What is Shaphard's Lemma?
what is its significance to economics?
Answer:
If you take the derivative of the expenditure function w.r.t the price of the good you are interested in you will get the amount a cost minimizing consumer will purchase. This time have to use the envelope theorem.
expenditures= p1*x1+P2*x2 X2 and X1 are the amount of goods that minimize expenditures with a given utility level. Take the derivative w.r.t. p1 and you get x1. Rembember this is x1 that solves the lagrangian of the expenditure minimization problem.
This is important because it is used for the slutsky decomposition
(m)
Shephard's lemma 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 expenditure or cost function are convex, then the cost minimizing point of a given good (i) with price pi is unique. The idea is that a consumer will buy a unique ideal amount of each item to miminize the price for obtaining a certain level of utility given the price of goods in the market. It was named after Ronald Shephard who gave a proof using the distance formula in a paper published in 1953, although it was already used by John Hicks (1939) and Paul Samuelson (1947).
Shephard's lemma 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 expenditure or cost function are convex, then the cost minimizing point of a given good (i) with price pi is unique. The idea is that a consumer will buy a unique ideal amount of each item to miminize the price for obtaining a certain level of utility given the price of goods in the market. It was named after Ronald Shephard who gave a proof using the distance formula in a paper published in 1953, although it was already used by John Hicks (1939) and Paul Samuelson (1947).
[edit] Definition
The lemma gives a precise formulation for the demand of each good in the market with respect to that level of utility and those prices: the derivative of the expenditure function (e(p,u)) with respect to that price:
where hi(u,p) is the Hicksian demand for good i, e(p,u) is the expenditure function, and both functions are in terms of prices (a vector p) and utility u.
Although Shephard's original proof used the distance formula, modern proofs of the Shephard's lemma use the envelope theorem.
Application
Shephard's lemma gives a relationship between expenditure (or cost) functions and Hicksian demand. The lemma can be re-expressed as Roy's identity, which gives a relationship between an indirect utility function and a corresponding Marshallian demand function.
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Answer:
If you take the derivative of the expenditure function w.r.t the price of the good you are interested in you will get the amount a cost minimizing consumer will purchase. This time have to use the envelope theorem.
expenditures= p1*x1+P2*x2 X2 and X1 are the amount of goods that minimize expenditures with a given utility level. Take the derivative w.r.t. p1 and you get x1. Rembember this is x1 that solves the lagrangian of the expenditure minimization problem.
This is important because it is used for the slutsky decomposition
(m)
Shephard's lemma 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 expenditure or cost function are convex, then the cost minimizing point of a given good (i) with price pi is unique. The idea is that a consumer will buy a unique ideal amount of each item to miminize the price for obtaining a certain level of utility given the price of goods in the market. It was named after Ronald Shephard who gave a proof using the distance formula in a paper published in 1953, although it was already used by John Hicks (1939) and Paul Samuelson (1947).
Shephard's lemma 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 expenditure or cost function are convex, then the cost minimizing point of a given good (i) with price pi is unique. The idea is that a consumer will buy a unique ideal amount of each item to miminize the price for obtaining a certain level of utility given the price of goods in the market. It was named after Ronald Shephard who gave a proof using the distance formula in a paper published in 1953, although it was already used by John Hicks (1939) and Paul Samuelson (1947).
[edit] Definition
The lemma gives a precise formulation for the demand of each good in the market with respect to that level of utility and those prices: the derivative of the expenditure function (e(p,u)) with respect to that price:
where hi(u,p) is the Hicksian demand for good i, e(p,u) is the expenditure function, and both functions are in terms of prices (a vector p) and utility u.
Although Shephard's original proof used the distance formula, modern proofs of the Shephard's lemma use the envelope theorem.
Application
Shephard's lemma gives a relationship between expenditure (or cost) functions and Hicksian demand. The lemma can be re-expressed as Roy's identity, which gives a relationship between an indirect utility function and a corresponding Marshallian demand function.
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