Solow Growth
Overview
Solow Production Function
{% Y = F(K(t), L(t)) %}
Assumptions
- Constant Returns to Scale : {% F(\lambda K, \lambda L) = \lambda F(K, L) %}
- Production factors exhibit diminishing returns
Next, the Solow model defines k = K/L and y = Y/L. That is k, is the capital per capita, and y is
production/income per capita.
Given constant returns to scale, we have
{% Y/L = F(K/L, L/L) = F(K/L, 1) %}
that is, production per capita is a function of capital divided by labor.
{% Y/L = y = f(k) %}
Lastly, the Solow model assumes a particular form for the rate of change of capital.
In particular, it is assumed that total capital depreciates at a given rate,{% \delta %}, and
new capital is added to the capital base as savings, given by the savings rate s.
{% \dot{K} = sY - \delta K %}
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This plot shows the assumption of a savings rate, which is proportional to the current national production / income.
The amount of production that is consumed, per worker, is then the income per worker (or production) minus
the amount of savings per worker.
{% c = (1-s) \times y %}
{% \dot{k} = \frac{\dot{K}}{L} - \frac{K}{L^2} \dot{L} %}
{% = sf(k) - (\delta + n)k %}
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Depreciation is proportional the total kapital (the x axis), whereas investment is proportional to production (the y axis).
There is a point where investment equals depreciation. At that point, growth is zero and the economy is in equilibrium.
Model Predictions
The most prominent prediction of the Solow growth model is that countries should converge to the same long term growth rate.
Example
