson Crusoe the consumer”. I “Robinson Crusoe the consumer” maximizes utility given prices (w, p) and subject to the budget constraint. 90 Ch 3.5: Robinson Crusoe economy Potato-producing economy in The Martian 91 Ch 3.5: Robinson Crusoe economy I The production possibility set of “Robinson Crusoe the producer” is Y = {( h, y) | 0  h  b and 0  y  h↵ }, where b > 0 and ↵ 2 (0, 1). I One way to think about this is that the production function is y(h) = h↵ but the firm can always waste some production. I We can represent Y graphically. 92 Ch 3.5: Robinson Crusoe economy I The utility function of “Robinson Crusoe the consumer” is u(h, y) = h1 y where 2 (0, 1). I Robinson Crusoe is endowed with T units of time and 0 coconuts. That is: ! = (T, 0). I Suppose that b > T. I Represent the budget set of “Robinson Crusoe the consumer” for some p, w > 0. 93 Ch 3.5: Robinson Crusoe economy I We want to find the Walrasian equilibrium. I That is, the prices (w⇤ , p⇤ ) and the allocation (h⇤ , y ⇤ ). I The plan is as follows: 1. Find the supply function and the profits of “Robinson Crusoe the producer”. 2. Find the demand function of “Robinson Crusoe the consumer”. 3. Set demand equal to supply in one market and find prices that achieve equality. 4. Obtain the allocation by using the prices derived above. I We will illustrate each of the steps 1-3 graphically. 94 Ch 3.5: Robinson Crusoe economy I Can (y c , hc , y f , hf ) be supported by prices in an equilibrium? I Convexity of Y is crucial! 95 Ch 3.6: Overview I We have extended the model to include production. I We have generalized the definition of Walrasian equilibrium. I We have shown that the main results (existence, FTWE, STWE) extend to the case with production. I We have studied the Robinson Crusoe economy. 96  Chapter 4: Time and uncertainty Jehle and Reny, Advanced Microeconomic Theory, Chapter 5.4 Varian, Chapter 19 97 Ch 4.1: Expanding the model I Our model so far was static — there was only one period so agents always consumed their entire budget. I But in reality, there are multiple periods in which consumption occurs and agents borrow and save to improve their utility. I In the model, the utility from a good, once it has been acquired, is guaranteed. I In reality, houses sometimes burn down, and people purchase insurance to protect themselves against this. I This chapter shows how our model can be expanded to include both time and uncertainty. 98 Ch 4.2: Time I In the very beginning of this class we mentioned that eggs today and eggs tomorrow are not the same commodity. I This is the basically how we can incorporate time into our model. I Suppose that the set of basic commodities is K = {1,... , k} and that the model exists over periods T = {1,... , t}. I We will consider each commodity-time combination as a distinct commodity. I Thus, denote the amount of commodity j allocated to consumer i in period ⌧ with xj,⌧ i. I If the consumers have preferences over all such commodity-time combination, we can simply apply all our previous results. 99 Ch 4.3: Uncertainty I Uncertainty is treated in the same way as time. I Suppose there are two (uncertain) states of the world — sun or rain. I There are also two goods, sunscreen and umbrellas. I The utility of that consumers derive from the goods depend on the state (sunscreen does not really help you stay dry). I Our model can capture this situation if we consider sunscreen when it rains and sunscreen when it is sunny to be two di↵erent commodities. I We can treat (much!) more general forms of uncertainty. 100 Chapter 5: The core of an economy Varian, Chapter 21.1 Jehle and Reny, Advanced Microeconomic Theory, Chapter 5.5 101 Ch 5.1: Other trading mechanisms I A Walrasian equilibrium is a particular way to trade. I There are other ways — e.g. bargaining. I What would be the equilibrium of these other trading mechanisms? I Assume we are back in the pure exchange economy. 102 Ch 5.1: Other trading mechanisms I Consider the following “market game” I the agents have some initial endowments !i I they randomly bump into each other and make tentative trades with each other I when no further trades can be made, all trades are carried out. I What might be a reasonable outcome of this game? I No two agents should be able to profitably trade with each other. I But also, no three agents should be able to profitably trade between themselves. I No group of agents, of any size, should be able to profitably trade between themselves. 103 Ch 5.2: The core Definition 5.1 (Blocking coalition) Let S ✓ N be a coalition of agents. We say that S blocks a feasible allocation x if there exists an allocation x0 such that P 0 P 1. i2S x i = i2S !i ; 2. ui (x0 i ) > ui (xi ) for all i 2 S. I One could alternatively define a weaker notion of a blocking allocation where one agent is strictly better o↵ and all other agents are not worse o↵. I This is the same as the di↵erence between strong and weak Pareto efficiency. I Like there, the two definitions of a blocking allocation are equivalent if utility is strongly increasing and continuous. 104 Ch 5.2: The core Definition 5.2 (The core of an exchange economy) The core of an exchange economy with endowment !, denoted C(!), is the set of all unblocked feasible allocations. 105 Ch 5.3: Core and equilibria I In this section we explore the relationship between core and equilibria. I We will show that a Walrasian equilibrium is always in the core and I that the core will shrink to the set of Walrasian equilibria as the economy grows. I Thus, no matter the exact nature of the trading mechanism, we will tend to end up in (or close to) a Walrasian equilibrium! 106 Ch 5.3: Core and equilibria Proposition 5.3 (Walrasian equilibirum is in the core) If (x⇤ , p) is a Walrasian equilibrium with initial endowments !, then x⇤ 2 C(!). I This gives us an easy way to guarantee that the core is not empty: if a Walrasian equilibrium exists, the core will be non-empty! I We will next show that only Walrasian equilibrium allocations remain in the core as the economy grows. 107 Ch 5.3: Core and equilibria Definition 5.4 (An r-fold replica economy) Take an exchange economy with a set of consumers N = {1,... , n}. An r-fold replica of this economy is an exchange economy where the set of consumers N r is such that there exists n disjoint sets N1r , N2r ,... , Nnr , where 1. N r = [ni=1 Nir , 2. |Nir | = r for all i, 3. for each consumer i 2 N all consumers in the set Nir have identical endowments and identical preferences. 108 Ch 5.3: Core and equilibria I Intuitively, an r-fold replica economy is created by duplicating r times each original consumer. I We will call each consumer in the set Nir a consumer of type i. I We will call the core of the r-fold replica economy the r-core. 109 Ch 5.3: Core and equilibria Proposition 5.5 (Equal treatment in the core) Suppose that preferences can be represented with strictly quasiconave, strongly increasing and continuous utility functions. Then, if x is an allocation in the core of an r-fold replica of the economy, then any two agents of the same type must receive the same bundle. I For simplicity, we provide a proof for the case when there are only two agent types. It suffices to provide the economic intuition. 110 Ch 5.3: Core and equilibria Proposition 5.6 (Shrinking core; or Edgeworth-Debreu-Scarf Theorem) Suppose that preferences can be represented with strictly quasiconave, strongly increasing and continuous utility functions. Further suppose that there is a unique Walrasian equilibrium x⇤ from initial endowments !. Then, if x 6= x⇤ there exists a replication r such that x is not in the r-core. I We will provide just a sketch of the proof. 111 Ch 5.3: Core and equilibria Herbert Scarf (1930-2015) Yale University 112 Ch 5.3: Core and equilibria Gérard Debreu (1921-2004) UC Berkeley Nobel Memorial Prize in Economics (1983) 113