As is the case with all economic models, the Basic Solow Growth Model consists of:
Variables: economic quantities of interest that we can calculate and measure, denoted by letters (or maybe two letters) like L for the number of workers in the labor force or Y for the amount of useful goods and services produced in a year—real GDP.
Parameters: determine which out of a broad family of potential models describes the behavior of the particular economic scenario at hand; the ability to work algebraically with parameters allows one to perform an enormous number of potential "what-if?" calculations very quickly and in a very small space
Behavioral relationships: relationships that (1) describe how humans, making economic decisions given their opportunities and opportunity costs, decide what to do, and (2) that thus determine the values of the economic variables, represented by equations that have an economic variable on the left hand side and, on the right, a set of factors that determine the value of the variable and a rule of thumb for what that value is currently.
Equilibrium conditions: conditions that tell us when the economy is in a position of balance, when some subset of the variables are "stable"—that is, are either constant are changing in simple and predictable ways, usually represented by solutions of some system of the equations that are behavioral relationships.
Accounting identities: statements about the relationships between variables that are automatically and necessarily true because of the way the variables are defined, represented by equations.
In any economic model there will be a key: a particular variable from which you can gain great insight if you manipulate the algebra to solve for how that variable is changing as a function of model parameters and of the economy's current state.
In the case of the Solow growth model, the key variable is the capital intensity of the economy, which we will write using a lower-case Greek letter kappa: $ \kappa $. We measure $ \kappa $ as the ratio of the capital stock of the economy $ K $, to the level of output and income $ Y $.
In every economic model economists proceed with their analysis by looking for an equilibrium: a point of balance, a condition of rest, a state of the system toward which the model will converge over time. Economists look for equilibrium for a simple reason: either an economy is at its (or one of its) equilibrium position(s), or it is moving—and probably (or hopefully?) moving rapidly—to an equilibrium position. The Solow growth model is no exception.
Once economists have found the equilibrium position toward which the economy tends to move, they then understand how the model will behave. And, if they have built the right model, it will tell you in broad strokes how the economy will behave.
In the Basic Solow Growth Model, the equilibrium economists look for is an equilibrium in which the economy’s capital-output ratio $ \kappa $ is contant, and thus the capital stock per worker, the level of income and output per worker, and the efficiency of labor are all three growing at exactly the same proportional rate—a rate that remains constant over time. We will call this rate g, for growth.
These files at nbviewer
weblog support: https://github.com/braddelong/public-files/blob/master/long-form-drafts/solow-model.ipynb