Ramsey Growth in Discrete and Continuous Time
This section solves a continuous-time version of the Ramsey/Cass-Koopmans (RCK ) model using the Hamiltonian method, and shows the relationship between that method and the discrete-time approach.
The problem is to choose a path of consumption per capita c \cons c { c } 0 ∞ \{c\}_{0}^{\infty} { c } 0 ∞
max { c } 0 ∞ ∫ 0 ∞ u ( c ) e − ϑ t \max_{\{c\}_{0}^{\infty}} \int_{0}^{\infty} \uFunc(\cons) e^{-\timeRate t} { c } 0 ∞ max ∫ 0 ∞ u ( c ) e − ϑt subject to
k ˙ = f ( k ) − c − ( ξ + δ ) k k > 0 ∀ t \begin{gathered}\begin{aligned}
\dot{\kap} & = \fFunc(\kap) - \cons - (\popGro+\depr)\kap
\\ \kap & > 0 ~\forall~t
\end{aligned}\end{gathered} k ˙ k = f ( k ) − c − ( ξ + δ ) k > 0 ∀ t where ϑ \timeRate ϑ ξ \popGro ξ δ \depr δ c ( t ) \cons(t) c ( t ) k ( 0 ) = ∙ \kap(0)=\bullet k ( 0 ) = ∙
Dealing with continuous-time problems, there is an implicit (t) argument that is never written out.
value of ∙ \bullet ∙
To emphasize the similarity between the continuous-time and the discrete-time solutions where we have typically
used the roman V V V V ( k ) \mathcal{V}(\kap) V ( k )
The current-value (discounted) Hamiltonian is
H ( k , c , λ ) = u ( c ) + ( f ( k ) − c − ( ξ + δ ) k ) λ \mathcal{H}(\kap,\cons,\lambda) = \uFunc(\cons) + (\fFunc(\kap)-c-(\popGro+\depr)\kap)\lambda H ( k , c , λ ) = u ( c ) + ( f ( k ) − c − ( ξ + δ ) k ) λ where k \kap k c \cons c
λ \lambda λ
λ \lambda λ
multiplier, so its value should be equivalent to the value of relaxing
the corresponding constraint by an infinitesimal amount. But the
constraint in question is the capital-accumulation constraint. Thus
λ \lambda λ
capital, V ( k + Δ k ) − V ( k ) Δ k \frac{\mathcal{V}(\kap+\Delta k)-\mathcal{V}(\kap)}{\Delta k} Δ k V ( k + Δ k ) − V ( k )
you can think of λ = V ′ ( k ) \lambda=\mathcal{V}'(\kap) λ = V ′ ( k )
The first necessary Hamiltonian condition for optimality is
This corresponds to u ′ ( c t ) = V ′ ( x t ) \uFunc^{\prime}(\cons_{t})=\VFunc^{\prime}(x_{t}) u ′ ( c t ) = V ′ ( x t )
∂ H ∂ c = 0 u ′ ( c ) = λ u ′ ( c ) = V ′ ( k ) . \begin{gathered}\begin{aligned}
\frac{\partial \mathcal{H}}{\partial c} & = 0 \\
\uFunc^{\prime}(\cons) & = \lambda %
\\ \uFunc^{\prime}(\cons) & = \mathcal{V}'(\kap) .
\end{aligned}\end{gathered} ∂ c ∂ H u ′ ( c ) u ′ ( c ) = 0 = λ = V ′ ( k ) . Note the similarity between (4)
usually obtain by using the Envelope theorem in the discrete-time
Basically only difference is x t x_{t} x t k t \kap_{t} k t k t \kap_{t} k t
problem,
u ′ ( c t ) = V ′ ( k t ) . \begin{gathered}\begin{aligned}
\uFunc^{\prime}(\cons_{t}) & = \VFunc^{\prime}(\kap_{t})
.
\end{aligned}\end{gathered} u ′ ( c t ) = V ′ ( k t ) . Thus, you can use the intuition you (should have) developed by now
about why the marginal utility of consumption should be equal to the
marginal value of extra resources to understand this Hamiltonian optimality
condition.
The second necessary condition is
This corresponds to V ′ ( x t ) = β R V ′ ( x t + 1 ) \VFunc^{\prime}(x_{t})=\Discount \Rfree \VFunc^{\prime}(x_{t+1}) V ′ ( x t ) = β R V ′ ( x t + 1 )
λ ˙ = ϑ λ − λ ( f ′ ( k ) − ( ξ + δ ) ) ( λ ˙ λ ) = ϑ − ( f ′ ( k ) − ( ξ + δ ) ) \begin{gathered}\begin{aligned}
\dot{\lambda} & = \timeRate\lambda - \lambda(\fFunc^{\prime}(\kap)-(\popGro+\depr)) %
\\ \left(\frac{\dot{\lambda}}{\lambda}\right) & = \timeRate-(\fFunc^{\prime}(\kap)-(\popGro+\depr))
\end{aligned}\end{gathered} λ ˙ ( λ λ ˙ ) = ϑ λ − λ ( f ′ ( k ) − ( ξ + δ )) = ϑ − ( f ′ ( k ) − ( ξ + δ )) which expresses the growth rate of λ \lambda λ
the interest rate r \rfree r ϑ \timeRate ϑ
To interpret this in terms of our discrete-time model, begin with
the condition
V ′ ( k t ) = R β V ′ ( k t + 1 ) . \begin{gathered}\begin{aligned}
\VFunc^{\prime}(\kap_{t}) & = \Rfree\Discount \VFunc^{\prime}(\kap_{t+1}).
\end{aligned}\end{gathered} V ′ ( k t ) = R β V ′ ( k t + 1 ) . The final necessary condition is just that the accumulation equation for
capital is satisfied,
k ˙ = f ( k ) − c − ( ξ + δ ) k . \dot{\kap} = \fFunc(\kap)-c-(\popGro+\depr)\kap. k ˙ = f ( k ) − c − ( ξ + δ ) k . This is the continuous-time equivalent of what we have previously called
the Dynamic Budget Constraint.
Emphasize that λ ˙ / λ \dot{\lambda}/\lambda λ ˙ / λ
Up to now in this course we haven’t thought very much about what the
time period is. Generally, we have expressed things in terms of
yearly rates, so that for example we might choose R = 1.04 \Rfree=1.04 R = 1.04
β = 1 / ( 1 + ϑ ) = 1 / ( 1.04 ) \Discount=1/(1+\timeRate)=1/(1.04) β = 1/ ( 1 + ϑ ) = 1/ ( 1.04 )
4 percent and a discount rate of 4 percent.
One of the attractive features of the time-consistent model we have
been using is that it generates self-similar behavior as the time interval is
changed. Thus if we wanted to solve a quarterly version of the model
we would choose R = 1.01 \Rfree=1.01 R = 1.01 β = 1 / 1.01 \Discount=1/1.01 β = 1/1.01
consumption of almost exactly 1/4 of the amount implied by the annual
model, so that four quarters of such behavior would aggregate to the
prediction of the annual model.
To put this in the most general form, suppose R \Rfree R β \Discount β
correspond to ‘annual rate’ values and we want to divide the year into
m m m
on a per-period basis would be R 1 / m \Rfree^{1/m} R 1/ m β 1 / m \Discount^{1/m} β 1/ m
discrete-time equation could be rewritten
V ′ ( k t ) = R 1 / m β 1 / m V ′ ( k t + 1 ) \begin{gathered}\begin{aligned}
\VFunc^{\prime}(\kap_{t}) & = \Rfree^{1/m}\Discount^{1/m} \VFunc^{\prime}(\kap_{t+1})
\end{aligned}\end{gathered} V ′ ( k t ) = R 1/ m β 1/ m V ′ ( k t + 1 ) where the time interval is now 1 / m 1/m 1/ m m m m
we’re talking weekly, so that period t + 1 t+1 t + 1
t t t e z ≈ 1 + z e^{z} \approx 1+z e z ≈ 1 + z
to note that this is approximately
V ′ ( k t ) ≈ [ e r ] 1 / m [ e − ϑ ] 1 / m V ′ ( k t + 1 ) = e ( 1 / m ) r e − ( 1 / m ) ϑ V ′ ( k t + 1 ) = e ( 1 / m ) ( r − ϑ ) V ′ ( k t + 1 ) = e ( 1 / m ) ( r − ϑ ) ( V ′ ( k t ) + Δ V ′ ( k t + 1 ) ) 1 = e ( 1 / m ) ( r − ϑ ) ( V ′ ( k t ) + Δ V ′ ( k t + 1 ) V ′ ( k t ) ) e ( 1 / m ) ( ϑ − r ) = ( V ′ ( k t ) + Δ V ′ ( k t + 1 ) V ′ ( k t ) ) V ′ ( k t ) ( e ( 1 / m ) ( ϑ − r ) − 1 ⏟ ≈ 1 + ( 1 / m ) ( ϑ − r ) − 1 ) = Δ V ′ ( k t + 1 ) Δ V ′ ( k t + 1 ) V ′ ( k t ) ≈ ( 1 / m ) ( ϑ − r ) m Δ V ′ ( k t + 1 ) V ′ ( k t ) ≈ ( ϑ − r ) . \begin{gathered}\begin{aligned}
\VFunc^{\prime}(\kap_{t}) & \approx [e^{\rfree}]^{1/m}[e^{-\timeRate}]^{1/m} \VFunc^{\prime}(\kap_{t+1})
\\ & = e^{(1/m)\rfree}e^{-(1/m)\timeRate} \VFunc^{\prime}(\kap_{t+1})
\\ & = e^{(1/m)(\rfree-\timeRate)} \VFunc^{\prime}(\kap_{t+1})
\\ & = e^{(1/m)(\rfree-\timeRate)} (\VFunc^{\prime}(\kap_{t})+\Delta \VFunc^{\prime}(\kap_{t+1}))
\\ 1 & = e^{(1/m)(\rfree-\timeRate)} \left(\frac{\VFunc^{\prime}(\kap_{t})+\Delta \VFunc^{\prime}(\kap_{t+1})}{\VFunc^{\prime}(\kap_{t})}\right)
\\ e^{(1/m)(\timeRate-\rfree)} & = \left(\frac{\VFunc^{\prime}(\kap_{t})+\Delta \VFunc^{\prime}(\kap_{t+1})}{\VFunc^{\prime}(\kap_{t})}\right)
\\ \VFunc^{\prime}(\kap_{t})(\underbrace{e^{(1/m)(\timeRate - \rfree)}-1}_{\approx 1+(1/m)(\timeRate-\rfree)-1}) & = \Delta \VFunc^{\prime}(\kap_{t+1})
\\ \frac{\Delta \VFunc^{\prime}(\kap_{t+1})}{\VFunc^{\prime}(\kap_{t})} & \approx (1/m)(\timeRate - \rfree)
\\ \frac{m\Delta \VFunc^{\prime}(\kap_{t+1})}{\VFunc^{\prime}(\kap_{t})} & \approx (\timeRate - \rfree)
%\\ \VFunc^{\prime}(\kap_{t+n})-\VFunc^{\prime}(\kap_{t}) & \approx \VFunc^{\prime}(\kap_{t})(\timeRate-\rfree)
.
\end{aligned}\end{gathered} V ′ ( k t ) 1 e ( 1/ m ) ( ϑ − r ) V ′ ( k t ) ( ≈ 1 + ( 1/ m ) ( ϑ − r ) − 1 e ( 1/ m ) ( ϑ − r ) − 1 ) V ′ ( k t ) Δ V ′ ( k t + 1 ) V ′ ( k t ) m Δ V ′ ( k t + 1 ) ≈ [ e r ] 1/ m [ e − ϑ ] 1/ m V ′ ( k t + 1 ) = e ( 1/ m ) r e − ( 1/ m ) ϑ V ′ ( k t + 1 ) = e ( 1/ m ) ( r − ϑ ) V ′ ( k t + 1 ) = e ( 1/ m ) ( r − ϑ ) ( V ′ ( k t ) + Δ V ′ ( k t + 1 )) = e ( 1/ m ) ( r − ϑ ) ( V ′ ( k t ) V ′ ( k t ) + Δ V ′ ( k t + 1 ) ) = ( V ′ ( k t ) V ′ ( k t ) + Δ V ′ ( k t + 1 ) ) = Δ V ′ ( k t + 1 ) ≈ ( 1/ m ) ( ϑ − r ) ≈ ( ϑ − r ) . We defined the interest rate and time preference
rate on an annual basis, but the time interval between t t t t + 1 t+1 t + 1
is only ( 1 / m ) (1/m) ( 1/ m ) m Δ V ′ ( k t + 1 ) m \Delta \VFunc^{\prime}(\kap_{t+1}) m Δ V ′ ( k t + 1 )
the speed of change in V ′ ( k t ) \VFunc^{\prime}(\kap_{t}) V ′ ( k t )
Recall that λ ˙ / λ \dot{\lambda}/\lambda λ ˙ / λ
Now, note that since the effective interest rate in this model is
f ′ ( k ) − ( ξ + δ ) \fFunc^{\prime}(\kap)-(\popGro+\depr) f ′ ( k ) − ( ξ + δ ) (10)
(6) λ = V ′ ( k ) \lambda = \mathcal{V}'(\kap) λ = V ′ ( k ) m \Delta \VFunc^{\prime}(\kap_{t+1}) = (d/dt)\mathcal{V}'(\kap) = \dot{\lambda}. Hence, the
second optimality condition in the Hamiltonian optimization method is
basically equivalent to the condition V ′ ( k t ) = R β V ′ ( k t + 1 ) \VFunc^{\prime}(\kap_{t})=\Rfree\Discount \VFunc^{\prime}(\kap_{t+1}) V ′ ( k t ) = R β V ′ ( k t + 1 )
from the discrete-time optimization method!
The final required condition (the transversality constraint)
This corresponds to the intertemporal budget constraint.
is
lim t → ∞ λ k e − ϑ t = 0 \begin{gathered}\begin{aligned}
\lim_{t \rightarrow \infty} \lambda \kap e^{-\timeRate t}& = 0
\end{aligned}\end{gathered} t → ∞ lim λk e − ϑt = 0 The translation of this into the discrete-time model is
lim t → ∞ β t u ′ ( c t ) k t = 0. \lim_{t \rightarrow \infty} \Discount^{t} \uFunc^{\prime}(\cons_{t}) \kap_{t} = 0. t → ∞ lim β t u ′ ( c t ) k t = 0. Consider the simple model with a constant gross interest rate R \Rfree R CRRA
utility. In that model, recall that {math}\cons_{t+1}= (\Rfree\Discount)^{1/\CRRA} \cons_{t}. Thus considered from time zero (12)
lim t → ∞ β t ( c 0 ( ( R β ) 1 / ρ ) t ) − ρ k t = 0 = c 0 − ρ β t [ ( R β ) t / ρ ] − ρ k t = c 0 − ρ β t β − t R − t k t = c 0 − ρ R − t k t → lim t → ∞ R − t k t = 0. \begin{gathered}\begin{aligned}
\lim_{t \rightarrow \infty} \Discount^{t} (\cons_{0} ((\Rfree\Discount)^{1/\CRRA})^{t})^{-\CRRA} \kap_{t} & = 0
\\ & = \cons_{0}^{-\CRRA} \Discount^{t}[(\Rfree\Discount)^{t/\CRRA}]^{-\CRRA} \kap_{t}
\\ & = \cons_{0}^{-\CRRA} \Discount^{t} \Discount^{-t} \Rfree^{-t} \kap_{t}
\\ & = \cons_{0}^{-\CRRA} \Rfree^{-t} \kap_{t}
\\ \rightarrow \lim_{t \rightarrow \infty} \Rfree^{-t} \kap_{t} & = 0
.
\end{aligned}\end{gathered} t → ∞ lim β t ( c 0 (( R β ) 1/ ρ ) t ) − ρ k t → t → ∞ lim R − t k t = 0 = c 0 − ρ β t [( R β ) t / ρ ] − ρ k t = c 0 − ρ β t β − t R − t k t = c 0 − ρ R − t k t = 0. What this says is that you cannot behave in such a way that you expect
k t \kap_{t} k t
forever.[1]
infinite-horizon version of the intertemporal budget constraint.
Among the infinite number of time paths of c \cons c k \kap k
will satisfy the first order conditions above, only one will also
satisfy this transversality constraint - because all the others imply
a violation of the intertemporal budget constraint.
Now differentiate (4)
c ˙ u ′ ′ ( c ) = λ ˙ \begin{gathered}\begin{aligned}
\dot{\cons} \uFunc^{\prime\prime}(\cons) & = \dot{\lambda}
\end{aligned}\end{gathered} c ˙ u ′′ ( c ) = λ ˙ and substitute this into equation (6)
c ˙ u ′ ′ ( c ) u ′ ( c ) = ( ϑ − ( f ′ ( k ) − ( ξ + δ ) ) ) c ˙ = − u ′ ( c ) u ′ ′ ( c ) ( f ′ ( k ) − ( ξ + δ ) − ϑ ) \begin{gathered}\begin{aligned}
\frac{\dot{\cons} \uFunc^{\prime\prime}(\cons)}{\uFunc^{\prime}(\cons)} & = (\timeRate-(\fFunc^{\prime}(\kap)-(\popGro+\depr))) \\
\dot{\cons} & = -\frac{\uFunc^{\prime}(\cons)}{\uFunc^{\prime\prime}(\cons)} (\fFunc^{\prime}(\kap)-(\popGro+\depr)-\timeRate)
\end{aligned}\end{gathered} u ′ ( c ) c ˙ u ′′ ( c ) c ˙ = ( ϑ − ( f ′ ( k ) − ( ξ + δ ))) = − u ′′ ( c ) u ′ ( c ) ( f ′ ( k ) − ( ξ + δ ) − ϑ ) using the fact
derived earlier that for a CRRA utility function
u ( c ) = c 1 − ρ / ( 1 − ρ ) , − u ′ ′ ( c ) c / u ′ ( c ) = ρ \uFunc(\cons)=c^{1-\CRRA}/(1-\CRRA), -\uFunc^{\prime\prime}(\cons) c/\uFunc^{\prime}(\cons) = \CRRA u ( c ) = c 1 − ρ / ( 1 − ρ ) , − u ′′ ( c ) c / u ′ ( c ) = ρ
= ( c / ρ ) ( f ′ ( k ) − ( ξ + δ ) − ϑ ) c ˙ / c = ρ − 1 ( f ′ ( k ) − ( ξ + δ ) − ϑ ) \begin{gathered}\begin{aligned}
& = (\cons/\CRRA) (\fFunc^{\prime}(\kap)-(\popGro+\depr)-\timeRate)
\\ \dot{\cons}/\cons & = \CRRA^{-1}(\fFunc^{\prime}(\kap)-(\popGro+\depr)-\timeRate)
\end{aligned}\end{gathered} c ˙ / c = ( c / ρ ) ( f ′ ( k ) − ( ξ + δ ) − ϑ ) = ρ − 1 ( f ′ ( k ) − ( ξ + δ ) − ϑ )