Graphical Representation of The Model
The model starts with a neoclassical production function Y/L = F(K/L), rearranged to y = f(k), which is the red curve on the graph. From the production function; output per worker is a function of capital per worker. The production function assumes diminishing returns to capital in this model, as denoted by the slope of the production function.
n = population growth rate
δ = depreciation (note, this is labeled d on the graph on the right)
k = capital per worker
y = output/income per worker
L = labor force
s = saving rate
Capital per worker change is determined by three variables:
- Investment (saving) per worker
- Population growth, increasing population decreases the level of capital per worker.
- Depreciation – capital stock declines as it depreciates.
When sy > (n + δ)k, in other words, when the savings rate is greater than the population growth rate plus the depreciation rate, when the green line is above the black line on the graph, then capital (k) per worker is increasing, this is known as capital deepening. Where capital is increasing at a rate only enough to keep pace with population increase and depreciation it is known as capital widening.
The curves intersect at point A, the "steady state". At the steady state, output per worker is constant. However total output is growing at the rate of n, the rate of population growth.
The optimal savings rate is called the golden rule savings rate and is derived below. In a typical Cobb–Douglas production function the golden rule savings rate is alpha.
Left of point A, point k1 for example, the saving per worker is greater than the amount needed to maintain a steady level of capital, so capital per worker increases. There is capital deepening from y1 to y0, and thus output per worker increases.
Right of point A where sy < (n + δ)k, point k2 for example, capital per worker is falling, as investment is not enough to combat population growth and depreciation. Therefore output per worker falls from y2 to y0.
Read more about this topic: Neoclassical Growth Model
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