micro2_slides_ch1-ch2.pdf

Microeconomics 2 Igor Letina Department of Economics University of Bern Fall 2024 Last updated: September 14, 2024 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 2024 Last updated: September 14, 2024 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