Solow-Swan Model
The Solow-Swan model, also called the exogenous growth model, is an economics model of national production.
Contents
Formulation
Assume that production is a Cobb-Douglas function of capital investment, labor, and technology: Y = f(K,L,A), specifically Y = Kα(AL)(1-α).
Some important notes about that formulation:
Technology augments the performance of labor, not capital. The term effective labor is adopted.
- Constant returns to scale are assumed: production increases proportional to the productive inputs.
This necessitates that the output elasticities of capital and effective labor are rivalrous: α and 1 - α. Really though, this is important for later model interpretation.
The interest of the model lies in growth of production, so re-phrase as a time-dependent function: Y(t) = K(t)α(A(t)L(t))(1-α).
Capital grows as under the Harrod-Domar model: dK(t)/dt = sY(t) - δK(t).
Further assume though that labor and technology are exogenous. This is not a dynamic system for these inputs. Labor is a function of time and a growth rate n: L(t) = L(0)etn; technology is a function of time and a growth rate r: A(t) = A(0)etr.
Some important notes about that formulation:
The partial derivative of ecx with respect to x is, of course, c ecx.
- Growth rates are relative, i.e. expressed as:
In summary, the labor supply grows at a rate of n; the effective labor supply grows at a rate of r + n.
The model can usefully be re-stated in terms of effective labor units:
That inner term of capital per effective labor units (or capital intensity) is typically written as k(t), therefore y(t) = k(t)α.
Some important notes about that formulation:
By splitting out that inner term, we have an easier derivation: dk/dt = sk(t)α - (n + g + δ)k(t).
This is why it is important to assume constant returns to scale.
That derivation suggests a steady state at sk(t)α = (n + g + δ)k(t), solved for k* as:
The interpretation is that capital investment will grow to a 'break-even point', and then cease growth. Therefore the key to long-term growth lies in the other productive factors, labor and technology. Since growth of the labor supply inherently has diminishing returns on production per effective labor unit (i.e., GDP per capita), the key to long-term growth per capita lies solely in technology.