2.1 The Basic Solow Model in Discrete Time

The Solow growth model is the foundational workhorse model of modern macroeconomics and the starting point for all serious growth theory. Its development was an intellectual breakthrough, moving beyond the restrictive assumptions of the earlier Harrod-Domar model. The core purpose of the Solow model is to provide a dynamic framework for understanding how three key forces—capital accumulation, population growth, and technological progress—interact to determine the path of an economy’s output over time. Its elegant simplicity allows us to isolate the roles of these factors and generate sharp, testable predictions about the growth process.

The economic environment of the basic Solow model is built on a set of core assumptions.

1. Production: There is a single final good produced using an aggregate production function, Y(t) = F[K(t), L(t)], where K is capital and L is labor. This function is assumed to exhibit constant returns to scale, meaning that doubling both inputs will exactly double output. It is also assumed to satisfy the Inada conditions (Assumption 2), which state that the marginal product of an input approaches infinity as the input goes to zero and approaches zero as the input becomes infinitely abundant. These conditions ensure that both capital and labor are essential for production and that the economy will converge to an interior steady state.
2. Households: Households are assumed to save a constant, exogenous fraction, s, of their income and consume the rest. This is a significant simplification, as it means saving behavior does not respond to changes in interest rates or economic policy.
3. Firms and Markets: Firms operate in perfectly competitive factor markets. They hire capital and labor, paying rental rates equal to their respective marginal products: the rental rate of capital is R(t) = FK and the wage rate is w(t) = FL. Due to the constant returns to scale assumption, Euler’s Theorem (Proposition 2.1) implies that total factor payments will exactly exhaust total output, meaning firms earn zero economic profits.

From these assumptions, we can derive the fundamental law of motion for the capital stock. Total savings in the economy is sY(t) = sF[K(t), L(t)], which is equal to total investment. A fraction δ of the existing capital stock depreciates each period. The capital stock in the next period, K(t+1), is therefore the sum of the undepreciated capital from this period, (1-δ)K(t), and new investment. This gives us the model’s central nonlinear difference equation: K(t+1) = sF[K(t), L(t)] + (1-δ)K(t)

To analyze living standards, it is essential to convert the model into per capita terms. We define the capital-labor ratio as k(t) = K(t)/L(t) and, thanks to constant returns to scale, can write the per capita production function as y(t) = f(k(t)), where y(t) = Y(t)/L(t). In this “intensive form,” and assuming a constant labor force for now (L(t) = L), the law of motion becomes: k(t+1) = sf(k(t)) + (1-δ)k(t)

A central concept in the Solow model is the steady state, a long-run equilibrium where key per capita variables are constant. A steady state is a point k* where the capital-labor ratio no longer changes, i.e., k(t+1) = k(t) = k*. Substituting this into the law of motion, we find that the unique positive steady state is determined by the condition: sf(k*) = δk* The economic intuition is clear and powerful: in the steady state, the amount of new investment per worker, sf(k*), is exactly equal to the amount of depreciation per worker, δk*. The capital stock is constant because new investment is just enough to replace the capital that wears out.

While the model’s saving rate s is exogenous, we can still ask what saving rate would be “optimal” from the perspective of maximizing long-run consumption. This leads to the concept of the “golden rule” level of capital accumulation. The golden rule capital stock, k*gold, is the steady-state level that maximizes steady-state consumption per capita. This level is characterized by a simple condition: f'(k*gold) = δ The intuition is that to maximize consumption, society should accumulate capital up to the point where the marginal product of the last unit of capital, f'(k*gold), is exactly equal to its depreciation rate, δ. Beyond this point, the extra output from more capital is less than the amount needed to maintain it, so consumption would fall. (Once we introduce population growth at rate n, this condition generalizes to f'(k*gold) = n+δ, which will facilitate later comparisons.)

The analysis of the steady state describes the long-run destination of the economy, but to understand phenomena like convergence, we must analyze the economy’s behavior outside of the steady state, a process known as transitional dynamics.

2.2 Transitional Dynamics and Technological Progress

Understanding transitional dynamics—how an economy behaves as it moves towards its steady state—is a primary function of the Solow model and the source of its key empirical predictions. The central question is whether an economy that starts away from its steady-state capital stock will automatically converge to it, and if so, at what speed.

Using a phase diagram (as conceptually described in Figure 2.2), we can visualize the dynamics of the capital-labor ratio, k. The diagram plots k(t+1) against k(t). The steady state k* is the point where the curve sf(k(t)) + (1-δ)k(t) intersects the 45-degree line. Due to the properties of the production function (specifically, diminishing returns), for any initial capital stock below the steady state (k(0) < k*), investment will exceed depreciation, causing k to monotonically increase toward k*. Conversely, for any k(0) > k*, depreciation will exceed investment, and k will monotonically decrease toward k*. This demonstrates that the steady state is globally stable: regardless of its starting point, the economy will always converge to the unique steady state k*. This convergence property is the theoretical foundation for the hypothesis of conditional convergence.

The basic model, however, cannot explain the sustained, long-run growth in living standards observed in many countries. To do this, we must introduce technological progress. Technological progress can be modeled in several ways:

• Hicks-neutral: Augments all factors symmetrically.
• Solow-neutral: Augments capital (capital-augmenting).
• Harrod-neutral: Augments labor (labor-augmenting).

A crucial result, known as Uzawa’s Theorem (Proposition 2.11), states that for an economy to exhibit a balanced growth path (BGP)—where key ratios like the capital-output ratio and factor income shares are constant—technological progress must be purely labor-augmenting (Harrod-neutral).

This is a powerful theoretical restriction with profound implications. For a balanced growth path to exist, key economic ratios like factor income shares must remain constant over time. If technological progress were, for instance, purely capital-augmenting, the “effective” capital stock would grow faster than the labor force. To prevent the share of capital in national income from rising toward unity, the price of capital would have to fall at just the right rate. This requires that the elasticity of substitution between capital and labor be exactly equal to 1 (the Cobb-Douglas case). Harrod-neutral (labor-augmenting) progress is the only form that allows for a balanced growth path with constant factor shares for any elasticity of substitution, making it a necessary condition for robust, general models of sustained growth.

Let’s now incorporate both population growth (at rate n) and Harrod-neutral technological progress (at rate g) into the model, formulated in continuous time for tractability. The level of technology, A(t), now grows at rate g. We analyze the economy in terms of “effective” units of labor, A(t)L(t). We define capital per effective worker as k(t) = K(t) / (A(t)L(t)). The fundamental differential equation governing the evolution of k becomes: k̇(t) = sf(k(t)) – (n + g + δ)k(t)

The new steady-state condition for this transformed variable k* is: sf(k*) = (n + g + δ)k* The intuition here is that investment must now cover three needs:

1. Replace depreciated capital (δk*).
2. Provide capital for new workers entering the labor force (nk*).
3. Provide capital for the now “more effective” workers resulting from technological progress (gk*). This process, where the amount of capital per worker rises to keep pace with technology, is known as capital deepening.

On the economy’s balanced growth path (BGP), the capital per effective worker, k, is constant. However, this masks underlying growth in per capita terms. On the BGP:

• Output per worker (Y/L) and capital per worker (K/L) both grow at the rate of technological progress, g.
• Total output (Y) and total capital (K) grow at the rate n + g.

This result is the key success of the Solow model: it explains sustained long-run growth in living standards as a direct result of technological progress. Its primary limitation, however, is that this technological progress is treated as an unexplained, exogenous process—it arrives like “manna from heaven.” Understanding the sources of this technological progress is the central task of modern endogenous growth theory.