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| * The derivative of ''e^cx^'' with respect to ''x'' is, of course, ''c e^cx^''. | * The partial derivative of ''e^cx^'' with respect to ''x'' is, of course, ''c e^cx^''. |
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 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.
Productiveness of capital investment and effective labor is relative: capital to a factor of α and effective labor to a factor of 1 - α.
We can estimate the α factor by measuring wages (the return of value on effective labor) and rent (the return of value on capital investment).
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).
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.