11.1 Macroeconomic Models
177
where the only difference to the previous model is the additional random process
for λ and the need to use the expectation value of the utility, expressed by the angle
brackets, in the definition of the objective functional.
The second extension we consider adds the effect of labor h t that is required to
produce the output y t . This extension describes the student’s dilemma, which is also
known as Hansen’s model. For the output y t at time step t we use a Cobb-Douglas
production function, given by
y t = λ f (k t , h t ) = λk
t h
1−
t
.
(11.14)
The dilemma of the student comes from the fact that she has to split her time between
work h t and leisure l t = 1 − h t to goof off. The leisure l t is valued in the utility
function, but the the work h t must be expended to achieve an objective, which now
values both profits p t and leisure l t and has the form V [ p, l] =
∞
j=0 β
j u( p t+ j , l t+ j ).
The complete dynamical system, including the objective functional is then given by
k t+1 = (1 − δ)k t + i t
p t = λ f (k t , h t ) − i t
(11.15)
l t = 1 − h t
V [ p, l] =
∞
j=0
β
j u( p t+ j , l t+ j ) ,
where u( p t , l t ) is often assumed to have a logarithmic dependence
u( p t , l t ) = log( p t ) + w log(l t )
(11.16)
with some weight w to assign a relative weight to the two contributions. Some people
value money higher than their spare time; they use a small value of w. Others do not
care much about money, but prefer to goof off instead; they use a large value of w.
In the third extension we illustrate how to make the evolution equations, which so
far use discrete time steps, time-continuous. This involves making the duration of a
time step t infinitely small, such that we can treat differences from period to period
as derivatives. Let us therefore denote the profit per unit time by p
= p t //t and
the other quantities λ
, δ
and i
are likewise the corresponding quantities, divided by
t. As before, β describes a factor to discount future utilities and p
0 is a reference
profit to make the argument of the logarithm unitless. The continuous-time version
of (11.6) and (11.7) is
˙
k = −δ
k + i
p
= λ
f (k) − i
(11.17)
V [ p
] =
∞
0
β
t log( p
(t)/ p
0 )dt ,
177
where the only difference to the previous model is the additional random process
for λ and the need to use the expectation value of the utility, expressed by the angle
brackets, in the definition of the objective functional.
The second extension we consider adds the effect of labor h t that is required to
produce the output y t . This extension describes the student’s dilemma, which is also
known as Hansen’s model. For the output y t at time step t we use a Cobb-Douglas
production function, given by
y t = λ f (k t , h t ) = λk
t h
1−
t
.
(11.14)
The dilemma of the student comes from the fact that she has to split her time between
work h t and leisure l t = 1 − h t to goof off. The leisure l t is valued in the utility
function, but the the work h t must be expended to achieve an objective, which now
values both profits p t and leisure l t and has the form V [ p, l] =
∞
j=0 β
j u( p t+ j , l t+ j ).
The complete dynamical system, including the objective functional is then given by
k t+1 = (1 − δ)k t + i t
p t = λ f (k t , h t ) − i t
(11.15)
l t = 1 − h t
V [ p, l] =
∞
j=0
β
j u( p t+ j , l t+ j ) ,
where u( p t , l t ) is often assumed to have a logarithmic dependence
u( p t , l t ) = log( p t ) + w log(l t )
(11.16)
with some weight w to assign a relative weight to the two contributions. Some people
value money higher than their spare time; they use a small value of w. Others do not
care much about money, but prefer to goof off instead; they use a large value of w.
In the third extension we illustrate how to make the evolution equations, which so
far use discrete time steps, time-continuous. This involves making the duration of a
time step t infinitely small, such that we can treat differences from period to period
as derivatives. Let us therefore denote the profit per unit time by p
= p t //t and
the other quantities λ
, δ
and i
are likewise the corresponding quantities, divided by
t. As before, β describes a factor to discount future utilities and p
0 is a reference
profit to make the argument of the logarithm unitless. The continuous-time version
of (11.6) and (11.7) is
˙
k = −δ
k + i
p
= λ
f (k) − i
(11.17)
V [ p
] =
∞
0
β
t log( p
(t)/ p
0 )dt ,