ECON100A_5 - DEMAND 1 WhatisDEMAND DEMAND is what we call...

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 1/4/2008 1 DEMAND
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 1/4/2008 2 What is DEMAND? DEMAND ” is what we call the solution functions x 1 *(p 1 ,p 2 , I), x 2 *(p 1 ,p 2 , I) of the utility maximization problem: I x p x p t s x x u x x = + 2 2 1 1 2 1 , . . ) , ( max 2 1
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 1/4/2008 3 Review: max u(x 1 , x 2 ) x 1 ,x 2.. s.t. p 1 x 1 + p 2 x 2 = I Utility - the function to be maximized: Budget constraint: Putting it all together: X 1 X 2 X 1 * X 2 *
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 1/4/2008 4 Two conditions for solution Tangency condition (x 1 *, x 2 *) solves: slope indif curve = slope budget line This yields : Graphical Approach: X 1 X 2 X 1 * X 2 *
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 1/4/2008 5 First-order conditions in constrained optimization problem: This yields Algebraically: X 1 X 2 X 1 * X 2 * c x x g x x x g x x x f x x x g x x x f = = = ) , ( . 3 ) , ( ) , ( . 2 ) , ( ) , ( . 1 2 1 2 2 1 2 2 1 1 2 1 1 2 1 λ λ
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 1/4/2008 6 “Demand” is what we call the solution functions x 1 *(p 1 ,p 2 , I), x 2 *(p 1 ,p 2 , I) of the utility maximization problem: Solution Case 1 Given nice properties, (differentiability, diminishing MRS) solution obtained from Tangency Condition And Budget condition: p 1 x 1 *+p 2 x 2 * =I Review Solution – Case 2 When we do not have nice properties (and we can’t take derivatives) we may get corner solutions. Use logic to work these out 2 1 2 1 2 2 1 1 p p *) x *, (x mu *) x *, (x mu = I x p x p t s x x u x x = + 2 2 1 1 2 1 , . . ) , ( max 2 1
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 1/4/2008 7 Why is the solution at the tangency? Suppose at (x 1 , x 2 ) MRS (x 1 , x 2 ) >p 1 /p 2 Say MRS (x 1 , x 2 ) =4, p 1 =10, p 2 =5 You can get a unit of good 1 for 2 units of good 2 in the marketplace. How many of good 2 would you be willing to give up for a unit of good 1? Answer: 4 MORE INUTITION FOR CASE 1 You are willing to give up 4 of good 2, but only need to give up 2 to get a unit of good 1, so: You would never choose this bundle! Get more good 1, less good 2 X 1 X 2 X 1 X 2
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 1/4/2008 8 Why is the solution at the tangency?
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