At its heart, the Solow model is about how an economy’s capital stock evolves over time. Capital—which includes everything from factories and machines to computers and infrastructure—is a critical input for producing goods and services. The model’s fundamental law of motion for capital can be expressed in a single, powerful equation.

The Core Equation

The entire dynamic of the basic model is captured by the following relationship:

K(t+1) = sF[K(t), L(t)] + (1-δ)K(t)

Let’s break this down piece by piece. Think of t as today and t+1 as tomorrow.

• K(t+1): This is the capital stock the economy will have tomorrow.
• K(t): This is the capital stock the economy has today.
• F[K(t), L(t)]: This is the aggregate production function. It’s a representation of the economy’s technology, showing how inputs—capital (K) and labor (L)—are combined to produce total output, or GDP (Y). So, Y(t) = F[K(t), L(t)].
• s: This is the economy’s saving rate. The model simplifies reality by assuming s is an exogenous, constant fraction of income that is saved and invested. While this is a simplification, it’s a “convenient starting point.” The total amount of new investment is therefore s * Y(t).
• δ: This is the depreciation rate. Each year, a fraction of the capital stock (δ) wears out or becomes obsolete. The term (1-δ)K(t) represents the amount of capital from today that survives into tomorrow.

In plain English, the equation says: Capital tomorrow equals the new investment we make today plus the capital from today that doesn’t wear out.

The Concept of the Steady State

The model’s central insight comes from analyzing its long-run behavior. Solow showed that an economy will converge toward a “steady state”—a point where key variables are constant. This concept of a ‘steady state’ is the model’s analytical core. It gives us a way to predict a country’s long-run income level based on a few key characteristics, turning a complex dynamic problem into a manageable equilibrium analysis.

To understand this, it’s easier to think in per-worker terms. Let k = K/L be capital per worker and y = Y/L be output per worker. The steady state is the point where the amount of new investment per worker is just enough to offset the amount of capital per worker lost to depreciation. In this equilibrium, investment, sf(k), is exactly equal to the amount of capital that needs to be “replenished,” δk.

If investment is greater than depreciation, the capital-per-worker ratio (k) grows. If investment is less than depreciation, k shrinks. The steady state is the point of balance where k is no longer changing.

The table below summarizes the key variables in the basic model’s steady state.

The “So What?” of the Basic Model

So what is the fundamental lesson from this simplified model? It proposes a direct link between a country’s habits—its saving rate—and its long-run prosperity. A country’s long-run level of income per worker is determined by factors that influence its steady state, namely its saving rate (s) and its level of technology (embedded in the production function f(k)). This implies that countries with higher saving rates will accumulate more capital per worker and, as a result, will be richer in the steady state.

This basic model gives us a powerful insight into the role of savings, but it also makes a stark prediction that clashes with reality: eventually, growth per person stops. To explain the sustained rise in living standards we see across the world, the model needs an engine for perpetual growth. That engine is technology.