ECON10A_5 (1)

# ECON10A_5 (1) - DEMAND 1 WhatisDEMAND DEMAND is what we...

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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
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 : MRS (x 1 *, x 2 *)=p 1 /p 2 Budget condition On budget line p 1 x 1 *+p 2 x 2 *= I Graphical Approach: X 1 X 2 X 1 * X 2 *
1/4/2008 5 First-order conditions in constrained optimization problem: This yields 1. mu 1 (x 1 *, x 2 *)=λp 1 2. mu 2 (x 1 *, x 2 *)=λp 2 3. p 1 x 1 *+p 2 x 2 *= I Note: Dividing 1. by 2. gives mu 1 (x 1 *, x 2 *)/mu 2 (x 1 *, x 2 *)= MRS (x 1 *, x 2 *)=p 1 /p 2 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

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1/4/2008 6 What About Multiple Goods? General formula for n goods Max u(x 1 ,x 2 ,…, x n ) s.t. p 1 x 1 +p 2 x 2 +…+p n x n =I Solution or 1. mu 1 (x 1 *, x 2 *, …,x n * )=λp 1 2. mu 2 (x 1 *, x 2 *, …,x n *)=λp 2 . . n. mu n (x 1 *, x 2 *, …,x n *)=λp n n+1. p 1 x 1 *+p 2 x 2 *+…+p n x n *=I j i, all for p *) x *,. .., x *, (x mu p *) x *,. .., x *, (x mu j n 2 1 j i n 2 1 i =
7 “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

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ECON10A_5 (1) - DEMAND 1 WhatisDEMAND DEMAND is what we...

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