Ramsey–Cass–Koopmans model

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The Ramsey–Cass–Koopmans model, or Ramsey growth model, is a neoclassical model of economic growth based primarily on the work of Frank P. Ramsey,^[1] with significant extensions by David Cass and Tjalling Koopmans.^[2][3] The Ramsey–Cass–Koopmans model differs from the Solow–Swan model in that the choice of consumption is explicitly microfounded at a point in time and so endogenizes the savings rate. As a result, unlike in the Solow–Swan model, the saving rate may not be constant along the transition to the long run steady state. Another implication of the model is that the outcome is Pareto optimal or Pareto efficient.^{[note 1]}

Originally Ramsey set out the model as a social planner's problem of maximizing levels of consumption over successive generations.^[4] Only later was a model adopted by Cass and Koopmans as a description of a decentralized dynamic economy with a representative agent. The Ramsey–Cass–Koopmans model aims only at explaining long-run economic growth rather than business cycle fluctuations, and does not include any sources of disturbances like market imperfections, heterogeneity among households, or exogenous shocks. Subsequent researchers therefore extended the model, allowing for government-purchases shocks, variations in employment, and other sources of disturbances, which is known as real business cycle theory.

Mathematical description

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,

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

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:

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

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:

We can rewrite the Ramsey rule aswhere 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

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]}

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