rogoff国际宏观经济学基础答案ch5ans
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∗ (s) = Y2w (s), s = 1, 2. C2 (s) + C2
Thus, we have four equations–eq. (2) and its Foreign analog, plus the state 1 ∗ (s)], and 2 equilibrium conditions–to determine the four unknowns [C2 (s), C2 s = 1, 2. If you play with these four equations, however, you will realize that the date 2 consumption allocation actually is indeterminate. To see this intuitively, imagine any equilibrium allocation of date consumption across states. Suppose now that Home were to reduce its state 1 consumption by 1/π (1) units and raise its state 2 consumption by 1/π (2) units. Constraint (2) would still be satisÞed and Home residents would feel no worse off. If Foreign were simultaneously to raise its state 1 consumption by 1/π (1) units 50
By Maurice Obstfeld (University of California, Berkeley) and Kenneth Rogoff (Princec MIT Press, 1996. ton University). ° 2 c °MIT Press, 1998. Version 1.1, February 27, 1998. For online updates and corrections, see http://www.princeton.edu/ObstfeldRogoffBook.html
1
"
#
48
Here, the last equality comes from eq. (17), Chapter 5. Using eq. (80) from the chapter, we see that the autarky price of the Arrow-Debreu security for state 1 relative to that of the state 2 security is p(1)a π (1)/Y2 (1) = . p(2)a π (2)/Y2 (2) Substitution of the preceding into the expression for p(1)B2 (1)/(1 + r), gives the required result. The result for B2 (2) follows from the identity CA1 = p(1) p(2) B2 (1) + B2 (2). 1+r 1+r
)
Thus by substitution of the Þrst-order condition,
µ
1 Y2 (s) 1+ C1 = E1 (1 + r)B1 + Y1 + , 1+r 1+r 1+r Y2 (s) C1 = E1 (1 + r)B1 + Y1 + . 2+r 1+r
( )
(
that is,
(3)
For the ∞-horizon case, we just get the usual “permanent-income” formula, essentially eq. (32) of Chapter 2 (suitably adapted). (b) The consumption formula of part a will be generally valid if the nonnegativity constraint on consumption never binds, that is, if, even when output 51
while lowering its state 2 consumption by 1/π (2) units–an action which is feasible for Foreign and does not lower its welfare–date 2 contingent commodity markets would still clear. Thus, the date 2 consumption allocation is not fully determined in the model. This indeterminacy result is, of course, a consequence of risk-neutrality with regard to date 2. People care about the expected values of second-period payoffs, but not about their distribution across states. This indifference explains why the national allocations of second-period consumption across the states of nature is not tied down. 3. (a) Ignoring nonnegativity, write the unconstrained maximization as a0 [(1 + r)B1 − B2 + Y1 ]2 2 ½ ¾ 1 a0 2 E1 [(1 + r)B2 + Y2 (s)] − [(1 + r)B2 + Y2 (s)] . + 1+r 2 max [(1 + r)B1 − B2 + Y1 ] −
Foundations of International Macroeconomics1
Workbook2
Maurice Obstfeld, Kenneth Rogoff, and Gita Gopinath
Chapter 5Biblioteka BaiduSolutions
1. Since consumption on date 2 is the sum of endowment and payments of contingent assets for the realized state, we have B2 (s) = C2 (s) − Y2 (s), s = 1, 2. For s = 1, premultiply by p(1)/(1 + r) and use eq. (16) in Chapter 5 to obtain the following: π(1)β p(1)Y2 (1) + p(2)Y2 (2) p(1) p(1) B2 (1) = Y2 (1) Y1 + − (1 + r) 1+β 1+r 1+r " # β p(1)Y2 (1) + p(2)Y2 (2) 1 = π (1) Y1 − π (1) 1+β 1+β 1+r " # p(1) p(1)Y2 (1) + p(2)Y2 (2) Y2 (1) + π(1) − 1+r 1+r " # p(1)Y2 (1) + p(2)Y2 (2) β 1 = π (1) Y1 − π (1) 1+β 1+β 1+r p(1)Y2 (1) p(2)Y2 (2) − π(2) + π (1) 1+r 1+r " # p(2)π (2)Y2 (1) π (1)/Y2 (1) p(1) = π (1)CA1 + − . 1+r π (2)/Y2 (2) p(2)
∗ C1 = C1 =
Y1w . 2
The equilibrium intertemporal budget constraint of a Home resident is C1 + π (1) π (2) π (1) π (2) C2 (1) + C2 (2) = Y1 + Y2 (1) + Y2 (2), 1+r 1+r 1+r 1+r
so that date 2 consumptions must obey π (1) π (1) π (2) Yw π (2) C2 (1) + C2 (2) = Y1 − 1 + Y2 (1) + Y2 (2); 1+r 1+r 2 1+r 1+r (2)
there is a similar equation for Foreign. Since utility is linear in date 2 consumption with weights π (1) and π (2), a Home resident is indifferent between any pair [C2 (1), C2 (2)] satisfying eq. (2). On date 2, however, goods-market equilibrium requires that
The statement of the exercise provides the intuition. 2. The necessary Þrst-order conditions are p(s) 0 0 u (C1 ) = π (s)β u [C2 (s)], s = 1, 2. 1+r For our utility function, u (C1 ) = 1/C1 and u [C2 (s)] = 1, so that the above conditions imply p(s) = π (s)β C1 , s = 1, 2. (1) 1+r ∗ (A similar relation holds for C1 , the initial consumption of Foreign residents.) If we divide eq. (1) for s = 1 by its analog for s = 2, we see that π (1) p(1) = , π (2) p(2) implying that equilibrium prices must be actuarially fair. Since p(1) + p(2) = 1, it follows that the Arrow-Debreu prices equal the respective probabilities of the state occurring: p(s) = π (s), s = 1, 2. Assuming that Home and Foreign share the same discount factor β , we may add eq. (1) for Home and for Foreign to obtain
B2
The Þrst-order condition for B2 is: C1 = E1 {C2 (s)} . The S + 1 budget constraints in the problem imply that E1
( ) ( )
C2 (s) C1 + 1+r
¶
= E1
Y2 (s) (1 + r)B1 + Y1 + . 1+r
∗ C1 + C1 = Y1w =
0 0
2p(1) . (1 + r)π (1)β
49
Because p(1) = π (1), we obtain an expression for the world interest rate 1+r = 2 . β Y1w
Using Euler eq. (1) again, but substituting in this expression for 1 + r, we see that equilibrium date 1 consumptions are:
Thus, we have four equations–eq. (2) and its Foreign analog, plus the state 1 ∗ (s)], and 2 equilibrium conditions–to determine the four unknowns [C2 (s), C2 s = 1, 2. If you play with these four equations, however, you will realize that the date 2 consumption allocation actually is indeterminate. To see this intuitively, imagine any equilibrium allocation of date consumption across states. Suppose now that Home were to reduce its state 1 consumption by 1/π (1) units and raise its state 2 consumption by 1/π (2) units. Constraint (2) would still be satisÞed and Home residents would feel no worse off. If Foreign were simultaneously to raise its state 1 consumption by 1/π (1) units 50
By Maurice Obstfeld (University of California, Berkeley) and Kenneth Rogoff (Princec MIT Press, 1996. ton University). ° 2 c °MIT Press, 1998. Version 1.1, February 27, 1998. For online updates and corrections, see http://www.princeton.edu/ObstfeldRogoffBook.html
1
"
#
48
Here, the last equality comes from eq. (17), Chapter 5. Using eq. (80) from the chapter, we see that the autarky price of the Arrow-Debreu security for state 1 relative to that of the state 2 security is p(1)a π (1)/Y2 (1) = . p(2)a π (2)/Y2 (2) Substitution of the preceding into the expression for p(1)B2 (1)/(1 + r), gives the required result. The result for B2 (2) follows from the identity CA1 = p(1) p(2) B2 (1) + B2 (2). 1+r 1+r
)
Thus by substitution of the Þrst-order condition,
µ
1 Y2 (s) 1+ C1 = E1 (1 + r)B1 + Y1 + , 1+r 1+r 1+r Y2 (s) C1 = E1 (1 + r)B1 + Y1 + . 2+r 1+r
( )
(
that is,
(3)
For the ∞-horizon case, we just get the usual “permanent-income” formula, essentially eq. (32) of Chapter 2 (suitably adapted). (b) The consumption formula of part a will be generally valid if the nonnegativity constraint on consumption never binds, that is, if, even when output 51
while lowering its state 2 consumption by 1/π (2) units–an action which is feasible for Foreign and does not lower its welfare–date 2 contingent commodity markets would still clear. Thus, the date 2 consumption allocation is not fully determined in the model. This indeterminacy result is, of course, a consequence of risk-neutrality with regard to date 2. People care about the expected values of second-period payoffs, but not about their distribution across states. This indifference explains why the national allocations of second-period consumption across the states of nature is not tied down. 3. (a) Ignoring nonnegativity, write the unconstrained maximization as a0 [(1 + r)B1 − B2 + Y1 ]2 2 ½ ¾ 1 a0 2 E1 [(1 + r)B2 + Y2 (s)] − [(1 + r)B2 + Y2 (s)] . + 1+r 2 max [(1 + r)B1 − B2 + Y1 ] −
Foundations of International Macroeconomics1
Workbook2
Maurice Obstfeld, Kenneth Rogoff, and Gita Gopinath
Chapter 5Biblioteka BaiduSolutions
1. Since consumption on date 2 is the sum of endowment and payments of contingent assets for the realized state, we have B2 (s) = C2 (s) − Y2 (s), s = 1, 2. For s = 1, premultiply by p(1)/(1 + r) and use eq. (16) in Chapter 5 to obtain the following: π(1)β p(1)Y2 (1) + p(2)Y2 (2) p(1) p(1) B2 (1) = Y2 (1) Y1 + − (1 + r) 1+β 1+r 1+r " # β p(1)Y2 (1) + p(2)Y2 (2) 1 = π (1) Y1 − π (1) 1+β 1+β 1+r " # p(1) p(1)Y2 (1) + p(2)Y2 (2) Y2 (1) + π(1) − 1+r 1+r " # p(1)Y2 (1) + p(2)Y2 (2) β 1 = π (1) Y1 − π (1) 1+β 1+β 1+r p(1)Y2 (1) p(2)Y2 (2) − π(2) + π (1) 1+r 1+r " # p(2)π (2)Y2 (1) π (1)/Y2 (1) p(1) = π (1)CA1 + − . 1+r π (2)/Y2 (2) p(2)
∗ C1 = C1 =
Y1w . 2
The equilibrium intertemporal budget constraint of a Home resident is C1 + π (1) π (2) π (1) π (2) C2 (1) + C2 (2) = Y1 + Y2 (1) + Y2 (2), 1+r 1+r 1+r 1+r
so that date 2 consumptions must obey π (1) π (1) π (2) Yw π (2) C2 (1) + C2 (2) = Y1 − 1 + Y2 (1) + Y2 (2); 1+r 1+r 2 1+r 1+r (2)
there is a similar equation for Foreign. Since utility is linear in date 2 consumption with weights π (1) and π (2), a Home resident is indifferent between any pair [C2 (1), C2 (2)] satisfying eq. (2). On date 2, however, goods-market equilibrium requires that
The statement of the exercise provides the intuition. 2. The necessary Þrst-order conditions are p(s) 0 0 u (C1 ) = π (s)β u [C2 (s)], s = 1, 2. 1+r For our utility function, u (C1 ) = 1/C1 and u [C2 (s)] = 1, so that the above conditions imply p(s) = π (s)β C1 , s = 1, 2. (1) 1+r ∗ (A similar relation holds for C1 , the initial consumption of Foreign residents.) If we divide eq. (1) for s = 1 by its analog for s = 2, we see that π (1) p(1) = , π (2) p(2) implying that equilibrium prices must be actuarially fair. Since p(1) + p(2) = 1, it follows that the Arrow-Debreu prices equal the respective probabilities of the state occurring: p(s) = π (s), s = 1, 2. Assuming that Home and Foreign share the same discount factor β , we may add eq. (1) for Home and for Foreign to obtain
B2
The Þrst-order condition for B2 is: C1 = E1 {C2 (s)} . The S + 1 budget constraints in the problem imply that E1
( ) ( )
C2 (s) C1 + 1+r
¶
= E1
Y2 (s) (1 + r)B1 + Y1 + . 1+r
∗ C1 + C1 = Y1w =
0 0
2p(1) . (1 + r)π (1)β
49
Because p(1) = π (1), we obtain an expression for the world interest rate 1+r = 2 . β Y1w
Using Euler eq. (1) again, but substituting in this expression for 1 + r, we see that equilibrium date 1 consumptions are: