Model setup

In the usual setup, time is continuous starting, for simplicity, at ${\displaystyle t=0}$ and continuing forever. By assumption, the only productive factors are capital ${\displaystyle K}$ and labour ${\displaystyle L}$, both required to be nonnegative. The labour force, which makes up the entire population, is assumed to grow at a constant rate ${\displaystyle n}$, i.e. ${\displaystyle {\dot {L}}={\tfrac {\mathrm {d} L}{\mathrm {d} t}}=nL}$, implying that ${\displaystyle L=L_{0}e^{nt}}$ with initial level ${\displaystyle L_{0}>0}$ at ${\displaystyle t=0}$. Finally, let ${\displaystyle Y}$ denote aggregate production, and ${\displaystyle C}$ denote aggregate consumption.

The variables that the Ramsey–Cass–Koopmans model ultimately aims to describe are ${\displaystyle c={\frac {C}{L}}}$, the per capita (or more accurately, per labour) consumption, as well as ${\displaystyle k={\frac {K}{L}}}$, the so-called capital intensity. It does so by first connecting capital accumulation, written ${\displaystyle {\dot {K}}={\tfrac {\mathrm {d} K}{\mathrm {d} t}}}$ in Newton's notation, with consumption ${\displaystyle C}$, describing a consumption-investment trade-off. More specifically, since the existing capital stock decays by depreciation rate ${\displaystyle \delta }$ (assumed to be constant), it requires investment of current-period production output ${\displaystyle Y}$. Thus,

$${\displaystyle {\dot {K}}=Y-\delta K-cL}$$

The relationship between the productive factors and aggregate output is described by the aggregate production function, ${\displaystyle Y=F(K,L)}$. A common choice is the Cobb–Douglas production function ${\displaystyle F(K,L)=AK^{1-\alpha }L^{\alpha }}$, but generally any production function satisfying the Inada conditions is permissible. Importantly, though, ${\displaystyle F}$ is required to be homogeneous of degree 1, which economically implies constant returns to scale. With this assumption, we can re-express aggregate output in per capita terms

$${\displaystyle F(K,L)=L\cdot F\left({\frac {K}{L}},1\right)=L\cdot f(k)}$$

For example, if we use the Cobb–Douglas production function with ${\displaystyle A=1,\alpha =0.5}$, then ${\displaystyle f(k)=k^{0.5}}$.

To obtain the first key equation of the Ramsey–Cass–Koopmans model, the dynamic equation for the capital stock needs to be expressed in per capita terms. Noting the quotient rule for ${\displaystyle {\tfrac {\mathrm {d} }{\mathrm {d} t}}\left({\tfrac {K}{L}}\right)}$, we have

a non-linear differential equation akin to the Solow–Swan model.

Maximizing welfare

If we ignore the problem of how consumption is distributed, then the rate of utility ${\displaystyle U}$ is a function of aggregate consumption. That is, ${\displaystyle U=U(C,t)}$. To avoid the problem of infinity, we exponentially discount future utility at a discount rate ${\displaystyle \rho \in (0,\infty )}$. A high ${\displaystyle \rho }$ reflects high impatience.

The social planner's problem is maximizing the social welfare function ${\displaystyle U_{0}=\int _{0}^{\infty }e^{-\rho t}U(C,t)\,\mathrm {d} t}$.

Assume that the economy is populated by identical immortal individuals with unchanging utility functions ${\displaystyle u(c)}$ (a representative agent), such that the total utility is:

$${\displaystyle U(C,t)=Lu(c)=L_{0}e^{nt}u(c)}$$

The utility function is assumed to be strictly increasing (i.e., there is no bliss point) and concave in ${\displaystyle c}$, with ${\displaystyle \lim _{c\to 0}u_{c}=\infty }$,^{[note 2]} where ${\displaystyle u_{c}}$ is marginal utility of consumption ${\displaystyle {\tfrac {\partial u}{\partial c}}}$.

Thus we have the social planner's problem:

${\displaystyle \max _{c}\int _{0}^{\infty }e^{-(\rho -n)t}u(c)\,\mathrm {d} t}$

${\displaystyle {\text{subject to}}\quad c=f(k)-(n+\delta )k-{\dot {k}}}$

where an initial non-zero capital stock ${\displaystyle k(0)=k_{0}>0}$ is given.

To ensure that the integral is well-defined, we impose ${\displaystyle \rho >n}$.

Solution

The solution, usually found by using a Hamiltonian function,^{[note 3][note 4]} is a differential equation that describes the optimal evolution of consumption,

the Keynes–Ramsey rule.^[5]

The term ${\displaystyle f_{k}(k)-\delta -\rho }$, where ${\displaystyle f_{k}=\partial _{k}f}$ is the marginal product of capital, reflects the marginal return on net investment, accounting for capital depreciation and time discounting.

Here ${\displaystyle \sigma (c)}$ is the elasticity of intertemporal substitution, defined by

$${\displaystyle \sigma (c)=-{\frac {u_{c}(c)}{c\cdot u_{cc}(c)}}=-{\frac {d\ln c}{d\ln(u'(c))}}}$$

It is formally equivalent to the inverse of relative risk aversion. The quantity reflects the curvature of the utility function and indicates how much the representative agent wishes to smooth consumption over time. If the agent has high relative risk aversion, then it has low EIS, and thus would be more willing to smooth consumption over time. It is often assumed that ${\displaystyle u}$ is strictly monotonically increasing and concave, thus ${\displaystyle \sigma >0}$. In particular, if utility is logarithmic, then it is constant:

$${\displaystyle u(c)=u_{0}\ln c\implies \sigma (c)=1}$$

We can rewrite the Ramsey rule as

$${\displaystyle \underbrace {{\frac {d}{dt}}\ln c} _{\text{consumption delay rate}}=\underbrace {\sigma (c)} _{{\text{EIS at current consumption level}}\quad }\underbrace {[f_{k}(k)-\delta -\rho ]} _{\text{marginal return on net investment}}}$$

where we interpret ${\displaystyle {\frac {d}{dt}}\ln c}$ as the "consumption delay rate", because if it is high, then it means the agent is consuming a lot less now compared to later, which is essentially what delayed consumption is about.

Graphical analysis in phase space

Phase diagram of the Ramsey model, for the case of ${\displaystyle f(k)=k^{0.5}}$, and ${\displaystyle n,\delta ,\rho ,\sigma =1,1,1.1,1}$.

Phase space graph (or phase diagram) of the Ramsey model. The blue line represents the dynamic adjustment (or saddle) path of the economy in which all the constraints present in the model are satisfied. It is a stable path of the dynamic system. The red lines represent dynamic paths which are ruled out by the transversality condition.

The two coupled differential equations for ${\displaystyle k}$ and ${\displaystyle c}$ form the Ramsey–Cass–Koopmans dynamical system.

A steady state ${\displaystyle (k^{\ast },c^{\ast })}$ for the system is found by setting ${\displaystyle {\dot {k}}}$ and ${\displaystyle {\dot {c}}}$ equal to zero. There are three solutions:

${\displaystyle f_{k}\left(k^{\ast }\right)=\delta +\rho \quad {\text{and}}\quad c^{\ast }=f\left(k^{\ast }\right)-(n+\delta )k^{\ast }}$

${\displaystyle (0,0)}$

${\displaystyle f(k^{*})=(n+\delta )k^{*}{\text{ with }}k^{*}>0,c^{*}=0}$

The first is the only solution in the interior of the upper quadrant. It is a saddle point (as shown below). The second is a repelling point. The third is a degenerate stable equilibrium.

By default, the first solution is meant, although the other two solutions are important to keep track of.

Any optimal trajectory must follow the dynamical system. However, since the variable ${\displaystyle c}$ is a control variable, at each capital intensity ${\displaystyle k}$, to find its corresponding optimal trajectory, we still need to find its starting consumption rate ${\displaystyle c(0)}$. As it turns out, the optimal trajectory is the unique one that converges to the interior equilibrium point. Any other trajectory either converges to the all-saving equilibrium with ${\displaystyle k^{*}>0,c^{*}=0}$, or diverges to ${\displaystyle k\to 0,c\to \infty }$, which means that the economy expends all its capital in finite time. Both achieve a lower overall utility than the trajectory towards the interior equilibrium point.

A qualitative statement about the stability of the solution ${\displaystyle (k^{\ast },c^{\ast })}$ requires a linearization by a first-order Taylor polynomial

${\displaystyle {\begin{bmatrix}{\dot {k}}\\{\dot {c}}\end{bmatrix}}\approx \mathbf {J} (k^{\ast },c^{\ast }){\begin{bmatrix}(k-k^{\ast })\\(c-c^{\ast })\end{bmatrix}}}$

where ${\displaystyle \mathbf {J} (k^{\ast },c^{\ast })}$ is the Jacobian matrix evaluated at steady state,^{[note 5]} given by

${\displaystyle \mathbf {J} \left(k^{\ast },c^{\ast }\right)={\begin{bmatrix}\rho -n&-1\\{\frac {1}{\sigma }}f_{kk}(k)\cdot c^{\ast }&0\end{bmatrix}}}$

which has determinant ${\displaystyle \left|\mathbf {J} \left(k^{\ast },c^{\ast }\right)\right|={\frac {1}{\sigma }}f_{kk}(k)\cdot c^{\ast }<0}$ since ${\displaystyle c^{*}>0}$ , ${\displaystyle \sigma }$ is positive by assumption, and ${\displaystyle f_{kk}<0}$ since ${\displaystyle f}$ is concave (Inada condition). Since the determinant equals the product of the eigenvalues, the eigenvalues must be real and opposite in sign.^[6]

Hence by the stable manifold theorem, the equilibrium is a saddle point and there exists a unique stable arm, or “saddle path”, that converges on the equilibrium, indicated by the blue curve in the phase diagram.

The system is called “saddle path stable” since all unstable trajectories are ruled out by the “no Ponzi scheme” condition:^[7]

${\displaystyle \lim _{t\to \infty }k\cdot e^{-\int _{0}^{t}\left(f_{k}-n-\delta \right)\mathrm {d} s}\geq 0}$

implying that the present value of the capital stock cannot be negative.^{[note 6]}