“Golden rule” of accumulation by E. Phelps. Neoclassical Solow model of economic growth and the golden rule of accumulation According to the Solow model, the golden rule is the rule
In the Solow model, a central place is given to technological progress which ensures continuous economic growth. Other models in this direction include one-factor Domar-Harrod model. In this model, product growth is associated with the rate of accumulation efficiency. The central equation of this model has the following form: y=av, where (1)
Y is the rate of growth of the product, a is the rate of accumulation, c is the efficiency of accumulation (capital productivity ratio).
When calculating the accumulation rate (a), it should be taken into account that, firstly, part of the accumulation is carried out at the expense of the depreciation fund and is used to compensate for the disposal of fixed capital, and secondly, the accumulation fund provides investment not only in fixed capital, but also in working capital, including reserves.
The neoclassical model, in conditions of equilibrium between supply and demand, takes into account variability capital productivity ratio . The capital-production relationship becomes flexible due to the fact that neoclassical models take into account not one, but two production factors and allow their interchangeability. By allowing different combinations of production factors, it is possible to achieve an increase in production volumes even with the same technology. Among the analytical tools of neoclassical models, the main place is occupied by the production function: Y = f (K, L), where Y is the product, and K and L are the costs of capital and labor. The volume and dynamics of the product are associated with the volume and dynamics of total costs and their efficiency: or Y = abk+ where d is a coefficient reflecting the ratio of the values of factors K and L to the value of product Y;
b and - function parameters characterizing the elasticity of volumes and dynamics of the product from the costs of production factors, i.e. parameters showing how much production volume will increase if any production factor increases by 1%;
K and P are the growth rates of capital and labor, respectively.
Solow model has the ability to describe these changes in dynamics, i.e. makes it look more like a film than a photograph. Solow growth model shows how savings, population growth, and technological progress affect output growth over time.
The model provides a framework with which to analyze one of the most important issues in economics: what part of the industrial product should be consumed today, and what part of it should be saved for future use . Since saving equals investment, saving determines the amount of capital the economy will have in the future.
The supply of goods in the Solow model is described using the well-known production function: Y=F (K,L), where K is capital, L is labor.
Those. The volume of production depends on the capital stock and the labor used. The Solow model assumes that production function has the property constant returns to scale.
A production function with constant returns to scale is convenient for this purpose because output per worker then depends on the amount of capital per worker.
The production function can be written as y=f(k), where f(k)=F (k,1). In Fig. This production function is depicted
f(k) Law of Diminishing Returns
Release (analogy).
for one
RTO employee
capital per worker K
The slope of this production function shows how much additional product per worker can be obtained if the capital-labor ratio is increased by one unit. This value is the marginal product of MKR capital. This can be written like this:
MKR = f(k + 1) - f(k). Note that as the capital-labor ratio increases, the graph of the production function becomes flatter, i.e. the angle of inclination decreases. This production function is characterized by a decreasing marginal productivity of capital: each additional unit of capital produces less output than the previous one. When the stock of capital per worker is small, each additional unit of capital produces a greater return. If the capital-labor ratio is high, then the additional unit of capital is less efficient and produces less additional output.
In the Solow model, demand comes from consumers and investors. In other words, the products produced by each worker are divided between consumption per worker and investment per worker: Y = c + I, where c is consumption, I is investment.
The Solow model assumes that consumption function takes the simple form C = (1 – S)·y, where the saving rate S takes values from 0 to 1. This function means that consumption is proportional to income. Every year part (1 – S) of income is consumed and part S is saved.
The role of this interpretation of consumption will become clear if we replace the value C with the value (1 – S) y in the identity of national accounts: y = (1 – S) y + I. After the transformation we obtain: I = S y. This equation shows that investment (like consumption) is proportional to income. If investment is equal to savings, the savings rate S shows what part of the output is allocated to capital investment.
Having presented the two main components of the Solow model - production function And consumption function, one can analyze how capital accumulation drives economic growth. Capital reserves may change for two reasons: 1. Investments lead to growth of capital reserves . 2. Part of capital wears out, that is, it is depreciated, which leads to reduction of capital reserves . In order to understand how capital stocks change, it is necessary to find the factors that determine the amount of investment and depreciation. Investment per worker is part of the product per worker (S·y). Replacing y expression of the production function, we represent investment per worker as a function of capital-labor ratio: I = S·f(k).
The higher the capital ratio k, the higher the output f(k) and the greater the investment I. This equation, which includes a production function and a consumption function, relates the existing stock of capital k to the accumulation of new capital i. The graph shows how the saving rate determines the division of product into consumption and investment for each value of k.
U Performance f(k)
capital-to-labor ratio k
The saving rate S determines the division of the industrial product into consumption and investment. For any level of capital-labor ratio k, output is f(k), investment is S·f(k), and consumption is f(k) – S·f (k).
Let us assume that a certain share of capital σ is retired annually. Let's call σ the retirement rate. For example, if capital is operated for an average of 25 years, then the disposal rate is 4% per year (σ = 0.04). Thus, the amount of capital that is retired each year is σ k . The graph shows how disposals depend on capital stocks.
σ K
Disposal
Capital ratio
The effect of investment and disposal on capital stock can be expressed using the following equation:
Change in capital reserves = investment - disposal, i.e. k=I-σк, where k is the change in capital reserves per employee per year. Since investments are equal to savings, the change in capital reserves can be written as follows: k = Sf (k) - σk. This equation shows that the change in capital stock is equal to investment Sf(k) minus capital disposal σk.
The higher the capital ratio, those higher output and investment per worker. However, the larger the capital stock, the more and the amount of disposal.
In Fig. shown, that there is only one level of capital ratio , at which investment equals depreciation . If exactly this level is reached in the economy, then it will not change over time, since the two forces acting on it (investment and disposal) are precisely balanced. Thus, at a given level of capital-to-labor ratio . Let's call this situation the state sustainable capital ratio and let's denote it k * .
Let us assume that capital stocks in the initial state exceed k *, for example, at point k 2. In this case, investment is less than disposal: capital is being retired faster than it is being added. Thus, the capital-labor ratio will decline, again approaching a sustainable level. At the point when the capital stock per worker reaches a sustainable level, investment will equal disposal, and the capital-labor ratio will neither rise nor fall.
Let us assume that the economy begins to develop, being in a steady state with the savings rate S 1 and capital reserves k 1 *. The saving rate then increases from S 1 to S 2 , causing a corresponding upward shift in the Sf(k) curve. With the initial level of savings S 1 and initial capital reserves k 1 *,
investments just compensate for the outflow of capital. Immediately after an increase in the saving rate, investment increases, but the capital stock, and therefore disposal, remains unchanged; as a result, investments exceed disposal. Capital will gradually increase until the economy reaches a new steady state k 2 * with a large capital-to-labor ratio and higher labor productivity than the previous steady state.
The Solow model shows that savings rate is key (defining) determinant of the sustainable capital ratio . If the savings rate is higher, then the economy will have, other things being equal, a larger stock of capital and a higher level of output.
Higher savings lead to faster growth, but this acceleration does not last forever. Increasing the saving rate ensures growth until the economy reaches a new steady state. If the economy maintains a high savings rate, then both the capital-labor ratio and productivity will be high, but it will not be possible to maintain high rates of economic growth forever.
According to the Solow model, a country that allocates a significant portion of its income to savings will have a high sustainable capital-labor ratio and, as a result, a high level of per capita income. Countries with high levels of investment (USA, Canada or Japan) usually have high per capita income, while countries with low levels of investment (Ethiopia, Zaire, Chad) tend to have low income per capita. International experience thus confirms the predictions of the Solow model that the saving rate is the most important determinant of a country's wealth or poverty.
Now let’s consider the question: what amounts of accumulation are optimal.
The level of capital accumulation that ensures a steady state with the highest level of consumption, is called the Gold level of capital accumulation, or " The golden rule" E. Phelps, and is denoted k ** .
The steady state level of consumption is the difference between the output and disposal of capital at steady state. It shows that an increasing capital-labor ratio has a dual effect on the amount of consumption: it contributes to an increase in output, but at the same time, a larger amount of output is required to compensate for the disposal of capital. In Fig. Steady state output and disposal are shown as a function of the steady state capital ratio. Steady state consumption is the difference between output and capital outflow. The figure shows that there is only one level of capital-labor ratio - the Golden Rule level k**, at which per capita consumption reaches its maximum.
If the capital-labor ratio is less than its level according to the Golden Rule, then an increase in capital reserves causes an increase in production that exceeds the increase in disposal. In this case, consumption increases. The production function curve slopes steeper than the σk** line, so that the distance between them (equal to consumption) increases as k* increases. On the other hand, if the amount of capital exceeds the level of the Golden Rule, a further increase in the capital-labor ratio will reduce consumption, since the increase in output will be less than the increase in capital disposal.
At the capital-labor ratio corresponding to the Golden Rule level, the production function and the σk * line have the same slope, and consumption reaches its maximum level.
If the stable stock of capital exceeds the level of the Golden Rule, then an increase in the volume of capital reduces consumption, since the marginal product of capital is less than the retirement rate. Therefore, the following condition constitutes the Golden Rule itself: MRC = σ. When the capital-labor ratio is at the level of the Golden Rule, the marginal product of capital is equal to the retirement rate. In other words, if the Golden Rule is satisfied, the marginal product minus the disposal rate, MRP = σ, is equal to zero.
The basic Solow model shows that in itself capital accumulation cannot explain continuous economic growth . A high saving rate temporarily increases the growth rate, but the economy eventually approaches a steady state in which the capital stock and output are constant. In order to explain the continuous economic growth that is observed in most countries of the world, the Solow model needs to be expanded to include two other sources of economic growth: population growth and technological progress.
An increase in the number of employees leads to a reduction in the capital intensity of each of them. The change in the capital stock per employee will be: k = I – σ·k – n·k. The three terms on the right side of this equation show the impact of investment, capital disposal and population growth on the capital-labor ratio. Investment increases k, while capital outflows and population growth decrease it. In order to use this equality, we replace I with S f(k) and rewrite it: k = S f(k) - (σ + n)·k. The effects of capital flight and population growth are now combined. The equation shows that population growth reduces the capital-labor ratio in the same way as retirement. Attrition reduces k by reducing the capital stock, while population growth reduces k by distributing capital among more workers.
In order for the economy to be in a stable state, investments S f(k) must compensate for the consequences of capital outflow and population growth – (σ + n)·k, which is shown in Fig. point of two curves.
Investments
k Capital ratio
Sustainable level
Population growth complements the original Solow model in three ways. Firstly, it allows us to get closer to explaining the causes of economic growth. In a steady state economy with a growing population, capital and output per worker remain unchanged, but as the number of workers grows at a rate n, capital and output also grow at a rate n. Consequently, population growth cannot explain long-term increases in living standards because output per worker remains constant at steady state. However, population growth can explain the continuous increase in gross output.
Second, population growth provides further explanation for why some countries are rich and others are poor.
So the Solow model predicts that countries with higher population growth rates will have lower GNP per capita.
Investments
Capital ratio
Third, population growth affects the rate of capital accumulation according to the Golden Rule. Recall that consumption per worker is equal to c = y - i. Since steady state output is f(k *) and steady state investment is (σ + n)·k *, the steady state level of consumption can be defined as c * = f(k *) - (σ+n)·k *. The level k * that maximizes consumption is such that MRC = σ + n, or, accordingly, MRC – σ = n. In the steady state, according to the Golden Rule, the marginal product of capital minus the retirement rate is equal to the population growth rate.
Now let's include Solow in the model technological progress– the third source of economic growth. Let's write the production function as follows: Y = F(K,L x E), where E represents a new variable, which we will call the labor efficiency of one worker. Labor efficiency depends on the health, education and qualifications of the workforce.
Describing technological progress through an increase in labor efficiency makes it similar to population growth.
Equation showing change To over time, now looks like this: A new element in this formula, g, the rate of technological progress, appears because To is the amount of capital per unit of labor with constant efficiency. If the value of g is large, then the total number of units of labor with constant efficiency grows quickly, and the increase in capital per such unit of labor is relatively small and may become negative.
Thus, given technological progress, our model can ultimately explain why living standards increase year after year. Thus we showed that technological progress can support continued growth in output per worker , while a high level of saving leads to high growth only until a steady state is reached. Once the economy reaches a steady state, the rate of growth of output per worker depends only on the rate of technological change. The Solow model shows that only technological progress can explain continuously growing standard of living .
The introduction of technological progress into the model also changes the conditions for fulfilling the Golden Rule. The golden rule for capital accumulation defines the sustainable level that maximizes consumption per unit of labor with constant efficiency. It should be said that the sustainable level of consumption per unit of labor with constant efficiency is: .
Sustainable consumption levels are maximized if:
MRC – σ + n + g, or MRC – σ = n + g. Thus, with a stock of capital according to the Golden Rule, the net marginal product of capital (MPC – σ) is equal to the growth rate of the volume of output n + g.
Control questions
In the AD-AS model, economic growth can be represented as:
a) shift to the left of the AS curve;
b) shift to the right of the AD curve;
c) shift to the left of the AD curve.
Mandatory
1. Agapova T. A., Seregina S. F. Macroeconomics: Textbook / Ed. ed. A. V. Sidorovich. – M.: Moscow State University Publishing House, 2001. – 416 p.
2. Dornbusch L., Fischer S. Macroeconomics / Translated from English. – M.: Moscow State University Publishing House; INFRA-M, 1997. –784 p.
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5. Microeconomics and macroeconomics / Col. auto ed. S. Budagovskaya. - Kiev: Fundamentals, 1998.
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7. Sachs D. Jeffrey, Larren B. Phillips. Macroeconomics. Global approach. - M.: Delo, 1996.
8. Samuelson Paul A., Nordgauz William D. Macroeconomics. – Kiev: Osnovi, 1995.
Additional
9. Agapova T. The concept of rational expectations and the effectiveness of macroeconomic policy // Russian Economic Journal.-1996.- No. 10.
10. Albegova I.M., Emtsov R.G., Kholopov A.V. State economic policy. – M.: DIS, 1998. – 380 p.
11. Bazilevich V. D., Balastrik L. O. Macroeconomics: Basic lecture notes. – K.: Chetverta Khvilya, 1997. – 275 p.
12. Baranovsky O. Groshova mass in the system of economic security of the state // Banking on the right. – 1996. – No. 4.
13. Borisova O. S. Regulation of the budget deficit of the Federal Republic of Germany // Finance. – 1992. – No. 2.
Equilibrium economic growth is compatible with various savings rates, but only the one that ensures economic growth with the maximum level of consumption will be optimal. The optimal rate of accumulation corresponds to the “golden rule of capital accumulation.”
In general, the answer to the question of what are the conditions for optimal economic growth for society was given by several economists (J. Mead, J. Robinson, etc.) in the early 1960s, but the American economist E. Phelps was the first to publish it. He also owns the term “golden rule of capital accumulation.”
Phelps asked how much capital a society on a balanced growth trajectory would want to have. If it is large enough, this will guarantee a high level of production, but an increasing part of it will go not for consumption, but for accumulation - society will not be able to enjoy the fruits of growth. If the amount of capital is too small, then almost everything that is produced can be consumed, but very little will be produced. Somewhere in the middle between the two extremes, obviously, there is an optimal point for society, at which the maximum volume of consumption is achieved.
Let To**- the level of capital-labor ratio corresponding to the rate of accumulation according to the “golden rule”, and c** - the level of consumption. All produced products are spent on consumption and investment. Substituting the values of each of the parameters that they took in a steady state, we obtain
From here it is easy to determine such a stable level of capital-labor ratio (k**), at which the volume of consumption (c**) is maximized and which corresponds to the “golden rule” (Fig. 13.4).
Rice. 13.4.
At the point E production function f(k*) and line d x k* have the same slope and consumption reaches its maximum level.
At the capital-to-labor level To** condition is met MRK=(an increase in the stock of capital by one unit gives an increase in output equal to the marginal product of capital, and increases the disposal of capital by the amount d).
If the factors of population growth and technological progress are taken into account, then the following condition is satisfied:
The Solow model and Phelps' "golden rule of accumulation" allow us to formulate some practical recommendations.
- 1. Increase or decrease the savings rate. If an economy develops with a capital stock greater than it would have under the Golden Rule, then it is necessary to implement policies aimed at reducing the savings rate. This in turn will lead to an increase in consumption and a corresponding decrease in investment and, therefore, a decrease in the sustainable level of the capital stock. If the economy develops with a lower capital-labor ratio than in a steady state according to the “golden rule,” then it is necessary to stimulate the growth of the savings rate in society. This will lead to a decrease in consumption, an increase in investment, and ultimately an increase in consumption.
- 2. Stimulating technical progress. As the Solow model suggests, a faster rate of population growth will have the effect of accelerating economic growth, but per capita output will decline at steady state. Another factor, an increase in the saving rate, will lead to higher per capita income and increase the capital-to-labor ratio, but will not affect the steady-state growth rate. Therefore, technological progress is the only factor ensuring economic growth in a steady state, i.e. increase in per capita income.
R. Solow's model of economic growth is a neoclassical model of economic growth that reveals the mechanism of influence of savings, growth of labor resources and scientific and technological progress on the standard of living of the population and its dynamics.
R. Solow's model was developed in 1956 and is intended to study equilibrium trajectories of economic growth; shows the relationship between savings and capital accumulation.
This is a simple continuous single-sector model of economic dynamics where only households and firms are represented.
R. Solow showed that the instability of dynamic equilibrium in the models of E. Domar and R. Harrod is a consequence of the lack of interchangeability of production factors. Instead of V. Leontiev's production function, he uses the Cobb-Douglas production function, where labor and capital are substitutes, and the sum of their elasticity coefficients for production factors is equal to one. In addition, the model is built on the following premises of the neoclassical school:
♦ perfect competition in the factor market and full employment;
♦ price flexibility in the goods market;
♦ constant returns to scale;
♦ diminishing productivity of capital;
♦ constant rate of capital retirement.
R. Solow's model consists of the following equations characterizing economic dynamics.
1. The volume of supply in the goods market is described by a production function with constant returns to scale:
For any positive Z the following is true:
where Y/L is the average labor productivity per employee (y); K t /L t capital-labor ratio (capital-labor ratio) of labor per employee (k t). Therefore we can write:
Thus, the volume of production per worker is a function of its capital ratio (Fig. 30.2).
Rice. 30.2. Graph of the production function per worker
2. The volume of demand for goods and services presented by consumers and investors, i.e., by the private sector without government orders and net exports: 
Then - investment per employee; - consumption per
one employee.
The equilibrium condition is the equality of I and S. Since the volume of investment is the share of savings in income:
In equilibrium, investment is equal to savings and proportional to income.
Capital stocks in the economy depend on the volume of investment (it) and capital outflow (dkt), therefore:
The capital stock at which investment (i t) is equal to capital outflow (dk t), and Ak t = 0, is called the sustainable level of capital-labor ratio (k*).
In a stable (stationary) state, a constant ratio of K/L and output per worker Y t /L t is established. At a capital-labor level corresponding to k*, the economy is in a state of long-term stable (stationary) equilibrium, to which it will always return.
The functioning of the Solow model can be illustrated graphically (Figure 30.3).
Rice. 30.3. Sustainable level of capital ratio
If the initial value k 4 is lower than k*, then sf(k) > dk.
If k 2 > k* - investment is less than depreciation. If the system deviates from the trajectory of equilibrium development, the economy, under the influence of endogenous mechanisms, will return to the equilibrium trajectory.
An increase in the savings rate from Sy 1 to Sy 2 shifts the investment curve upward. Now at the previous steady state point, investment exceeds disposal. The economy will strive to achieve a new steady state with greater capital and labor productivity (Figure 30.4).
From the foregoing, the following conclusions can be drawn:
♦ an increase in the savings rate in the short term leads to an acceleration of the growth rate of national income (from k 4 * to k 2 *);
♦ in the long run, a new long-term equilibrium state is established, while the level of capital and labor productivity per worker increases.
3. The country's population growth is increasing at a constant rate. Thanks to the flexibility of prices in the factor market, full employment is constantly maintained, i.e., the number of employees grows at the same rate as the population in the country.
In this case, capital reserves may change because:
♦ investments lead to an increase in capital reserves;
♦ part of the capital is depreciated, which leads to a decrease in capital reserves;
♦ part of the capital goes to newly recruited workers.
Capital accumulation will therefore be:
Rice. 30.4. Increase in savings rate
where k t is the change in capital reserves per employee; i t - investments per employee; dk t - depreciation per employee; nk t is capital growth due to population growth and employment in the economy.
The product nk t shows the need for additional capital per worker so that the capital-labor ratio remains constant.
Since yt = f(k), then the condition for stable equilibrium in the economy with constant capital-labor ratio:
In order for the capital-labor ratio to remain constant with population growth, it is necessary to increase capital at the same rate as the population. In addition, output and population should grow at the same rate:
Let's consider the economic consequences of increasing population growth rates and their slowdown for the country's economy.
1. The population growth rate increased from n to n" at the same rate of accumulation (Fig. 30.5).
In Fig. Figure 30.5 shows that an increase in the population growth rate shifts the line (d + n)k up and to the left.
The initial steady state of the economy corresponds to point c. As the population growth rate increases, capital per worker will decrease until the economy reaches a new steady state at point C with a lower capital-labor ratio. A lower level of capital-labor ratio corresponds to lower labor productivity (from point y 0 * to point y 1 **). At the same time, the equilibrium growth rate of national income increases.
2. Slowdown in population growth rates from n to n" at the same rate of accumulation (Fig. 30.6).
From Fig. 30.6 it follows that the slowdown in population growth shifts the line (d + n)k down and to the right, from point k* the capital-labor ratio per worker begins to grow until the economy reaches the desired steady state at point C with a higher capital-labor ratio and, accordingly labor productivity.
At the same time, the equilibrium rate of economic growth slows down. In the first case, rapid population growth at a given level of savings determines a low level of per capita income. The level of savings of the population is insufficient for the growth of capital-labor ratio. In the second case, the level of per capita income increases.
The foundations of this model were laid in his work “Contribution to the Theory of Economic Growth” (1956). The scientist came to the conclusion that the main reason for the instability of the economy in the Harrod-Domar model is the fixed value of capital intensity (a), which reflects the rigid ratio between production factors - labor and capital (K / b). However, one of these factors often remains “underutilized”. In accordance with the principles of neoclassical theory, the proportions between capital and labor should be variable (this is precisely the neoclassical nature of the growth theory of R.-M. Solow). they are determined by manufacturers who minimize costs depending on the prices of these factors. Therefore, instead of a fixed (K/L), Solow included a linearly homogeneous production function in his model:
Dividing all terms by b and denoting income per worker (Y / L) by y, and capital intensity K / L by we obtain:
y = LF (k, 1) Lf (k).
As in the Harrod-Domar model, it is assumed that the population grows at a constant rate, and investment constitutes a constant share of income, determined by the saving rate a:
Fundamental Solow Equation- the increase in the capital-to-worker ratio of one worker provides the remainder of specific investments (savings), formed after providing capital goods to all additional workers.
If sf (k) = nk, then the capital-labor ratio remains the same (dk = 0), that is, the economy grows without any structural changes in the relationship between factors. This is balanced growth.
In the Solow model (as opposed to the Harrod-Domar model), the balanced growth trajectory is sustainable, as shown by the graph (Figure 5).
Rice. 5. Solow model
The direct pc on this graph shows how much each worker must save and invest from his income in order to provide capital goods for future workers (including his own children). The sf(k) curve shows the level of his actual savings depending on the achieved level of capital-labor ratio. As the capital-to-labor ratio increases, the growth rate of investment (savings) naturally falls. The vertical distance between the curve and the straight line means, according to the fundamental Solow equation, a differential change in the capital-to-work ratio dk. At point k * (for example, k1) the capital-labor ratio rises, and at all points to the right of k * (for example, k2) it falls, so that the economy constantly shifts towards k *, and the trajectory of balanced growth is sustainable.
In the Solow model, the savings rate s matters only when the economy reaches a sustainable development path: the greater the value of s, the higher the 8k graph and, accordingly, the level of k *. But once growth is replenished, its further rate depends only on population growth and technological progress.
The following main conclusions follow from the Solow model:
a) it shows that the rate of savings in the economy determines the size of the capital stock, and, accordingly, the volume of production. The higher the savings rate, the higher the capital ratio and high productivity;
b) an increase in the saving rate causes a period of rapid growth until a new steady state is reached. In the long run, an increase in the savings rate does not affect the growth rate. Continued productivity growth depends on technological progress;
C) economic policy makers often claim that the rate of capital accumulation should be increased. Increasing government savings and tax incentives for private savings are ways to accelerate capital accumulation;
d) the rate of population growth also affects the standard of living. The higher the population growth rate, the lower the output per worker.
From the Solow model it turned out that the higher the savings rate, the higher the capital-labor ratio of the worker in a state of balanced growth, and therefore, the higher the rate of balanced growth. But growth is not an end in itself. Therefore, the next logical step was to determine the conditions for optimal economic growth for society. This was done simultaneously and independently of each other by several economists (including Nobel laureates J. Mead, M.-F.-C. Allais) in the early 60s of the 20th century, but the first to publish the answer to the question was the American professor E. Phelps . He also owns the term “golden rule of capital accumulation”, introduced into scientific circulation.
Golden rule level- a level of capital-to-weight ratio that ensures the largest volume of consumption.
At this level, the net marginal product of capital equals the rate of growth of production. Estimates made for real economies (the US economy) indicate that capital stocks are well below the golden rule level. To achieve it, an increase in investment is required and, accordingly, a decrease in the level of consumption of current generations.
The use of the "golden rule" in practice was limited due to the rather inflated output predictions, but it made it possible to formulate conclusions regarding real economic growth. The Solow model and the “golden rule” turned out to be quite simple and very convenient analytical tools to use. With their help, it became possible to study the impact on economic growth of various modifications of the production function, technological progress, changes in the rate of savings and taxation, and the like. Through the efforts of R.-M. Solow, J. Mead and other economists, the Solow model was disintegrated: the production of consumer and investment goods was taken into account separately. Models were also created that took into account the “age” of capital goods, since their different generations have different productivity. The work of J. Tobin was introduced into the theory of economic growth of the money supply (more precisely, government obligations that citizens have on an equal basis with capital).
In the 70s of the XX century. interest in the theory of economic growth has fallen. This was caused primarily by sharp cyclical fluctuations in the Western economy, as well as the fact that after the invention of the Solow model and the “golden rule”, progress in this area followed the path of increasing the complexity of mathematical technology without breakthroughs in the economic sense.
Until the 80s, economists were unable to introduce into the model the main factor of economic growth - technical progress, which remained exogenous. The innovations (also highly mathematized) of growth theory made in the 1980s envision positive externalities (externalities) of economic growth that provide a source of increasing returns for the economy. Increasing social returns are provided (according to P. Romer) by expenditures on research and experimental design work (R&D), and according to the opinion of R. Lucas1, investments in human rather than physical capital, although in different individual cases this is not necessarily “necessary” One of the conclusions of the Romer and Lucas models is that an economy with greater resources of human capital and scientific advances has a better chance of growth in the long run than one that lacks these advantages.
The Solow model is still relevant today. Experts note the theoretical elegance of her econometric estimates. The model allows us to analyze one of the most important questions of economics: what part of the produced product should be consumed now and what part should be stored for use in the future.
Study R.-M. Solow, the production function became the basis for the development of intra-industry balances of economic development, which, contrary to the conclusions of Keynesian theory, are based on the principle of automatic self-regulation of the economic system through the formation of a rational production structure. The indicators that were introduced into the function were more stable, and the connections between them were less elastic. its use for this purpose has proven effective.
Proposed by S.-S. Kuznets methods for determining national income use statistics (double counting of national income as the sum of costs and as the sum of income). His methods for calculating national income, national product and other important indicators are used not only in official reporting in the United States, but also in statistical publications of other countries.
The modern theory of economic growth has become the logical culmination of the earlier works of S.-S. Kuznets, devoted to the study of national income and its components. Currently, the term “gross national product” (GNP) is generally accepted, but at the beginning of the last century it was ignored. S.-S. Kuznets was not the first to study this issue, but it was his work that was so clear and understandable that it became a guide in this area. He more accurately assessed the output of the final product, the formation of capital and savings, and the distribution of income between different segments of the population. His legacy, which met the new requirements of the economy, laid the foundation for the assessment of GNP and its components by the US Federal Government, influenced further studies of economic growth, and made it possible to develop a unified methodology for calculating national income and GNP for all countries.
6.3.1 Models of economic growth R. Solow
R. Solow (b. 1924), winner of the Nobel Prize in Economics in 1987, developed two models: a factor analysis model of the sources of economic growth and a model showing the influence of savings, labor force growth and scientific and technical progress on the standard of living of the population and its dynamics.
The basis of the first model was the Cob-ba-Douglas production function, modified by introducing another factor - the level of technology development:
Solow concluded that a change in technology will lead to an equal increase in the marginal product K and L, i.e. Q = Tf(K, L).
Thus, the increase in output depends proportionally on the increase in technology, the increase in fixed capital and the increase in invested labor.
If the shares of labor and capital in output are measured on the basis of labor productivity, capital-labor ratio per worker and capital productivity, then the contribution of technical progress is presented as the remainder after subtracting from the increase in output the share obtained due to the increase in labor and capital. This is the so-called Solow residual, which expresses the share of economic growth due to technological progress, or “advancement in knowledge.”
Another Solow model shows the relationship between saving, capital accumulation and economic growth. If we denote the production per employee q, the amount of capital per employee k (capital or capital-labor ratio), then the production function will take the form: q = Tf(k).
As the capital-labor ratio increases, q increases, but to a lesser extent, since the marginal productivity of capital (capital productivity) falls.
In the Solow model, output is determined by investment (I) and consumption (C). It is assumed that the economy is closed from the world market and domestic investments (I) are equal to national savings, or the volume of gross accumulation (S).
The dynamics of output volume in this case depends on the capital ratio, which changes under the influence of the disposal of fixed capital or investment.
In turn, investments depend on the rate of gross accumulation, which is a relative value and is calculated as the ratio of gross accumulation to the created product. The savings rate determines the division of the product into investment, savings and consumption. With an increase in the rate of accumulation (savings), investments increase, exceeding disposal. At the same time, production assets increase. In the short term, the acceleration of economic growth depends on the rate of accumulation.
Subsequently, developing his model, Solow introduced new factors that, along with investment and disposal, affect the capital-labor ratio: labor force growth and technological progress. It is believed that technological changes are labor-saving, promoting advanced training, development of professional skills, and raising the educational level of workers.
(Materials are based on: E.A. Maryganova, S.A. Shapiro. Macroeconomics. Express course: textbook. - M.: KNORUS, 2010. ISBN 978-5-406-00716-7)
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