8105-11MidtermKey - 1 MACROECONOMIC THEORY T J KEHOE ECON...

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Unformatted text preview: 1 MACROECONOMIC THEORY T. J. KEHOE ECON 8105 FALL 2011 ANSWERS TO MIDTERM EXAMINATION 1. (a) With an Arrow-Debreu markets structure futures markets for goods are open in period 0 . Consumers trade futures contracts among themselves. An Arrow-Debreu equilibrium is sequence of prices 1 2 ˆ ˆ ˆ , , , p p p and consumption levels 1 1 1 1 2 ˆ ˆ ˆ , , , c c c ; 2 2 2 1 2 ˆ ˆ ˆ , , , c c c such that Given 1 2 ˆ ˆ ˆ , , , p p p , consumer i , 1,2 i , chooses 1 2 ˆ ˆ ˆ , , , i i i c c c to solve max log t i t t c ˆ ˆ s.t. i i t t t t t t p c p w i t c . 1 2 1 2 ˆ ˆ t t t t c c w w , 0,1, t . (b) With sequential market markets structure, there are markets for goods and bonds open every period. Consumers trade goods and bonds among themselves. A sequential markets equilibrium is sequences of interest rates 1 2 3 ˆ ˆ ˆ , , , r r r , consumption levels 1 1 1 1 2 ˆ ˆ ˆ , , , c c c ; 2 2 2 1 2 ˆ ˆ ˆ , , , c c c , and bond holdings 1 1 1 1 2 ˆ ˆ ˆ , , ,... b b b ; 2 2 2 1 2 ˆ ˆ ˆ , , ,... b b b such that Given 1 2 3 ˆ ˆ ˆ , , , r r r , the consumer i , 1,2 i , chooses 1 2 ˆ ˆ ˆ , , , i i i c c c ; 1 2 ˆ ˆ ˆ , , ,... i i i b b b to solve max log t i t t c 1 ˆ s.t. (1 ) i i i i t t t t t c b w r b , 0,1, t i t c , i t b B i b . Here i t b B , where B is chosen large enough, rules out Ponzi schemes but does not otherwise bind in equilibrium. 1 2 1 2 ˆ ˆ t t t t c c w w , 0,1, t 1 2 ˆ ˆ t t b b , 0,1, t 2 (c) Proposition 1: Suppose that 1 2 ˆ ˆ ˆ , , , p p p ; 1 1 1 1 2 ˆ ˆ ˆ , , , c c c ; 2 2 2 1 2 ˆ ˆ ˆ , , , c c c is an Arrow- Debreu equilibrium. Then 1 2 3 ˆ ˆ ˆ , , , r r r ; 1 1 1 1 2 ˆ ˆ ˆ , , , c c c ; 2 2 2 1 2 ˆ ˆ ˆ , , , c c c ; 1 1 1 1 2 ˆ ˆ ˆ , , ,... b b b ; 2 2 2 1 2 ˆ ˆ ˆ , , ,... b b b is a sequential markets equilibrium where 1 ˆ ˆ 1 ˆ t t t p r p 1 ˆ ˆ ˆ ˆ (1 ) i i i i t t t t t b w r b c , 0,1, t . Proposition 2: Suppose that 1 2 3 ˆ ˆ ˆ , , , r r r ; 1 1 1 1 2 ˆ ˆ ˆ , , , c c c ; 2 2 2 1 2 ˆ ˆ ˆ , , , c c c ; 1 1 1 1 2 ˆ ˆ ˆ , , ,... b b b ; 2 2 2 1 2 ˆ ˆ ˆ , , ,... b b b is a sequential markets equilibrium. Then 1 2 ˆ ˆ ˆ , , , p p p ; 1 1 1 1 2 ˆ ˆ ˆ , , , c c c ; 2 2 2 1 2 ˆ ˆ ˆ , , , c c c is an Arrow-Debreu equilibrium where ˆ 1 p 1 1 ˆ ˆ (1 ) t t s s p r , 1,2, t . (d) Using the two consumers’ first order conditions t i t i t p c , 1,2 i , we can write 1 2 2 1 t t c c ....
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This note was uploaded on 01/29/2012 for the course ECONOMICS 101 taught by Professor Tikk during the Spring '11 term at University of Toronto.

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8105-11MidtermKey - 1 MACROECONOMIC THEORY T J KEHOE ECON...

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