Harrod Model - Domara

The Harrod – Domar model is a neo-Keynesian model of economic growth that explains the growth of the economy, provided that the coefficients of capital intensity and the propensity to save in the long run are constant. Animation and acceleration processes were integrated for the first time in the model. The model combines the work of Roy F. Harrod , who first proposed his model of guaranteed growth in 1939 , and Eusei Domar , who in 1946 expanded the conditions for the short-term Keynesian equilibrium in the long-term period. Subsequent polemics and criticisms of the Harrod-Domar model led to the creation of the neoclassical economic growth model of Solow .

Creation History

The first work of Roy Harrod, "An Essay on the Theory of Dynamics" ^[1] on economic dynamics, was published in 1939, after the war in 1948, Harrod's lectures, delivered in 1946-1947 at the University of London , in the work " On the Theory of Economic Dynamics " were published. In 1973, Harrod, publishing the book "Theory of Economic Dynamics", containing a holistic presentation of his theory of economic dynamics, supplemented a number of concepts with clearer definitions of the Harrod model ^[2] .

Eusei Domar in 1941-1942, based on the scheme in the book of Alvin Hansen's "Budget Policy and Business Cycles" ( Fiscal Policy and Business Cycles , 1941), which demonstrated the influence of a constant flow of investments on national income, came to the conclusion that such a flow leads to growing income, and the result was the article “ The Burden of the Debt and the National Income , 1944) ^[3] , which, in turn, led to other articles, including“ Expansion of capital, pace growth and employment ”( Capital Expansion, Rate of Growth and Employment , 1946 ^[4] ). In these articles, Domar used the growth rate as an analytical tool in the study of specific economic problems, reprinting them in Essays on the Theory of Economic Growth in 1957, forming the Domar model ^[3] .

In 1956, Robert Solow was one of the first to combine these two models (two approaches) into one, calling it the Harrod – Domar model in his article “Contribution to the Theory of Economic Growth ^[5] ). The model itself continues to be in demand today in a number of studies of international organizations ^[6] , as well as its individual parts, the Harrod model ^[7] and the Domar model, are still used ^[8] .

Assumptions

The model has a number of prerequisites ^[9] ^[10] ^[11] :

Capital intensity (ratio of capital to product $K/Y$ ${\ displaystyle K / Y}$ ) is constant.
The rate of expansion of labor supply $n$ ${\ displaystyle n}$ and rate of increase in labor productivity $\delta$ ${\ displaystyle \ delta}$ exogenous and permanent.
Penchant for saving $s$ ${\ displaystyle s}$ constant, that is, with an increase in GDP income by $\Delta Y$ ${\ displaystyle \ Delta Y}$ amount of savings $S$ ${\ displaystyle S}$ will increase proportionally and will be equal to $s\Delta Y$ ${\ displaystyle s \ Delta Y}$ .
Investments $I$ ${\ displaystyle I}$ equal to savings $I=S$ ${\ displaystyle I = S}$ , that is, an increase in savings due to an increase in income leads to an increase in investments, increasing the capital used to produce GDP.
Additional capital adds GDP income to capital productivity ratio $c$ ${\ displaystyle c}$ and then GDP growth again increases the volume of savings, etc.
Issue depends on one resource - capital $K$ ${\ displaystyle K}$ .
Investment lag is zero.

Mechanism

The increase in aggregate demand depends on the growth in investment, which depends on savings ^[11] :

\Delta Y=m\Delta Y=1/(1-c)\Delta Y=1/s\Delta I

{\ displaystyle \ Delta Y = m \ Delta Y = 1 / (1-c) \ Delta Y = 1 / s \ Delta I}

where

m

{\ displaystyle m}

- cost multiplier,

c

{\ displaystyle c}

- marginal propensity to consume,

s

{\ displaystyle s}

- marginal propensity to save.

Accelerator and Capital Ratio

Accelerator Model Harrod-Domar

M. Blaug demonstrates the effect of the accelerator ^[12] :

G=s/z

{\ displaystyle G = s / z}

,

Where $G=\Delta Y/Y$ ${\ displaystyle G = \ Delta Y / Y}$ - income growth rate, $s$ ${\ displaystyle s}$ - average propensity to save, $S/Y=\Delta S/\Delta Y$ ${\ displaystyle S / Y = \ Delta S / \ Delta Y}$ - marginal propensity to save, $z=\Delta K/\Delta Y$ ${\ displaystyle z = \ Delta K / \ Delta Y}$ - accelerator, incremental rate of return on assets.

With monotonous growth, planned savings are equal to planned investments, the load of the created production capacities is full, and so does income by I / z, and investment rates grow by I / zY or s' / z percent in the next period. Equilibrium income $Y$ ${\ displaystyle Y}$ takes place at the intersection of the savings function S (Y) and the investment function $I=f(\Delta Y)$ ${\ displaystyle I = f (\ Delta Y)}$ function $P=\Delta K/\Delta Y=z$ ${\ displaystyle P = \ Delta K / \ Delta Y = z}$ Is a constant, and is a coefficient of performance. $OY_{1}$ ${\ displaystyle OY_ {1}}$ - initial income at full capacity utilization, entails a flow of investment $S_{1}Y_{1}=\Delta K$ ${\ displaystyle S_ {1} Y_ {1} = \ Delta K}$ due to which power grows on $S_{1}P_{1}=Y_{1}Y_{2}=\Delta Y$ ${\ displaystyle S_ {1} P_ {1} = Y_ {1} Y_ {2} = \ Delta Y}$ with an increase in the rate of capital return $S_{1}Y_{1}/S_{2}Y_{2}=z$ ${\ displaystyle S_ {1} Y_ {1} / S_ {2} Y_ {2} = z}$ , which leads to an increase in capacity with $OY_{1}$ ${\ displaystyle OY_ {1}}$ before $OY_{2}$ ${\ displaystyle OY_ {2}}$ and to increase investment from I 'to I "- the effect of the accelerator, which leads to investment $S_{2}Y_{2}$ ${\ displaystyle S_ {2} Y_ {2}}$ and capacity growth at $S_{2}P_{2}=Y_{2}Y_{3}$ ${\ displaystyle S_ {2} P_ {2} = Y_ {2} Y_ {3}}$ to new income $OY_{3}$ ${\ displaystyle OY_ {3}}$ . The gain of the second year is greater than the first $OY_{3}-OY_{2}>OY_{2}-OY_{1}$ ${\ displaystyle OY_ {3} -OY_ {2}> OY_ {2} -OY_ {1}}$ Thus, production capacities are growing, and their growth is increasing in absolute value, attracting investment growth in order to avoid excess capacity. Income, investment, savings and consumption are growing exponentially ^[12] .

Accelerator (refers to the income of the first period) is equal $S_{2}Y_{2}/Y_{1}Y_{2}=S_{3}Y_{3}/Y_{2}Y_{3}$ ${\ displaystyle S_ {2} Y_ {2} / Y_ {1} Y_ {2} = S_ {3} Y_ {3} / Y_ {2} Y_ {3}}$ inverse to the productivity coefficient (refers to the income of the next period), which is the slope of the function P ', and is equal to $S_{1}Y_{1}/Y_{1}Y_{2}=S_{2}Y_{2}/Y_{2}Y_{3}$ ${\ displaystyle S_ {1} Y_ {1} / Y_ {1} Y_ {2} = S_ {2} Y_ {2} / Y_ {2} Y_ {3}}$ ^[12] .

Keynesian multiplier inverse of marginal propensity to save (inverse of the slope of the function $S(Y)$ ${\ displaystyle S (Y)}$ ) is equal $\Delta Y/\Delta I$ ${\ displaystyle \ Delta Y / \ Delta I}$ , shifting the function from I 'to I ", multiplies the growth of investment in additional income $Y_{2}Y_{3}$ ${\ displaystyle Y_ {2} Y_ {3}}$ , generating unplanned savings that must be invested. And the accelerator accelerates $Y_{2}Y_{3}$ ${\ displaystyle Y_ {2} Y_ {3}}$ in additional investments, ensuring the equality of planned savings and planned investments as a condition for establishing equilibrium. The propensity to save, the accelerator and the coefficient of capital productivity are constants ^[12] .

Equilibrium state

The combined Harrod — Domar model is called the “knife-edge” model, because if the equilibrium is upset, then no built-in stabilizers return the economy to an equilibrium state. The model explains the guaranteed growth rate ^[10] :

g=as

{\ displaystyle g = as}

,

Where

g=\Delta Y/Y

{\ displaystyle g = \ Delta Y / Y}

S=I=\Delta K=\Delta Y/c=sY\to g=\Delta Y/Y=as

{\ displaystyle S = I = \ Delta K = \ Delta Y / c = sY \ to g = \ Delta Y / Y = as}

.

Deviation from equilibrium leads to even greater deviation from equilibrium: if planned investments are more than effective savings, then expected demand is less than effective, which means that firms, due to lack of savings relative to planned investments, reduce investments, reducing supply, making the gap even bigger between real supply and effective demand, causing inflation. And if planned investments are less than effective savings, then a recession occurs ^[10] .

For a regional economy, equilibrium conditions, the sum of savings S and import M is equal to the sum of investments I and exports X ^[10] :

S+M=I+X

{\ displaystyle S + M = I + X}

(s_{i}+m_{i})Y_{i}=I_{i}+x_{i}Y_{i}

{\ displaystyle (s_ {i} + m_ {i}) Y_ {i} = I_ {i} + x_ {i} Y_ {i}}

(s_{i}+m_{i}-x_{i})Y_{i}=I_{i}

{\ displaystyle (s_ {i} + m_ {i} -x_ {i}) Y_ {i} = I_ {i}}

\Delta Y_{i}/Y_{i}=(s_{i}+m_{i}-x_{i})a_{i}=g_{i}

{\ displaystyle \ Delta Y_ {i} / Y_ {i} = (s_ {i} + m_ {i} -x_ {i}) a_ {i} = g_ {i}}

,

Where $m=M/Y$ ${\ displaystyle m = M / Y}$ - the tendency to import capital, $x=X/Y$ ${\ displaystyle x = X / Y}$ - propensity to export capital, $I=\Delta K=\Delta Y/a$ ${\ displaystyle I = \ Delta K = \ Delta Y / a}$ .

Dynamics

The model describes the dynamics of income Y (t) ^[11] :

Y(t)=C(t)+I(t)

{\ displaystyle Y (t) = C (t) + I (t)}

,

where C is the consumption function , I is the investment function, t is time.

The growth rate of national income is equal to the growth rate of investment:

g=\Delta Y/Y=\Delta I/I=s/k

{\ displaystyle g = \ Delta Y / Y = \ Delta I / I = s / k}

,

where k is the capital intensity, s is the savings rate, g is the growth rate.

The growth rate is determined simultaneously by the rate of savings and output per unit of capital cost or per unit of investment (an indicator of investment efficiency). The growth rate of national income is the faster, the greater the volume of savings and investments in the economy. The real growth rate, achievable at a given level of savings and investment, is predetermined by the increase in the product, which gives one additional unit of investment. The equilibrium income growth rate (at which production capacities are fully used) is directly proportional to the rate of savings and marginal productivity of capital. To maintain equilibrium, investment and national income must grow at the same and constant pace. Such a dynamic equilibrium is rarely stable, because the growth rate of investment in the private sector may deviate from the level set by the model ^[11] .

Harrod supplemented the model with an endogenous investment function. Any change in income causes a corresponding change in investment is directly proportional to the change in income ^[11] :

I_{t}=z(Y_{t}-Y_{t-1})

{\ displaystyle I_ {t} = z (Y_ {t} -Y_ {t-1})}

,

where z is the accelerator, I is the investment.

Guaranteed Growth

Under equilibrium conditions, aggregate demand is equal to aggregate supply, and then in the current period growth rates are maintained ^[9] :

\Delta Y_{t}/Y_{t-1}=s/(z-s)

{\ displaystyle \ Delta Y_ {t} / Y_ {t-1} = s / (zs)}

.

Ratio $s/(z-s)$ ${\ displaystyle s / (zs)}$ , providing an equilibrium level in the economy (full use of production capacity) is called a guaranteed growth rate .

If the actual growth rate is lower than guaranteed, then the economic system is moving away from the state of equilibrium. If the guaranteed growth rate is higher than natural (provides “full employment”), then due to a lack of labor resources, the actual growth rate will be lower than the guaranteed one, and producers who are disappointed in their expectations will reduce output and investment: there will be a state of depression. If the guaranteed growth rate is less than natural, then the actual rate may exceed the guaranteed one, since the existing surplus of labor resources will create the preconditions for investment growth: a boom will begin. Only with the equality of guaranteed, natural and actual growth rates is the ideal development of the national economy achieved. Since any deviation of investments from the conditions of a guaranteed growth rate takes the system out of balance, the Harrod model is unstable ^[11] .

Production Function

The production technology of the model is described by the Leontief production function with constant marginal productivity of capital, provided that labor is not a scarce resource. The existing excess supply in the labor market determines a constant price level ^[11] :

Y_{t}=min(AK_{t},BL_{t})

{\ displaystyle Y_ {t} = min (AK_ {t}, BL_ {t})}

,

Where $A$ ${\ displaystyle A}$ and $B$ ${\ displaystyle B}$ - exogenous production parameters, $K$ ${\ displaystyle K}$ - capital $L$ ${\ displaystyle L}$ - labor.

Leontief production function

Harrod-Domar Model

R. Barro and H. Sala-i-Martin note that the Leontief production function (function with fixed proportions), which corresponds to the CES function ^[13] :

Y=A{\cdot }{\Big (}\alpha (b{\cdot }K)^{\Psi }+(1-\alpha )((1-b){\cdot }L)^{\Psi }{\Big )}^{\frac {1}{\Psi }}

{\ displaystyle Y = A {\ cdot} {\ Big (} \ alpha (b {\ cdot} K) ^ {\ Psi} + (1- \ alpha) ((1-b) {\ cdot} L) ^ {\ Psi} {\ Big)} ^ {\ frac {1} {\ Psi}}}

but where $\Psi \to -\infty$ ${\ displaystyle \ Psi \ to - \ infty}$ :

Y=F(L,K)=min(AK,BL)

{\ displaystyle Y = F (L, K) = min (AK, BL)}

,

Where $A>0$ ${\ displaystyle A> 0}$ and $B>0$ ${\ displaystyle B> 0}$ Are constants.

Thus, when $AK=BL$ ${\ displaystyle AK = BL}$ - all workers and machines are loaded; at $AK>BL$ ${\ displaystyle AK> BL}$ - capital is used in volume $(B/A)L$ ${\ displaystyle (B / A) L}$ , and the remaining is not claimed; at $AK<BL$ ${\ displaystyle AK <BL}$ - the volume of labor is used in volume $(A/B)K$ ${\ displaystyle (A / B) K}$ and the rest remains unemployed. The assumption of the absence of interchangeability between capital and labor leads to the fact that there is either an infinite increase in unemployment or simple equipment.

When per capita, the production function has the form ^[13] :

y=min(Ak,B)

{\ displaystyle y = min (Ak, B)}

,

Where $y=Y/L$ ${\ displaystyle y = Y / L}$ , $k=K/L$ ${\ displaystyle k = K / L}$ .

At $k<B/A$ ${\ displaystyle k <B / A}$ capital is fully utilized and $y=Ak$ ${\ displaystyle y = Ak}$ , and the production function curve intersects zero and has a slope $A$ ${\ displaystyle A}$ .

At $k>B/A$ ${\ displaystyle k> B / A}$ capital is constant and $Y=BL$ ${\ displaystyle Y = BL}$ , $y=B$ ${\ displaystyle y = B}$ . At $k\to \infty$ ${\ displaystyle k \ to \ infty}$ marginal product $f^{'}(k)=0$ ${\ displaystyle f ^ {'} (k) = 0}$ , which means the Inada condition is satisfied, the production function does not generate endogenous growth.

At $k<B/A$ ${\ displaystyle k <B / A}$ savings curve shape $sf(k)/k$ ${\ displaystyle sf (k) / k}$ - direct at the level $s A$ ${\ displaystyle sA}$ , and when $k>B/A$ ${\ displaystyle k> B / A}$ the savings curve tends to zero at $k\to \infty$ ${\ displaystyle k \ to \ infty}$ .

Depreciation curve $(n+\delta )$ ${\ displaystyle (n + \ delta)}$ has the shape of a horizontal line at $(n+\delta )$ ${\ displaystyle (n + \ delta)}$ .

At a low rate of savings $sA<n+\delta$ ${\ displaystyle sA <n + \ delta}$ the savings curve does not cross the depreciation curve, so the steady state $k^{*}$ ${\ displaystyle k ^ {*}}$ no, the capital growth rate is negative, the economy is shrinking $k,y,c\to \infty$ ${\ displaystyle k, y, c \ to \ infty}$ , it has ever-increasing unemployment .

At a high savings rate $sA>n+\delta$ ${\ displaystyle sA> n + \ delta}$ the savings curve approaches zero at $k\to \infty$ ${\ displaystyle k \ to \ infty}$ and crosses the depreciation curve at a point of steady stationary value $k^{*}>B/A>$ ${\ displaystyle k ^ {*}> B / A>}$ so that the capital growth rate is negative at $k>k^{*}$ ${\ displaystyle k> k ^ {*}}$ , and when $k<k^{*}$ ${\ displaystyle k <k ^ {*}}$ positive. At $k^{>}*B/A$ ${\ displaystyle k ^ {>} * B / A}$ equipment is idle, part of the capital is not in demand and is growing monotonously, but there are no unemployed workers. Because $k$ ${\ displaystyle k}$ Is a constant in a stationary state, then the growth rate $K$ ${\ displaystyle K}$ equal to the growth rate $L$ ${\ displaystyle L}$ and is equal $n$ ${\ displaystyle n}$ . The share of equipment used is constant, the number of unclaimed equipment is growing at a pace $n$ ${\ displaystyle n}$ . A stationary state in which capital and labor are fully in demand in production, $sA=n+\delta$ ${\ displaystyle sA = n + \ delta}$ ^[13] .

Model

Under the condition of stationarity, where the growth rate of capital is equal to the growth rate of output, investments may not be equal to savings (the difference is covered by net imports). And if net imports (the difference between region’s imports and exports (MX)) are negative, eating up the region’s savings, the region’s revenue growth rate will be zero. As the share of net imports increases, the region’s revenue growth rate will increase. Therefore, regions with positive net exports are growing faster, as they receive additional savings from other regions. Equilibrium growth can be achieved due to the influx of workers from the regions in conditions of labor shortages, and in case of excess, eliminated due to the outflow of workers to other regions. Differences in regional growth rates remain due to the initial difference in growth, the difference in growth rates will only increase over time, but if poor regions are net importers of capital, then equilibrium paths converge regional growth rates, thus poorer regions will grow with higher the pace. Regions with a high propensity for saving and efficient use of capital, or with an industry structure characterized by low capital intensity, will have a higher growth rate ^[10] .

Criticism of the Model

The model explains the high growth rates of the regions, which initially have insignificant domestic savings and the ratio of capital to output, negative trade balance due to the import of capital. However, the following disadvantages of the Harrod — Domar model can be distinguished:

a closed economy, that is, the model does not explain the occurrence of flows of capital and labor between regions in case of imbalance;
the model does not demonstrate the possibility of convergence — divergence between regions when the region is a net importer of capital or a net exporter of labor;
the model also does not explain the attractiveness of poor regions for investments, which become net exporters of capital ^[10] ;
increasing savings and investment is a necessary but not sufficient condition for accelerated growth;
the need for structural, institutional and cultural prerequisites (developed commodity and money markets, well-trained personnel, the prevailing psychology of entrepreneurship, an effective administrative system) for the model to work;
instability of the trajectory of balanced growth, the economy does not have stabilizers that can absorb external influences ^[11] ;
the unrealistic assumption that there is no interchangeability between labor and capital in the long run due to the possibility of organizational and technological changes, as well as a constant rate of capital to issue for various industries in the long run, the constancy of the savings propensity rate in the long run with significant changes in the quality of life ^{[ 14]} ;
ignored the role of government, foreign trade, non-economic factors ^[14] .

Notes

↑ Harrod RF An Essay in Dynamic Theory // Economic Journal. - 1939. - March ( No. 49 ). - S. 14–33 .
↑ Manevich V.E., Nikolaev L.K., Ovsienko V.V. Harrod's Theory of Economic Dynamics and Analysis of the Russian Economy / R. Harrod. - Theory of economic dynamics. - M .: CEMI RAS, 2008. - S. 1. - ISBN 978-5-8211-0464-9 .
↑ ¹ ² Blaug M. 100 great economists after Keynes . - SPb. : Economics, 2009 .-- S. 97-99 . - ISBN 978-5-903816-03-3 .
↑ Domar E. Capital Expansion, Rate of Growth and Employment // Econometrica. - 1946. - April ( t. 14 , No. 2 ). - S. 137-147 .
↑ Solow RM A Contribution to the Theory of Economic Growth // The Quarterly Journal of Economics. - 1956.- February Vol.70, No.1. - P. 65-94.
↑ Easterly W. The ghost of financing gap: how the Harrod-Domar growth model still haunts development economics // Research working paper, Washington, DC: World Bank. - 1997. - No. WPD 1807 . - P. 13.
↑ Hoover KD Was Harrod Right? // GREDEG WP No. 2013-02. - 2008.
↑ Boianovsky M. Modeling economic growth: Domar on moving equilibrium // CHOPE Working Paper No. 2015-10. - 2015. - October.
↑ ¹ ² Hemberg D. Early growth theory: Domar and Harrod models / Ed .: Afanasyeva V.S. and Entova R.M. - Modern economic thought. - M .: Progress, 1981.
↑ ¹ ² ³ ⁴ ⁵ ⁶ Limonov L.E. Regional economics and spatial development . - M .: Publishing house Yurayt, 2015. - T. 1. - S. 167-173. - ISBN 978-5-9916-4444-0 .
↑ ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ Nureev R.M. Development Economics: Models of the Formation of a Market Economy . - M .: Norma, 2008 .-- S. 26-29. - ISBN 978-5-468-00159-2 .
↑ ¹ ² ³ ⁴ Blaug M. Economic thought in retrospect . - M .: Delo Ltd., 1994 .-- S. 153-155. - ISBN 5-86461-151-4 .
↑ ¹ ² ³ Barro R.J. , Sala-i-Martin H. Economic growth. - M .: Binom, 2010 .-- S. 97-100. - ISBN 978-5-94774-790-4 .
↑ ¹ ² TR Jain, Balbir Kaur Bojaj. Development Economics . - VK Publications. - 2008 .-- S. 112-128. - ISBN 9788187344056 . Archived February 7, 2016 on the Wayback Machine

Links

[:0-1] Harrod RF An Essay in Dynamic Theory // Economic Journal. - 1939. - March ( No. 49 ). - S. 14–33 .

[:1-2] Manevich V.E., Nikolaev L.K., Ovsienko V.V. Harrod's Theory of Economic Dynamics and Analysis of the Russian Economy / R. Harrod. - Theory of economic dynamics. - M .: CEMI RAS, 2008. - S. 1. - ISBN 978-5-8211-0464-9 .

[:2-3] ¹ ² Blaug M. 100 great economists after Keynes . - SPb. : Economics, 2009 .-- S. 97-99 . - ISBN 978-5-903816-03-3 .

[:8-4] Domar E. Capital Expansion, Rate of Growth and Employment // Econometrica. - 1946. - April ( t. 14 , No. 2 ). - S. 137-147 .

[:11-5] Solow RM A Contribution to the Theory of Economic Growth // The Quarterly Journal of Economics. - 1956.- February Vol.70, No.1. - P. 65-94.

[:12-6] Easterly W. The ghost of financing gap: how the Harrod-Domar growth model still haunts development economics // Research working paper, Washington, DC: World Bank. - 1997. - No. WPD 1807 . - P. 13.

[:13-7] Hoover KD Was Harrod Right? // GREDEG WP No. 2013-02. - 2008.

[:14-8] Boianovsky M. Modeling economic growth: Domar on moving equilibrium // CHOPE Working Paper No. 2015-10. - 2015. - October.

[:3-9] ¹ ² Hemberg D. Early growth theory: Domar and Harrod models / Ed .: Afanasyeva V.S. and Entova R.M. - Modern economic thought. - M .: Progress, 1981.

[:4-10] ¹ ² ³ ⁴ ⁵ ⁶ Limonov L.E. Regional economics and spatial development . - M .: Publishing house Yurayt, 2015. - T. 1. - S. 167-173. - ISBN 978-5-9916-4444-0 .

[:6-11] ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ Nureev R.M. Development Economics: Models of the Formation of a Market Economy . - M .: Norma, 2008 .-- S. 26-29. - ISBN 978-5-468-00159-2 .

[:5-12] ¹ ² ³ ⁴ Blaug M. Economic thought in retrospect . - M .: Delo Ltd., 1994 .-- S. 153-155. - ISBN 5-86461-151-4 .

[:7-13] ¹ ² ³ Barro R.J. , Sala-i-Martin H. Economic growth. - M .: Binom, 2010 .-- S. 97-100. - ISBN 978-5-94774-790-4 .

[:9-14] ¹ ² TR Jain, Balbir Kaur Bojaj. Development Economics . - VK Publications. - 2008 .-- S. 112-128. - ISBN 9788187344056 . Archived February 7, 2016 on the Wayback Machine