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Theory of Consumer Behaviour

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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. Complete: Either , A B or A ~ B is satisfied A (B完備性or完全性) 2. Transitive : and B C then A C (推移性) 3. Reflexive : A A (連続性or反射性) Public Economy 8 Preference (3) – indifference curve x2 Question Are these two lines satisfy the y three assumptions? x x2 0 x1 Indifference curve (無差別曲線) x n C(x) y R y ~ x y z 0 x1 x, y and z would be indifferent, Public Economy which is obviously inconsistent9 Preference (4) – indifference curve Question Are these two lines satisfy the three assumptions? x2 x2 0 x1 x 0 x1 Public Economy 10 Preference (5) • Additional assumptions regarding to preference A(x , x ) 1 2 1. Monotonicity (単調性): x1 x1 A B B(x1, x2 ) (凸性) 2. Convexity x2 Better bundles Average bundle x Worse bundles 0 x1 Complete, Transitive and Reflexive There exist Utility Function Public Economy 11 Utility Function • What is utility function? Definition x,y Rn ,x y There exist u : Rn R that satisfies u(x) u(y) Theorem If the preference satisfies complete, transitive, reflexive and monotonicity, then there exist utility function that satisfies x,y Rn ,x y There exist that satisfies u(x) u(y) Public Economy 12 Utility function and Indifference curve u(x) u u(x) 2 x2 u(x) u3 x2 u3 y u2 x u(x) u1 u1 0 x1 x1 Indifference curve is expressed as a contour line (等高線) of an utility function Public Economy 13 Example of utility function x2 u(x) 0.5log x1 0.5log x2 0 x1 Public Economy 14 • Draw a graph and indifference curve of the following utility functions u1(x) 0.1log x1 0.1log x2 u2 (x) x1x2 2 2 u3 (x) x1 x2 Public Economy 15 Various Utility Function • Ordinal utility (序数的効用) Only the order of the utilities is meaningful u(x , x ) x x – Cobb-Douglas type 1 2 1 2 or u(x1, x2 ) ln x1 ln x2 – Linear u(x1, x2 ) x1 x2 – Leontief type u(x1, x2 ) minx1,x2 – CES type u(x1, x2 ) ax1 bx2 • Cardinal Utility (基数的効用) The value of the utilities is also meaningful Can we express the value of utility correctly? Public Economy 16 Choice (選択) • Consumers are assumed to choose most preferable bundle from their budget set x2 The utility of the right upper indifference curve is higher Budget Set We should find a bundle B whose utility is maximum 0 x 1 among a given budget set Public Economy 17 Consumer Behaviour Model (消費者行動モデル) max u(x1, x2 ,..., xn ) x subject to n pi xi I i1 x 2 xi 0 i 1,...,n Monotonical utility function Budget Set B 0 x1 Public Economy 18 Consumer Behaviour Model (消費者行動モデル) max u(x1, x2 ,..., xn ) x subject to n pi xi I i1 x2 Budget Set B 0 x1 Public Economy 19 First order condition (一階条件) n L(x,) u(x1, x2 ,..., xn ) pi xi I i1 L xi 0: u xi pi n L 0: pi xi I i1 Public Economy 20 Marginal Rate of Substitution (MRS;限界代替率) x2 Dx Slope = 2 = marginal rate of Dx1 substitution Dx2 Dx1 0 x1 MRS measures the rate at which the consumer is just willing to substitute one good for the other Public Economy 21 Graphic illustration of first order condition x2 marginal rate of substitution (限界代替率) = market rate of substitution (市場代替率) u x p i i u x j p j 0 x1 Public Economy 22 Demand (需要) • The consumer’s demand function give the optimal amount of each of the goods as a function of the prices and income faced by the consumer Solution of the Consumer behaviour model xi p1, p2 ,..., pn , I p (Marshallian) demand function (マーシャルの)需要関数 Public Economy x1 23 Example • Find demand function with Cobb-Douglas type utility function u(x , x ) ln x ln x 1 2 1 2 u(x1, x2 ) x1 x2 max subject to p1x1 p2 x2 I Hint: It is easier to solve if we assume Cobb-Douglas type utility function as Public Economy 24 Answer • First order condition L aln x1 bln x2 p1x1 p2x2 I L a p 0 x x 1 L 1 1 p x p x I 0 L b 1 1 2 2 p 0 x x 2 2 2 • Rewrite above equations using MRS a x p 1 1 b x2 p2 p1x1 p2 x2 I 0 Public Economy 25 Answer (Cont) • By solving those equations a I x1p1, p2 , I a b p1 b I x2 p1, p2 , I a b p2 Public Economy 26 Homogeneous function Definition If a function f(x) satisfies following feature, then f is said to be homogeneous of degree k; (k次同次関数) t 0,xRn , f tx t k f x Public Economy 27 Question Assume Cobb-Douglas type utility function and proof following propositions 1. Demand function is homogeneous with degree 0 2. Demand function is monotonic decrease (単調 減少) with regard to price and monotonic increase (単調減少) with regard to income Public Economy 28 Answer a I x p , p , I 1 1 2 a b p 1. a tI 1 x1tp1,tp2 ,tI x1p1, p2 , I a b tp1 2. It can easily be shown from above demand function Public Economy 29 Indirect Utility Function (間接効用関数) • A consumer’s indirect utility function v(p,I) gives the consumer’s maximal utility when faced with a price p and an amount income I. It represents the consumer’s preference over market conditions. vp1,..., pn , I max ux1,..., xn x n subject to pi xi I i1 Identity (恒等式) vp1,..., pn , I ux1p1,..., pn , I,..., xn p1,..., pn , I Public Economy 30 Example 1. Find indirect utility function whose utility function is Cobb-Douglas type 2. Proof following identity between indirect utility function and demand function vp1,..., pn , I pi x1p1,..., pn , I vp1,..., pn , I I Roy’s identity (ロイの恒等式) Public Economy 31 Answer 1. From the identify, a b a I b I aabb I ab vp , p , I 1 2 a b p a b p a bab pa pb 1 2 1 2 2. (In case n=2) We can get derivative of indirect utility function as v u x u x v u x u x following; 1 2 and 1 2 p1 x1 p1 x2 p1 I x1 I x2 I Then, if we differentiate the budget constraint (p 1 x 1 p 2 x 2 I ) x1 x2 with respect to p1 (with fixed I), we can get x1 p1 p2 0 p1 p2 x1 x2 Also, by differentiating by I, we can get p1 p2 1 I I From those equations and the condition of utility maximisation ( u x 1 u x 2 p 1 p 2 ), we can obtain v p x p , p , I 1 Q.E.D. 1 1 2 v I Public Economy 32 Income change and Demand • Superior good (上級財) – Good whose demand increases as income increase – Example: Luxury goods • Intermediate good (中間財) – Good whose demand is stable against income – Example: Tissue paper • Inferior good (下級財) – Good whose demand decreases as income increase – Example: Substitution of rice (such as potato) Public Economy 33 Price change and Demand • Ordinal good (正常財) – Good whose demand decrease as it’s price increase – Example: Beer • Giffen good (ギッフェン財) – Good whose demand increase as it’s price increase – Example: Substitution of rice (such as potato) Giffen good Ordinal good x2 x2 (Good 1) Price x1 Price x1 decrease Reduction in decrease Public Economy demand for good 1 34 Another approach describing Consumer Behaviour x2 x2 Set satisfying a certain utility level Budget Set B 0 x1 0 x1 Public Economy 35 Expenditure Minimisation Problem (支出最小化問題) n min pi xi x i1 subject to ux1,..., xn u • Expenditure function (支出関数) ep,u • Hicksian demand (ヒックスの需要関数) hp,u Public Economy 36 Expenditure Function Expenditure Minimisation Problem n ep1,..., pn ,u min pi xi x i1 Expenditure usxubject1,..., x nto u Function First Order Condition p ux ,..., x x i 1 n i p j ux1,..., xn x j ux1,..., xn u Public Economy 37 Hicksian Demand Function • Solution of Expenditure Minimisation Problem hi p,u • Identity (恒等式) hi p,vp, I xi p, I u vp, I xi p,ep, I hi p,u x2 x1 ep,vp, I I vp,ep,u u h2 ep,vp, I I h1 Public Economy 38 Feature of Expenditure Function and Hicksian Demand Function • Expenditure Function ( e p , u ) is homogenous of degree 1 with regard to p.
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