個體經濟學一. Total utility of consuming (x, y), denoted as u(x, y), is the total level of total satisfaction of consuming(x, y).

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1 個體經濟學一 M i c r o e c o n o m i c s (I) CH2 The analysis of consumer behavior *Untility function Total utility of consuming (x, y), denoted as u(x, y), is the total level of total satisfaction of consuming(x, y). Cardinal utility analysis (number itself is meaningful) Ordinal utility analysis (number is used to rank bundles) In Ordinal utility analysis, higher value of TU is associated with higher level of satisfaction. utility function is a function from bundles to a real number u(x1, y1) > u(x2, y2) (x1, y1) > (x2, y2) such that u(x1, y1) = u(x2, y2) (x1, y1) ~ (x2, y2) EX: (2, 3) (3, 7) (5, 10) u : (x, y) R u(x, y) total utility *Marginal utility (of X, of Y) u = u(x, y) MU x = u(x,y) x MU y = u(x,y) y ( u(x,y) ) ( u(x,y) y ) The law of Diminishing marginal utility x MU X ( 2 u(x,y) 2 < 0 ) y MU Y ( 2 u(x,y) y 2 < 0 )

2 Any positively monotonic transformation of a utility function is also a utility function representing the same preference. *Cardinal v.s. Ordinal utility analysis EX: x : # of toast y: # of ham u(x, y) = min x, y positively monotonic 3 2 transformation *Marginal utility min{ x 3, y 2 } => same preference. MU x = u(x,y) x or u(x,y) MU y = u(x,y) y or u(x,y) y or U x (x, y) or U y (x, y) (or simple U x ) (or simple U y ) *Marginal utility is the slope of the utility function If more is better (assumption of nonsatiation) MU x > 0, MU y > 0 But Law of Diminishing marginal utility x, MU x y, MU y Figure1 8:

3 {(x, y) (x, y)~ (x 1, y 1 )for all (x, y) S} Given (x1, y1) S, we have total utility u(x1, y1) : {(x, y) u(x, y) = u(x 1, y 1 )for all (x, y) S} MRS xy = Δy Δx MRS xy depends on (x, y) note that u(x, y) = u(x,y) x Figure19 : Indifference curve (given an indifference curve) (du(x, y) = u(x,y) x + u(x,y) y + u(x,y) y du(x, y) = Mu x + Mu y dy y dy) On an indifference curve, a change in X and Y must satisfy Mu x + Mu y dy = 0 dy = Mu x Mu y Figure20 :

4 (note: MRS xy = Δy Δx of an indifference curve) MRS xy (x, y) = Mu x(x,y) Mu y (x,y) suppose Mu x (x,y) is diminishing on X Mu y (x,y) is diminishing on y } Diminishing marginal utility. Can we have diminishing MRS xy? (MRS xy as x or y ) x => Mu x (Diminishing Mu x ) y => Mu y (stays on the same IC) Since MRS xy = Mu x as x Mu y Figure21:Diminishing MRS xy dmrs xy > < 0? We hope dmrs xy < 0. Mu x = u x Mu y = u y Mu x (x,y) x Mu x (x,y) y Mu y (x,y) x Mu y (x,y) y or u x = u xx or u x y = u xy or u y = u yx = u xy or u y y = u yy

5 dmrs xy = d( Mu x Mu y ) = d( u x u y ) = u du x y u x u y 2 du y note that du x u x = u ux y + u x y dy u x u y y + u x y dy u 2 y a c u x (given y) c b u x and dy Figure22 :Diminishing MRS xy (note that dy = u x u y (= Mu x Mu y )) u y u xx u xy ux uy u x u yx u yy u x uy u y 2 u y 2 u xx u xy u x uy u xu y u yx u yy u x uy u 3 y u y 2 u xx 2u x u y u xy +u x 2 u yy u y 3 < 0 none satisfaction u x,u y > 0 u xx <0 u yy <0 } Diminishing Mu x, Mu y Diminishing marginal utility is not sufficient to imply diminishing MRS xy, we need to know u xy >0 <0

6 *Budget Constraint X, Y Quantity:x, y price: P x, income: m Expenditure of(x, y) = P x x + y Budget constraint: (x, y) P x x + y m Budget line: (x, y) P x x + y = m P x x + y = m y = m P x x y = m P x x Figure23 :Budget constraint MRSxy = Δy Δx on an indifference curve = subject exchange rate. P x = Δy Δx on an budget line = object exchange rate. 1. 所得 (m) 改變, m Figure24 :Budget constraint when income change

7 2. P x, chang P x, fixed, m fixed Figure25 :Budget constraint when price of X change(increase) *Special case 1.Quantity Discount on X X X 0, P x X > X 0, P x 2 m: income } as usual Py:Price of gppd Y Figure26 :Budget constraint with special case

8 y 0 = m P xx 0 (x 0, y 0 ) is on D, B Price of X =0.5 P x = }on D, B income = m m P x X 0 0.5P x X 0 + = m 0.5P x X 0 + m P x X 0 = m m 0.5P x X 0 = m m m = 0.5P x X 0 Distance between A.D= m 0.5P xx 0 D.B budget line = (x, y) 0.5P x X + Y = m 0.5P x X 0 X > X 0 2.Quota on X Figure27 :Budget constraint with quota 3. WIC(Woman.infant.children) Food coupon X 0 :good X of food coupon

9 Figure28 : Budget constraint with food coupon 4. Endowment 稟賦 (x0, y0)~m= P x x 0 + y 0 P x, P x P x, P x >P x m = P x X 0 + Y 0 Figure29 :Budget constraint with endowment

10 Figure30 :Interior solution of consumer equilibrium *Consumer equilibrium A consumer maximizes his or her utility subjected to his/her budget constraint. (X*, Y*) is a consumer equilibrium where indifference curve is tangent to the budget line(indifference curve touches the budget line at e) or slope of the indifference curve at e = slope of the budget line. MRSxy(X*, Y*) = P x or one more condition: Pxx*+Pyy* = m corner solution 角解 e: x*>0, y*=0 or x* = 0, y* = 0 P x x * + y * = m P x x * = m, x * = m P x x * = m P x, y* = 0, corner solution at e. MRS xy P x

11 Figure31 :Corner solution of consumer equilibrium another corner solution: x * = 0, y* = m, MRS xy P x *In the interior solution case: if MRS xy > P x subject change object exchange rate ( 主觀 ) 個人偏好 ( 客觀 ) 價錢比 x, y Figure32 :

12 a c c has more X => c is better than a. c e MRS xy = P x 同理, MRS xy < P x => x, y *WIC case Figure33 :Consumer equilibrium with food coupon s type budget constraint *X and Y are perfect complements Figure34 :Interior solution under the preference is perfect complements

13 *To max u(x, y) s.t P x x + y = m u(x, y) is some continuously differentiable function How to solve the consumer s problem? General problem: max f(x. y) s.t. g(x, y) = 0 (min) L(x, y, λ) = u(x, y) + λ(m P x X Y) Lagrange multiplier method = f(x, y) + λg(x, y) constrained problem non-constrained problem and apply FOC to the non- constrained problem. 1 L(x,y,λ) u(x,y) 2 L(x,y,λ) y u(x,y) y 3 L(x,y,λ) λ = u(x,y)+λ m P xx Y = λp x -- 1 = u(x,y)+λ m P xx Y y = λp x -- 2 = u(x,y)+λ m P xx Y λ = u(x,y) λp x = 0 = u(x,y) λp y y = 0 = m P x x y = budget constraint 1 ' / 2 ' = Mu x Mu y = P x -- 4 MRS xy = Mu x Mu y 4 MRS xy = Mu x Mu y = P x 在同樣的 $1 上比較 Mu x P x = Mu y Mu x P x = u(x,y) x expenditure x

14 *Special case 1. X and Y are perfect complement. 3 units of X is always consumed with 2 units of Y in fixed proportion. P x =$2, =$5, m=$80. u(x, y) = min x 3, y 2 Optimal combination of X&Y: x 3 = y 2 y x = 2 3 2x x = x = 80 3 x = 15, y = 10 Generalized to min x a, y b 最不浪費的比例 s.t. P x x+ y=m Ex. X and Y are perfect complements. A consumer always uses 3 units of X with 2 units of Y in the fixed proportion. u(x, y) = min x 3, y 2 u(x, y) = min x a, y b max u(x, y) = min x 3, y 2 s.t. P x x+ y=m x a = y b y = b a x budget constraint: P x x+ y=m b P x x + x = m a b P x + x = m a m x = b P x + a P x =$2, =$5 a=3, b=2, m=80 x am = = 15 ap x + b y = b am bm = = 10 a ap x + b ap x + b

15 Ex. Cobb-Douglas utility function u(x, y) total utility = x α y β, α, β > 0 Mu x = u(x,y) Mu y = u(x,y) y = α y β = αx α 1 y β > 0 no satiation point (no bliss point) = α y β y = βxα y β 1 > 0 Mu x = α(α 1)xα 2 y β α < 1 Mu y y = β(β 1)xα y β 2 β < 1 Diminishing Marginal utility MRS xy = Mu x = αxα 1 y β Mu y βx α y β 1 = α y β x x 越大, y 越小 dmrs xy For any α, β>0 and for all x and y dmrs xy < 0 i.e. MRS xy is diminishing (with x) i.e. indifference curve is always convex < 0 In equilibrium, MRS xy = P x α y β x = P x y(expenditure on Y) = β α P xx(expenditure on X) P x x + y = m P x x + β α P xx = m α + β P α x x = m e x (expenditure of x) = P x x = α α + β m(short of e x) e y (expenditure of y) = y = β α + β m(short of e y) x = α m α + β P x y = β m α + β