The Ramsey/Cass-Koopmans (RCK) Model

Ramsey Ramsey (1928), followed much later by Cass Cass (1965) and Koopmans Koopmans (1965), formulated the canonical model of optimal growth for an economy with exogenous ‘labor-augmenting’ technological progress.

1The Budget Constraint¶

The economy has a perfectly competitive production sector that uses a Cobb-Douglas aggregate production function

\begin{gathered}\begin{aligned} % \Inc_{t} & = \ProdFunc_{t}(\Kap,\Labor) = \Kap^{\kapShare}(\PtyLab_{t} \Labor)^{1-\kapShare} \Inc & = \ProdFunc(\Kap,\Labor) = \Kap^{\kapShare}(\PtyLab \Labor)^{1-\kapShare} \end{aligned}\end{gathered}

(1)

to produce output using capital and labor.^[1]

\begin{gathered}\begin{aligned} \dot{\Labor}/\Labor & = \popGro \end{aligned}\end{gathered}

(2)

and $\PtyLab$ is an index of labor productivity that grows at rate

\begin{gathered}\begin{aligned} \dot{\PtyLab }/\PtyLab & = \ptyGro. \end{aligned}\end{gathered}

(3)

Thus, technological progress allows each worker to

produce perpetually more as time goes by with the same amount of

physical capital.^[2]

units’ of labor in the economy.

Aggregate capital accumulates according to

\begin{gathered}\begin{aligned} \dot{\Kap} & = \Inc - \Cons - \depr K . \end{aligned}\end{gathered}

(4)

Lower case variables are the upper case version divided by efficiency units, i.e.

\begin{gathered}\begin{aligned} \inc & = \Inc/(\PtyLab \Labor) \\ & = \Kap^{\kapShare}(\PtyLab \Labor)^{1-\kapShare}/(\PtyLab \Labor) \\ & = (\Kap/\PtyLab \Labor)^{\kapShare} \\ & = \kap^{\kapShare} . \end{aligned}\end{gathered}

(5)

Note that

\begin{gathered}\begin{aligned} \dot{\kap} \equiv \left(\frac{d\kap}{dt}\right) & = \left(\frac{\dot{\Kap}\PtyLab \Labor - K(\dot{\PtyLab }L + \PtyLab \dot{\Labor})}{(\PtyLab \Labor)^{2}}\right) \\ & = \dot{\Kap}/\PtyLab \Labor - k(\dot{\PtyLab }/{\PtyLab } + \dot{\Labor}/\Labor) \\ & = \dot{\Kap}/\PtyLab \Labor - (\ptyGro +\popGro)k \end{aligned}\end{gathered}

(6)

which means that (4) can be divided by $\PtyLab \Labor$ and becomes

\begin{gathered}\begin{aligned} \dot{\Kap}/\PtyLab \Labor & = \inc - \cons - \depr \kap \\ \dot{\kap}+(\ptyGro +\popGro)k & = \prodFunc(\kap) - \cons - \depr \kap \\ \dot{\kap} & = \prodFunc(\kap) - \cons - (\ptyGro +\popGro+\depr) \kap . \end{aligned}\end{gathered}

(7)

A steady-state will be a point where $\dot{\kap} = 0$ .

Equation (7) yields a first candidate for an optimal

steady-state of the growth model: It seems reasonable to argue that the best

possible steady-state is the one that maximizes $\cons$ . This is the

“golden rule” optimality condition of

Phelps Phelps (1961), an article well worth reading; this is

one of the chief contributions for which Phelps won the Nobel prize.

Now suppose that there is a social planner whose goal is to maximize the discounted

sum of CRRA utility from per-capita consumption:

\max \int_{0}^{\infty} \left(\frac{(\Cons/\Labor)^{1-\CRRA}}{1-\CRRA}\right) e^{-\timeRate t} .

(8)

But $\Cons/\Labor = \PtyLab (\Cons/\PtyLab \Labor)= \PtyLab c$ . Recall that for a variable growing at rate $\ptyGro$ ,

\begin{gathered}\begin{aligned} \PtyLab _{t} & = \PtyLab _{0}e^{\ptyGro t} \end{aligned}\end{gathered}

(9)

so if the economy started off in period 0 with productivity $\PtyLab_{0}$ , by date $t$ we can rewrite

\begin{gathered}\begin{aligned} \Cons/\Labor & = c\PtyLab \\ & = c\PtyLab _{0}e^{\ptyGro t}. \end{aligned}\end{gathered}

(10)

Using (10) and the other results above, we can rewrite the social planner’s objective function as

\begin{gathered}\begin{aligned} \int_{0}^{\infty} \left(\frac{(\PtyLab c)^{1-\CRRA}}{1-\CRRA}\right) e^{-\timeRate t} & = \PtyLab _{0}^{1-\CRRA}\int_{0}^{\infty} \left(\frac{\cons^{1-\CRRA}}{1-\CRRA}\right) e^{-\timeRate t}e^{(1-\CRRA)\ptyGro t} \\ & = \PtyLab _{0}^{1-\CRRA}\int_{0}^{\infty} \util(c) e^{\left((1-\CRRA)\ptyGro -\timeRate \right) t} . \end{aligned}\end{gathered}

(11)

Thus, defining $\nu = \timeRate - (1-\CRRA) \ptyGro$ and normalizing the initial

level of productivity to $\PtyLab _{0}=1$ , the complete optimization problem

can be formulated as

\max \int_{0}^{\infty} \util(c) e^{-\nu t}

(12)

subject to

\dot{\kap} = \kap^{\kapShare}-\cons-(\popGro+\ptyGro+\depr)k,

(13)

which has a discounted Hamiltonian representation

\Ham(\kap,c,\lambda) = \util(c) + (\kap^{\kapShare} - \cons - (\popGro+\ptyGro+\depr)\kap)\lambda.

(14)

The first discounted Hamiltonian optimization condition requires {math}\partial \Ham/\partial \cons = 0:

\begin{gathered}\begin{aligned} c^{-\CRRA} & = \lambda \\ -\CRRA c^{-\CRRA-1}\dot{\cons} & = \dot{\lambda} . \end{aligned}\end{gathered}

(15)

The second discounted Hamiltonian optimization condition requires:

\begin{gathered}\begin{aligned} \dot{\lambda} & = \nu\lambda - \partial \Ham/\partial \kap \\ & = \nu\lambda - \lambda (\prodFunc^{\prime}(\kap) - (\popGro+\ptyGro+\depr)) \\ \dot{\lambda}/{\lambda} & = (\nu+(\popGro+\ptyGro+\depr)-\prodFunc^{\prime}(\kap)) \\ \left(\frac{-\CRRA c^{-\CRRA-1}\dot{\cons}}{\cons^{-\CRRA}}\right) & = (\nu+(\popGro+\ptyGro+\depr)-\prodFunc^{\prime}(\kap)) \\ \CRRA \dot{\cons}/\cons & = (\prodFunc^{\prime}(\kap)-\nu-(\popGro+\ptyGro+\depr)) \\ \dot{\cons}/\cons & = \CRRA^{-1}(\underbrace{\prodFunc^{\prime}(\kap)-(\popGro+\ptyGro+\depr)}_{\mbox{$\equiv \acute{r}$}}-\nu) \end{aligned}\end{gathered}

(16)

where the definition of $\acute{r}$ is motivated by thinking of

$\prodFunc^{\prime}(\kap)-(\popGro+\ptyGro+\depr)$ as the interest rate net of depreciation and

dilution.

This is called the “modified golden rule” (or sometimes the

“Keynes-Ramsey rule” because it was originally derived by

Ramsey with an explanation attributed to Keynes).

Thus, we end up with an Euler equation for consumption growth that is

just like the Euler equation in the perfect foresight partial equilibrium consumption model, except that now the relevant interest rate can vary over

time as $\prodFunc^{\prime}(\kap)$ varies.

Substituting in the modified time preference rate gives

\begin{gathered}\begin{aligned} \dot{\cons}/\cons & = \CRRA^{-1}(\prodFunc^{\prime}(\kap)-(\popGro+\depr+\ptyGro)-\timeRate +(1- \CRRA) \ptyGro) \\ & = \CRRA^{-1}(\prodFunc^{\prime}(\kap)-(\popGro+\depr)-\timeRate - \CRRA \ptyGro), \end{aligned}\end{gathered}

(17)

and finally note that defining per capita consumption $\chi = \Cons/\Labor$

so that $\cons = \chi \PtyLab ^{-1}$ ,

\begin{gathered}\begin{aligned} \dot{\cons} & = \dot{\biggl(\chi \PtyLab ^{-1}\biggr)} = \dot{\chi}\PtyLab ^{-1}-(\chi \PtyLab ^{-1})\dot{\PtyLab }/\PtyLab \\ \dot{\cons}/\cons & = \dot{\biggl(\chi \PtyLab ^{-1}\biggr)}/(\chi \PtyLab^{-1}) = \dot{\chi}/\chi - \ptyGro \end{aligned}\end{gathered}

(18)

and since (17) can be written

\begin{gathered}\begin{aligned} \dot{\cons}/\cons & = \CRRA^{-1}(\prodFunc^{\prime}(\kap)-(\popGro+\depr)-\timeRate ) - \ptyGro, \end{aligned}\end{gathered}

(19)

we have

\begin{gathered}\begin{aligned} \dot{\chi}/\chi & = \CRRA^{-1}(\prodFunc^{\prime}(\kap)-(\popGro+\depr)-\timeRate ) \end{aligned}\end{gathered}

(20)

so the formula for per capita consumption growth (as a function of

$\kap$ ) is identical to the model with no growth (equation

(17) with $\ptyGro=0$ ). Any important differences between

the no-growth model and the model with growth therefore must come

through the channel of differences in $\kap$ .

3The Steady State¶

The assumption of labor augmenting technological progress was made

because it implies that in steady-state, per-capita consumption, income,

and capital all grow at rate $\ptyGro$ .^[3]

$\dot{\cons}/\cons = 0$ implies that at the steady-state value of $\check{\kap}$ ,

\begin{gathered}\begin{aligned} \prodFunc^{\prime}(\check{\kap}) & = \timeRate +\popGro+\depr+\CRRA \ptyGro \\ \kapShare \check{\kap}^{\kapShare-1} & = \timeRate +\popGro+\depr+\CRRA \ptyGro \\ \check{\kap} & = \left(\frac{\timeRate +\popGro+\depr+\CRRA \ptyGro}{\kapShare}\right)^{\frac{1}{\kapShare-1}} \\ \check{\kap} & = \left(\frac{\kapShare}{\timeRate +\popGro+\depr+\CRRA \ptyGro}\right)^{\frac{1}{1-\kapShare}} \end{aligned}\end{gathered}

(21)

Thus, the steady-state $\kap$ will be higher if capital

is more productive ( $\kapShare$ is higher), and will be lower if

consumers are more impatient, population growth is faster,

depreciation is greater, or technological progress occurs more rapidly.

4A Phase Diagram¶

While the RCK model has an analytical solution for its steady-state,

it does not have an analytical solution for the transition to the

steady-state. The usual method for analyzing

models of this kind is a phase diagram in $\cons$ and $\kap$ .

The first step in constructing the phase diagram is to take the differential

equations that describe the system and find the points where they are

zero. Thus, from (7) we have that $\dot{\kap}=0$ implies

\begin{gathered}\begin{aligned} \cons & = \prodFunc(\kap) - (\ptyGro +\popGro+\depr) \kap \end{aligned}\end{gathered}

(22)

and we have already solved for the (constant) $\check{\kap}$ that characterizes

the $\dot{\cons}/\cons=0$ locus. These can be combined to generate the borders between the phases in the phase diagram, as illustrated in Figure 1.

\dot{\cons}/\cons =0 and \dot{\kap}=0 Loci — Figure 1: $\dot{\cons}/\cons =0$ and $\dot{\kap}=0$ Loci

5Transition¶

Actually, as stated so far, the solution to the problem is very simple: The

consumer should spend an infinite amount in every period. This solution is

not ruled out by anything we have yet assumed (except possibly the fact that

once $\kap$ becomes negative the production function is undefined).

Obviously, this is not the solution we are looking for. What is missing is

that we have not imposed anything corresponding to the intertemporal

budget constraint. In this context, the IBC takes the form of a “transversality

condition,”

\begin{gathered}\begin{aligned} \lim_{t \rightarrow \infty} \lambda_{t} e^{(\ptyGro +\popGro) t - \int_{0}^{t} r_{\tau} d\tau }k_{t} & = 0. \end{aligned}\end{gathered}

(23)

The intuitive purpose of this unintuitive equation is basically to prevent

the capital stock from becoming negative or infinity as time goes by.

Obviously a capital stock that was negative for the entire future could not

satisfy the equation. And a capital stock that is too large will have

an arbitrarily small interest rate, which will result in the LHS of the

TVC being a positive number, again failing to satisfy the TVC.

Figure 2 shows three paths for $\cons$ and $\kap$

that satisfy (17) and (7). The topmost path,

however, is clearly on a trajectory toward zero then negative

$\kap$ , while the bottommost path is heading toward an infinite

$\kap$ . Only the middle path, labelled the “saddle path,” satisfies

both (17) and (7) as well as the TVC

(23).

6Interactive Notebooks¶

An explicit numerical solution to the Ramsey problem, with a description of a solution method and its

mathematical/computational underpinnings, is available here.

7Appendix: Numerical Solution¶

The RCK model does not have an analytical solution, which means that

numerical methods must be used to find out the model’s quantitative

implications for transition paths.

The method of solution of these kinds of models is not important for

the purposes of first year graduate macroeconomics; this appendix

has been written as a reference for more advanced students who might

be beginning their research on growth models.

The most straightforward method of numerical solution for perfect foresight

models of this kind is called the

‘time elimination’ method. It starts from the fact that

\begin{gathered}\begin{aligned} \left(\frac{d\cons/dt}{d\kap/dt}\right) &= d\cons/d\kap. \end{aligned}\end{gathered}

(24)

Note from (17) that we can write

\begin{gathered}\begin{aligned} \dot{\cons} & = \CRRA^{-1}(\prodFunc^{\prime}(\kap)-\timeRate -(\popGro+\depr)-\CRRA \ptyGro)\cons , \end{aligned}\end{gathered}

(25)

so we can obtain

\begin{gathered}\begin{aligned} d\cons/d\kap & = \left(\frac{(\kapShare \kap^{\kapShare-1}-\timeRate -(\popGro+\depr)-\CRRA \ptyGro)\cons} {\CRRA(\kap^{\kapShare} - \cons - (\ptyGro +\popGro+\depr) \kap)}\right) \end{aligned}\end{gathered}

(26)

which is a differential equation with no analytical solution. Many numerical math packages can solve differential equations numerically, yielding a numerical version of the $\cFunc(\kap)$ function.

There is one problem, however, which is that at the steady-state values of $\cons$ and $\kap$ both numerator and denominator of this equation are zero. The alternative is to solve the differential equation twice: Once for a domain extending from $\kap=0$ to $\check{\kap}-\epsilon$ , yielding $\cFunc_{-}(\kap)$ , and once for a domain from $\check{\kap}+\epsilon$ to some large value of $\kap$ , yielding $\cFunc_{+}(\kap)$ . The true consumption policy function can then be approximated by interpolating between the upper endpoint of $\cFunc_{-}(\kap)$ and the lower endpoint of $\cFunc_{+}(\kap)$ .

For further details of the numerical solution of this model, see this Jupyter notebook, or clone the repo and, in the cloned directory, run the corresponding python program: ipython RamseyCassKoopmans.py.

Footnotes¶

All Roman variables are functions of time, but putting time subscripts on everything would clutter the notation, so we do it only where necessary for clarity. Labor hours (the same as population) increases exogenously at a constant rate^[4]
↩
The ancient Greek philosophers captured eternal truths; therefore, Greek letters represent constants whose value never changes.
↩
This is the definition of ‘labor-augmenting’ (Harrod-neutral) productivity growth; with a Cobb-Douglas production function, it turns out to be essentially the same as ‘capital-augmenting’ productivity growth, also known as Hicks-neutral, as well as output-neutral (‘Solow-neutral’) progress. The quantity $\PtyLab \Labor$ is known as the number of ‘efficiency
↩
See Grossman et al. (2016) for a discussion of the realism of this requirement.
↩

References¶

Ramsey, F. P. (1928). A Mathematical Theory of Saving. Economic Journal, 38(152), 543–559. 10.2307/2224098
Cass, D. (1965). Optimum Growth in an Aggregative Model of Capital Accumulation. Review of Economic Studies, 32, 233–240. 10.2307/2295827
Koopmans, T. C. (1965). On the concept of optimal economic growth. In (Study Week on the) Econometric Approach to Development Planning (pp. 225–287). North-Holland Publishing Co., Amsterdam.
Phelps, E. S. (1961). The Golden Rule of Accumulation. American Economic Review, 638–642.
Grossman, G. M., Helpman, E., Oberfield, E., & Sampson, T. (2016). Balanced Growth Despite Uzawa (Working Paper No. 21861; Working Paper Series). National Bureau of Economic Research. 10.3386/w21861