Turnpike theory refers to a set of economic theories about the optimal path of accumulation (often capital accumulation

Capital accumulation

The accumulation of capital refers to the gathering or amassing of objects of value; the increase in wealth through concentration; or the creation of wealth. Capital is money or a financial asset invested for the purpose of making more money...

) in a system based depending on the initial and final levels. In the context of a macroeconomic exogenous growth model

Exogenous growth model

The neoclassical growth model, also known as the Solow–Swan growth model or exogenous growth model, is a class of economic models of long-run economic growth set within the framework of neoclassical economics...

, for example, it says that if an infinite optimal path is calculated, and an economic planner wishes to move an economy from one level of capital to another, as long as the planner has sufficient time, the most efficient path is to quickly move the level of capital stock to a level close to the infinite optimal path, and to allow capital

Capital (economics)

In economics, capital, capital goods, or real capital refers to already-produced durable goods used in production of goods or services. The capital goods are not significantly consumed, though they may depreciate in the production process...

 to develop along that path until it is nearly the end of the desired term and the planner must move the capital stock to the desired final level. The name of the theory refers to the idea that a turnpike

Toll road

A toll road is a privately or publicly built road for which a driver pays a toll for use. Structures for which tolls are charged include toll bridges and toll tunnels. Non-toll roads are financed using other sources of revenue, most typically fuel tax or general tax funds...

 is the fastest route between two points which are far apart, even if it is not the most direct route.

Origins

Although the idea can be traced back to John von Neumann

John von Neumann

John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...

 in 1945, Lionel W. McKenzie

Lionel W. McKenzie

Lionel Wilfred McKenzie was the Wilson Professor Emeritus of Economics at the University of Rochester. He was born in Montezuma, Georgia. He completed undergraduate studies at Duke University in 1939 and subsequently moved to Oxford that year as a Rhodes Scholar...

 traces the term to Robert Dorfman

Robert Dorfman

Robert Dorfman was emeritus professor of political economy at Harvard University. Dorfman made great contributions to the fields of economics, group testing and in the process of coding theory....

, Paul Samuelson

Paul Samuelson

Paul Anthony Samuelson was an American economist, and the first American to win the Nobel Memorial Prize in Economic Sciences. The Swedish Royal Academies stated, when awarding the prize, that he "has done more than any other contemporary economist to raise the level of scientific analysis in...

, and Robert Solow

Robert Solow

Robert Merton Solow is an American economist particularly known for his work on the theory of economic growth that culminated in the exogenous growth model named after him...

's "Linear Programming and Economics Analysis" in 1958, referring to an American English word for a Highway:

Thus in this unexpected way, we have found a real normative significance for steady growth - not steady growth in general, but maximal von Neumann growth. It is, in a sense, the single most effective way for the system to grow, so that if we are planning long-run growth, no matter where we start, and where we desire to end up, it will pay in the intermediate stages to get into a growth phase of this kind. It is exactly like a turnpike paralleled by a network of minor roads. There is a fastest route between any two points; and if the origin and destination are close together and far from the turnpike, the best route may not touch the turnpike. But if the origin and destination are far enough apart, it will always pay to get on to the turnpike and cover distance at the best rate of travel, even if this means adding a little mileage at either end. The best intermediate capital configuration is one which will grow most rapidly, even if it is not the desired one, it is temporarily optimal.

Variations

McKenzie in 1976 published a review of the idea up to that point. He saw three general variations of turnpike theories.

1. In a system with a set initial and terminal capital stock where the objective of the economics planner is to maximize the sum of utility over the finite accumulation period, then so long as the accumulation period is long enough, most of the optimal path will be within some small neighborhood of an infinite path that is optimal. This often implies that
2. If a finite optimal path starts on (or near) the infinite path, it hugs that path for most of the time, regardless of the desired capital stock at the end.
3. The theorem also generalized for infinite paths, where the basic result is that optimal paths converge to each other, regardless of initial capital stocks.

Applications

This theorem has many applications in optimal control and in a general equilibrium context. In the general equilibrium context, the variation involving infinite capital accumulation paths can be applied. In a system with many infinitely lived agents with the same (small) discount rates on the future, regardless of initial endowments, the equilibrium allocations of all agents converge.