The Ramsey/Cass-Koopmans (RCK) Model

Ramsey (1928), followed much later by Cass (1965) and 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

to produce output using capital and labor.^[1] Labor hours (the same as population) increases exogenously at a constant rate^[2]

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

Thus, technological progress allows each worker to produce perpetually more as time goes by with the same amount of physical capital.^[3] The quantity $\PtyLab \Labor$ is known as the number of ‘efficiency units’ of labor in the economy.

Aggregate capital accumulates according to

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

Note that

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

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 (1961), an article well worth reading; this is one of the chief contributions for which Phelps won the Nobel prize.

2The Social Planner’s Problem¶

Now suppose that there is a social planner whose goal is to maximize the discounted sum of CRRA utility from per-capita consumption:

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

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

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

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

subject to

which has a discounted Hamiltonian representation

The first discounted Hamiltonian optimization condition requires $\partial \Ham/\partial \cons = 0$:

The second discounted Hamiltonian optimization condition requires:

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

and finally note that defining per capita consumption $\chi = \Cons/\Labor$ so that $\cons = \chi \PtyLab ^{-1}$,

and since (17) can be written

we have

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 $\ptyGro$.^[4]

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

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

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.

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,”

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

Note from (17) that we can write

so we can obtain

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.