Solow Model
NOTES ON
NEOCLASSICAL (SOLOW) GROWTH MODEL
Neoclassical Growth model shows why growth rate of per capita income cannot be maintained through continuous saving and investment. The reason is that as capital per labor rises, marginal productivity of capital runs into diminishing returns. Let the production function be : Y = output, K = capital stock and L = labor force (population). This function is assumed to be constant returns to scale type ie if you multiply each input by a factor λ output is also multiplied by the same factor. Setting λ = 1/L: Letting
output per labor and
= capital per labor:
This means that output per worker depends only on capital per worker. The key is, in the neoclassical constant returns to scale production function, as capital per worker rise output per worker rises at a diminishing rate. ie marginal productivity of capital falls.
Next we consider how net capital stock grows over time. The assumption is it grows by the extent of net investment, the difference between gross investment and depreciation of the existing capital stock. Letting depreciation rate be : Gross investment equals total savings S:
Savings rate s is assumed to be a constant: Combining (4), (5), (6): Dividing by K: Next we use the growth formula that growth rate of a ratio is the growth rate of the numerator minus the growth rate of the denominator. Recalling that
:
Multiplying through by
:
Substituting for growth rate:
from (8) and letting n =
= constant labor force (population)
Simplifying (note that
):
+ n) k (12)
This is called the fundamental equation of change of the neoclassical growth model. The intuition is straight forward. Suppose the savings rate is zero and therefore there is no investment. The equation then says capital per worker will fall at the rate of + n), exactly what we
would expect since K is
NEOCLASSICAL (SOLOW) GROWTH MODEL
Neoclassical Growth model shows why growth rate of per capita income cannot be maintained through continuous saving and investment. The reason is that as capital per labor rises, marginal productivity of capital runs into diminishing returns. Let the production function be : Y = output, K = capital stock and L = labor force (population). This function is assumed to be constant returns to scale type ie if you multiply each input by a factor λ output is also multiplied by the same factor. Setting λ = 1/L: Letting
output per labor and
= capital per labor:
This means that output per worker depends only on capital per worker. The key is, in the neoclassical constant returns to scale production function, as capital per worker rise output per worker rises at a diminishing rate. ie marginal productivity of capital falls.
Next we consider how net capital stock grows over time. The assumption is it grows by the extent of net investment, the difference between gross investment and depreciation of the existing capital stock. Letting depreciation rate be : Gross investment equals total savings S:
Savings rate s is assumed to be a constant: Combining (4), (5), (6): Dividing by K: Next we use the growth formula that growth rate of a ratio is the growth rate of the numerator minus the growth rate of the denominator. Recalling that
:
Multiplying through by
:
Substituting for growth rate:
from (8) and letting n =
= constant labor force (population)
Simplifying (note that
):
+ n) k (12)
This is called the fundamental equation of change of the neoclassical growth model. The intuition is straight forward. Suppose the savings rate is zero and therefore there is no investment. The equation then says capital per worker will fall at the rate of + n), exactly what we
would expect since K is