The Solow Growth Model
The Solow Growth Model illustrates how saving money, growth in the labor force, and technical progresses affect an economy's capital accumulation and output in the long term. As capital stock grows and the economy output increases, more economic growth occurs.
The Supply of Goods and the Production Function
In this model, the supply of goods is determined by the production function, which states that the level of output (Y) depends on the capital stock (K), the labor force (L), and the efficiency of labor (E): Y = FK , L × E.
The efficiency of labor reflects society's knowledge about production methods. As technology improves, the efficiency of labor rises, and each hour of work produces more output. Since L × E takes into account the number of actual workers and the efficiency of each worker, it can be interpreted as the effective number of workers.
We assume that the production function has constant returns to scale, meaning that if both capital and labor increase by a factor of z, output will increase by the same factor. That is, z⁢Y=F⁡z⁢K,z⁢L, for any z > 0.
This assumption allows us to analyze all quantities relative to the size of the effective labor force by setting z = 1L× E. Now, YL ×E = FKL× E, 1, and since 1 is a constant, it can be ignored.
Using lowercase letters, we can write the production function in per effective worker terms: y = fk, where y = YL × E represents the output per effective worker and k = KL × E represents the capital per effective worker.
The slope of this function is the marginal product of capital (MPK), or the amount of extra output a worker produces when given an additional unit of capital. MPK = Δ yΔ k = ⅆⅆ k fk.
The Demand for Goods and the Consumption Function
In this closed-economy model, the demand for goods depends only on consumption and investment expenditures. Output per effective worker (y) must be allocated between consumption per effective worker (c) and investment per effective worker (i). As a result, y = c + i.
We assume that people save a certain fraction (s) of their income and consume what is remaining. So, the consumption function can be expressed as c=1−s⁢y, where the saving rate (s) is between 0 and 1.
Therefore, rewriting the equation from above, obtains: y = 1−sy + i, which can be rearranged to i=s⁢y. This implies that the rate of saving is also a fraction of the output devoted to investment.
Changes in the Capital Stock and the Steady State
For any given level of capital (k), the production function, fk, determines how much output the economy produces. Meanwhile, the saving rate (s) determines the allocation of output between consumption and investment. Increasing the rate of saving increases the level of investment, and as the capital stock grows, so too does the amount of capital per effective worker.
There are three factors which decrease the capital per effective worker:
1) The depreciation rate (δ), which accounts for the proportion of the capital stock that wears out each year.
2) The labor force growth rate (n), which reduces k by spreading the existing capital stock more thinly among a larger number of workers.
3) The growth of labor efficiency (g), which is a labor-augmenting form of the technological progress. It causes output to increase as though the labor force had grown by g%. As a result, g is known as the rate of labor-augmenting technical progress.
The term δ+n+g⁢k defines the break-even investment. This amount of investment is needed to keep the capital per effective worker constant. The term δ⁢k is needed to replace the depreciated capital. Furthermore, the term n k is needed to provide capital for new workers, and lastly, the term g k is needed to provide capital created by advances in technology for the new "effective workers".
It is known that investment increases the capital stock; while depreciation, labor force growth, and technological progress reduce it. As a result, the impact of these opposing forces on k can be mathematically expressed as: Δ k = i −δ+n+g⁢k = s fk − δ + n + gk.
Steady state represents the equilibrium of the economy in the long term. Equilibrium occurs exactly when the investment equals the break-even investment. As a result, capital stock does not change.
For given values of s, δ, n, and g, there is only one level of k for which Δ k = 0. This is known as the steady state level of capital stock per effective worker, k∗. Thus, steady state occurs mathematically when s fk∗ = δ + n + gk∗.
The Golden Rule Steady State
Of all the steady states an economy can reach, only one can provide the highest level of consumption for citizens. This particular steady state is known as the Golden Rule Steady State. It represents the steady state level of capital per effective worker, which maximizes consumption of the Golden Rule level of capital, kgold∗ .
Rearranging the demand equation from above, we obtain c = y − i. In the steady state, we know that i = s fk∗ = δ + n + gk∗. As a result, the steady state level of consumption can be expressed as c∗ = fk∗ − δ + n + gk∗.
Graphically, c∗ is depicted by the distance between the production function (the output curve) and the break-even investment line. This distance is maximized when the two curves have the same slope of δ + n + g. As mentioned earlier, the slope of the production function is equal to the marginal product of capital, so kgold∗ can be found by solving MPK = δ + n + g . There is only one saving rate at which the Golden Rule Steady State can be attained. To obtain a steady state with maximum consumption, the investment function must cross the break-even investment line precisely when k = kgold∗ .
The following graph shows the production function, investment function, and break-even investment line for an economy in the long term. On the left hand side, use the sliders to adjust the factors affecting capital accumulation and economic growth. Click "Show Current Steady State" or "Show Golden Rule Steady State" to display each state on the graph. Additionally, click "Steady-State Level of Consumption" and "Steady-State Level of Investment" individually or simultaneously to see how they affect the Solow Model of Economic Growth.
Factors Affecting Long Term Economic Growth
Depreciation Rate (d )
Population Growth Rate (n)
Rate of Labor-Augmenting Technological Progress (g)
Saving Rate (s)
MathApps/Finance and Economics
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