Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: THEend8_
LECTURE 2
Consumer Theory
CONSUMER CHOICE: UTILITY MAXIMIZATION Part 1 ECA5101LECTURE 2 2 Where are we? LECTURE 2 ECA5101 Semester 1 AY 2021/2022 3 • Preference • Budget constraint • Consumer’s optimal choice – On the budget line – On the highest indifference curve • The optimal choice is the point of tangency – Tangency condition + budget line – Or the Lagrangian method • Optimal basket is not always a point of tangency What is the optimal basket? LECTURE 2 ECA5101 Semester 1 AY 2021/2022 4 • Suppose the consumer has utility function • Price of food is 1, price of clothing is 2, consumer’s income is 10 • The utility maximization problem is U(F,C) = FC +10F max F,C FC +10F s.t. F + 2C =10 What is the optimal basket? Cont’ LECTURE 2 ECA5101 Semester 1 AY 2021/2022 5 • Solving it, we get the two equations • The solution is F=15, C=-2.5 • Is it the optimal basket? F + 2C =10 C +10 F = 1 2 Rewriting the Utility Maximization Problem LECTURE 2 ECA5101 Semester 1 AY 2021/2022 6 • In fact, there should be two more constraints to any utility maximization problem – The consumption of each good cannot be negative • The true utility maximization problem (problem A) is max F,C FC +10F s.t. F + 2C =10 F ≥ 0 C ≥ 0 What is the difference? • What is the difference between problem A and the following problem (problem B)? • If the solution to B satisfies F>=0 and C>=0 – It is also the solution to A • If the solution to B does not satisfy F>=0 or C>=0 – It is not the solution to A LECTURE 2 ECA5101 Semester 1 AY 2021/2022 7 max F,C FC +10F s.t. F + 2C =10 The Intuitive Way • Solve problem B and check – Whether the solution to B indeed satisfies F>=0 and C>=0 • If yes, we are done – Solution to problem B is also the solution to problem A • Unfortunately in our example the solution to B is F=15, C=-2.5 – Violates C>=0 LECTURE 2 ECA5101 Semester 1 AY 2021/2022 8 The Intuitive Way Cont’ • This means the constraint C>=0 must bind – I.e., it holds with equality, C=0 – As C=-2.5 is not possible, C=0 is the best/closest we can get • Thus the solution to A is F=10, C=0 • When there are inequality constraints, the constraints may or may not bind – In this example, the constraint C>=0 binds while the constraint F>=0 does not bind LECTURE 2 ECA5101 Semester 1 AY 2021/2022 9 How do we know F=10, C=0 is optimal? LECTURE 2 ECA5101 Semester 1 AY 2021/2022 10 • At this basket, consumer spends all the money on food • Comparing the per dollar marginal utilities at this point • If possible, consumer wants to buy more F and less C to increase utility • But consumption of C is already 0 MUF PF = C +10 PF =10 > MUCPC = F PC = 10 2 = 5 The Scenario in Graph LECTURE 2 ECA5101 Semester 1 AY 2021/2022 11 F C 0 10 5 U1 U2 H J I U3 Corner Solution LECTURE 2 ECA5101 Semester 1 AY 2021/2022 12 • At optimal basket, it is not always true that both (all) goods are consumed • Corner solution is an optimal basket at which the consumption of at least one good is 0 – Optimal basket either on the horizontal or vertical axis • An optimal basket in which both goods are consumed is an interior solution • At corner solutions – Indifference curve may not be tangent to the budget line The Formal Way • If we want to solve it formally, we can rewrite problem A as • The solution to this problem is F=10, C=0 • But in most cases we do not need to use the formal method LECTURE 2 ECA5101 Semester 1 AY 2021/2022 13 max F,C,a,b FC +10F s.t. F + 2C =10 F − a2 = 0 C − b2 = 0 Application: Back-To-School Vouchers • NTUC offers back-to-school education vouchers to low-income families – $125 voucher per school child to be spent on school-related goods • Similar program – US food stamps • What is the effect of the voucher on – Consumer’s choice – Consumer’s utility LECTURE 2 ECA5101 Semester 1 AY 2021/2022 14 Impact of Voucher on Consumer 1 LECTURE 2 ECA5101 Semester 1 AY 2021/2022 15 School(x) Others(y) y1 A 0 Cy2 x2x1125Px I Px I Py I +125 Px B C is the new optimal basket with voucher y is called a composite good as it is the composite of all other goods Impact of Voucher on Consumer 2 LECTURE 2 ECA5101 Semester 1 AY 2021/2022 16 School(x) Others(y) y1 A 0 x1 125Px I Px I Py I +125 Px B B is the new optimal basket with voucher How about a cash subsidy of $125? LECTURE 2 ECA5101 Semester 1 AY 2021/2022 17 School(x) Others(y) y1 A 0 Cy2 x2x1125Px I Px I Py I +125 Px B Consumer 1: C is still the new optimal basket with cash Cash Gives Consumer 2 Higher Utility! LECTURE 2 ECA5101 Semester 1 AY 2021/2022 18 School(x) Others(y) y1 A 0 x1 125Px I Px I Py I +125 Px B Consumer 2: D is the new optimal basket with cash x2 y2 D Why not just use cash? • The lump sum principle – Cash subsidy gives consumers higher or the same utility compared to subsidizing specific goods – Income tax leaves consumers with higher or the same utility compared to tax on specific goods • Why not use cash instead of the back-to-school vouchers? LECTURE 2 ECA5101 Semester 1 AY 2021/2022 19 REVEALED PREFERENCE Part 2 ECA5101 Semester 1 AY 2021/2022LECTURE 2 20 What is revealed preference? LECTURE 2 ECA5101 Semester 1 AY 2021/2022 21 • What we have been doing so far – Given preference (indifference curves/utility functions) – Given budget constraint – We can find consumer’s optimal choice • Can we go the other way round? – Given budget constraint – Given consumer’s optimal choice – Can we get any information on preference? • Revealed preference is the analysis that enable us to infer preference based on observed prices and choices Strictly Preferred vs. Weakly Preferred LECTURE 2 ECA5101 Semester 1 AY 2021/2022 22 • A is strictly preferred to B • A is weakly preferred to B if – Either – Or – We use A B A B A ≈ B A ≥ B From Choice to Preference LECTURE 2 ECA5101 Semester 1 AY 2021/2022 23 • Suppose we observe the budget constraint of a consumer • We also know the optimal basket chosen given the budget constraint • But we do not know his preference – We know his preference satisfies the four assumptions – We also know his preference does not change with prices or income • Our goal – To infer preference – how he ranks different baskets A vs. Other Points on the Budget Line 0 24 y LECTURE 2 ECA5101 Semester 1 AY 2021/2022 x A B Since A is optimal, no other point on the budget line is strictly preferred to A C 10 20 5 10 Can we conclude that A is strictly preferred to any other basket on the budget line? A vs. Other Points on the Budget Line Cont’ 0 25 y LECTURE 2 ECA5101 Semester 1 AY 2021/2022 x A B A is revealed to be weakly preferred to any other basket on the budget line C 10 20 5 10 Depending on the consumer’s preference, A may not be the only optimal basket A vs. Other Points below the Budget Line 0 26 y LECTURE 2 ECA5101 Semester 1 AY 2021/2022 x A B A is revealed to be strictly preferred to any other basket in the budget set (but not on the budget line) C C B A ≥C ⇒ A B 10 20 5 10 Another Way to Understand Revealed Preference LECTURE 2 ECA5101 Semester 1 AY 2021/2022 27 • Suppose basket A=(xA, yA) is the optimal basket given prices Px, Py, and income I – Basket A must be on the budget line • No other affordable basket is strictly preferred to A • Therefore, if basket B=(xB, yB) is strictly preferred to basket A, it must be that PxxA +PyyA = I PxxB +PyyB > PxxA +PyyA = I Another Way to Understand Revealed Preference Cont’ LECTURE 2 ECA5101 Semester 1 AY 2021/2022 28 • Similarly, if basket C=(xC, yC) is indifferent to basket A, it must be that • To summarize – If A is the optimal basket given the budget constraint – Any basket that is strictly preferred to A cannot be affordable – Any basket that is indifferent to A cannot cost less than A PxxC +PyyC ≥ PxxA +PyyA = I DEMAND FUNCTION AND INDIRECT UTILITY FUNCTION Part 3 ECA5101 Semester 1 AY 2021/2022LECTURE 2 29 Consumer Theory Roadmap • Consumer choice – Utility maximization problem – Expenditure minimization problem • Demand – Demand function and indirect utility function – Compensated demand function and expenditure function • Revealed preference • Slutsky equation • Consumer welfare LECTURE 2 ECA5101 Semester 1 AY 2021/2022 30 From Optimal Baskets to Individual Demand Curve • Assume the consumer chooses food and clothing • Suppose the price of food changes – The price of clothing and income are fixed • Consumer’s optimal consumption of food will change • Consumer’s individual demand curve for food is the optimal consumption level of food as a function of the price of food LECTURE 2 ECA5101 Semester 1 AY 2021/2022 31 Demand Curve for Food LECTURE 2 ECA5101 Semester 1 AY 2021/2022 32 F F C PF PF =10PF = 20PF = 30 30 20 10 D 20 28 50 20 28 50 Price-consumption curveA B C Demand curve for food Demand Curve • What is demand curve? – Optimal (utility maximizing) quantity of a good the consumer is willing to buy as a function of its price – Holding income and other prices fixed • Law of demand – Demand curve is downward sloping – Higher price, lower quantity demanded LECTURE 2 ECA5101 Semester 1 AY 2021/2022 33 Example: Deriving Individual Demand Curve LECTURE 2 ECA5101 Semester 1 AY 2021/2022 34 • Suppose the consumer has utility function • Suppose price of clothing is 2, income is 10 • What is the demand curve for food? • The consumer solves U(F,C) = FC max F,C FC s.t. PFF + 2C =10 What if income changes? LECTURE 2 ECA5101 Semester 1 AY 2021/2022 35 F F C I I = 350I = 200I =100 100 20 25 45 20 25 45 Income-consumption curve A B C Suppose prices are fixed 200 350 Engel Curve Engel Curve LECTURE 2 ECA5101 Semester 1 AY 2021/2022 36 • Engel curve of a good is the curve that shows the relationship between income and optimal consumption – Holding other factors fixed • If the good is normal – Engel curve is upward sloping • If the good is inferior – Engel curve is downward sloping Demand Function LECTURE 2 ECA5101 Semester 1 AY 2021/2022 37 • Quantity demanded depends on – Price of the good – Income – Prices of other goods • Can we write down a general formula? – Quantity demanded as a function of all parameters (income and all prices) – I.e., • Demand function for a good is quantity demanded as a function of income and all prices F(PF,PC, I ), C(PF,PC, I ) Example: Demand Function LECTURE 2 ECA5101 Semester 1 AY 2021/2022 38 • Using the same utility function U(F,C)=FC • The consumer solves • We get • Demand functions are PF PC = C F PFF +PCC = I F = I2PF , C = I2PC max F,C FC s.t. PFF +PCC = I Cobb-Douglas Utility Function LECTURE 2 ECA5101 Semester 1 AY 2021/2022 39 0 x y U(x, y) = Axαyβ ,A > 0,α > 0,β > 0 MUx = Aαxα−1yβ MUy = Aβxαyβ−1 MRSx,y = MUx MUy = Aαxα−1yβ Aβxαyβ−1 = αy βx Demand Function for Cobb-Douglas Utility Function LECTURE 2 ECA5101 Semester 1 AY 2021/2022 40 • The consumer solves • The tangency condition is • Tangency condition can be written as αy βx = Px Py Pyy = β α Pxx max x,y Axαyβ s.t. Pxx +Pyy = I Demand Functions with Cobb-Douglas Utility Function Cont’ LECTURE 2 ECA5101 Semester 1 AY 2021/2022 41 • The budget line is • Demand functions for x and y are • For Cobb-Douglas utility functions – Demand for one good does not depend on the price of the other good Pxx +Pyy = I x = α α +β × I Px , y = β α +β × I Py Indirect Utility Function • Demand function tells us how consumption of a good changes with prices and income • How about consumer’s utility? – When prices or income change, optimal consumption will change – Maximum utility will change • Maximum utility depends indirectly on prices and income • Indirect utility function is maximum utility as a function of prices and income LECTURE 2 ECA5101 Semester 1 AY 2021/2022 42 V (Px,Py, I ) =U(x(Px,Py, I ), y(Px,Py, I )) Example: Indirect Utility Function • Suppose a consumer solves the following problem • The demand functions are • The indirect utility function is LECTURE 2 ECA5101 Semester 1 AY 2021/2022 43 max F,C FC s.t. PFF +PCC = I F = I2PF , C = I2PC V (PF,PC, I ) =U( I 2PF , I2PC ) = I 2 4PFPC Re-examine the Lagrange Multiplier • If we were to solve the problem using Lagrangian, the multiplier would be • Using the example from the previous page • Differentiating the indirect utility function w.r.t. income, we also get LECTURE 2 ECA5101 Semester 1 AY 2021/2022 44 λ = MUF PF = MUC PC λ = MUF PF = C PF = I 2PFPC ∂V (PF,PC, I ) ∂I = I 2PFPC Envelope Theorem • Suppose we solve the following problem – where a is a parameter • The Lagrangian is • The solution is LECTURE 2 ECA5101 Semester 1 AY 2021/2022 45 max x,y f (x, y,a) s.t. g(x, y,a) = 0 Λ(x, y,λ,a) = f (x, y,a)+λg(x, y,a) x*(a), y*(a), λ*(a) Envelope Theorem Cont’ • If we plug the optimal values back into the objective function, we get • The envelope theorem states that – When the parameter changes, the rate of change in the maximal value of the objective function can be found by partially differentiating the Lagrangian function w.r.t. the parameter and evaluating the derivative at the optimal point LECTURE 2 ECA5101 Semester 1 AY 2021/2022 46 f *(a) df *(a) da = ∂Λ ∂a x=x*,y=y*,λ=λ* Proof by Envelope Theorem
- 经济Economics
- 金融Finance
- 会计Accounting
- 工商管理Business Management
- 计算机computer science
- 商业分析business analysis
- 法律Law
- 管理Management
- 市场营销Marketing
- 哲学Philosophy
- 心理学Psychology
- 物理Physical
- 统计学statistics
- 电气工程electric engineering
- 化学chemistry
- 生物biology
- 数学mathematics
- 语言学linguistics
- 土木工程civil engineering
- 工程力学engineering mechanics
- 艺术art
PROFESSIONAL HELP!
在线咨询
立即下单
Wechat：THEend8_ 
Email：