11.1 Macroeconomic Models
175
investigate the optimum by calculating the derivative of V t [ p] with respect to k t+1 ,
where we have to express all profits p t through the values of the capital k t via
p t = λ f (k t ) + (1 − δ)k t − k t+1 . Using k t+1 instead of p t is irrelevant, because both
contain the information about how much of the output y t = λ f (k t ) is carried over
into the next time period. We find conditions for the p t or equivalently for the k t+1
by setting the derivative to zero
0 =
∂u (λ f (k t ) + (1 − δ)k t − k t+1 )
∂k t+1
+ β
∂ V t+1 [ p]
∂k t+1
(11.9)
= −u
(λ f (k t ) + (1 − δ)k t − k t+1 )
+βu
(λ f (k t+1 ) + (1 − δ)k t+1 − k t+2 )
λ f
(k k+1 ) + 1 − δ
= −u
( p t ) + βu
( p t+1 )
λ f
(k k+1 ) + 1 − δ
where u
( p) denotes the derivative of the utility function with respect to its argument.
Note that V t+1 [ p] contains the k t shifted by one period, which accounts for the inner
derivative in the square bracket in the second equality. The last equality only reexpresses the arguments of the utility functions in terms of the profits, rather than
the capital in order to make the equation easier to read.
Using the recursive description of the objective functional, we could now progress
to find an equation for the next period and by repeating this procedure we arrive at
an infinite sequence of equations like (11.9). Even taking the initial capital k 0 into
account, it is still impossible to solve this infinite sequence of equations. We can,
however, close the equations by assuming to take out all capital as profit at a far-away
time. This terminates the infinite regression and leads to a finite-dimensional set of
non-linear and coupled equations, which is difficult to solve. Therefore, instead of
analyzing the transient system, we continue with a discussion of the steady-state
solution that the system approaches asymptotically.
In the steady state, we will have k t = k t+1 = ¯
k and therefore also u( p t ) =
u( p t+1 ), which allows us to simplify the last line in (11.9) to
0 = −1 + β
λ f
( ¯
k) + 1 − δ
or
1
β
− 1 + δ = λ f
( ¯
k) .
(11.10)
If we assume a Cobb-Douglas production function f (k) = k
, we can solve for the
equilibrium capital base ¯
k, the profit ¯
p and the investment ¯
i rate.
¯
k =
1
λλ
1
β
− 1 + δ
1
−1 , ¯
p = λ ¯
k
− δ ¯
k , and ¯
i = δ ¯
k
(11.11)
These equations allow us to evaluate, or example, the effect of the deprecation δ
on the steady-state profits. This provides us with information on how the failure
rate of the equipment affects profits. We therefore plot ¯
k, ¯
p, and ¯
i as a function of
δ on the left-hand side in Fig. 11.2. Here we use β = 0.8, = 0.7, and λ = 1 in
the simulation. We observe that the sustainable capital ¯
k, shown as the solid black
175
investigate the optimum by calculating the derivative of V t [ p] with respect to k t+1 ,
where we have to express all profits p t through the values of the capital k t via
p t = λ f (k t ) + (1 − δ)k t − k t+1 . Using k t+1 instead of p t is irrelevant, because both
contain the information about how much of the output y t = λ f (k t ) is carried over
into the next time period. We find conditions for the p t or equivalently for the k t+1
by setting the derivative to zero
0 =
∂u (λ f (k t ) + (1 − δ)k t − k t+1 )
∂k t+1
+ β
∂ V t+1 [ p]
∂k t+1
(11.9)
= −u
(λ f (k t ) + (1 − δ)k t − k t+1 )
+βu
(λ f (k t+1 ) + (1 − δ)k t+1 − k t+2 )
λ f
(k k+1 ) + 1 − δ
= −u
( p t ) + βu
( p t+1 )
λ f
(k k+1 ) + 1 − δ
where u
( p) denotes the derivative of the utility function with respect to its argument.
Note that V t+1 [ p] contains the k t shifted by one period, which accounts for the inner
derivative in the square bracket in the second equality. The last equality only reexpresses the arguments of the utility functions in terms of the profits, rather than
the capital in order to make the equation easier to read.
Using the recursive description of the objective functional, we could now progress
to find an equation for the next period and by repeating this procedure we arrive at
an infinite sequence of equations like (11.9). Even taking the initial capital k 0 into
account, it is still impossible to solve this infinite sequence of equations. We can,
however, close the equations by assuming to take out all capital as profit at a far-away
time. This terminates the infinite regression and leads to a finite-dimensional set of
non-linear and coupled equations, which is difficult to solve. Therefore, instead of
analyzing the transient system, we continue with a discussion of the steady-state
solution that the system approaches asymptotically.
In the steady state, we will have k t = k t+1 = ¯
k and therefore also u( p t ) =
u( p t+1 ), which allows us to simplify the last line in (11.9) to
0 = −1 + β
λ f
( ¯
k) + 1 − δ
or
1
β
− 1 + δ = λ f
( ¯
k) .
(11.10)
If we assume a Cobb-Douglas production function f (k) = k
, we can solve for the
equilibrium capital base ¯
k, the profit ¯
p and the investment ¯
i rate.
¯
k =
1
λλ
1
β
− 1 + δ
1
−1 , ¯
p = λ ¯
k
− δ ¯
k , and ¯
i = δ ¯
k
(11.11)
These equations allow us to evaluate, or example, the effect of the deprecation δ
on the steady-state profits. This provides us with information on how the failure
rate of the equipment affects profits. We therefore plot ¯
k, ¯
p, and ¯
i as a function of
δ on the left-hand side in Fig. 11.2. Here we use β = 0.8, = 0.7, and λ = 1 in
the simulation. We observe that the sustainable capital ¯
k, shown as the solid black