DOCSLIB.ORG

Chapter Two Consumption Choice Sets Budget Constraints Budget

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Chapter Two Consumption Choice Sets Budget Constraints Budget Consumption Choice Sets • A consumption choice set is the collection of all consumption choices available to the Chapter Two consumer. • What constrains consumption choice? Budgetary and Other Constraints – Budgetary, time and other resource limitations. on Choice Budget Constraints Budget Constraints • A consumption bundle containing x 1 units • Q: When is a consumption bundle of commodity 1, x 2 units of commodity 2 (x 1, … , x n) affordable at given prices and so on up to x n units of commodity n is p1, … , p n? denoted by the vector (x 1, x 2, … , x n). • Commodity prices are p 1, p 2, … , p n. 1 Budget Constraints Budget Constraints • Q: When is a bundle (x 1, … , x n) affordable • The bundles that are only just affordable at prices p 1, … , p n? form the consumer’s budget constraint. • A: When This is the set p x + … + p x ≤ m 1 1 n n ≥ ≥ 0 where m is the consumer’s (disposable) { (x 1,…,x n) | x 1 0, …, x n and = income. p1x1 + … + p nxn m }. Budget Set and Constraint for Budget Constraints Two Commodities x2 • The consumer’s budget set is the set of all Budget constraint is affordable bundles; m /p 2 p1x1 + p 2x2 = m. B(p 1, … , p n, m) = ≥ ≥ Not affordable { (x 1, … , x n) | x 1 0, … , x n 0 and ≤ Just affordable p1x1 + … + p nxn m } • The budget constraint is the upper Affordable boundary of the budget set. m /p 1 x1 2 Budget Set and Constraint for Budget Set and Constraint for Two Commodities Two Commodities x2 x2 Budget constraint is p1x1 + p 2x2 = m is m /p 2 m /p 2 p1x1 + p 2x2 = m. x2 = -(p 1/p 2)x 1 + m/p 2 so slope is -p1/p 2. the collection of all affordable bundles. Budget Budget Set Set m /p 1 x1 m /p 1 x1 Budget Constraint for Three Budget Constraints Commodities • If n = 3 what do the budget constraint and x2 the budget set look like? p1x1 + p 2x2 + p 3x3 = m m /p 2 m /p 3 x3 m /p 1 x1 3 Budget Set for Three Budget Constraints Commodities • For n = 2 and x 1 on the horizontal axis, x ≥≥≥ ≥≥≥ ≥≥≥ the constraint’s slope is -p /p . What 2 { (x 1,x 2,x 3) | x 1 0, x 2 0, x 3 0 and 1 2 ≤≤≤ does it mean? m /p 2 p1x1 + p 2x2 + p 3x3 m} p m x ===−−− 1 x +++ 2 p 1 p m /p 3 2 2 x3 • Increasing x 1 by 1 must reduce x 2 by p1/p 2. m /p 1 x1 Budget Constraints Budget Constraints x x 2 2 Opp. cost of an extra unit of Slope is -p1/p 2 commodity 1 is p 1/p 2 units foregone of commodity 2. And the opp. cost of an extra -p1/p 2 +1 unit of commodity 2 is +1 -p /p 2 1 p2/p 1 units foregone of commodity 1. x1 x1 4 Budget Sets & Constraints; Higher income gives more Income and Price Changes choice x2 New affordable consumption • The budget constraint and budget set choices depend upon prices and income. What Original and happens as prices or income change? new budget constraints are parallel (same Original slope). budget set x1 How do the budget set and Budget Constraints - Income budget constraint change as Changes x2 income m decreases? • Increases in income m shift the constraint Consumption bundles outward in a parallel manner, thereby that are no longer enlarging the budget set and improving affordable. choice. Old and new • Decreases in income m shift the New, smaller constraints constraint inward in a parallel manner, budget set are parallel. thereby shrinking the budget set and reducing choice. x1 5 Budget Constraints - Income Budget Constraints - Price Changes Changes • No original choice is lost and new choices • What happens if just one price decreases? are added when income increases, so • Suppose p 1 decreases. higher income cannot make a consumer worse off. • An income decrease may (typically will) make the consumer worse off. How do the budget set and Budget Constraints - Price budget constraint change as p 1 Changes x2 decreases from p ’ to p ”? 1 1 • Reducing the price of one commodity m/p 2 New affordable choices pivots the constraint outward. No old choice is lost and new choices are added, Budget constraint so reducing one price cannot make the -p ’/p pivots; slope flattens 1 2 consumer worse off. from -p1’/p 2 to Original -p ”/p -p1”/p 2 budget set 1 2 m/p 1’ m/p 1” x1 6 Budget Constraints - Price Uniform Ad Valorem Sales Changes Taxes • Similarly, increasing one price pivots the • An ad valorem sales tax levied at a rate of constraint inwards, reduces choice and 5% increases all prices by 5%, from p to may (typically will) make the consumer (1+0 .05)p = 1 .05p. worse off. • An ad valorem sales tax levied at a rate of t increases all prices by tp from p to (1+ t)p. • A uniform sales tax is applied uniformly to all commodities. Uniform Ad Valorem Sales Uniform Ad Valorem Sales Taxes x2 Taxes • A uniform sales tax levied at rate t m changes the constraint from p1x1 + p 2x2 = m p1x1 + p 2x2 = m p2 to m p1x1 + p 2x2 = m/(1+ t) + (1+ t)p 1x1 + (1+ t)p 2x2 = m ()1++ t p2 i.e. p1x1 + p 2x2 = m/(1+ t). m m x + 1 ()1 ++ t p1 p1 7 Uniform Ad Valorem Sales Uniform Ad Valorem Sales Taxes x Taxes x 2 2 A uniform ad valorem m m sales tax levied at rate t p Equivalent income loss p is equivalent to an income 2 is 2 t m m tax levied at rate . −−− m === t +++ + m m + 1 t ()1++ t p2 1+++ t 1+++ t ()1++ t p2 m m x m m x + 1 + 1 ()1 ++ t p1 p1 ()1 ++ t p1 p1 Other Taxes (Subsidies) The Food Stamp Program • Quantity tax: p1+t • Food stamps are coupons that can be • Lump sum tax : m-t legally exchanged only for food. • How does a commodity-specific gift such • Rationing: x1<Q as a food stamp alter a family’s budget constraint? 8 The Food Stamp Program The Food Stamp Program G F + G = 100: before stamps. • Suppose m = $100, p F = $1 and the price of “other goods” is p = $1. G 100 Budget set after 40 food • The budget constraint is then stamps issued. F + G =100. The family’s budget set is enlarged. 40 100 140 F The Food Stamp Program The Food Stamp Program G F + G = 100: before stamps. • What if food stamps can be traded on a 120 black market for $0.50 each? 100 Budget constraint after 40 food stamps issued. Budget constraint with black market trading. 40 100 140 F 9 Budget Constraints - Relative The Food Stamp Program G Prices F + G = 100: before stamps. 120 • “Numeraire” means “unit of account”. 100 Budget constraint after 40 • Suppose prices and income are measured food stamps issued. in dollars. Say p 1=$2, p 2=$3, m = $12. Black market trading Then the constraint is makes the budget 2x 1 + 3x 2 = 12. set larger again. 40 100 140 F Budget Constraints - Relative Budget Constraints - Relative Prices Prices • If prices and income are measured in • The constraint for p 1=2, p 2=3, m=12 2x + 3x = 12 cents, then p 1=200, p 2=300, m=1200 and 1 2 the constraint is is also 1x 1 + (3/2)x 2 = 6, the constraint for p =1, p =3/2, m=6. 200x 1 + 300x 2 = 1200, 1 2 the same as Setting p 1=1 makes commodity 1 the numeraire and defines all prices relative 2x 1 + 3x 2 = 12. • Changing the numeraire changes neither to p1; e.g. 3/2 is the price of commodity the budget constraint nor the budget set. 2 relative to the price of commodity 1. 10 Budget Constraints - Relative Budget Constraints - Relative Prices Prices • p =2, p =3 and p =6 ⇒ • Any commodity can be chosen as the 1 2 3 numeraire without changing the budget set • price of commodity 2 relative to or the budget constraint. commodity 1 is 3/2, • price of commodity 3 relative to commodity 1 is 3. • Relative prices are the rates of exchange of commodities 2 and 3 for units of commodity 1. Shapes of Budget Constraints Shapes of Budget Constraints • Q: What makes a budget constraint a • But what if prices are not constants? straight line? • E.g. bulk buying discounts, or price • A: A straight line has a constant slope and penalties for buying “too much”. the constraint is • Then constraints will be curved. p1x1 + … + p nxn = m so if prices are constants then a constraint is a straight line. 11 Shapes of Budget Constraints - Shapes of Budget Constraints Quantity Discounts with a Quantity Discount x • Suppose p is constant at $1 but that 2 m = $100 2 Slope = - 2 / 1 = - 2 p =$2 for 0 ≤ x ≤ 20 and p =$1 for x >20. 100 1 1 1 1 (p =2, p =1) Then the constraint’s slope is 1 2 ≤ ≤ - 2, for 0 x1 20 -p /p = Slope = - 1/ 1 = - 1 1 2 { - 1, for x 1 > 20 (p 1=1, p 2=1) and the constraint is 20 50 80 x1 Shapes of Budget Constraints Shapes of Budget Constraints with a Quantity Discount with a Quantity Penalty x2 m = $100 x2 100 Budget Budget Constraint Constraint Budget Set Budget Set 20 50 80 x1 x1 12 Shapes of Budget Constraints - Shapes of Budget Constraints - One Price Negative One Price Negative • Commodity 1 is stinky garbage.
Load more

Recommended publications
  • Lecture 4 " Theory of Choice and Individual Demand
    Lecture 4 - Theory of Choice and Individual Demand David Autor 14.03 Fall 2004 Agenda 1. Utility maximization 2. Indirect Utility function 3. Application: Gift giving –Waldfogel paper 4. Expenditure function 5. Relationship between Expenditure function and Indirect utility function 6. Demand functions 7. Application: Food stamps –Whitmore paper 8. Income and substitution e¤ects 9. Normal and inferior goods 10. Compensated and uncompensated demand (Hicksian, Marshallian) 11. Application: Gi¤en goods –Jensen and Miller paper Roadmap: 1 Axioms of consumer preference Primal Dual Max U(x,y) Min pxx+ pyy s.t. pxx+ pyy < I s.t. U(x,y) > U Indirect Utility function Expenditure function E*= E(p , p , U) U*= V(px, py, I) x y Marshallian demand Hicksian demand X = d (p , p , I) = x x y X = hx(px, py, U) = (by Roy’s identity) (by Shepard’s lemma) ¶V / ¶p ¶ E - x - ¶V / ¶I Slutsky equation ¶p x 1 Theory of consumer choice 1.1 Utility maximization subject to budget constraint Ingredients: Utility function (preferences) Budget constraint Price vector Consumer’sproblem Maximize utility subjet to budget constraint Characteristics of solution: Budget exhaustion (non-satiation) For most solutions: psychic tradeo¤ = monetary payo¤ Psychic tradeo¤ is MRS Monetary tradeo¤ is the price ratio 2 From a visual point of view utility maximization corresponds to the following point: (Note that the slope of the budget set is equal to px ) py Graph 35 y IC3 IC2 IC1 B C A D x What’swrong with some of these points? We can see that A P B, A I D, C P A.
    [Show full text]
  • 2. Budget Constraint.Pdf
    Engineering Economic Analysis 2019 SPRING Prof. D. J. LEE, SNU Chap. 2 BUDGET CONSTRAINT Consumption Choice Sets . A consumption choice set, X, is the collection of all consumption choices available to the consumer. n X = R+ . A consumption bundle, x, containing x1 units of commodity 1, x2 units of commodity 2 and so on up to xn units of commodity n is denoted by the vector xx =(1 ,..., xn ) ∈ X n . Commodity price vector pp =(1 ,..., pn ) ∈R+ 1 Budget Constraints . Q: When is a bundle (x1, … , xn) affordable at prices p1, … , pn? • A: When p1x1 + … + pnxn ≤ m where m is the consumer’s (disposable) income. The consumer’s budget set is the set of all affordable bundles; B(p1, … , pn, m) = { (x1, … , xn) | x1 ≥ 0, … , xn ≥ 0 and p1x1 + … + pnxn ≤ m } . The budget constraint is the upper boundary of the budget set. p1x1 + … + pnxn = m 2 Budget Set and Constraint for Two Commodities x2 Budget constraint is m /p2 p1x1 + p2x2 = m. m /p1 x1 3 Budget Set and Constraint for Two Commodities x2 Budget constraint is m /p2 p1x1 + p2x2 = m. the collection of all affordable bundles. Budget p1x1 + p2x2 ≤ m. Set m /p1 x1 4 Budget Constraint for Three Commodities • If n = 3 x2 p1x1 + p2x2 + p3x3 = m m /p2 m /p3 x3 m /p1 x1 5 Budget Set for Three Commodities x 2 { (x1,x2,x3) | x1 ≥ 0, x2 ≥ 0, x3 ≥ 0 and m /p2 p1x1 + p2x2 + p3x3 ≤ m} m /p3 x3 m /p1 x1 6 Opportunity cost in Budget Constraints x2 p1x1 + p2x2 = m Slope is -p1/p2 a2 Opp.
    [Show full text]
  • 3 Lecture 3: Choices from Budget Sets
    3 Lecture 3: Choices from Budget Sets Up to now, we have been rather demanding about the data that we need in order to test our models. We have made two important assumptions: that we observe choices from all possible choice sets, and that we observe choice correspondences (i.e. we see all the options that a decision maker would be ‘happy with’). In many cases, we may not be so lucky with our data. Unfortunately, without these two properties, conditions and are no longer necessary or sufficient to guarantee a utility representation. Consider the following example of an incomplete data set. Example 1 Let = and say we observe the following (incomplete) choice correspondence { } ( )= { } { } ( )= { } { } ( )= { } { } This choice correspondence satisfies properties and trivially. is satisfied because we do not observe any choices from sets that are subsets of each other. is satisfied because we never see two objects chosen from the same set. However, there is no way that we can rationalize these choices with a complete preference relation. The first observation implies that ,thesecond  that and the third that 3. Thus, any binary relation that would rationalize these choices   would be intransitive. In fact, in order for theorem 1 to hold, we don’t have to observe choices from all subsets of , but we do have to need at least all subsets of that contain two and three elements (you should go back and look at the proof of theorem 1 and check that you agree with this statement). What about if we drop the assumption that we observe a choice correspondence, and instead observe a choice function? For example, we could ask the following question: Question 1 Let :2 ∅ be a choice function.
    [Show full text]
  • EC9D3 Advanced Microeconomics, Part I: Lecture 2
    EC9D3 Advanced Microeconomics, Part I: Lecture 2 Francesco Squintani August, 2020 Budget Set Up to now we focused on how to represent the consumer's preferences. We shall now consider the sour note of the constraint that is imposed on such preferences. Definition (Budget Set) The consumer's budget set is: B(p; m) = fx j (p x) ≤ m; x 2 X g Francesco Squintani EC9D3 Advanced Microeconomics, Part I August, 2020 2 / 49 Budget Set (2) 6 x1 2 L = 2 X = R+ c c c c c c c c c c c c B(p; m) c c c c c c c - x2 Francesco Squintani EC9D3 Advanced Microeconomics, Part I August, 2020 3 / 49 Income and Prices The two exogenous variables that characterize the consumer's budget set are: the level of income m the vector of prices p = (p1;:::; pL). Often the budget set is characterized by a level of income represented by the value of the consumer's endowment x0 (labour supply): m = (p x0) Francesco Squintani EC9D3 Advanced Microeconomics, Part I August, 2020 4 / 49 Utility Maximization The basic consumer's problem (with rational, continuous and monotonic preferences): max u(x) fxg s:t: x 2 B(p; m) Result If p > 0 and u(·) is continuous, then the utility maximization problem has a solution. Proof: If p > 0 (i.e. pl > 0, 8l = 1;:::; L) the budget set is compact (closed, bounded) hence by Weierstrass theorem the maximization of a continuous function on a compact set has a solution. Francesco Squintani EC9D3 Advanced Microeconomics, Part I August, 2020 5 / 49 First Order Condition Result If u(·) is continuously differentiable, the solution x∗ = x(p; m) to the consumer's problem is characterized by the following necessary conditions.
    [Show full text]
  • Senior Secondary Course ECONOMICS (318)
    Senior Secondary Course ECONOMICS (318) 2 Course Coordinator Dr. Manish Chugh fo|k/kue~ loZ/kua iz/kkue~ NATIONAL INSTITUTE OF OPEN SCHOOLING (An Autonomous Institution under MHRD, Govt. of India) A-24-25, Institutional Area, Sector-62, NOIDA-201309 (U.P.) Website: www.nios.ac.in, Toll Free No. 18001809393 Printed on 60 GSM Paper with NIOS Watermark. © National Institute of Open Schooling May, 2015 (,000 Copies) Published by the Secretary, National Institute of Open Schooling, A-24-25, Institutional Area, Sector-62, Noida and printed at M/s #1PGQHHUGV Printers, 5/3, Kirti Nagar, Industrial Area, New Delhi. ADVISORY COMMITTEE Prof. C.B. Sharma Dr. Kuldeep Agarwal Dr. Rachna Bhatia Chairman Director (Academic) Assistant Director (Academic) NIOS, NOIDA (UP) NIOS, NOIDA (UP) NIOS, NOIDA (UP) CURRICULUM COMMITTEE Dr. O.P. Agarwal Sh. J. Khuntia Dr. Padma Suresh (Former Director of the Eco. Deptt.) Associate Professor (Economics) Associate Professor (Economics) NREC College, Meerut University School of Open Learning Sri Venkateshwara College Khurja (UP) Delhi University, Delhi Delhi University, Delhi Prof. Renu Jatana Sh. H.K. Gupta Sh. A.S. Garg Associate Professor, MLSU, Rtd.PGT, From NCT Rtd.Vice Principal, From NCT Udaipur (Rajasthan) New Delhi. New Delhi. Sh. Ramesh Chandra Dr.Manish Chugh Retd. Reader (Economics) Academic Officer (Economics), NCERT, Delhi. NIOS, NOIDA LESSON WRITERS Sh. J. Khuntia Dr. Anupama Rajput Ms.Sapna Chugh Associate Professor (Eco.), Associate Professor, PGT (Economics) School of Open Learning, Janki Devi Memorial College S.V. Public School, Jaipur Univ. of Delhi Univ. of Delhi Dr. Bhawna Rajput Sh. A.S. Garg Dr.
    [Show full text]
  • A Theoretical Exposition of Consumers' Response To
    fite, A THEORETICAL EXPOSITION OF CONSUMERS'RESPONSE TO ALTERNATIVE FOOD POLICIES WAITE LIBRARY DEPT. OF AG & APPLIED ECONOMICS 1994 BUFORD AVE. - 232 COB UNIVERSITY OF MINNESOTA ST. PAUL, MN 55108 U.S.A. DEPARTMENT OF AGRICULTURAL AND RESOURCE ECONOMICS DIVISION OF AGRICULTURE AND NATURAL RESOURCES UNIVERSITY OF CALIFORNIA AT BERKELEY WORKING PAPER NO.615 A THEORETICAL EXPOSITION OF CONSUMERS'RESPONSE TO ALTERNATIVE FOOD POLICIES . by G.Mythili WAITE MEMO s r-- " DEPT. OF AG.AND Avr, 1994 BUFORD Av. UNIVERSITY OF N. ST. PAUL,MN 5C This paper was written while the author was a Ford Foundation Post Doctoral Fellow in the Department of Agricultural and Resource Economics, University of California at Berkeley. The author wishes to thank Brian Wright for his helpful comments and suggestions. California Agricultural Experiment Station Giarmini Foundation ofAgricultural Economics August, 1991 37X -79`i 4L3"/55 (//'--6/5 A THEORETICAL EXPOSITION OF CONSUMERS'RESPONSE TO ALTERNATIVE FOOD POLICIES 1.Introduction &his study is an attempt to relate alternative food subsidy programs with reference to the implication for consumer theories. The Government's goal is set on raising the nutritional standard of those who are underfed rather than redistributional aspects. The rationale for such goal assumes away consumer sovereignty. This study is focussed only on consumer sector and ignores production sector merely to avoid complexities involved in the theoretical formulation, though it is recognized that in. most of the thirdworld countries where a significant proportion of the population live on farming and consume their own produce, the linkage between production and consumption decisions does influence the behavioral pattern of an individual as a consumer.
    [Show full text]
  • The Budget Constraint Consumption Sets N a Consumption Set Is the Collection of All Consumption Choices Available to the Consumer
    The Budget Constraint Consumption Sets n A consumption set is the collection of all consumption choices available to the consumer. n What constrains consumption choice? n Budgetary, time and other resource limitations. Budget Constraints n A consumption bundle containing x1 units of commodity 1, x2 units of commodity 2 and so on up to xn units of commodity n is denoted by the vector (x1, x2, … , xn). n Commodity prices are p1, p2, … , pn. Budget Constraints n Q: When is a consumption bundle (x1, … , xn) affordable at given prices p1, … , pn? Budget Constraints n Q: When is a bundle (x1, … , xn) affordable at prices p1, … , pn? n A: When p1x1 + … + pnxn ≤ m where m is the consumer’s (disposable) income. Budget Constraints n The bundles that are only just affordable form the consumer’s budget constraint. This is the set { (x1,…,xn) | x1 ≥ 0, …, xn ≥ 0 and p1x1 + … + pnxn = m }. Budget Constraints n The consumer’s budget set is the set of all affordable bundles; B(p1, … , pn, m) = { (x1, … , xn) | x1 ≥ 0, … , xn ≥ 0 and p1x1 + … + pnxn ≤ m } n The budget constraint is the upper boundary of the budget set. Budget Set and Constraint for x2 Two Commodities Budget constraint is m /p2 p1x1 + p2x2 = m. m /p1 x1 Budget Set and Constraint for x2 Two Commodities Budget constraint is m /p2 p1x1 + p2x2 = m. m /p1 x1 Budget Set and Constraint for x2 Two Commodities Budget constraint is m /p2 p1x1 + p2x2 = m. Just affordable m /p1 x1 Budget Set and Constraint for x2 Two Commodities Budget constraint is m /p2 p1x1 + p2x2 = m.
    [Show full text]
  • Eco-Mechanisms Within Economic Evolution – Schumpeterian Approach
    Agnieszka Lipieta Cracow University of Economics Poland Andrzej Malawski Cracow University of Economics Poland Eco-mechanisms within economic evolution – Schumpeterian approach Corresponding author Agnieszka Lipieta Department of Mathematics Cracow University of Economics Rakowicka 27, 31 – 510 Cracow, Poland +48 12 2935 270 e-mail: [email protected] https://orcid.org/0000-0002-3017-5755 Eco-mechanisms within economic evolution – Schumpeterian approach Abstract Supporting eco-innovations and economic eco-activities is one of a important challenge to decision makers. The above is related to the problem of specification of mechanisms resulting in introducing eco-innovations. The original vision of the economic evolution determined by innovation was firstly presented by Joseph Schumpeter who identified essential innovative changes as well as indicated different mechanisms governing the economic evolution. The aim of the paper is to suggest a cohesive topological approach maintained in the stream of Schumpeter’s thought, to study changes in the economy, which result in the elimination of at least one harmful commodity or technology from the market, by incorporating Hurwicz apparatus in a suitably modified competitive economy. Qualitative properties of mechanisms which can occur within the economic evolution are also analyzed. Keywords: eco-innovation; eco-mechanism; economic evolution JEL: D41, D50, O31 1. Introduction “Eco-innovation is any innovation resulting in significant progress towards the goal of sustainable development, by reducing the impacts of our production modes on the environment, enhancing nature’s resilience to environmental pressures, or achieving a more efficient and responsible use of natural resources” (Europa.eu› eu › pubs › pdf › factsheets › ec). In the interest of each community is introducing eco-innovations and eco-activities and the above should be the most important task and the challenge to decision makers.
    [Show full text]
  • Chapter 3 Consumer Behavior
    Chapter 3 Consumer Behavior Questions for Review 1. What are the four basic assumptions about individual preferences? Explain the significance or meaning of each. (1) Preferences are complete: this means that the consumer is able to compare and rank all possible baskets of goods and services. (2) Preferences are transitive: this means that preferences are consistent, in the sense that if bundle A is preferred to bundle B and bundle B is preferred to bundle C, then bundle A is preferred to bundle C. (3) More is preferred to less: this means that all goods are desirable, and that the consumer always prefers to have more of each good. (4) Diminishing marginal rate of substitution: this means that indifference curves are convex, and that the slope of the indifference curve increases (becomes less negative) as we move down along the curve. As a consumer moves down along her indifference curve she is willing to give up fewer units of the good on the vertical axis in exchange for one more unit of the good on the horizontal axis. This assumption also means that balanced market baskets are generally preferred to baskets that have a lot of one good and very little of the other good. 2. Can a set of indifference curves be upward sloping? If so, what would this tell you about the two goods? A set of indifference curves can be upward sloping if we violate assumption number three: more is preferred to less. When a set of indifference curves is upward sloping, it means one of the goods is a “bad” so that the consumer prefers less of that good rather than more.
    [Show full text]
  • Indirect Utility and Duality
    EC 701, Fall 2005, Microeconomic Theory October 20, 2005 page 181 4. Duality in Consumer Theory Definition 4.1. For any utility function U (x),the corresponding indirect utility function is given by: V (p, w) max U (x) x 0, px w ≡ x { | ≥ ≤ } max U (x) x Bp,w , ≡ x { | ∈ } so that if x ∗ is the solution to the UMP, then V (p, w)=U (x ∗). Note that V (p, w) max U(x) x 0,px w ≡ x { | ≥ ≤ } and x(p, w) argmax U(x) x 0,px w , ≡ x { | ≥ ≤ } so that V (p, w) U(x(p, w)). ≡ EC 701, Fall 2005, Microeconomic Theory October 20, 2005 page 182 Example 4.1. Find the demand correspondence and the indirect utility function for the linear utility function U = x + y. With the given utility function, x and y are perfect substitutes • and the MU s are both 1 so the consumer will buy only the cheaper good. Let pm =min px , py . Demand for the cheaper good will be • { } w/pm and demand for the more expensive good will be 0. If px = py then demand for the goods can be any combination • such that expenditures add up to w. EC 701, Fall 2005, Microeconomic Theory October 20, 2005 page 183 The consumer will always buy w/p units of the cheaper good, so • m his utility must also be w/pm. Therefore, the indirect utility function is w v (px,py,w)= min p ,p { x y} p2 pp21= increasing V Buy x1 Vppw(,12 ,)= u Buy x2 p 0 1 ¿¿Quasiconcave?? • EC 701, Fall 2005, Microeconomic Theory October 20, 2005 page 184 Proposition 4.1.
    [Show full text]
  • Radner Equilibrium: Definition and Equivalence with Arrow-Debreu
    Radner Equilibrium: Definition and Equivalence with Arrow-Debreu Equilibrium Econ3030 Fall2020 Lecture 11, November 12 Outline 1 Sequential Trade and Arrow Securities 2 Radner Equilibrium 3 Equivalence between Arrow-Debreu and Radner Equilibria 4 Assets and Asset Markets Timing of Trades in Arrow-Debreu Remark In an Arrow-Debreu economy, all decisions are made at date 0. Individuals exchange promises to deliver and receive quantities of the goods according to the state that is realized. Afterwards, as time and uncertainty unfold, these promises are carried through exactly as planned, without changes. What if tomorrow there were also markets for the L goods? Is there an incentive to trade in these spot markets? Spot Markets Do Not Matter in Arrow-Debreu Fact Given an Arrow-Debreu equilibrium, there are no incentives to trade in spot markets. Proof. Suppose not: consumers can find a mutually beneficial trade in some period t. Thus, there exist a feasible allocation that all consumers pefer weakly to the equilibrium bundle, with at least one consumer preference being strict. consumers trade only if they do better. This new allocation is feasible, and Pareto dominates the Arrow-Debreu equilibrium. But this is impossible, because the First Welfare Theorem holds. This conclusion is disappointing as most real world markets are spot markets, not the forward markets imagined by an Arrow-Debreu economy. Spot and Forward Markets Remark Most real world markets are spot, not forward. Objective Reconcile this observation with a general equilibrium model with uncertainty similar to Arrow-Debreu. Idea The main role of state contingent commodities is to allow welfare transfers across states..
    [Show full text]
  • Mathematical Economics Final, December 13, 2001
    Mathematical Economics Final, December 13, 2001 1. Consider the function f(x; y) = x3 + xy + y2 − x=2 + y. Find and classify (maximum, minimum, saddlepoint) all critical points of f. Answer: The derivative is df = (3x2 +y −1=2; x+2y +1). Setting this to zero, we find 3x2 +y −1=2 = 0 and x + 2y = −1. The second equation can be rewritten y = −(1 + x)=2. Substituting in the first equation yields 3x2 − x=2 − 1 = 0. This has solutions x = 2=3 and x = −1=2. The critical points are then (2=3; −5=6) and (−1=2; −1=4). The Hessian is: 6x 1 H = d2f = : 1 2 Thus H1 = 6x and H2 = 12x − 1. When x = 2=3, H1 = 4 > 0 and H2 = 7 > 0. The Hessian is positive definite, and so (2=3; −5=6) is a local minimum. (There is no global maximum or minimum since 3 f(x; 0) = x − x=2 is unbounded above and below.) When x = −1=2, H1 = −3 < 0 and H2 = −2 < 0. The Hessian is indefinite, and so (−1=2; −1=4) is a saddlepoint. 2. A consumer has utility function u(x; y) = ln x + ln y. The consumer consumes non-negative quantities of both goods, subject to two budget constraints: 3x + 2y ≤ 6 and 2x + 3y ≤ 6. Find (x∗; y∗) that maximizes utility subject to the above four constraints. Be sure to check the constraint qualification and second-order conditions. Answer: This sort of problem can arise when one of the goods is rationed via rationed coupons, and there is a market for ration coupons where the relative price for coupons is different than for goods.
    [Show full text]