Lecture 6 _Marshallian Demand_

Lecture 6 _Marshallian Demand_ - Marshallian Demand...

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Unformatted text preview: Marshallian Demand (Lecture 6) Marshallian Demand Example Indirect Utility and Expenditures Function Properties of the Marshallian Changes in Income Changes in the Price of one Good Deriving the Marshallian Demand Marshallian Demand (Lecture 6) Nicholson (2007), Chapter 5 January 19, 2011 George Georgiou - Intermediate Microeconomics 1/25 Marshallian Demand (Lecture 6) Marshallian Demand Example Indirect Utility and Expenditures Function Properties of the Marshallian Changes in Income Changes in the Price of one Good Deriving the Marshallian Demand Marshallian Demand Turns out the demand that we have been talking about until now has a name. And the name is Marshallian Demand . The marshallian demand is the solution we get to our familiar utility maximization problem: max x 1 , x 2 u ( x 1 , x 2 ) s.t. p 1 x 1 + p 2 x 2 = m The two demands for good 1 and 2 we will get by solving this problem will be a function of the three things we always know: p 1 , p 2 and m . x * i = x i ( p 1 , p 2 , m ) , i = 1 , 2 George Georgiou - Intermediate Microeconomics 2/25 Marshallian Demand (Lecture 6) Marshallian Demand Example Indirect Utility and Expenditures Function Properties of the Marshallian Changes in Income Changes in the Price of one Good Deriving the Marshallian Demand Marshallian Demand: Example Lets repeat an example with a Cobb-Douglas utility but this time without numbers for prices and income and lets get the marshallian demand as a function of the prices of the two goods and income. max x 1 , x 2 u = x a 1 x b 2 where a + b = 1 s.t. p 1 x 1 + p 2 x 2 = m Lets use the tangency condition to solve this problem.- p 1 p 2 = MRS - p 1 p 2 =- ax a- 1 1 x b 2 bx a 1 x b- 1 2 George Georgiou - Intermediate Microeconomics 3/25 Marshallian Demand (Lecture 6) Marshallian Demand Example Indirect Utility and Expenditures Function Properties of the Marshallian Changes in Income Changes in the Price of one Good Deriving the Marshallian Demand Marshallian Demand: Example Continuing: p 1 p 2 = a b x- 1 1 x 2 p 1 p 2 = a b x 2 x 1 p 1 x 1 = a b p 2 x 2 (1) Now plugging (1) into the budget constraint and remembering that a + b = 1: m = p 1 x 1 + p 2 x 2 = a b p 2 x 2 + p 2 x 2 = p 2 x 2 a b + 1 p 2 x 2 a + b b = m p 2 x 2 1 b = m x * 2 = bm p 2 (2) George Georgiou - Intermediate Microeconomics 4/25 Marshallian Demand (Lecture 6) Marshallian Demand Example Indirect Utility and Expenditures Function Properties of the Marshallian Changes in Income Changes in the Price of one Good Deriving the Marshallian Demand Marshallian Demand: Example Now plugging (2) into the budget constraint again: m = p 1 x 1 + p 2 x 2 = p 1 x 1 + p 2 bm p 2 m = p 1 x 1 + bm m- bm = p 1 x 1 (1- b ) m = p 1 x 1 am = p 1 x 1 x * 1 = am p 1 We have gloriously found the two marshallian demands , x * 1 = am p 1 , x * 2 = bm p 2 ....
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Lecture 6 _Marshallian Demand_ - Marshallian Demand...

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