HW1-Solutions

HW1-Solutions - Homework Assignment 1 Solutions Econ 382...

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Unformatted text preview: Homework Assignment 1 Solutions Econ 382, Professor Platt, BYU Do not distribute — only for current students 1 Warmup 1. Alex and Bob are new college roommates. On arrival at their dorm, they discovered that both had brought cereal and ramen noodles from their parents’ pantry. Suppose that Alex brought 12 boxes of cereal and 20 packs of noodles, while Bob brought 10 boxes of cereal and 10 packs of noodles. Alex has a utility function U a ( c a ,n a ) = c . 75 a n . 25 a , and Bob’s utility function U b ( c b ,n b ) = c . 5 b n . 5 b . (a) (2 pts) Set up the consumer’s problem for each roommate. Solution: max c a ,n a c . 75 a n . 25 a s.t. p c c a- p n n a ≤ 12 p c + 20 p n max c b ,n b c . 5 b n . 5 b s.t. p c c b- p n n b ≤ 10 p c + 10 p n (b) (2 pts) List the conditions that must be satisfied for a particular set of allocations and prices to be a competitive equilibrium. Solution: A competitive equilibrium is an allocation ( c * a ,n * a ,c * b ,n * b ) and prices ( p * c ,p * n ) such that i. For all i , ( c * i ,n * i ) solves person i ’s maximization problem. ii. Markets clear: c * a + c * b ≤ 22 and n * a + n * b ≤ 30. (c) (2 pts) Solve for Alex’s demand for cereal and noodles. Solution: £ A = c . 75 a n . 25 a + λ (12 p c + 20 p n- p c c a- p n n a ) ∂ £ A ∂c a = 3 4 n a c a . 25- λp c = 0 ∂ £ A ∂n a = 1 4 c a n a . 75- λp n = 0 ∂ £ A ∂λ = 12 p c + 20 p n- p c c a- p n n a = 0 1 By substituting for λ with the first to FOCs: 3 4 p c n a c a . 25 = 1 4 p n c a n a . 75 . With cross multiplication: 3 p n n a = p c c a . Substitute this into the third FOC: 12 p c + 20 p n = 4 p n n a = ⇒ n a = 3 p c +5 p n p n . Substitute n a into either the third FOC or two lines above: c a = 3(3 p c +5 p n ) p c . (d) (2 pts) Solve for Bob’s demand for cereal and noodles. Solution: £ B = c . 5 b n . 5 b + λ (10 p c + 10 p n- p c c b- p n n b ) ∂ £ B ∂c b = 1 2 n b c b . 5- λp c = 0 ∂ £ B ∂n b = 1 2 c b n b . 5- λp n = 0 ∂ £ B ∂λ = 10 p c + 10 p n- p c c b- p n n b = 0 By substituting for λ with the first to FOCs: 1 2 p c n b c b . 5 = 1 2 p n c b n b . 5 . With cross multiplication: p n n b = p c c b . Substitute this into the third FOC: 10 p c + 10 p n = 2 p n n b = ⇒ n b = 5 p c +5 p n p n . Substitute n b into either the third FOC or two lines above: c b = 5 p c +5 p n p c . (e) (2 pts) Solve for equilibrium prices. Solution: Always remember that you get to normalize one price. Let’s choose p * c = 1. You may then choose either market clearing condition and substitute in the demand equations just derived: c a + c b = 12 + 10 3(3 p c + 5 p n ) p c + 5 p c + 5 p n p c = 22 3(3 + 5 p n ) + 5 + 5 p n = 22 20 p n = 8 = ⇒ p * n = 0 . 4 or, using the market clearing condition for noodles: n a + n b = 20 + 10 3 p c + 5 p n p n + 5 p c + 5 p n p n = 30 3 + 5 p n + 5 + 5 p n = 30 p n 20 p n = 8 = ⇒ p * n = 0 . 4 2 The fact that we get the same answer either way is a nice double check on our...
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This note was uploaded on 03/17/2010 for the course ECON 388 taught by Professor Mcdonald,j during the Spring '08 term at BYU.

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HW1-Solutions - Homework Assignment 1 Solutions Econ 382...

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