Solow Model Develompent Economics

The Solow model

Definition

The standard neoclassical model of economic growth was developed in 1956 by Robert Solow. It has three basic sources: labor (L), capital (K), and knowledge (A). The Solow model retains the assumption of diminishing returns but explores another property of the neoclassical production function: the property of constant returns to scale. If capital expands at the same rate as population, total output will grow at the same rate, and per capita income will remain constant, rather than declining towards a low-level subsistence trap.

In general, production functions may also exhibit decreasing returns to scale (output grows less than proportionally compared to inputs) and increasing returns to scale (output grows more than proportionally compared to inputs). The key assumption of the neoclassical growth model is constant returns to scale (CRS), meaning if one increases the use of all inputs by a positive proportional factor, output will rise in the same proportion. The law of diminishing returns states that increasing the use of one input while holding the other constant will lead to output growing less than proportionally.

Assumptions

Consider a closed economy with no government, where a large number of small firms produce a single homogeneous good, Y, using two inputs: labor (N) and capital (K), which can be produced and accumulated. Population and the labor force are the same. Inputs are hired from households, who are also the owners of the firms and the consumers in this economy. Households spend a fraction (1-s) of their income on consumption and save the remaining to buy new capital. The capital stock depreciates at a constant rate.

In this model, the ability to accumulate capital (via savings) prevents output per capita from declining when population increases. Output is produced by a neoclassical production function Y = F(K, L) that has usual properties: diminishing marginal product and constant returns to scale. The economy is closed, meaning savings are equal to investments (S ≡ I). Population growth is given exogenously (outside of the model).

The production function has diminishing returns to per capita capital. As capital increases, output falls due to a shortage of labor. It represents the technical knowledge of the economy. The net national product Y is a function of capital K and labor L, Y = F(K, L). This aggregate production function is fixed; how the product depends on capital and labor does not change as time passes. The production function exhibits constant returns to scale; doubling the capital and labor doubles the output. Depreciation increases at a constant rate as the capital stock increases. The more capital you have, the more capital depreciation you have.

Capital accumulation

Where does money for capital accumulation come from? From saving and investments. Depreciation is growing at the same rate as the capital stock grows. Each unit of capital creates an equal amount of depreciation. When investment (I) is greater than depreciation, the capital stock is growing. As the capital stock grows, investment and depreciation intersect at one point, the steady-state level of capital, where I = depreciation.

When investment is less than depreciation, some of the capital stock needs repair, but there isn’t enough investment to do all the needed repairs, causing the capital stock to shrink and pushing it back to the steady state. In words, since the economy generates savings (and hence new investment) larger than the amount needed to keep the amount of capital per worker constant, the capital-labor ratio will increase.

The figure displays the production function in the intensive form, per capita savings, and the break-even investment line. The steady state occurs when the break-even investment line crosses the schedule of per capita savings. If the economy starts out on the left (right) of the steady state, per capita savings will exceed (be less than) the required amount to keep the capital-labor ratio constant, causing the capital to increase (decrease).

If the economy starts out on the right of the steady state, per capita savings will be less than required to keep the capital-labor ratio constant, causing the capital to decrease.

Transitional dynamics

An important feature of the Solow model is that if the economy is not in the steady state, it will converge to the steady state. However, the economy will not jump instantaneously from one steady state to the other: since capital accumulation is bounded by the availability of savings, there will be an adjustment period, during which the transition occurs.