消費者剩餘 - 經濟

這題和上面討論有相關

※ 引述《taiyaki35 (小俠)》之銘言：
: 來源: 黃貞穎老師考古題
: 科目: 個經
: 問題:
: Consumer's surplus: A consumer has the utility function
: U(x,y) =e^((ln(X)+Y)^1/3)
: where X is the good in concern and Y is the money
: that can be spent on all other goods. (So the price of Y is normalized to
: be 1). The income of this consumer is 100.
: (a) (10pts) Derive the demand function of x for this consumer. Make sure that
: at every price of x, the consumer always has enough income to buy the amount
: of x as indicated by hiss demand function.
: (b) (10pts) Calculate the price elasticity of the demand function in (a).
: Is it true that the absolute value of the elasticity of the demand decreases
: as the amount of x increases?
: (c) (10pts) Suppose price of x decreases from 2 to 1. Calculate the change
: in consumer's surplus.
: (d) (10pts) Suppose price of x decreases from 2 to 1. Calculate the
: compensating variation of this price change.
: (e) (10pts) Suppose price of x decreases from 2 to 1. Calculate the
: equivalent variation of this price change .

令G(x,y)=(ln(X)+Y) 則U=V(G)=e^(G^1/3), 只要同樣的G就會產生同樣的U
dU = V'(G) dG = V'(G) (1/x dX + dY)
又C=PX + Y => dC = P dX + DY
相切時有 V'(G)/x /P = V'(G)/1 消掉V'(G)
1/x / P = 1/1 =t 此例直接解出t=1
則切點 x=1/P
(a)任意價格P任意預算下最佳消費 X=1/P
(b) d(lnX)/d(lnP) = -1 則 elasticity=1
(c) X dP = 1/P dP = d(lnP), consumer's surplus: ln2-ln1 = ln2
(d) 先限C求U(求G即可):
X=1/P , Y= C-PX = C-1=99 代入 G = ln(1/P) + Y = 99-lnP
P由2 to 1 則 Gs 由 99-ln2 變 Ge = 99
然後再限G求C: X=1/P Y=G-lnX= G+lnP 代入 C=PX+Y=1+G+lnP
compensating variation: 用原等效用線G=Gs, C=1+Gs+lnP=1+99-ln2+lnP
delta C =delta lnP => 答案為 ln2
(e)equivalent variation用新等效用線G=Ge=99, C=1+99+lnP
delta C =delta lnP => 答案為 ln2


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