Unit 14 - Economic Growth


The Solow Model

Fundamental to any discussion of economic issues is the topic of growth. Growth entails the expansion of an economy's productive capacity, its output, its income, and the living standards of its citizens. In this topic we will discuss the basic concept of growth and how additions to capital, population, savings, and technology impact growth. An important part of the discussion will deal with the convergence theorem, which says that rapidly growing less developed economies will eventually converge to living standards enjoyed in developed countries such as the United States and Japan.

The concept of growth discussed in this section is different from the changes in GDP that an economy experiences due to fluctuations in aggregate demand during the business cycle. Here we focus on long-run economic growth. This relates to the expansion of an economy's productive capacity, which is considered relatively constant from year to year, independent of fluctuations in aggregate demand and GDP. This was shown graphically by an expansion in the production possibilities frontier and by a rightward shift in the aggregate supply curve.

The Production Function

 14.1)

Equation 14.1 shows the basic relationship between output growth and changes in inputs of capital and labor and technological improvements. The equation indicates that output growth depends on labor growth, plus capital growth and technical progress. The relationship between inputs and output, for a given state of technology, is summarized in the production function.

The production function is used to relate how inputs produce goods and services. We can write a simple aggregate production function as:

(14.2) Y = AF(K, L)

In equation 14.2, F denotes a functional relationship showing how the level of aggregate economic output, Y, changes with variations in an economy's capital (K) and labor (L) inputs (1). We assume that the coefficients of K and L are positive; thus, increases in either or both will also increase output. A is used to denote changes in technology, which here are assumed to be exogenous, or outside the influences of change in capital and labor inputs. In other words, changes in our capital stock, K, do not affect technological progress (2). As you would expect, technology is also assumed to have a positive correlation with output; technological improvements lead to higher levels of output, Y, given capital and labor inputs. As a result, better technology, such as improvements in the quality of the existing capital stock, makes the labor inputs more productive and increases the output per worker (with a fixed supply of labor and capital).


(1) Although the interpretation of how changes in K or L actually influence changes in Y is not important for this course, the functional relationship, F, describes the relationship of existing technology. With an increase in either K or L, there may be constant, decreasing, or increasing returns to scale.

(2) The assumption that the rate of technological advancement and capital accumulation are independent or exogenous, arising outside the model is currently an important area of economic research. Models of production are being revised to show that with greater levels of capital (especially human capital) not only does output grow at a faster rate, but the pace of technological improvement also quickens, giving a further catalyst to economic growth.


 

 

 


We begin by assuming the total population and aggregate labor input, L, and the level of technology, A, remain constant. Figure 14-1 shows the relationship between the capital input and output. The slope of the curve at any point represents the marginal product of capital. The marginal product of capital represents the incremental gain in output as additional capital is added to the capital stock, holding the labor input fixed.

The slope in the graph decreases because of the diminishing marginal productivity of capital. This implies that as an additional unit of capital is added to a fixed labor supply, the gain in output is positive but less than the extra output generated from the addition of the previous unit of capital (3).


(3) We are working with the neoclassical growth model which assumes an economy of perfect competition, where output increases with the addition of labor and capital inputs, and the economy exhibits diminishing returns in the labor and capital inputs.


Additions to the Capital Stock

Again assume that the population and thus labor supply and technology are fixed. We also assume some portion of the capital stock depreciates or wears out each year. To maintain a constant capital stock, depreciated capital must be replaced each year. The annual depreciation of the capital stock, K, is given by the Greek symbol delta. The depreciation rate equals:

,

where delta has a value greater than zero. We next specify s as equal to the fraction of output (or income) that is saved. Thus, total saving equals sY. In this case, savings equals the part of output that is not consumed and is available for investment.

In this model savings is used for investment, part of which is used to replace depreciated capital. Gross investment equals new additions to the capital stock plus depreciation. Or put another way, the net change in the capital stock equals gross investment minus the replacement of the depreciated capital stock. Relating gross saving, sY, to investment yields:

(14.3)

Equation 14.3 says that the change in the capital stock (net investment) equals total savings minus the savings necessary to replace the part of the capital stock that has depreciated. Any savings greater than that required to replace depreciated capital adds to the capital stock in the form of new investment.

 

 

 


Using the right part of equation 14.3 we have another expression for gross savings, sF(K,L), which equals gross investment as well. We can use this expression to illustrate graphically the determination of net investment. Figure 14-2 shows the relationship between total savings and net investment. Holding the depreciation rate, delta, constant, the amount of capital that depreciates (delta x K) increases with the total capital stock along the horizontal axis. When gross savings exceeds the amount of savings required to replace depreciated capital, at a point to the left of K* such as K(0), there are net additions to capital. Due to diminishing marginal returns on capital, this function becomes flatter as more capital is added. At K*, all savings goes toward the replacement of depreciated capital, leaving no extra money for net additions to capital.

Equation 14.3 and Figure 14-2 give a fundamental base for economic growth. Savings rates sufficient to allow net additions to the capital stock are essential for economic growth. As we discussed in Topic 1, net investment is the catalyst for long-run economic growth. If we are at point K(0) in the graph, then the change in K is positive, increasing the capital stock, and the economy expands. The net addition to capital is represented by a rightward movement to K*. As long as total savings exceeds the amount required to replace depreciated capital, the economy continues to add to its capital stock and move towards K*. At K*, net investment is zero and all savings go towards the replacement of capital that has worn out. This is an important point known as the steady state, a concept we will soon discuss in more detail. At the steady state shown here, net investment equals zero and the capital stock remains constant at K*.

The production function, Y = AF(K, L), relates the capital stock to the level of aggregate output, Y. Assuming that we hold our labor input constant, output has a positive relationship to changes in capital, K. As our economy moves from K(0) to K*, there are net additions to the capital stock in the transition and thus additional output is generated. At K*, Y reaches its steady-state value, Y* = AF(K*, L). The relationship between additions to K and changes in the growth rate of Y described here sets the foundation for economic growth theory, which will be developed in this topic.

Changes in the Savings Rate

Now we consider the effects on the economy when the savings rate increases. Consider the impact in equation 14.3 when the savings rate rises. If we hold the pace of depreciation constant, this creates additional that which can be used to further build up the capital stock.

 

 

 


We begin by assuming a steady-state equilibrium at K*. From this point in Figure 14-3, the savings rate out of income rises from s1 to s2. This is shown graphically by a shift upward of the sF(K,L) curve. The increase in savings eliminates the old steady-state output level and the economy moves from K* towards K'. In this transition, there are net additions to the capital stock and increased growth rates of output, Y. Output growth continues to exceed the steady-state rate until the new level of saving, s2F(K,L), is just sufficient to replenish the capital stock at K'.

To summarize the results of an increase in the savings rate:


As a result of the increase in the savings rate, the economy moves from K* to K'. It is important to note:

  1. Both K* and K' are points that yield a constant steady-state output growth.
  2. The transition between K* and K' involves a higher economic growth rate as there is positive net investment. However, diminishing marginal returns reduce the incremental output as additional capital is added to the capital stock.
  3. Although the output growth rate at K' is identical to the rate at K*, K' is consistent with a greater total level of output, consumption, and standard of living.

Changes in the Population and the Labor Force

So far we have looked at the effect on output growth when there is an increase in net investment and capital, holding the labor force constant. Now we add a level of increased realism by allowing for population growth. Along with population growth come increases in the supply of labor. We will assume that population, and thus the labor force, grows at some constant rate, 1% annually, for example.

Allowing for increasing the labor supply requires a modification of our notation:

We can modify equation 14.3 to incorporate an increasing labor supply:

(14.4)

and we make the assumption of constant returns to scale, which gives the result that output increases proportionally to the increase in the labor input. This outcome is conditional upon a constant capital-labor ratio, k. In other words, each new entrant into the labor force receives the same capital to work with as the workers already employed. If the new worker is employed in construction, he receives hammers and nails, just the same as his co-workers. Or if the new employee is hired by a firm, she is provided a computer and local area network access so that she is just as productive as other workers engaged in a similar task. Her employer will not just hand her a slide rule and tab of paper when she can use a computer.

Using equation 14.4 and extending the assumption of a constant capital-labor ratio implies that the capital stock must increase proportionally to the increase in the labor supply to maintain k at a constant level. Thus, we have the result that K and Y increase proportionately to L, and that k and y are independent of L at the steady state. In the steady state, the output per worker, y, and the capital-labor ratio, k, remain constant, while total output and capital increase proportionally to the labor supply growth rate (4).


(4) Consider the case if the labor force remains constant. The model used here implies that the economy stagnates at zero economic growth. To reach the conclusion that zero labor force growth leads to no economic growth we need to assume that technology remains constant and the presence of diminishing returns to capital.



Figure 14-4 illustrates the steady-state when population and labor growth are present. As you can see, graphically little has changed from before. The important modification involves the depreciation of the existing capital stock to incorporate the additional capital required to maintain a stable capital-labor ratio. Since the rate of depreciation and labor growth rate are assumed to change at a fixed annual rate, the required investment to maintain a consistent capital stock per worker is (n + delta)k, where n represents the rate of population (and labor force) growth.

In the case presented here, to maintain the steady state, savings need to be sufficient to replace depreciated capital, plus equip new workers with sufficient capital to maintain a constant level of k. At the steady state, output and economic growth will equal the rate of increase in the population.

When total savings and investment diverge from the amount required to maintain an economic steady state, two conditions can exist:


(5) The economy ends up at a new steady state despite a falling marginal product of labor, because this is eventually offset by an increasing marginal product of capital as the capital-labor ratio falls. This of course assumes a constant savings rate. It is not hard to construct a scenario where savings rates also decline with capital, since poor nations have a difficult time generating positive savings. If savings rates are falling, a continued economic contraction is the possible result, with decreasing GDP and falling living standards.


Technological Advances

The last layer we add to the Solow growth model is to allow for improvements in technology. We consider technological advances to be labor-improving, increasing labor productivity for a given level of capital. In this case, the quality of capital is better and workers using this capital are more efficient. This result can be represented by an increase in output per worker. The relationship of changes in the capital stock to replenishment savings is shown in equation 14.7, where a represents the rate of technological progress.

(14.7)

Equation 14.7 shows that net investment is possible when total savings are sufficient to replace depreciated capital, maintain a constant capital-labor ratio for new workers, and keep up with technological advances. For example, if the rate of technological improvement is 2% a year, then 2% of the capital stock must be updated annually to ensure that workers are as efficient as possible. If new technology is not adopted in the workplace, then improvements in technology yield no gains in output per worker.

Technological improvement is a double-edged sword. In this case the annual change in output equals the rate of growth in the labor force plus the rate of technological advance (6), which improves output per worker. So technological improvement gives an extra kick to economic growth, but it also requires a greater share of savings devoted to maintaining a capital stock which utilizes the technology. This leaves less potential surplus savings available for net investment.


(6) In this example, labor force growth, n, is assumed to equal 1% a year while improvements in technology are assumed to equal 2%, yielding 3% annual growth is savings is equal to the replenishment rate of offsetting depreciation plus equipping new workers with capital and upgrading the existing capital stock to keep pace with technological improvements.


The relationships between total savings and replenishment (which equals the term in ( ) in equation 14.7 x k) is the same as discussed before. If total savings exceeds replenishment, then there is positive net investment, and higher economic growth rate than the sum of labor and technology growth.

As noted earlier, the neoclassical growth model developed here assumes that technology is outside or exogenous to the system. The implication of this assumption is that additions to the capital stock do not lead to technological advances. In reality, firms undertake research and development to improve existing capital and production techniques. By allowing technology to be endogenous or within the system, we can see how additions to the capital stock reinforce technological advances, and also achieve increasing returns on capital.

The Steady-State

The steady-state refers to an economy where total savings equals the amount required to offset depreciation, maintain a constant capital-labor ratio for new workers and to keep the capital stock tuned to improvements in technology. In this case, net investment equals zero, and annual economic growth will equal to the n + a terms in equation 14.7, the labor force (population) growth rate, plus the pace of technological change. Of course there are many variations on the steady-state which can be used to explain differential growth rates of various global economies and how changes in public policy and society affect growth.

Increasing Returns to Scale

We have covered the basic neoclassical economic growth model developed by Robert Solow and Trevor Swan. The model assumes a perfectly competitive economy where output grows in response to increases in capital or labor used in production or improvements in technology. The presence of diminishing returns implies that absent net additions to the capital stock, a mature economy will reach a steady-state rate of growth determined by labor force growth and the pace of technological innovation. With diminishing returns and favorable conditions present, less developed countries will achieve rapid growth, eventually converging at the standard of living presently enjoyed in today's wealthy countries. In the future, today's wealthy countries together with tomorrow's newly wealthy countries, will walk hand-in-hand down the steady-state growth path.

To achieve this utopian world, or at least one of relative economic equality across international borders, the presence of diminishing returns in production is critical. Diminishing returns assumes that as additional capital is used in production, output per worker increases, but at a decreasing rate. Certainly an acceptable hypothesis for an industrial society. As another unit of capital is added to the assembly line, output increases, but by less than the addition of the previous capital item. In the absence of technological improvements, the model used here implies that for zero population and labor force growth nations, the economy stagnates at zero economic growth. Clearly, this is not true for some countries actually exhibiting zero population growth. However, for many, growth remains positive.

To reach the conclusion that zero labor force growth leads to no economic growth we need to assume that technology remains constant and the presence of diminishing returns to capital. But instead, if capital exhibits increasing returns then additional capital increases worker output by a greater proportion that the previous addition of physical capital. If gains in worker productivity are significant when the capital stocks increases, than an economy can exhibit positive growth even if the labor force and technology present are stagnant. This will be the likely case when we include human capital in our definition of capital. Adding new physical capital leads to greater worker efficiency as workers learn how to use the capital and find ways to innovate to increase their output.

As an example of increasing returns, consider the computer software industry. Software is often a complex collection of computer code instructing the computer how to perform operations such as assembling a spreadsheet, connecting to another computer, or viewing documents on the world-wide web. Before ever being sold, writing and testing software takes a good deal of time and of labor - known as a fixed cost. But once completed, the cost of replicating the additional copies of the software program is practically zero - a variable cost. As a result, the software industry displays increasing returns to scale in production.

Consider the company that spends $1 million to develop a personal finance computer program. If one copy is sold for $50, then the loss is substantial. If one million copies are sold at $50 each then profits are strong and continue to rise for each additional copy of the software sold. In the example here, increasing returns implies that for every additional unit produced, revenues and profits rise as the cost to produce that item falls. For computer software, with relatively constant variable costs of production, the share of fixed costs attributed to each additional unit produced will fall.

As higher technology increasingly becomes a part of modern society and the economy, the potential for sustained increasing returns in production becomes a greater possibility. As modern industry adopts to the information technology revolution of high powered computers, software and networks, the production of goods and services may enjoy increasing returns over substantial ranges of output. We are not just talking about software and computers, but the industries that adapt to them such as banking, manufacturing and many others. In terms of the growth model presented here, increasing returns allow for prolonged economic growth above the steady-state level determined by labor force growth and technological advances. The constraint to long-run growth comes on the demand side, not as the steady state model concludes, from the supply-side (7).


(7) This leads to an interesting public-policy debate. In our study of the business cycle in Topic 4, the conclusion was that the government could affect short-run economic growth through aggregate demand, but changes in aggregate supply were considered outside the active policymakers domain. Aggregate supply changes were considered to remain constant from year to year and determined by long-run circumstances. With the possibility of increasing returns we may have the situation where aggregate demand growth can influence the long-run growth rate of aggregate supply.


Increasing returns leads to the conclusion that long-run economic growth will not be restricted by steady state constraints. Growth above steady-state levels will persist as long as production exhibits increasing returns to scale. Furthermore, with increasing returns, the convergence of less-developed countries to economic living standards equal to today's developed countries may not occur, or at least take significantly longer to achieve. The absence or delay of convergence should not be viewed as negative. With proper conditions, LDCs will grow rapidly, easily surpassing the living standards enjoyed today in the United States, Japan and in other developed countries. However, it may take much longer for them to catch-up at any given point in time.

Copyright 2002, Jay Kaplan
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