Modern Microeconomics
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Theory of Consumer Behaviour
Theory of Consumer Behaviour Public Economy 1 What is Consumer Behaviour? • Suppose you earn 12,000 yen additionally – How many lunch with 1,000 yen (x1) and how many movie with 2,000 yen (x2) you enjoy? (x1, x2) = (10,1), (6,3), (3,4), (2,5), … – Suppose the price of movie is 1,500 yen? – Suppose the additional bonus is 10,000 yen? Public Economy 2 Consumer Behaviour • Feature of Consumer Behaviour • Consumption set (Budget constraint) • Preference • Utility • Choice • Demand • Revealed preference Public Economy 3 Feature of Consumer Behaviour Economic Entity Firm(企業), Consumer (家計), Government (経済主体) Household’s income Capital(資本), Labor(労働),Stock(株式) Consumer Firm Hire(賃料), Wage(賃金), Divided(配当) Goods Market (財・サービス市場) Price Demand Supply Quantity Consumer = price taker (価格受容者) Public Economy 4 Budget Set (1) • Constraint faced by consumer Possible to convert – Budget Constraint (income is limited) into monetary unit – Time Constraint (time is limited) under the given wage rate – Allocation Constraint Combine to Budget Constraint Generally, only the budget constraint is considered Public Economy 5 Budget Set (2) Budget Constraint Budget Constraint (without allocation constraint) (with allocation constraint) n pi xi I x2 i1 x2 Income Price Demand x1 x1 Budget Set (消費可能集合) B B 0 x1 0 x1 n n n n B x R x 0, pi xi I B x R x 0, pi xi I, x1 x1 i1 i1 Public Economy 6 Preference (1) • What is preference? A B A is (strictly) preferred to B (A is always chosen between A and B) A B A is preferred to B, or indifferent between two (B is never chosen between A and B) A ~ B A and B is indifferent (No difference between A and B) Public Economy 7 Preference (2) • Assumption regarding to preference 1. -
CONSUMER CHOICE 1.1. Unit of Analysis and Preferences. The
CONSUMER CHOICE 1. THE CONSUMER CHOICE PROBLEM 1.1. Unit of analysis and preferences. The fundamental unit of analysis in economics is the economic agent. Typically this agent is an individual consumer or a firm. The agent might also be the manager of a public utility, the stockholders of a corporation, a government policymaker and so on. The underlying assumption in economic analysis is that all economic agents possess a preference ordering which allows them to rank alternative states of the world. The behavioral assumption in economics is that all agents make choices consistent with these underlying preferences. 1.2. Definition of a competitive agent. A buyer or seller (agent) is said to be competitive if the agent assumes or believes that the market price of a product is given and that the agent’s actions do not influence the market price or opportunities for exchange. 1.3. Commodities. Commodities are the objects of choice available to an individual in the economic sys- tem. Assume that these are the various products and services available for purchase in the market. Assume that the number of products is finite and equal to L ( =1, ..., L). A product vector is a list of the amounts of the various products: ⎡ ⎤ x1 ⎢ ⎥ ⎢x2 ⎥ x = ⎢ . ⎥ ⎣ . ⎦ xL The product bundle x can be viewed as a point in RL. 1.4. Consumption sets. The consumption set is a subset of the product space RL, denoted by XL ⊂ RL, whose elements are the consumption bundles that the individual can conceivably consume given the phys- L ical constraints imposed by the environment. -
Slutsky Equation the Basic Consumer Model Is Max U(X), Which Is Solved by the Marshallian X:Px≤Y Demand Function X(P, Y)
Slutsky equation The basic consumer model is max U(x), which is solved by the Marshallian x:px≤y demand function X(p; y). The value of this primal problem is the indirect utility function V (p; y) = U(X(p; y)). The dual problem is minx fp · x j U(x) = ug, which is solved by the Hicksian demand function h(p; u). The value of the dual problem is the expenditure or cost function e(p; u) = p · H(p; u). First show that the Hicksian demand function is the derivative of the expen- diture function. e (p; u) = p · h(p; u) ≤p · h(p + ∆p; u) e (p + ∆p; u) = (p + ∆p) · h(p + ∆p; u) ≤ (p + ∆p) · h(p; u) (p + ∆p) · ∆h ≤ 0 ≤ p · ∆h ∆e = (p + ∆p) · h(p + ∆p; u) − p · h(p; u) ∆p · h(p + ∆p; u) ≤ ∆e ≤ ∆p · h(p; u) For a positive change in a single price pi this gives ∆e hi(p + ∆p; u) ≤ ≤ hi(p; u) ∆pi In the limit, this shows that the derivative of the expenditure function is the Hicksian demand function (and Shephard’s Lemma is the exact same result, for cost minimization by the firm). Also ∆p · h(p + ∆p; u) − ∆p · h(p; u) = ∆p · ∆h ≤ 0 meaning that the substitution effect is negative. Also, the expenditure function lies everywhere below its tangent with respect to p: that is, it is concave in p: e(p + ∆p; u) ≤ (p + ∆p) · h(p; u) = e(p; u) + ∆p · h(p; u) The usual analysis assumes that the consumer starts with no physical en- dowment, just money. -
Chapter 5 Online Appendix: the Calculus of Demand
Chapter 5 Online Appendix: The Calculus of Demand The chapter uses the graphical model of consumer choice to derive the demand for a good by changing the price of the good and the slope of the budget constraint and then plotting the relationship between the prices and the optimal quantities as the demand curve. Figure 5.7 shows this process for three different prices, three budget constraints, and three points on the demand curve. You can see that this graphical approach works well if you need to plot a few points on a demand curve; however, getting the quantity demanded at every price requires us to apply the calculus of utility maximization. Before we do the calculus, let’s review what we know about the demand curve. The chapter defines the demand curve as “the relationship between the quantity of a good that consumers demand and the good’s price, holding all other factors constant.” In the appendix to Chapter 4, we always Figure 5.7 Building an Individual’s Demand Curve (a) (b) Mountain Dew Mountain Dew P = 4 (2 liter bottles) (2 liter bottles) G PG = 1 Income = $20 P = 2 10 10 G PG = $1 PMD = $2 4 3 3 U U 1 2 U U 1 3 2 0 14 20 0 3105 8 14 20 Quantity of grape juice Quantity of grape juice (1 liter bottles) (1 liter bottles) Price of Price of grape juice grape juice ($/bottle) ($/bottle) Caroline’s $4 demand for 2 grape juice $1 1 0 0 14 3 8 14 Quantity of grape juice Quantity of grape juice (1 liter bottles) (1 liter bottles) (a) At her optimal consumption bundle, Caroline purchases (b) A completed demand curve consists of many of these 14 bottles of grape juice when the price per bottle is $1 and quantity-price points. -
List of Paradoxes 1 List of Paradoxes
List of paradoxes 1 List of paradoxes This is a list of paradoxes, grouped thematically. The grouping is approximate: Paradoxes may fit into more than one category. Because of varying definitions of the term paradox, some of the following are not considered to be paradoxes by everyone. This list collects only those instances that have been termed paradox by at least one source and which have their own article. Although considered paradoxes, some of these are based on fallacious reasoning, or incomplete/faulty analysis. Logic • Barbershop paradox: The supposition that if one of two simultaneous assumptions leads to a contradiction, the other assumption is also disproved leads to paradoxical consequences. • What the Tortoise Said to Achilles "Whatever Logic is good enough to tell me is worth writing down...," also known as Carroll's paradox, not to be confused with the physical paradox of the same name. • Crocodile Dilemma: If a crocodile steals a child and promises its return if the father can correctly guess what the crocodile will do, how should the crocodile respond in the case that the father guesses that the child will not be returned? • Catch-22 (logic): In need of something which can only be had by not being in need of it. • Drinker paradox: In any pub there is a customer such that, if he or she drinks, everybody in the pub drinks. • Paradox of entailment: Inconsistent premises always make an argument valid. • Horse paradox: All horses are the same color. • Lottery paradox: There is one winning ticket in a large lottery. It is reasonable to believe of a particular lottery ticket that it is not the winning ticket, since the probability that it is the winner is so very small, but it is not reasonable to believe that no lottery ticket will win. -
APEC 5152 - Handout 1
APEC 5152 - Handout 1 February 16, 2017 Micro-Preliminaries Contents 1 Consumer preferences 2 1.1 The indirect utility function . 4 1.2 The expenditure function . 6 1.3 Aggregation . 7 2 Production technologies 8 2.1 The cost function . 9 2.2 The value-added function . 11 2.3 Aggregation . 13 2.3.1 The aggregate cost function and value added function . 13 2.3.2 The aggregate gross national product function . 15 3 Appendix 16 3.1 The Primal-Dual Problem (Envelope Theorem) . 16 3.2 Elasticities and homogenous functions . 18 1 Introduction - microeconomic foundations Throughout these notes, the following notation denotes factor endowments, factor rental rates and output prices. Sectors are indexed by j 2 f1; 2; 3g ; and denote the quantity of sector-j's output by 3 the scalar Yj: Corresponding output prices are denoted p = (p1; p2; p3) 2 R++, with the scalar pj representing the per-unit price of sector-j output. We consider three factor endowments: labor, L; capital, K; and land Z. The corresponding factor rental rates are: w is the wage rate, r is the rate of return to capital, and τ is the unit land rental rate. Represent the vector of factor rental rates by w = (w; r; τ) : 1 Consumer preferences The economy is composed of a large number of atomistic households. Each household faces the η η η η 3 same vector of prices p and the same vector of factor rental rates w. Let υ = (L ;K ;Z ) 2 R++ denote the level of factor endowments held by household-η; with Lη;Kη and Zη representing the η 3 household's endowment of labor, capital and land. -
Public Economics Lecture Notes
Public Economics Lecture Notes Matteo Paradisi 1 Contents 1 Section 1-2: Uncompensated and Compensated Elas- ticities; Static and Dynamic Labor Supply 4 1.1 Uncompensated Elasticity and the Utility Maximization Problem . 4 1.2 Substitution Elasticity and the Expenditure Minimization Problem . 6 1.3 Relating Walrasian and Hicksian Demand: The Slutsky Equation . 6 1.4 StaticLaborSupplyChoice .................................. 7 1.5 Dynamic Labor Supply . 10 2 Section 2: Introduction to Optimal Income Taxa- tion 12 2.1 TheIncomeTaxationProblem ................................ 12 2.2 Taxation in a Model With No Behavioral Responses . 12 2.3 TowardstheMirrleesOptimalIncomeTaxModel . 13 2.4 Optimal Linear Tax Rate . 13 2.5 OptimalTopIncomeTaxation ................................ 15 3 Section 3-4: Mirrlees Taxation 17 3.1 TheModelSetup........................................ 17 3.2 OptimalIncomeTax ...................................... 19 3.3 Diamond ABC Formula . 20 3.4 Optimal Taxes With Income Effects ............................. 21 3.5 Pareto Efficient Taxes . 23 3.6 A Test of the Pareto Optimality of the Tax Schedule . 23 4 Section 5: Optimal Taxation with Income Effects and Bunching 28 4.1 Optimal Taxes with Income Effects.............................. 28 4.2 Bunching Estimator . 31 5 Section 6: Optimal Income Transfers 34 5.1 OptimalIncomeTransfersinaFormalModel . 34 5.2 Optimal Tax/Transfer with Extensive Margin Only . 35 5.3 Optimal Tax/Transfer with Intensive Margin Responses . 36 5.4 Optimal Tax/Transfer with Intensive and Extensive Margin Responses . 37 6 Section 7: Optimal Top Income Taxation 38 6.1 TrickleDown: AModelWithEndogenousWages . 38 6.2 Taxation in the Roy Model and Rent-Seeking . 39 6.3 Wage Bargaining and Tax Avoidance . 40 7 Section 8: Optimal Minimum Wage and Introduc- tion to Capital Taxation 44 7.1 Optimal Minimum Wage . -
IMVH Notation List
INTERMEDIATE MICROECONOMICS VIDEO HANDBOOK NOTATION LIST 11/18/14 MATHEMATICAL AND BASIC CONCEPTS x, ƒ(x) change in value of x, change in value of ƒ(x) ƒ(x), ƒ(x) first, second derivative of single-variable function ƒ(x) ƒ(x1,...,xn)/xi or ƒi (x1,...,xn) partial derivatives of multivariate function ƒ(x1,…,xn) y,x or E y,x elasticity of y with respect to x generic parameter in an optimization problem x*(), x1*(),…,xn*() solutions function(s) to a general optimization problem () optimal value function for a general optimization problem Lagrange multiplier P generic price for supply-demand diagram Q generic quantity for supply-demand diagram S, D supply and demand for supply-demand diagram CONSUMPTION AND DEMAND xi consumption level of commodity i (x1,…,xn) consumption bundle , , weak preference, strict preference and indifference U(x1,…,xn) utility function MUi (x1,…,xn) or Ui (x1,…,xn) marginal utility of commodity i MRSij(x1,…,xn) marginal rate of substitution between commodities i and j pi price of commodity i I consumer’s income * xi (p1,…,pn,I ) ordinary (“Marshallian”) demand function V(p1,…,pn,I ) indirect utility function c – xi (p1,…,pn,u) compensated (“Hicksian”) demand function – E(p1,…,pn,u) expenditure function EV, CV equivalent variation, compensating variation Le consumer’s leisure La consumer’s labor supply I 0 consumer’s nonlabor income w wage rate i or r interest rate ct consumption in period t Mt income in period t st saving in period t PRODUCTION AND COST L labor input K capital input q or Q output ƒ(L,K) or F(L,K) -
Advanced Microeconomics Comparative Statics and Duality Theory
Advanced Microeconomics Comparative statics and duality theory Harald Wiese University of Leipzig Harald Wiese (University of Leipzig) Advanced Microeconomics 1 / 62 Part B. Household theory and theory of the …rm 1 The household optimum 2 Comparative statics and duality theory 3 Production theory 4 Cost minimization and pro…t maximization Harald Wiese (University of Leipzig) Advanced Microeconomics 2 / 62 Comparative statics and duality theory Overview 1 The duality approach 2 Shephard’slemma 3 The Hicksian law of demand 4 Slutsky equations 5 Compensating and equivalent variations Harald Wiese (University of Leipzig) Advanced Microeconomics 3 / 62 Maximization and minimization problem Maximization problem: Minimization problem: Find the bundle that Find the bundle that maximizes the utility for a minimizes the expenditure given budget line. needed to achieve a given utility level. x2 indifference curve with utility level U A C B budget line with income level m x1 Harald Wiese (University of Leipzig) Advanced Microeconomics 4 / 62 The expenditure function and the Hicksian demand function I Expenditure function: e : R` R R, ! (p, U¯ ) e (p, U¯ ) := min px 7! x with U (x ) U¯ The solution to the minimization problem is called the Hicksian demand function: χ : R` R R` , ! + (p, U¯ ) χ (p, U¯ ) := arg min px 7! x with U (x ) U¯ Harald Wiese (University of Leipzig) Advanced Microeconomics 5 / 62 The expenditure function and the Hicksian demand function II Problem Express e in terms of χ and V in terms of the household optima! Lemma For any α > 0: χ (αp, U¯ ) = χ (p, U¯ ) and e (αp, U¯ ) = αe (p, U¯ ) . -
ECN 3040 Quiz 4 Winter 2019 1. the Marshallian Demand Function Is
ECN 3040 Quiz 4 Winter 2019 1. The Marshallian demand function is obtained by A) solving the expenditure minimization problem for the optimal value of the good conditional on prices and income. B) solving the expenditure minimization problem for the optimal value of the good conditional on prices and utility. C) solving the utility maximization problem for the optimal value of the good conditional on prices and utility. D) solving the utility maximization problem for the optimal value of the good conditional on prices and income. 2. The Hicksian demand function for good X states opitmal demand for X as a function of A) money income and utitlity. B) prices and utility. C) the inverted Lagrange multiplier. D) prices and money income. 3. For a normal good, the good's Hicksian demand curve is _____________ the good's Marshallian demand curve. A) the inverse of B) unrelated to C) steeper than D) flatter than 4. Billie Jean has utility function U XY0.6 0.4 . She has $100 of income to spend, the price of X is PX $10 and the price of Y is PY $8 . At her optimal consumption bundle, Billie Jean A) chooses {X 2, Y 10}and enjoys U1 10.6 units of utility. B) chooses {X 4, Y 6}and enjoys U1 4.6 units of utility. C) chooses {X 9, Y 8}and enjoys U1 8.6 units of utility. D) chooses {X 6, Y 5} and enjoys U1 5.6 units of utility. Page 1 5. Given the consumer has Cobb-Douglas utility function U XY1 , the Hicksian demand function for good X is I A) X H PX (1 )[I P ] B) X H Y PX 1 H PX C) X U 1 PY 1 H PY D) X U 1 PX 6. -
Money-Metrics in Applied Welfare Analysis: a Saddlepoint Rehabilitation∗
Money-Metrics in Applied Welfare Analysis: A Saddlepoint Rehabilitation∗ Edward E. Schleey r M. Ali Khanz September 13, 2020 Abstract Once a popular tool to estimate welfare changes, the money metric of McKenzie (1957) and Samuelson (1974) gradually faded from use after welfare theorists and practioners argued that it led to inegalitarian recommendations. We present a rehabilitation articulated through Uzawa's saddle-point theorem of concave programming. Allowing for non-ordered preferences, we (i) prove that any competitive equilibrium allocation maximizes the money- metric sum at the associated competitive equilibrium price; and (ii) connect the results to Radner's local welfare measure and the behavioral welfare proposals of Bernheim-Rangel (2009). As a foundation for its use in local cost-benefit analysis, we also give conditions for the money-metric to be differentiable in consumption, without either transitivity or convexity of preferences. Key words and phrases: Money-metric, saddlepoint inequalities, first fundamental welfare theorem, incom- plete preferences, intransitive preferences, non-standard decision theories, behavioral welfare economics, benefit function, distance function, cost-benefit in the small JEL Classification Numbers: D110, C61, D610. ∗We have benefited substantially from the encouragement and comments of Beth Allen, Salvador Barbara, Robert Becker, Gabriel Carroll, Luciano de Castro, Eddie Dekel, Tsogbadral Galaabaatar, Koichi Hamada, Peter Hammond, Michael Jerison, Elliot Lipnowski, Kevin Reffett, Larry Samuelson, Ludvig Sinander, V. Kerry Smith, Metin Uyank, John Weymark, Nicholas Yannelis, and Itzhak Zilcha. A first draft of this work was written during Khan's visit to the Department of Economics at Arizona State University (ASU), March 1-6, 2017 and during Schlee's visit to the Department of Economics at Ryerson University as Distinguished Lecturer, May 1-5, 2017. -
Notes on Microeconomic Theory
Notes on Microeconomic Theory Nolan H. Miller August 18, 2006 Contents 1 The Economic Approach 1 2ConsumerTheoryBasics 5 2.1CommoditiesandBudgetSets............................... 5 2.2DemandFunctions..................................... 8 2.3ThreeRestrictionsonConsumerChoices......................... 9 2.4AFirstAnalysisofConsumerChoices.......................... 10 2.4.1 ComparativeStatics................................ 11 2.5Requirement1Revisited:Walras’Law.......................... 11 2.5.1 What’stheFunnyEqualsSignAllAbout?.................... 12 2.5.2 Back to Walras’ Law: Choice Response to a Change in Wealth . 13 2.5.3 TestableImplications............................... 14 2.5.4 Summary:HowDidWeGetWhereWeAre?.................. 15 2.5.5 Walras’Law:ChoiceResponsetoaChangeinPrice.............. 15 2.5.6 ComparativeStaticsinTermsofElasticities................... 16 2.5.7 WhyBother?.................................... 17 2.5.8 Walras’ Law and Changes in Wealth: Elasticity Form . 18 2.6 Requirement 2 Revisited: Demand is Homogeneous of Degree Zero. 18 2.6.1 Comparative Statics of Homogeneity of Degree Zero . 19 2.6.2 AMathematicalAside................................ 21 2.7Requirement3Revisited:TheWeakAxiomofRevealedPreference.......... 22 2.7.1 CompensatedChangesandtheSlutskyEquation................ 23 2.7.2 Other Properties of the Substitution Matrix . 27 i Nolan Miller Notes on Microeconomic Theory ver: Aug. 2006 3 The Traditional Approach to Consumer Theory 29 3.1BasicsofPreferenceRelations............................... 30 3.2FromPreferencestoUtility...............................