Solow's Growth
Total Production Function¶
The model starts with a standard Cobb-Douglas production function, which assumes constant returns to scale to capital (\(K\)) and labor (\(L\)).
where \(\alpha\) is the capital share of income, with \(0 < \alpha < 1\).
Intensive Form of Production Function¶
The model is analyzed in "per worker" terms, which simplifies the dynamics. This is done by dividing the production function by the labor force (\(L\)). We assume the entire population is in the labor force and that the labor force grows at a constant rate, \(n\).
- \(y = Y/L\) (output per worker)
- \(k = K/L\) (capital per worker)
The production function in intensive form becomes:
The model assumes diminishing returns to capital per worker, meaning that as \(k\) increases, output per worker (\(y\)) also increases, but at a decreasing rate. - FOC: \(\dfrac{\partial{y}}{\partial{k}}=\alpha k^{\alpha-1}\) (increasing function) - SOC: \(\dfrac{\partial^2{y}}{\partial{k^2}} = (\alpha)(\alpha-1)k^{(\alpha-2)}\) (decreasing function)
Capital Accumulation¶
\(K = sy - \delta k\)
Intensive form of Capital Accumulation¶
\(\dot{k} = sk^\alpha - (n+\delta)k\)
Comparative Statics¶
- Savings Rate
- Population Growth
Technology¶
Harrod Neutral¶
\(Y = F(K,AL)\) Labor augmenting
Solow Neutral¶
\(Y = F(AK,L)\) Capital augmenting
Hicks Neutral¶
\(Y = A F(K,L)\) **
The Basic Solow Model without Technology¶
The Solow-Swan growth model, or simply the Solow model, is a neoclassical economic model that explains long-run economic growth by looking at the accumulation of capital, labor, and exogenous technological progress. It is a refinement of the [[Harrod Domar Growth model|Harrod-Domar model]] because it introduces the concept of diminishing returns to capital.
1. The Production Function
2. Intensive Form
The Capital Accumulation Equation¶
The central equation of the Solow model describes the change in the capital stock per worker (\(\dot{k}\)). This change is determined by two opposing forces:
- Investment: The increase in capital per worker, which is a portion of output (\(sY\)) and thus a portion of output per worker (\(sy\)).
- Depreciation and Population Growth: The decrease in capital per worker due to depreciation (\(\delta k\)) and the need to equip new workers entering the labor force with capital (\(nk\)).
The change in capital per worker is the difference between these two forces:
Substituting the intensive production function (\(y=k^\alpha\)) gives:
- \(sk^\alpha\) represents investment per worker (the addition to capital stock per worker).
- \((n + \delta)k\) represents the break-even investment (the amount needed to keep the capital-labor ratio constant).
Steady State Equilibrium¶
A steady state is reached when the capital per worker (\(k\)) is constant. At this point, the change in capital per worker (\(\dot{k}\)) is zero.
This steady state is stable. If \(k\) is below the steady state, investment is greater than break-even investment, so \(k\) will rise. If \(k\) is above the steady state, investment is less than break-even investment, so \(k\) will fall.
Key implication: Without technological progress, the Solow model predicts that economies will converge to a steady state where growth in per capita output ceases. This is because of the diminishing returns to capital.
Adding Technological Progress¶
To explain sustained long-run growth, the Solow model incorporates exogenous technological progress, often assumed to be labor-augmenting (Harrod neutral).
1. Augmented Production Function: Technology (\(A\)) enhances labor productivity, so the effective labor force is \(AL\). \(\(Y = F(K, AL) = K^\alpha (AL)^{1-\alpha}\)\)
2. Intensive Form with Technology: To find the steady state, we work in terms of capital and output per effective worker (\(\tilde{k} = K/AL\) and \(\tilde{y} = Y/AL\)). The capital accumulation equation becomes: \(\(\dot{\tilde{k}} = s\tilde{y} - (n + g + \delta)\tilde{k}\)\) where \(g\) is the rate of technological progress (\(\dot{A}/A\)).
3. The New Steady State: A new steady state is reached when \(\dot{\tilde{k}} = 0\). Along this new balanced growth path: * Capital per effective worker (\(\tilde{k}\)) is constant. * Output per effective worker (\(\tilde{y}\)) is constant. * Output per worker (\(y\)) grows at the rate of technological progress (\(g\)). * Total output (\(Y\)) grows at the sum of the rates of population growth and technological progress (\(n+g\)).
This shows that sustained growth in per capita income is only possible due to technological progress, as capital accumulation alone is limited by diminishing returns.
Convergence Theory¶
The Solow model predicts conditional convergence. This means that countries with similar underlying economic fundamentals—such as savings rates, population growth, and access to technology—should converge to the same steady state. A poorer country with less capital per worker will initially have a higher marginal product of capital and thus a higher growth rate than a richer country. This allows it to "catch up." However, countries with different steady states (due to differences in savings, technology, or population growth) will not converge. This explains why rich and poor countries don't automatically converge,