wickens宏观经济理论动态一般均衡方法习题答案
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(a) derive the optimal long-run solution. (b) Analyse the short-run solution.
2.3.
Consider
the
CES
production
来自百度文库
function
|w
=
1
D[nw
1
+
(1
1
)qw
1
]
1 1
1
(a) Show that the CES function becomes the Cobb-Douglas function as $ 1.
where |w is output, fw is consumption, lw is investment, nw is the capital stock and the objective is
to maximize
X 4 Yw = vX (fw+v)
v=0
derive the optimal solution under hyperbolic discounting and comment on any dierences with
0
Exercises
Chapter2
2.1. We have assumed that the economy discounts v periods ahead using the geometric (or exponential) discount factor v = (1 + )v for {v = 0> 1> 2> ===}. Suppose instead that the economy uses the sequence of hyperbolic discount factors v = {1> *> *2> *3> ===} where 0 ? * ? 1.
(a) Compare the implications for discounting of using geometric and hyperbolic discount factors.
(b) For the centrally planned model
|w = fw + lw nw+1 = lw nw
Macroeconomic Theory: A Dynamic General Equilibrium Approach
Mike Wickens University of York Princeton University Press
Exercises and Solutions
Completed January 2010
and ow is leisure (ow + qw = 1).
(a) Derive expressions from which the long-run solutions for consumption, labour and capital
may be obtained.
(b) What are the implied long-run real interest rate and wage rate?
(b) Verify that the CES function is homogeneous of degree one and hence satisfies I (nw> qw) =
Iq>wqw + In>wnw.
1
2.4. Consider the following centrally-planned model with labor
|w = fw + lw
nw+1 = lw nw
|w
=
1
D[nw
1
+
(1
1
)qw
1
]
1
1
1
where the objective is to maximize
X 4 Yw = v[ln fw+v + * ln ow+v]>
1 =
1+
v=0
where |w is output, fw is consumption, lw is investment, nw is the capital stock, qw is employment
(b) Show that although the solution for the basic centrally-planned economy of Chapter 2 is a saddlepath, it can be approximately represented by a stable autoregressive process.
(c) Comment on the implications for labor of having an elasticity of substitution between
capital and labor dierent from unity
(d) Obtain the long-run capital-labor ratio.
the solution based on geometric discounting.
2.2. Assuming hyperbolic discounting, the utility function X (fw) = ln fw and the production function |w = Dnw,
2.6. In continuous time the basic centrally-planned economy problem can be written as: max-
imize
R04hwx(fw)gw
2.5. (a) Comment on the statement: "the saddlepath is a knife-edge solution; once the economy departs from the saddlepath it is unable to return to equilibrium and will instead either explode or collapse."
2.3.
Consider
the
CES
production
来自百度文库
function
|w
=
1
D[nw
1
+
(1
1
)qw
1
]
1 1
1
(a) Show that the CES function becomes the Cobb-Douglas function as $ 1.
where |w is output, fw is consumption, lw is investment, nw is the capital stock and the objective is
to maximize
X 4 Yw = vX (fw+v)
v=0
derive the optimal solution under hyperbolic discounting and comment on any dierences with
0
Exercises
Chapter2
2.1. We have assumed that the economy discounts v periods ahead using the geometric (or exponential) discount factor v = (1 + )v for {v = 0> 1> 2> ===}. Suppose instead that the economy uses the sequence of hyperbolic discount factors v = {1> *> *2> *3> ===} where 0 ? * ? 1.
(a) Compare the implications for discounting of using geometric and hyperbolic discount factors.
(b) For the centrally planned model
|w = fw + lw nw+1 = lw nw
Macroeconomic Theory: A Dynamic General Equilibrium Approach
Mike Wickens University of York Princeton University Press
Exercises and Solutions
Completed January 2010
and ow is leisure (ow + qw = 1).
(a) Derive expressions from which the long-run solutions for consumption, labour and capital
may be obtained.
(b) What are the implied long-run real interest rate and wage rate?
(b) Verify that the CES function is homogeneous of degree one and hence satisfies I (nw> qw) =
Iq>wqw + In>wnw.
1
2.4. Consider the following centrally-planned model with labor
|w = fw + lw
nw+1 = lw nw
|w
=
1
D[nw
1
+
(1
1
)qw
1
]
1
1
1
where the objective is to maximize
X 4 Yw = v[ln fw+v + * ln ow+v]>
1 =
1+
v=0
where |w is output, fw is consumption, lw is investment, nw is the capital stock, qw is employment
(b) Show that although the solution for the basic centrally-planned economy of Chapter 2 is a saddlepath, it can be approximately represented by a stable autoregressive process.
(c) Comment on the implications for labor of having an elasticity of substitution between
capital and labor dierent from unity
(d) Obtain the long-run capital-labor ratio.
the solution based on geometric discounting.
2.2. Assuming hyperbolic discounting, the utility function X (fw) = ln fw and the production function |w = Dnw,
2.6. In continuous time the basic centrally-planned economy problem can be written as: max-
imize
R04hwx(fw)gw
2.5. (a) Comment on the statement: "the saddlepath is a knife-edge solution; once the economy departs from the saddlepath it is unable to return to equilibrium and will instead either explode or collapse."