2003-2007 midterm

2003-2007 midterm - Macroeconomics 702, Second Midterm...

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Macroeconomics 702, Second Midterm Suggested solution by Ahu and Vivian Question 1 (a) (10 points: 3 for consumption possibility set and production set, 3 for consumer’s problem and f rm’s problem, 4 for de f ning AD equilibrium) Let ε t =( s A,t ,s B,t ) Z 2 Ξ and H = Ξ × Ξ ×··· where Ξ is a support of mood shocks to agents. And an element of H , h t ε 0 1 , ..., ε t ) Ξ t is a history of shocks up to period t. We denote the probability of h t conditional on information at 0 is π ( h t ) . Then, the commodity space is: L = { x | x t ( h t )=( x 1 t ( h t ) ,x 2 t ( h t ) 3 t ( h t )) R 3 , t, h t and k x k < a.s. } Consumption possibility set for agent of type i is X i = { x i L | { c i t ( h t ) ,a i t +1 ( h t ) } t =0 0 such that x i 1 t ( h t )= a i t +1 ( h t )+ c i t ( h t ) , x i 2 t ( h t ) [0 , 1] i 2 t ( h t ) a i t ( h t 1 ) t , h t and a 0 given } where x i 1 t ( h t ) is the produced f nal goods, x i 2 t ( h t ) is the capital service input, for agent of type i at time t after history h t . Production set for f rm is Y i = { y i L | 0 y i 1 t ( h t ) F [ y i 3 t ( h t ) ,y i 2 t ( h t )] , t, h t . } For a given price system p ( x ) , the type i consumer’s optimization problem is: max x X i X t =0 X h t H t β t π ( h t ) ( c t ) 1 σ 1 σ (1) such that 3 X j =1 X t X h t p j,t ( h t ) x i j,t ( h t ) 0 And f rm’s optimization problem: max y Y i 3 X j =1 X t X h t p j,t ( h t ) y j,t ( h t ) (2) An AD competitive equilibrium is a triad ( p A B ) that satis f es 1. Given p i solves type i consumer’s problem, i = A, B. 2. y i solves f rm’s problem. 3. market clears x A 1 + x B 1 = y A 1 + y B 1 (3) x A 3 + x B 3 = y A 3 + y B 3 (4) x A 2 = y A 2 B 2 = y B 2 (5) . 1
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Remark 1 Because goods can be transported freely across islands, people’s in- vestments have the same rate of return. Therefore, there is only one variable, x i 3 ( h t ) for saving in each agent’s consumption possibility set. And f rms uses the capital that people invest to. Market clearing condition for capital is (4). But since labor cannot move, market clearing condition is (5). (b) Theorem 2 First Basic Welfare Theorem: If the preferences of consumers are nonsatiated ( { x n } X that converges to x X such that U ( x n ) >U ( x ) ), an allocation ( x ,y ) of an ADE ( p ,x ) is PO. Theorem 3 Second Basic Welfare Theorem: If (i) X is convex, (ii) preference is convex (for x, x 0 X, if x 0 <x ,then x 0 < (1 θ ) x 0 + θx for any θ (0 , 1) ), (iii) U ( x ) is continuous, (iv) Y is convex, (v)Y has an interior point, then with any PO allocation ( x ) such that x is not a satiation point, there exists a continuous linear functional p such that ( x ,p ) is a Quasi-Equilibrium ((a) for x X which U ( x ) U ( x ) implies p ( x ) p ( x ) and (b) y Y implies p ( y ) p ( y ) ) Lemma 4 If, for ( x ) in the theorem above, the budget set has cheaper point than x ( x X such that p ( x ) <p ( x ) ), then ( x ) is a ADE.
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2003-2007 midterm - Macroeconomics 702, Second Midterm...

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