個體經濟學一 M i c r o e c o n o m i c s (I) CH 5 More on the analysis of consumer behavior Figure74 An increase in the price of X, P x P x1 P x2, P x2 > P x1 Assume = 1 and m are fixed. m =e(p X2,, u 1 ) m=e(p X1,, u 1 )= e(p X2,, u 2 ) m = e(p X1,, u 2 ) CV= e(p X2,, u 1 )-m= e(p X2,, u 1 )- e(p X2,, u 2 ) distance between u 1 and u 2 in terms of new prices EV= e(p X1,, u 2 ) m= e(p X1,, u 2 ) e(p X1,, u 1 ) distance between u 2 and u 1 in terms of old prices
Figure70 EX: X &Y are perfect complement. u(x,y)= min{ x, y } 2 (P x1,, m)=(0.5,1,100) (P x2,, m)=(1,1,100) most efficient x,y ratio: x 2 = y => y=1.5x e 1 satisfies y=1.5x => x=50 0.5x+y=100 y=75 u(x,y)= min{ x, y } => u 2 1 = 25 e 2 satisfies y=1.5x => x=40 x+y=100 y=60 u(x,y)= min{ x, y } => u 2 2 = 20 since e = e 1 = (50,75) => m =1*50+1*75=125 CV=125-100=25 since e 4 = e 2 = (40,60) => m =0.5*40+1*60=80 EV=80-100=-20 ordinary demand functions max x,y (min{ x 2, y })
s.t. P x x + y = m x = y 2 P x x + y = m P x x + 1.5x = m (P x + 1.5 )x = m => y=1.5x P x x + y = m x*= m = 2m P x +1.5 2P x + y*= 1.5m P x +1.5 = m 2P x + Indirect utility function V(P x,, m)=u(x*, y*)= m 2P x + Compensated demand functions min x,y P x x + y s.t. min{ x 2, y }=u x 2 = y = u => xh = 2u y h = u independent of P x, Expenditure function e(p x,, u)= P x x h + y h =P x 2u + u=(2p x + )u u 1 = V(P x1,, m) = V(0.5,1,100) = 100 2 0.5+ 1 = 25 u 2 = V(P x2,, m) = V(1,1,100) = 100 2 1+ 1 = 20 CV= e(p X2,, u 1 )-m = e(1,1,25)-100 = (2*1+*1)*25-100=25 EV= e(p X1,, u 2 ) m= e(0.5,1,20)-100=(2*0.5+*1)*20-100=-20
EX: u(x.y)=x 2 y (P x1,, m)=(0.5,1,90) (P x2,, m)=(1,1,90) ordinary demand functions max x,y x 2 y s.t. P x x + y = m x = 2 y = 1 m m = 2 2+1 P x P x m = 1 m 2+1 Indirect utility function m V(P x,, m) = x 2 y =( 2 ) 2 ( 1 P x m ) u 1 = V(0.5,1,90)=42000 x 1 = 120, y 1 = 0 u 2 = V(1,1,90) = 108000 x 2 = 60, y 2 = 0 Compensated demand functions min x,y P x x + y s.t. x 2 y = u Foc MRS xy = P x 1 x 2 y=u 2 LHS of 1 MRS xy = Mu x = 2xy = 2y Mu y x 2 x 1 => 2y x = P x => y= P xx 2 2 => x 2 P x 2 x = u P x 2 x = u x = 2 P x u x h = 2 u P x y h = P x 2 u 2 P x = P x 2 4P2 u y
Expenditure function e(p X1,, u 2 )= P x x + y = P x 2 u P x e(p X2,, u 1 )= e(1,1,42000)= 864000 = 2 108000 = 108000= 90 4 + 108000 + P x 2 4P2 u y + 108000 e(p X1,, u 2 )= e(0.5,1,108000) = 0.5 108000 CV=90 4 = 0 2 90 + 15 2 = 45 2 EV=45 2 90 (P x1,, m) x, y old equilibrium V(P x1,, m)= u(x, y )=u 1 (P x2,, m) x, y new equilibrium V(P x2,, m)= u(x, y )=u 2 e(p x,, u) is the expenditure function min x,y P x x + y s.t. u(x,y)= u x h = x(p x,, u) e(p x,, u)= P x x h + y h e(p X1,, u 2 ) e(p X2,, u 1 ) e(p X1,, u 1 )=m e(p X2,, u 2 )=m = 2P 2 x u + 1 108000 16 (x h, y h ) compensated demand function y h = y(p x,, u) + 0.25P 2 x u Figure71 結論 : CV= e(p X2,, u 1 )-m= e(p X2,, u 1 )- e(p X2,, u 2 ) EV= e(p X1,, u 2 ) -m= e(p X1,, u 2 ) - e(p X1,, u 1 )
EX: u(x,y)= x + y max x,y x + y s.t. P x x + y = m Foc MRS xy = P x 1 x, y P x x + y = m 2 1 LHS MRS xy = Mu 1 x 2 = x 1 2 = 1 Mu y 1 2 x 1 2 = 1 2x 0.5 1 1 2x 0.5 = P x x = P 2 y 4P2 x note that in the process of finding compensating demand function, we have the FOC: MRS xy = P x 1 x h x + y = u 2 y h 1 => x h = 2 4P x 2 x*=x h PE = SE + IE In this example, x 2 = x =>PE = SE, there is no income effect. Figure72
2 => P x 2 + P 4 y = m x 2 y = m 2 4P x 2 y = m 4P x Figure7 P x1 = 0.5, = 1, m = 90 P x2 = 1, = 1, m = 90 V(P x,, m) = 2 4P x 2 + m Py 4P x = 2P x + m Py 4P x = m Py + 4P x u 1 = V P x1,, m = V(0.5, 1, 90) = 90.5 x* = 1, y* = 89.5 u 2 = V P x2,, m = V(1, 1, 90) = 90.25 x = 0.25, y = 89.75 e(p x,, u) =? min x,y P x x + y s.t. x + y = u
FOC. MRS xy = P x 1 x h x + y = u 2 y h 1 => x h = x = 2 4P x 2 u 不影響 2 => 2P x + y = u y h = u 2P x e(p x,, u) = P x 2 4P x 2 + (u 2P x ) = 2 P y2 + P 4 u x 2P x = u 2 4P x CV = e(p x2,, u 1 ) m = e(1, 1, 90.5) m = (90.5 0.25) 90 = 0.25 EV = e(p x1,, u 2 ) m = e(0.5, 1, 90.25) m = (90.25 0.5) 90 = -0.25 CV = -EV *quasi-linear function quasi linear in y utility function u(x, y) = y + f(x) CV = -EV *Quasi-linear utility function u(x, y) = y + f(x) given u(x, y) = u 1 an indifference curve {(x, y) u(x, y) = u 1 } ={(x, y) y + f(x) = u 1 } ={(x, y) y = u 1 f(x)} u(x, y) = u 2 another indifference curve {(x, y) y = u 2 f(x)}
Figure74 Equivalent variation(ev) and compensating variation(cv) the vertical distance between those two difference curve (suppose u 2 > u 1 ) y 2 y 1 = u 2 u 1 (f(x) canceled) Figure75 given a X MRS xy = MU x = f (x) = f (x) no y MU y 1 given a X, f(x) = MRSxy is independent of Y. Since MRSxy = f (x) From the FOC, MRS xy = P x f (x) = P x => x* = x h is a function of P x and
(no m, no income effect) PE = SE IE = 0 *Consumer Surplus, CS A consumer is willing to pay for X 1 units of X 1 units of X The consumer pays Px 1 * X 1 for X 1 units of X Figure76 x CS = 1 - = = P 0 x xdx P x1 X 1 an increase of price of X from P x1 to P x2, P x2 > P x1 Figure77 Change in consumer surplus with an increase of price of x
*Relationship among CV, EC & CS a ~ b ~ c ~ d (0, y 0 ) ~ (1, y 1 ) ~ (2, y 2 ) ~ (, y ) the consumer is willing to pay (y 0 - y 1 ) for the 1st unit of X 2nd (y 1 y 2 ) Figure78 = 1, y :other expenditure. MRSxy at x=1 = y 0 - y 1 at x=2 = y 1 y 2 at x= = y 2 y in equilibrium, MRS xy = P x = 1 => MRS xy = P x (or MVx)
compensated demand function. not an ordinary demand function. given u(x, y) = u 1 Figure79 CS =? base on X* CS base on x h (u 1 ) =? CV CS base on x h (u 2 ) =? EV Figure80 結論 : P x1 P x2, P x2 > P x1 X is normal => CV > CS > EV Quasi-linear u(x, y) => CV = CS = EV
*Revealed Preference The theory of revealed preference. Bundle (x 1, y 1 ) is revealed preferred to bundle (x 2, y 2 ) if 1 both bundles are affordable/ 2 (x 1, y 1 ) is chosen (but not (x 2, y 2 )) ( 1 P x x 1 + y 1 P x x 2 + y 2 ) *The principle of the revealed preference. Suppose a consumer is rational and (x 1, y 1 ) is revealed preferred to (x 2, y 2 ), then (x 1, y 1 ) must be preferred to (x 2, y 2 ) (Axiom) The weak axiom of revealed prefernence. (WARP) Suppose (x 1, y 1 ) is revealed preferred to (x 2, y 2 ) statement A then (x 1, y 1 ) cannot be revealed preferred to (x 2, y 2 ) statement B statement A is true => according to the principle of the revealed preference, we have (x 1, y 1 ) > (x 2, y 2 ) statement B is true => an inconsistency in the consumer s preference. EXAMPLE 某人將全部所得用於買 A 物及 B 物 當 A 物價格 P A 為 2 元, 而 B 物價格也為 2 元時, 他以 80 元的所得購買 20 單位的 A 物, 當 P A = 4, P B = 2 時, 他以 120 元的所得購買 25 單位的 A 物 請問其消費行為是否符合經濟理性? 試加比較說明之 (Definition) (x 1, y 1 ) is revealed preferred to (x 2, y 2 ) if (1) both (x 1, y 1 ) and (x 2, y 2 ) are affordable (2) (x 1, y 1 ) is chosen if (x 1, y 1 ) is chosen at (P x1, P x2 ), P x x 1 + y 1 P x x 2 + y 2 The principle of the revealed preference If consumer is rational Then (x 1, y 1 ) is revealed preferred to (x 2, y 2 ) implies (x 1, y 1 ) is preferred to (x 2, y 2 ) (x 1, y 1 ) is revealed preferred to (x 2, y 2 )
then (x 2, y 2 ) cannot be revealed preferred to (x 1, y 1 ) P x1 x 1 + 1 y 1 P x1 x 2 + 1 y 2 and (x 1, y 1 ) is chosen at (P x2, 2 ), (x 2, y 2 ) is chosen, but not (x 1, y 1 ) P x2 x 1 + 2 y 1 P x2 x 2 + 2 y 2 Example (a) (P x1, 1 ) = (20, 1) (x 1, y 1 ) = (2, 40) m 1 = 80 (P x2, 2 ) = (20, 4) (x 2, y 2 ) = (, 25) m 2 = 160 (b) (P x1, 1 ) = (20, 1) (x 1, y 1 ) = (, 20) m 1 = 80 (P x2, 2 ) = (20, 4) (x 2, y 2 ) = (2,0) m 2 = 160 sol. (a) P x1 x 1 + 1 y 1 = 20 2 + 1 40 = 80 P x1 x 2 + 1 y 2 = 20 + 1 25 = 85 =>(x 1, y 1 ) is not revealed preferred to (x 2, y 2 ) (1) sol. P x2 x 2 + 2 y 2 = 20 + 4 25 = 160 P x2 x 1 + 2 y 1 = 20 2 + 4 40 = 200 => (x 2, y 2 ) is not revealed preferred to (x 1, y 1 ) (2) (1), (2) => doesn t violate the WARP. (b) P x1 x 1 + 1 y 1 = 20 + 1 20 = 80 P x1 x 2 + 1 y 2 = 20 2 + 1 0 = 70 =>(x 1, y 1 ) isrevealed preferred to (x 2, y 2 ) (1) P x2 x 2 + 2 y 2 = 20 2 + 4 0 = 160 P x2 x 1 + 2 y 1 = 20 + 4 20 = 140 => (x 2, y 2 ) is revealed preferred to (x 1, y 1 ) (2) (1), (2) => violate the WARP. a not RP to b b not RP to a c RP to d d RP to c WARP ok! violate to WARP
Figure81: at (P x1, 1 ), e is chosen f is affordable, e is RP to f at (P x2, 2 ), f is chosen e is not affordable, f is not RP to e Figure82 =>e, f don t violate WARP at (P x1, 1 ), g is chosen h is not affordable, g is not RP to h at (P x2, 2 ), h is chosen g is affordable, h is RP to g h>g =>g, h don t violate WARP
(a, b) (a, c) (a, d) (a, e) ok! not ok Figure8 Income is fixed at m at (P x1, 1 ) the consumer chooses a at (P x2, 2 ) the consumer chooses b a b price effect of a decrease in price of X (P x1 P x2, P x2 < P x1 ) Figure89 How to decompose PE into SE and IE? slutsky substitution effect. If a consumer would have chosen c after a slutsky income subsidy, note that at P x1,, and m, a and c are affordable, a is chosen =>a is RP to c at P x2,, and m, a and c are both affordable, c is chosen =>c is RP to a check a, c is OK with WARP At (P x1,, m), a is chosen but c is not affordable => a is not RP to c. At (P x2,, m), c is chosen,a and c are both affordable
=> c is RP to a => doesn t violate WARP P x, slutsky substitution effect a c x > x 1 (X is cheaper, X substitutes for Y) After slutsky subsidy, P x2 x 1 + 2 y 1 = P x2 x + 2 y = m 1 original bundle a is c is chosen after affordable at new price slutsky subsidy =>c is RP to a a cannot RP to c => c is not affordable at (P x, ) m = P x1 x 1 + 1 y 1 > P x1 x + 1 y 2 1-2 (P x2 P x1 )x 1 + (2 1 )y 1 > (P x2 P x1 )x + (2 1 )y (P x2 P x1 )(x x 1 ) + 2 1 (y y 1 ) < 0 2 = 1 = fixed =>(P x2 P x1 ) (x x 1 ) < 0 =>P x2 < P x1 =>x x 1 > 0 x > x 1 *Tax Tax on gasoline (X) $t tax on each unit of X P x P x + t =>equilibrium e 1 e 2 consumer is worse off e 1 is revealed pregerred to e 2 (e 1 & e 2 are affordable before imposing a $t unit tax) $t * X => a tax return to the consumer =>new equilibrium e
Figure84 A: P x x + y = m B: (P x + t)x + y = m The budget line after tax return: (P x + t)x + y = m + tx =>P x x + y = m same as A new income: m = m + tx new price: P x + t & slope of the budget line (after tax and tax return) => P x+t (a steeper budget line) (P x + t)x + y = m + tx => both e 1 and e are affordable at (P x => e 1 is revealed preferred to e =>consumer is worse off at e Suppose new equilibrium were at e the new budget line is D. the expenditure of e 1 at (P x + t) is less than m => e is revealed preferred to e (both are on A, and e 1 is chosen before tax) Based on new budget line C e 1 is not affordable after tax and tax return e is not revealed preferred to e 1
conclusion: X after a tax and tax return e 必在 e 1 左上方 *Price Index and welfare based period : 0 current period : t at period 0 : P x0, 0 X 0, Y 0 at period t : P xt, t X t, Y t Compare welfare between periods 0 and t the consumer is better off in period 0 ((X 0,Y 0 ) is RP to X t,y t ) if P x0 x 0 + 0 y 0 > P x0 x t + 0 y t (1) the consumer is better off in period t ((X t,y t ) is RP to X 0,Y 0 ) if P xt x t + t y t > P xt x 0 + t y 0 (2) m t = P xt x t + t y t mt (1) Paasche price index 巴氏指數 P xt x t +t y t P x0 x 0 +0 y 0 P xtx t +t y t P x0 x t +0 y t price index weighted by current consumption bundle Index p m t m 0 the consumer is better off in period O. (2) m 0 Lasperes price index 拉氏指數 P xt x t + t y t P x0 x 0 + 0 y 0 P x t x 0 + t y 0 P x0 x 0 + 0 y 0 price index weighted by based period consumption bundle CPI is one of Lasperes price index Index L m t m 0 the consumer is better off in period t.