The Legacy of Frank Ramsey

February 22, 2022

Share this article:

In history, there have been numerous influential scholars in the field of economics and econometrics. Think for instance of John Maynard Keynes, who is known for his influence in macroeconomics through the Keynesian School. Or think of Adam Smith, who introduced the ‘Invisible Hand’, a metaphor for an invisible force that drives the behaviour of individuals in a free market. However, in this article, I want to introduce you to Frank Ramsey, a British economist who had great influence on macroeconomic modelling.

The life of Frank Ramsey

From a young age, it was clear that Ramsey was going to be an active scholar. Attending Cambridge and obtaining his bachelor in mathematics at the age of 19, he specialized in mathematical logic, philosophy and economics. Examples of this are that he became great friends with the late Wittgenstein and also introduced the Ramsey Theorem in mathematical logic.

Nevertheless, most interesting about Ramsey are his very influential papers in economics. First, “A Mathematical Theory of Saving” (1928) is an article that tries to describe a simple question, namely what fraction of its income should a country save? Ramsey employs the so called calculus of variations (which should ring a bell for second year EOR students!) to determine to optimal evolution of savings and consumption over time to maximize future wellbeing. Second, “A Contribution to the Theory of Taxation” (1927), is a seminal article that goes into public finance. Namely, the paper deals with problems regarding a monopolist who faces two problems, as it wants to both cover its costs and maximize the consumer surplus. This was a problem introduced to Ramsey by another famous economist, Arthur Pigou. As a solution, the concept of Ramsey pricing is introduced, which states that a monopolist has to set the price in such a way that the monopolist can cover both of the problems mentioned above. Finally, “Truth and Probability” (1926) is an article on the notion of probability. Where his mentor Keynes in “In A Treatise on Probability” (1921) argued that epistemic (knowledge based) probabilities are not subjective but objective, Ramsey thought of probabilities as not being related to all knowledge, but restricted to each individual. As suggested by Ramsey, beliefs on the individual level, determined by individual knowledge, determine probabilities.

Sadly, Ramsey died at the very young age of 26. However, he still appears from time to time in modern economics classes. For instance, the Keynes-Ramsey rule is a condition that follows from optimization when dealing with an optimal control problem. These days, Ramsey is mostly know for his contribution to the Ramsey-Cass-Koopmans (RCK) model of economic growth, which we will discuss in the next section.

The Ramsey-Cass-Koopmans Model

Nowadays, the RCK model is discussed in every advanced macroeconomics course. Building upon the work by Ramsey, economists David Cass and Tjalling Koopmans incorporated a more microeconomics based (adding rules to the model that govern consumption behaviour of households) foundation to a model of economic growth. The original model by Ramsey described a social planner who had to set out a consumption path over time to maximize the future wellbeing, whereas the RCK model features a dynamic economy with households. Here, I will discuss the model and the optimal policy.

First, we assume that production takes place via a function F(K,L), which depends on capital, K, and labour, L. In per capita (say per individual in the economy) terms, assuming that the function is homogenous of degree one (which means that doubling both capital and labour by one unit also doubles final output by one unit), the function becomes F(K,L)=LF(\frac{K}{L},1)=Lf(k). Here, k=\frac{K}{L} denotes capital per capita. The dynamics of the capital stock are then given by:

(1)   \begin{equation*}\frac{\text{d} k(t)}{\text{d} t} = f(k(t)) - (n+\delta)k(t)-c(t).\end{equation*}


This equation can be interpreted as follows. First, the left-hand side of the equation denotes the change in capital per capita. It equals how much is produced in the economy, f(k), minus how much capital per capita changes due to population growth n and depreciation of capital \delta, minus consumption per capita c. The social planner has the objective to maximize social welfare over time. Thus, the objective function will be:

(2)   \begin{equation*}U=\int_0^{\infty}e^{-\rho t}U(C(t)) \:\text{d}t.\end{equation*}


The equation above can be seen as a discounted stream of utilities. The term e^{-\rho t} serves as an exponential discount factor and 0< \rho < 1 indicates time preferences of consumption. When \rho is small, you value consumption in the future as much as you do today, whereas when \rho gets closer to 1, you value consumption today more over consumption in the future. Assuming that the economy consists of individuals that are identical, we can state it in terms of an agent that lives forever and is representative of the economy. The utility function becomes U(C(t))=Lu(c(t))=L_0 e^{nt} u(c(t)), where we assume the economy starts off with a population of L_0 (in this problem normalized to one), which grows at rate (n).
The maximization problem can then be stated as follows:

(3)   \begin{equation*} max_c U = \int_0^{\infty}e^{-(\rho-n)t}u(c(t)) d t,   \end{equation*}

subject to c(t)=f(k(t))-(n+\delta)k(t)-k'(t).
This problem, which features an objective function and a dynamic constraint, can be solved using the so-called Hamiltonian function. The Hamiltonian for this problem is given by:

(4)   \begin{equation*} H=e^{-\rho t}u(c) +\mu[f(k)-(n+\delta)k-c]. \end{equation*}


As the reader can verify at home, first-order conditions of maximizing the Hamiltonian imply the following equation, which is known as the Keynes-Ramsey rule:

(5)   \begin{equation*} \frac{dc}{dt}=\frac{1}{\sigma}[f'(k)-\delta-\rho]c, \end{equation*}


where \sigma denotes the intertemporal elasticity of substitution, which measures how much an agent is willing to obtain a smooth consumption path over time. The equilibrium of the two equations, which can be found by setting \frac{dc}{dt}=\frac{dk}{dt}=0, is given by:

(6)   \begin{align*}\frac{1}{\sigma}[f'(k)-\delta-\rho]c&=0 \implies f'(k^*)=\delta+\rho \\ c-f(k)+(n+\delta)k&=0 \implies c^*=f(k^*)-(n+\delta)k^*\end{align*}


The stability of the solution (k^*,c^*) can be found by examining the eigenvalues of the Jacobian matrix evaluated at this point. The Jacobian is defined as follows:
\textbf{J}(k^*,c^*)=\begin{bmatrix}\frac{\partial \dot k}{\partial k} & \frac{\partial \dot k}{\partial c} \\\frac{\partial \dot c}{\partial c} & \frac{\partial \dot c}{\partial k}\end{bmatrix}=\begin{bmatrix}f'(k^*)-(n+\delta) & -1 \\ \frac{1}{\sigma}f''(k^*)c^* & \frac{1}{\sigma}[f'(k^*)-\delta-\rho] \end{bmatrix} = \begin{bmatrix} \rho - n & -1 \\ \frac{1}{\sigma}f''(k^*)c^* & 0\end{bmatrix}
The determinant of the Jacobian evaluated at the equilibrium is given by:

(7)   \begin{align*}|\textbf{J}(k^*,c^*)|=(\rho -n) \times 0 - -1 \times \frac{1}{\sigma}f''(k^*)c^*=\frac{1}{\sigma}f''(k^*)c^*\underbrace{<}_{f''(k^*)<0}0.\end{align*}


As the determinant is equal to the product of the eigenvalues, one eigenvalue is positive and one is negative. Thus, the system is saddle path stable.
I hope this article gave you an insight into the life of Frank Ramsey and into the application of the theoretical concepts developed in your studies!

Sources:

Ramsey, F.P., 1928. A mathematical theory of saving. The economic journal, 38(152), pp.543-559.

Ramsey, F.P., 1927. A Contribution to the Theory of Taxation. The economic journal, 37(145), pp.47-61.

Sydsæter, K., Hammond, P., Seierstad, A. and Strom, A., 2008. Further mathematics for economic analysis. Pearson education.

Read more