Question The Deterministic Neoclassical Growth Model: Theo

07 Jun Question The Deterministic Neoclassical Growth Model: Theo

Question
The Deterministic Neoclassical Growth Model: Theory and Computation, and I just need some help with Matlab codes. I can compute parts 4, 5, 6, 7 and 8, but I need to know how to do the policy function in Matlab so I can derive the optimal capital and consumption paths.

Homework # 2
Dirk Krueger

1

The Deterministic Neoclassical Growth Model:
Theory and Computation

Consider the social planner problem associated with the neoclassical growth
model
max 1

t

ln(ct )

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fct ;kt+1 gt=0

1
X

s.t.

ct + kt+1

t=0

Akt

ct ; kt+1

0

k0 given

1. Write down the Euler conditions and the transversality condition for this
problem. Compute the steady state values of c and k:

2. Write down the functional equation associated with this problem. Guess
that the value function has the form
v(k) = a0 + a1 ln(k)

Calculate the value function (i.e. nd a0 ; a1 ) and the policy function.

3. Try to approximate v(k) : Guess that v0 (k) = 0 for all k and use the
iterative updating rule
vn+1 (k) =

max

0 k0 Ak

fln (Ak

k 0 ) + vn (k 0 )g

Th

Calculate analytically the functions v1 ; v2 ; v3 and compare your result with
2:

4. Let = 0:3; = 0:6 and A = 1. Let capital take values on the discrete
grid K = f0:04; 0:08; 0:12; 0:16; 0:2g: Make the original guess v0 (k) = 0 for
all k and perform the rst three steps of the value function iteration for
all k 2 K
vn+1 (k)

=

k

s.t.
k 0 k 0:3

0

k0

max ln k 0:3
0

2

K

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k 0 + 0:6vn (k 0 )

5. Perform value function iterations until
max jvn+1 (k)
k2K

vn (k)j < 10

6

Report the nal value and policy functions that you obtain (Hint: you
probably want to use a computer; if you do, report your code).
6. Repeat part 5: with a grid of capital stocks K = f0:04; 0:0408 : : : 0:1992; 0:2g
(Hint: you have to use a computer; again report your code). Compare
your results with part 2:

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7. Suppose the planner has an initial capital stock k0 = 0:04: Provide a table
with the rst 50 elements of the optimal capital and consumption levels
fct ; kt+1 g50 the planner should and would choose. Note: you can do this
t=0
by hand, but I would not advise so.
8. Repeat questions 6: and 7: for the parameter values = 0:3; = 0:75 and
A = 1 and the same grid of capital stocks K = f0:04; 0:0408 : : : 0:1992; 0:2g
and try to explain the di¤erences (and similarities) in your ndings.

2

The Deterministic Neoclassical Growth Model
with Labor Supply

Now consider the social planners problem associated with the neoclassical growth
model with endogenous labor supply
max

fct ;nt ;kt+1 g1
t=0

s.t.

ct + kt+1

ct ; nt ; kt+1

1
X

t

U (ct ; nt )

t=0

Akt n1
t
0

Th

k0 given

1. Derive the intertemporal optimality condition (Euler equation) relating
marginal utilities of consumption at period t and t + 1; and the intratemporal optimality condition relating the marginal utility of consumption
and the marginal disutility of work to each other. For this question you
can assume that the nonnegativity constraints do not bind. Also note that
for this question I do NOT impose the constraint nt 1:
2. Write down the functional equation associated with this problem.
3. Suppose that
U (ct ; nt ) = log(ct

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nt )

where

> 0 is a parameter.1 Guess that the value function has the form
v(k) = a0 + a1 ln(k)

Calculate the value function (i.e. nd a1 ; I spare you a0 since it is a mess)
and the policy functions c = C(k) and n = N (k).
4. Now suppose consumption goods and capital goods are produced in two
di¤erent sectors. The production functions in both sectors are given by
=

k1t n1
1t

kt+1

=

k2t n1
2t

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ct

Capital and labor inputs in both sections have to satisfy
k1t + k2t

=

kt

n1t + n2t

=

nt :

The planner can allocate both capital and labor freely across the two
sectors in the current period, that is, no time is required to relocate either
capital or labor across sectors. Repeat question 2.

5. Everything is as in the previous question, but now capital in both sectors
is predetermined; that is, if the planner decides to move capital between
the two sectors today, this change takes e¤ect at the beginning of next
period. Labor can still be reallocated instantaneously. Repeat question 2.

6. Augment your code from part 1 of this homework to numerically calculate
the value function for the functional equation in question 2 of this part,
with the utility function
U (ct ; nt ) = log(ct

nt )

Set
= 0:3; A = 1 and = 0:6: In the code you have to take care of
potential corner solutions.

Th

7. Suppose the planner has an initial capital stock k0 = 0:04: Provide a table
with the rst 50 elements of the optimal capital and consumption levels
fct ; nt ; kt+1 g50 .
t=0

3

The Deterministic Neoclassical Growth Model
with Two Classes of Households

Consider the following extension of the neoclassical growth model. The representative rm operates a Cobb-Douglas technology given by
Yt = Kt L1
t

1 These preferences are sometimes called GHH preferences, named after Jeremy Greenwood,
Zvi Hercovitz and Greg Hu¤man who invented them.

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where Kt is the aggregate capital stock, Lt is aggregate labor supply and 2
(0; 1) is a parameter. Capital depreciates at rate 2 (0; 1]:
There are two groups of households i = 1; 2: Group i has i members, each
i
member has an initial endowment of capital k0 > 0; one unit of time to spend
on labor or leisure, and preferences represented by a utility function of the form
1
X

t i

i
u (ci ; lt )
t

t=0

i
lt

where is labor supply of household i at time t: Unless otherwise stated you
1
2
cannot assume that 1 = 2 ; k0 = k0 or u1 = u2 :

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1. Dene a sequential markets equilibrium.

2. Dene a recursive competitive equilibrium.
3. Now suppose that
question

1

=

2

= 1=2 and that for the remainder of the
ci
t

i
lt

1

i

1
:
1
i
Compute the steady state allocation(s) of this model. If you cannot do so
in closed form, provide the sharpest characterization possible.
i
ui (ci ; lt ) =
t

4. Now suppose that 1 = 2 = 1=2; 1 = 2 = 1 (i.e. log utility),
=
1
2
0; k0 = k0 = k0 > 0 and = 1: Characterize as fully as possible the
sequential markets equilibrium of this model, given an arbitrary initial
condition k0 > 0:

4

The Neoclassical Growth Model with Habit
Persistence

Consider the neoclassical growth model. A representative rm produces output
according to the production function
Yt = F (Kt ; Nt )

Th

Capital depreciates at a constant rate
0: The aggregate resource constraint
reads as
Yt = Ct + Kt+1 (1
)Kt
where Ct is aggregate consumption.
There is a large number of identical households with total mass equal to 1:
Each household is endowed with k0 = K0 units of capital and one unit of time
in every period. The household has preferences over individual consumption
streams fct g1 representable by the lifetime utility function
t=0
1
X

t

U (ct ; ct

t=0

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1)

Note that each household chooses fct g1 ; and treats c
t=0
condition.

1

as an exogenous initial

1. State the social planner problem recursively. Clearly identify the state
and control variables.
2. Use the rst order condition and the envelope condition to derive the Euler
equation of the social planner problem.
3. Dene a recursive competitive equilibrium.

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4. Use the rst order condition and the envelope condition to derive the Euler
equation that the representative household faces.

Th

5. For the purpose of this question you can assume that there exists a unique
competitive equilibrium. Is this equilibrium Pareto e¢ cient? You don
t
need to provide a formal proof, but use the answers to 2. and 4. to explain
your answer, and give some intuition for your answer.

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