The Golden Rule of Saving - a hypothetical trajectory of balanced economic growth proposed by Phelps, according to which each generation saves for future generations the same part of the national income that the previous generation leaves to it.

E. Phelps' golden rule of accumulation is satisfied when marginal product minus the disposal rate is equal to zero: MPK – σ = 0.

If an economy begins to grow with a capital stock greater than the Golden Rule, policies must be implemented to lower the savings rate in order to reduce the sustainable level of the capital stock.

This will cause an increase in the level of consumption and a decrease in the level of investment. Capital expenditure will be less than capital disposal. The economy is leaving a steady state. Gradually, as capital stocks decline, output, consumption, and investment will also decline to a new steady state. The level of consumption will be higher than before. And vice versa.

Capital accumulation alone cannot explain continued economic growth. A high saving rate temporarily increases the growth rate, but the economy eventually approaches a steady state in which capital stocks and output are constant.

Population growth is included in the model. We will assume that the population in the economy under consideration is equal to the labor force and is growing at a constant rate n. Population growth complements the original model in 3 ways:

1. 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 because the number of workers grows at a rate n, capital and output also grow at a rate n.

Population growth explains the increase in gross output.

2. Population growth provides an additional explanation for why some countries are rich and others are poor. An increase in the population growth rate reduces the capital-to-labor ratio, and productivity also decreases. Countries with higher population growth rates will have lower levels of GNP per capita.

3. Population growth affects the level of capital accumulation according to salary. MPK - σ = n.

where E is the labor efficiency of 1 employee.

It depends on health, education and qualifications. The L*E component represents labor measured in units of labor with constant efficiency.

The volume of production depends on the number of units of capital and on the number of efficient units work force. Labor efficiency depends on the health, education and qualifications of the workforce.

Technological progress causes an increase in labor efficiency at a constant rate g. This form of technological progress is called labor-saving. Because the labor force grows at a rate of n and the return from each unit of labor grows at a rate of g, the total number of effective units of labor L*E grows at a rate of (n+g).

The Solow model shows that only technological progress can explain the continuously increasing standard of living. It changes and Golden Rule: MPK = σ + n + g.

The state should encourage scientific research, protect copyright, and provide tax incentives.

The optimal rate of capital accumulation should ensure economic growth with a maximum level of consumption. The level of capital accumulation that ensures a steady state with highest level consumption is called gold savings level ( denoted byk**).

From the equation for the steady state (13) it follows that when the saving rate changes, the sustainable level of capital-labor ratio also changes, and, accordingly, the sustainable consumption per capita also changes.

The change in consumption when the saving rate changes depends on the initial state of the economy. Sustainable per capita consumption increases with growth s at low savings rates and falls at high rates. Per capita consumption at a steady-state capital-labor ratio is found as the difference between income and savings :

c*=f(k*(s))-sf(k*(s)). Considering that sf(k*)=(n+d)k*, can be output:

(14)c*=f(k*(s))-(n+d)k*(s).

By maximizing (14) over s, we find: Since , the expression in parentheses must be equal to zero. The capital ratio at which the expression in brackets is equal to zero is called capital-labor ratio corresponding to the golden rule and is denoted by:

Condition (15), which determines the stationary level k that maximizes stationary consumption c, is called the golden rule of capital accumulation. Thus, the savings rate that ensures the maximum amount of sustainable consumption per capita can be found from the condition:

where is the solution to equation (15). So, if we maintain the same level of consumption for all living now and for all future generations, that is, if we treat future generations as we would like them to do with us, then this maximum level stationary consumption per capita that can be provided.

The Golden Rule can be represented graphically. Saving rate s g in Figure 2 corresponds to the golden rule, since sustainable capital k g such that the slope f(k) at a point is equal (n+d). As can be seen from the figure, when the savings rate increases to or decreases to sustainable per capita consumption falls compared to : And .

Rice. 85. The golden rule of capital accumulation.

If the savings rate in the economy exceeds and, accordingly, the sustainable capital-labor ratio is higher than under the golden rule, then the distribution of resources in such an economy is dynamically inefficient. By reducing the saving rate to , it would be possible to achieve an increase in per capita consumption in the long run, The change in per capita consumption is shown schematically in Figure 85.

At the moment when the saving rate decreases, per capita consumption increases sharply and then falls monotonically to . Taking into account the fact that , we find that even during the transition to a new steady state, the economy at each point in time has higher per capita consumption than the initial level.

Thus, an economy with a saving rate exceeding , saves too much and, as a result, the allocation of resources is dynamically inefficient.

Rice. 85. Dynamics of consumption per capita when the savings rate decreases from the level to .

If the saving rate in the economy is less than , then by increasing the saving rate to , it would be possible to achieve a higher sustainable capital-labor ratio, but in the transition period consumption would be lower than at present. Thus, in in this case It cannot be stated unequivocally that such a distribution of resources is ineffective, since everything depends on how society values future consumption relative to current consumption, that is, on intertemporal preferences.

Sustainable capital-to-labor ratio depends on the following parameters: saving rates, depreciation rates and population growth rates.

1. Change in savings rate.

If the state manages to somehow achieve an increase in the savings rate, then the graph of the function sf(k)/k will move upward and sustainable capital will increase, as shown in Figure 85.

Rice. 86. Change in capital-labor ratio as a result of an increase in the saving rate from to

As follows from Figure 86, an increase in the saving rate is followed by a jump in the growth rate of capital-labor ratio, then, as the capital-labor ratio increases, the distance between the curves sf(k)/k And (n+d) contracts and goes to zero. Thus, immediately following an increase in the saving rate, the growth rate of capital becomes higher than the population growth rate, and as the new steady state is approached, the growth rates of K and L converge again.

From here, we can conclude that a change in the saving rate does not affect the long-term growth rate of output, but affects the growth rate in the process of moving towards a steady state. Thus, an increase in the saving rate leads to a sharp increase in the growth rate of labor productivity, however, as it approaches a steady state, this effect disappears.

Fig.88. Dynamics of the output growth rate with an increase in the population growth rate from n 1 to n 2

The growth rate of labor productivity will first become negative and then increase until it returns to zero. In this case, the growth rate of output itself in the new steady state will be higher than in the initial one, as shown in Figure 88.

In a closed economy, where more savings actually means more investment, stimulating savings (for example, by reducing taxes on securities income) could promote economic growth. On the other hand, the state could stimulate investment directly, for example, through investment tax credits.

Another component of economic growth is scientific technical progress and the accumulation of human capital, that is, knowledge and experience. Thus, the government should pursue policies aimed at stimulating education, research and development by subsidizing these areas directly or by rewarding firms that actively invest in human capital through various tax incentives.

There are basic, fairly simple models that explain the essence and possibility of using macroeconomic production functions.

In addition to one or another combination of production factors, the flexibility of the production function is provided by special coefficients. They are called elasticity coefficients. These are power coefficients of factors of production, showing how the volume of production will increase if the factor of production increases by one. The elasticity coefficient is found empirically by solving a special system of equations obtained from the original production function model.

The literature distinguishes between production functions with both constant and variable elasticity coefficients. Constant ratios mean that the product grows in the same proportion as the factors of production.

The simplest two-factor model: capital K and labor L.

If the elasticity coefficients are constant, then the function is written as follows:

Where Y- national product;

L - labor (man-hours or number of employees);

K is the capital of the entire society (machine hours or amount of equipment);

Elasticity coefficient;

A is a constant coefficient (found by calculation).

When analyzing the aggregate demand and aggregate supply (AD-AS) model, it was assumed that the only variable factor of production was labor, and capital and technology were considered constant. These assumptions cannot be considered adequate for long-term analysis, since in the long term there is both a change in the capital stock and the presence of technical progress. Thus, with a change in capital and technology, the level of full employment will also change, which means that the aggregate supply curve will shift, which will inevitably affect equilibrium output. However, an increase in output does not mean that the country's population has become richer, since the population also changes along with output. Economic growth usually refers to the growth of real GDP per capita.

N. Kaldor (in 1961), studying economic growth in developed countries, came to the conclusion that there are certain patterns in changes in output, capital and their ratios in the long term. The first empirical fact is that the growth rate of employment is less than the growth rate of capital and output or, in other words, the capital-to-employment ratio (capital-labor ratio) and the output-to-employment ratio (labor productivity) are growing. On the other hand, the ratio of output to capital showed the absence of a significant trend, that is, output and capital changed at approximately the same pace.

Kaldor also looked at the dynamics of returns to factors of production. It was noted that the real wage shows a steady upward trend, while the real interest rate does not have a definite trend, although it is subject to continuous fluctuations. Empirical research also shows that productivity growth rates vary significantly between countries.

The question of what factors influence economic growth remains one of the central issues in macroeconomics, and debate over the sources of economic growth continues to this day. However, most economists, following Robert Solow's classic 1957 work, identify the following key drivers of economic growth: technological progress, capital accumulation and labor force growth.

In order to describe the contribution of each of these factors to economic growth, consider output Y as a function of the capital stock ( K), labor resources used ( L):

The volume of production depends on the capital stock and the labor used. The production function has the property constant return from scale.

For simplicity, let’s correlate all values with the number of employees (L):

Y/ L = F (K/ L, 1).

This equation shows that output per worker is a function of capital per worker.

Let's denote:

y = Y/ L – output per 1 worker (labor productivity, output);

k = K/ L – capital-labor ratio.

This function, according to neoclassical ideas, should illustrate the following: if the amount of social capital used per worker increases, then the product per worker (marginal labor productivity) grows, but to a lesser extent.

Graphically, this means that the function f(K) has a first derivative that is greater than zero f (K)>0. The second derivative of the function is f (K)<0. Это означает, что хотя функция и является положительной, она убывает по мере прироста продукта и производительности труда (рис.12.2).

Rice. 12.2 Neoclassical production function

Capital and labor are rewarded based on their respective marginal factors of production. The remuneration of capital is determined by the tangent of the angle of inclination to the curve f(K) at point P - the marginal productivity of capital. Then, WN is the share of capital in the total product; OW – share of wages in the product; OW – the whole product.

In the Solow model, demand for goods and services comes from consumers and investors. Those. The output produced by each worker is divided between consumption per worker and investment per worker:

The model assumes that the consumption function takes the simple form:

c = (1 – s) * y,

where the savings rate s takes values 0 – 1.

This function means that consumption is proportional to income.

Let’s replace the value – c – with the value (1 – s)* y:

y = (1 – s) * y + i.

After transformation we get: i = s*y.

This equation shows that investment (like consumption) is proportional to income. If investment equals saving, then the saving rate (s) also shows how much of the output is allocated to investment.

Capital reserves can change for 2 reasons:

Investments lead to an increase in inventories;

Part of the capital wears out, i.e. is depreciated, which reduces inventories.

∆k = i – σk,

change in capital stock = investment - disposal,

σ - disposal rate; ∆k – change in capital reserves per 1 employee per year.

If there is a single level of capital-to-labor ratio at which investment equals depreciation, then the economy will reach a level that will not change over time. This is a situation of sustainable capital ratio.

The level of capital accumulation that ensures a stable state with the highest level of consumption is called the Golden level of capital accumulation.

In 1961 American economist E. Phelps developed the rule of accumulation, called the “golden” rule. In general, the golden rule of accumulation can be formulated as follows: the level of capital accumulation that ensures the highest consumption of society and a stable state of the economy is called the golden level of capital accumulation, i.e. the optimal equilibrium level of the economy will be achieved provided that capital income is fully invested.

The Golden Rule of Saving - a hypothetical trajectory of balanced economic growth proposed by Phelps, according to which each generation saves for future generations the same part of the national income that the previous generation leaves to it.

E. Phelps' golden rule of accumulation is satisfied when the marginal product minus the disposal rate is equal to zero:

If the economy begins to develop from a capital reserve greater than the Golden Rule, It is necessary to implement policies aimed at reducing the saving rate in order to reduce the sustainable level of the capital stock.

Population growth explains the increase in gross output.

3. Population growth affects the level of capital accumulation according to salary.

where E is the labor efficiency of 1 employee.

It depends on health, education and qualifications. The L*E component represents labor measured in units of labor with constant efficiency.

The volume of production depends on the number of units of capital and on the number of effective units of labor. Labor efficiency depends on the health, education and qualifications of the workforce.

The Solow model shows that only technological progress can explain the continuously increasing standard of living. This also changes the Golden Rule:

MPK = σ + n + g.

The state should encourage scientific research, protect copyright, and provide tax incentives.

The “golden rule” of accumulation was formulated by the American economist E. Phelps in 1961. According to the rule, per capita consumption in a growing economy reaches its maximum at the moment when the marginal product of capital becomes equal to the rate of economic growth.

At the optimal rate of capital accumulation (&**), corresponding to the “golden rule”, the condition must be met: the marginal product of capital is equal to depreciation (capital retirement), i.e.:

and if we take into account the rate of population growth and technological progress, then:

Now let’s assume that the economy is in a state of equilibrium, but does not correspond to the “golden rule” and the government has to determine a growth policy and develop a program for achieving maximum per capita consumption.

In this case, two options for the state of the economy are possible.

1. The economy has a greater stock of capital than is necessary to comply with the “golden rule”.

2. The capital stock does not reach the “golden rule” level.

Determining the capital stock that corresponds to the “golden rule” means solving the problem of choosing the optimal rate of accumulation.

Let's consider the first option for economic development. A decrease in the savings rate leads to an increase in consumption and a decrease in investment. At the same time, the economy goes out of equilibrium.

The new equilibrium will correspond to the "golden rule" with a higher level of consumption, since the initial stock of capital is excessively high, with a reduction in income and the level of investment.

The second option for economic development requires a responsible choice of politicians, since the decisions they make affect the vital interests of different generations. An increase in the savings rate leads to a decrease in consumption and an increase in investment. As capital accumulates, production, consumption and investment begin to increase until a new steady state with a higher level of consumption is reached. But a high level of consumption will be preceded by a transition period with a decrease in consumption. This period can span the life of an entire generation, providing the benefits of economic growth to subsequent generations.

The winners of the Nobel Prize in Economics in 2004 were American Edward Prescott and Norwegian Finn Kydland living in the United States. Scientists Award

awarded for "their contributions to dynamic macroeconomics: the timing of economic policies and driving forces within business cycles." A press release published on the Nobel Prize website states: “...The driving forces and fluctuations within business cycles and the design of economic policy are key areas of macroeconomic research. Finn Kydland and Edward Prescott have made fundamental contributions to these important areas, not only in terms of macroeconomic analysis, but also in terms of practice in monetary and fiscal policy in many countries."

The study carried out by scientists combined the analysis of long-term economic growth and short-term economic fluctuations. Scientists use R. Solow's model of economic growth. The contribution of the most important factor of long-term economic growth - technical progress - is determined by the so-called “Solow residual”. Technological progress can cause short-term cyclical fluctuations, as total factor productivity increases under the influence of a technology shock. The laureates created an entire scientific field called “real economic cycles,” according to which the source of cyclical fluctuations is supply-side shocks. This theory uses the following provisions: a) price flexibility in the short term; b) changes in real indicators depend on real changes in the economy: technological shifts and changes in fiscal policy.

As a result of an increase in labor productivity, wages increase, which causes an increase in the supply of labor in a given period of time and capital productivity. Kydland and Prescott consistently develop the neoclassical idea of the ability of a market economy to self-regulate without government intervention. In their opinion, the fall in output is only the result of temporary deviations in economic growth rates.

"The Golden Rule of Saving"

In the simplest model of accumulation, three sectors are distinguished: enterprises, the state and the population. For each sector, monetary accumulation is expressed as the difference between income and investment expenditure.

For industrial enterprises, the main sources of capital accumulation are funds in the form of temporarily free capital. For the production process, the accumulation of money is necessary to ensure continuity, expand production, and limit it from various fluctuations in demand and supply. Businesses typically account for up to 20% of total cash accumulation.

Government funds represent government reserves and act as the difference between tax revenues and expenditures of the central government and local governments. The main prerequisites for such accumulation are: the state of the state budget, investment expenses, which require the preliminary accumulation of funds.

The public sector also includes the accumulation of monetary capital, carried out through state pension insurance funds. Although the source of funds in these funds is mainly the income of the population, the capital is managed by the state. The state's share in the total capital accumulation accounts for about 10%.

3. Savings of the population represent that part of wages that is not used for current needs and is set aside for unforeseen cases or provision in old age, for the purchase of durable items, expensive goods. In the economic literature, four motives for such accumulation are identified: income-related, commercial motive, precautionary motive, speculative (P. Samuelson and M. Friedman).

The growth of household savings as the main source of accumulation is a characteristic process for all countries. The indicator of this growth is both the absolute value and the savings rate.

The growth in the savings rate can be described using a function called "golden rule of accumulation":

SY = PCR + YR+DU + RR + GPP,

Where S.Y.- share of savings in income;

PCR- rate of change in consumer prices;

YR- rate of change in real income;

D.U.- differences in unemployment rates;

R.R.- real interest rate;

GPP- rate of change in government consumption.

The accumulation process is influenced by the following factors:

With income growth consumption of durable goods increases, which requires preliminary cash savings;

changes in the structure of population consumption;

3) influence tax system and social insurance.

The higher the income taxes, the lower the disposable income and, therefore, the savings. The role of the social insurance system is twofold. On the one hand, it reduces income and savings, and on the other, it makes it possible to increase national economic accumulation;

inflation, the meaning of which is also ambiguous. According to one theory, money depreciates, so it moves to other assets (real estate, gold), but in fact, people, having even small amounts, begin to save more for a rainy day. The second point of view connects the change in savings with inflation expectations, which leads to an increase in savings, since the precautionary motive plays a role in this;

cyclical economic development, in the process of which, during the recovery, savings decrease as a favorable environment weakens the precautionary motive and the speculative motive (interest rates decrease). During a crisis, both of these motives manifest themselves quite clearly, which leads to an increase in savings.

non-cash payment of wages, which leads to some savings (reduced costs of going to the bank) and the ability of the bank to use account balances in the form of loan capital.

In general, there are three main forms of savings: deposits in the credit system, purchase of securities, deposits in insurance companies. Nevertheless, different actors prefer certain forms of accumulation.