Ch4. 個體經濟學一 M i c r o e c o n o m i c s (I) Comparative Statics and Demand EX: u(, y) = 2 + y, Ma,y 2 + y s. t. P + P y y = m FOC MRS y = P P y P + P y y = m MRS y = MU = 2 0.5 0.5 = P MU y 1 P y 1 0.5 = P P y = P y 2 P 2 ------ 1 P 2 y P + P y y = m Figure44 :Engel curve P y y = m - P y 2 P 1 2 y = m P y 0.5 y = m P y P y P ------ 2
d y d = 0.5 0.5 d 2 y d 2 = 0.25 1.5 > 0 Figure45 :Price consumer curve not a Giffen good X & Y are substitutes From the demand curves. Figure46 Demand curve that X&Y substitutes = P y 2 independent of income. & P 2 P y, P, y = m P y P y P P, y & P y, y & m, y Y is a normal good. Market demand = Sum of the individual demand
EX: Inverse demand function: Demand curve consumer A: P = 10-2X A, X A = 5 0.5P B: P = 5-3X B, X B = 5 3 1 3 P C: P = 12-6X C, X CB = 2 1 6 P Figure47 Horizontal aggregation in Demand curve 12 P X A = 0, X B = 0, X C = 0, = X A + X B + X C = 0 10 P 1 X A = 0, X B = 0, X C = 2 1 6 P, = X C = 2 1 6 P 5 P 10 X A = 5 1 2 P, X B = 0, X C = 2 1 6 P, = X A + X C = 7 2 3 P 0 P 5 X A = 5 1 2 P, X B = 5 3 1 3 P, X C = 2 1 6 P, = 26 3 P *Conclusion: 1. Market demand "curve" is the "horizontal" sum of the individual demand curves. 2. Individual demand curves are downward sloping. market demand curve is downward sloping.
*Property of the market demand curve: Downward sloping (usually) slope represents quantity demanded depends on price. price, quantity demanded sensitivity of the change of the quantity demanded with the change with the change in price ( price of other good, income,... other determinants) *Price Elasticity of demand (of good X) ε p = P = P percentage change in quantity demanded of X percentage change in price of X = P P slope of the demand function ε p d depends on, P, dp determinants : 1. property of the good X (necessity?) 2. substitute availability 3. the proportion of the ependiture of X with respect to the income. 4. time frame. ε p = 1 unit elastic ε p > 1 elastic ε p < 1 inelastic. if P 0, ε p = P P
( P 2 2 ) Figure48 Elasticity concept (change in X response to P) ε p = 2 1 P 2 P 1 P 1 1 (point elasticity of demand) ε p = 2 1 P 2 P 1 P1+P2 2 1+2 = 2 1 P 2 P 1 2 P1+P2 2 1+2 = 2 1 2 P 1 +P ( arc elasticity of demand) P 2 P 1 1 + ε p = P P > 0 P X P < 0 P X P X P changes Total ependiture of X doesn't change if ε p = 1 P and ε p > 1 ependiture of X P and ε p < 1 ependiture of fied X e = P (P,P y, m,..) de = dp = P d dp dp dp + dp dp = P d dp + = ( d dp P + 1) = ( 1 ε p ) de = dp dp dp < 0 if ε p > 1 elastic
de = dp dp dp > 0 if ε p < 1 inelastic. de = dp = 0 if ε p dp dp = 1 unit elastic *Special case 1.Horizontal demand curve ε p at all points on the demand curve. 2.Vertical demand curve Figure49 Perfect elastic Demand 非用不可的商品 e. 癌症治療藥物 ε p = 0 Figure50 perfect inelastic Demand
3.Linear demand curve P = a-bx ε p at d = d dp P X = a b P b = 1 b a bx oa note that b= = fa oc fd = fd fa of og = og fa of og = dc da Figure51:Elasticity of linear Demand curve ε p = 1 if dc = da i.e. d is eactly the middle point of the demand curve.
Figure52 Elasicity of linear demand curve ε p at d = d dp P = 1 af of og fd = fd af of og = og af of og = of af = dc da ε p at d = d c d a > dc da Figure53 Price elasticity at demand
*Demand curve with ε p = 1 at all points in the demand curve. P 1 X 1 = P 2 X 2 P X = constant = e X = e P = ep 1 ε p = d dp P = e P 2 = 1 = 1 P e P Figure54 Demand curve *another formula of ε p dln = 1 d dln = 1 d dlnp dp = 1 P dlnp = 1 P dp 1 dln = d 1 = d P dlnp P dp dp => ε p = dln dlnp dln dlnp = ε dln dlnp = ε Let ln=s lnp =t dln = ds dlnp dt ds dt = s = ln dt
ε p = ε some constant at all points on a demand curve. = ep ε = e P ε ε p = dln = dlnep ε = d(lne εlnp ) dlnp d(lnp ) dlnp = εdlnp = ε = ε dlnp Suppose ε p = ε Demand function =? ε p = dln dlnp = dln dlnp constant dln dlnp = ε dln dlnp = ε dln dlnp dlnp = εdlnp ln = εdlnp + C ln = e εdlnp +C X = e εdlnp e c = e c e lnp ε = e c P ε *Income elasticity of demand ε m = m 0 m m = m m ε m = m m X = (P, P y, m) X = (mjp, P y ) Engel curve given Figure55 use engel curve to demostrate income elasticity
ε m at d =? ε m = m m at d = fd fa of og = og fa of og = fo fa = dc da Figure56 use engel curve to demostrate income elasticity m *Definition: ε < 0 => inferioi good m, * 0 < ε m < 1 => necessity m ε > 1 => luury *Cross Elastivity of Demand X = (P, P y, m) ε y = Py Py P y 0 ε y = P y P y = P y P y ε y > 0 => X & Y are substitutes. ε y < 0 => X & Y are complements.
Budget constraint P + P y y = m e + e y = m => e m + e y m = 1 share of ependiture S + S y = 1 P + P y m y = m = 1 m m P X m m + P yy y m = e ε m m m y m m + e y ε m y m = 1 S ε m + S y ε m y = 1 ε m < 0 inferior good => ε m y > 0 furthermore ε m y > 1 => Y is luury. 0 < ε m < 1 necessity => ε m y > 1 => Y is luury. *Homogeneous Function Def: f(, y) is homogeneous od degree k in X and Y if f(t, ty) = t k f(, y) for all t > 0 n variable, o<m<n Def:f( 1, 2,, n ) is homogeneous of degree k in 1, 2,, m if f(t 1, t 2,, t m, m+1,, n ) = t k f( 1, 2,, n ) for all t > 0 Eample 1: u(, y) = min{ 3, y 2 } u(t, ty) = min{ t 3, ty 2 } = t min{ 3, y 2 } = t u(, y) k = 1 => u(, y) = min{, y } is homogeneous of degree 1 in X and Y 3 2 Eample 2: u(, y) = 24 + 6y # of boes of Coke # of si pack Coke u(t, ty) = 24t + 6ty = t(24 + 6y) =t u(, y) k = 1 => u(, y ) = 24 + 6y is homogeneous of degree 1 in X and Y.
Eample 3: Cobb-Douglas utility function u(, y) = 0.2 y 0.4 u(t, ty) = (t) 0.2 (ty) 0.4 = t 0.6 0.2 y 0.4 = t 0.6 u(, y) 0.2 y 0.4 is homogeneous of degree 0.6 inx and Y. Eample 4: u(, y) = 0.5 + y u(t, ty) = (t) 0.5 + ty = t 0.5 ( 0.5 +t 0.5 y) u(, y) can t be arranged in t k ( 0.5 + y) form => u(, y ) = 0.5 + y isn t a homogeneous fct. Eample 5: Demand function * = (P, P y, m) (tp, tp y, tm) =? t k (P, P y, m) y* = y(p, P y, m) y(tp, tp y, tm) =? t k y(p, P y, m) Are demand function homogeneous? If yes, in which rariables? (1) ma,y u(, y) s.t. P + P y y = m FOC => MRS y (= Mu Mu y ) = P P y P + P y y = m => * = (P, P y, m) y* = y(p, P y, m) (2) ma,y u(, y) s.t. tp + tp y y = tm FOC => MRS y (= Mu Mu y ) = P P y tp + tp y y = tm => P + P y y = m * = (tp, tp y, tm) = (P, P y, m) y* = y(tp, tp y, tm) = y(p, P y, m) solved in (2) solved in (1) t k = 1 => k = 0 Demand function (P, P y, m), y(p, P y, m) are homogeneous of degree 0 in P, P y and m. Money illusion 貨幣幻覺
suppose f(, y) is homogeneous of degree k in X and Y f(t, ty) = t k f(, y) for all t>0 df(t,ty) dt = dtk f(,y) dt dt f + f dty dt y = dt ktk 1 f(, y) f X + f y Y = kt k 1 f(, y) Let t = 1, f X + f y Y = kf(, y) Euler Teorem Equation An application of Euler Teorem * = (P, P y, m) are homogeneous of degree k y* = y(p, P y, m) in P, P y and m According to Euler Teorem, we have P P + P P y + m = k = 0 y m 0, P P + P y P y y + m m = 0 ε p + ε y + ε m = 0 P P = P P