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'''Hằng đẳng thức Roy''' (đặt theo tên nhà kinh tế học người Pháp [[René Roy]]) là [[công thức]] giúp tính được [[hàm cầu Marshall]] bằng cách lấy [[đạo hàm]] của [[hàm thỏa dụng gián tiếp]] theo giá cả chia cho đạo hàm của hàm thỏa dụng gián tiếp theo [[thu nhập]].
==Tham khảo==
{{Đang dịch}}
'''Roy's identity''' (named for [[France|French]] [[economist]] [[Rene Roy]]) is a major result in [[microeconomics]] having applications in [[consumer]] choice and the [[theory of the firm]]. The lemma states that if [[indifference curves]] of the [[indirect utility function]] are [[convex]] in [[prices]], then the cost minimizing point of a given [[good (economics)|good]] (<math>i</math>), with price <math>p_i</math>, is [[unique]]. The idea is that a consumer will have an ideal amount of each item to minimize the price for obtaining a certain level of utility given the price of goods in the market.
== Derivation of Roy's identity ==
[[Roy's identity]] reformulates [[Shephard's lemma]] in order to get a [[Marshallian demand function]] for an individual and a good (<math>i</math>) from some indirect utility function.
First, we obtain a trivial identity by substituting the expenditure function for [[wealth]] or [[income]] (<math>m</math>)in the indirect utility function (<math>\psi\ (m, u)</math>, at a utility of <math>u</math>):
:<math>\psi\ ( e(p, u), p) = u </math>
This says that the indirect utility function evaluated in such a way that minimizes the cost for achieving a certain utility given a set of prices (a vector <math>p</math>) is equal to that utility when evaluated at those prices.
Taking the partial derivative of both sides of this equation with respect to the price of a single good <math>p_i</math> and a constant utility level we have:
:<math>\frac{ \partial \psi\ [e(u,p),p]}{\partial m} \frac{\partial e(u,p)}{\partial p_i} + \frac{\partial \psi\ [e(u,p),p]}{\partial p_i} = 0</math>.
Rearranging, we obtain
:<math>\frac{\partial e(u,p)}{\partial p_i}=-\frac{\frac{\partial \psi\ [e(u,p),p]}{\partial p_i}}{\frac{\partial \psi\ [e(u,p),p]}{\partial m}}=x_i(m,p)</math>
== Application ==
This gives a method of deriving the [[Marshallian demand function]] of a good for some consumer from the indirect utility function of that consumer. It is also fundamental in deriving the [[Slutsky Equation]].
[[Category:Microeconomics]]
[[Category:Underlying principles of microeconomic behavior]]
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