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Unformatted text preview: Lecture 4: Demand Perlo/ Chapter 4 Vladimir Petkov VUW 12 March 2009 Vladimir Petkov (VUW) Lecture 4: Demand 12 March 2009 1 32 Graphical Representation Of Duality A S R Vladimir Petkov (VUW) Lecture 4: Demand 12 March 2009 2 / 32 Duality De&ned Consider the utility maximization problem max q 1 , q 2 U ( q 1 , q 2 ) s.t. p 1 q 1 + p 2 q 2 = Y . Suppose that its solution is q & 1 , q & 2 . Now &x utility at U & = U ( q & 1 , q & 2 ) and consider the expenditure minimization problem min q 1 , q 2 p 1 q 1 + p 2 q 2 s.t. U ( q 1 , q 2 ) = U & . The solution to this problem will be exactly q & 1 , q & 2 ! Vladimir Petkov (VUW) Lecture 4: Demand 12 March 2009 3 / 32 Duality (Continued) Similarly, suppose we solve the following expenditure minimization problem: min q 1 , q 2 p 1 q 1 + p 2 q 2 s.t. U ( q 1 , q 2 ) = U . Let the solution to this problem be q && 1 , q && 2 and &x consumer income at Y && = p 1 q && 1 + p 2 q && 2 . Now consider the utility maximization problem max q 1 , q 2 U ( q 1 , q 2 ) s.t. p 1 q 1 + p 2 q 2 = Y && . The solution to this problem will be exactly q && 1 , q && 2 ! So the expenditure minimization problem and the utility maximization problem are closely related: they are known as dual problems . Vladimir Petkov (VUW) Lecture 4: Demand 12 March 2009 4 / 32 Uncompensated Demand By de&nition demand tells us how much a consumer is willing to buy at a given price, holding constant other factors (such as tastes and preferences, income, and prices of complements and substitutes). Take the utility maximization problem analyzed in the previous lecture. We solved for the optimal quantities as functions of prices and income. That is, we solved for the consumers uncompensated demand functions for these goods: q 1 = Z ( p 1 , p 2 , Y ) q 2 = B ( p 1 , p 2 , Y ) . In the CobbDouglas example, q 1 = Y / p 1 and q 2 = ( 1 & ) Y / p 2 . The two goods are neither complements nor substitutes: the demand depends only on the goods own price. Vladimir Petkov (VUW) Lecture 4: Demand 12 March 2009 5 / 32 Uncompensated Demand (Continued) Suppose that we double all prices and income. What will happen to uncompensated demand? The utility maximization problem becomes max q 1 , q 2 U ( q 1 , q 2 ) s.t. 2 p 1 q 1 + 2 p 2 q 2 = 2 Y . Note that " 2 " cancels out from the budget constraint. So this problem is equivalent to the original one. The two problems will have identical solutions. Thus, if we multiply all prices and income by some number, demand will not change! Consider our CobbDouglas example: q 1 = 2 Y 2 p 1 = Y p 1 , q 2 = ( 1 & ) 2 Y 2 p 2 = ( 1 & ) Y p 2 . Vladimir Petkov (VUW) Lecture 4: Demand 12 March 2009 6 / 32 Graphical Interpretation of Uncompensated Demand Now we use a graphical approach to construct the uncompensated demand curve....
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This note was uploaded on 05/24/2011 for the course ECON 201 taught by Professor Paulclacott during the Fall '10 term at Victoria Wellington.
 Fall '10
 PaulClacott
 Microeconomics

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