Solow–Swan model

While “neoclassical growth model” redirects here, it mayalso refer to the Ramsey–Cass–Koopmans model.

The Solow–Swan model is an exogenous growthmodel, an economic model of long-run economic growthset within the framework of neoclassical economics. Itattempts to explain long-run economic growth by look-ing at capital accumulation, labor or population growth,and increases in productivity, commonly referred toas technological progress. At its core is a neoclassi-cal aggregate production function, usually of a Cobb–Douglas type, which enables the model “to make con-tact with microeconomics”.[1]:26 The model was devel-oped independently by Robert Solow and Trevor Swan in1956,[2][3] and superseded the post-Keynesian Harrod–Domar model. Due to its particularly attractive mathe-matical characteristics, Solow–Swan proved to be a con-venient starting point for various extensions. For in-stance, in 1965, David Cass and Tjalling Koopmansintegrated Frank Ramsey’s analysis of consumer opti-mization, thereby endogenizing the savings rate—see theRamsey–Cass–Koopmans model.

1 Background

The neo-classical model was an extension to the 1946Harrod–Domar model that included a new term: pro-ductivity growth. Important contributions to the modelcame from the work done by Solow and by Swan in 1956,who independently developed relatively simple growthmodels.[2][3] Solow’s model fitted available data on USeconomic growth with some success.[4] In 1987 Solowwas awarded the Nobel Prize in Economics for his work.Today, economists use Solow’s sources-of-growth ac-counting to estimate the separate effects on economicgrowth of technological change, capital, and labor.[5]

1.1 Extension to the Harrod–Domarmodel

Solow extended the Harrod–Domar model by:

• Adding labor as a factor of production;

• And capital-labor ratios are not fixed as they are inthe Harrod–Domar model. These refinements allowincreasing capital intensity to be distinguished fromtechnological progress.

1.2 Short-run implications

In the short run, growth is determined by moving to thenew steady state which is created only from the changein the capital investment, labor force growth and depre-ciation rate. The change in the capital investment is fromthe change in the savings rate.

1.3 Long-run implications

The standard Solow model predicts that in the long run,growth will be equal to the new steady state. This isthe biggest weaknesses of the model because it meansthat, in the long run, there is no growth. The idea thata country reaches steady state and stays there forever isconsidered by some Economists to be unrealistic, and toallow a continued growth condition in the long-term theSolow Romer model is used. By combining the Solowand Romer models, economists are able to predict a longrun situation that includes sustained growth. However,there are natural limits to economic growth within theplanetary boundaries that are not addressed by the SolowRomer approach.

1.4 Assumptions

The key assumption of the neoclassical growth model isthat capital is subject to diminishing returns in a closedeconomy.→Given a fixed stock of labor, the impact on output ofthe last unit of capital accumulated will always be lessthan the one before.→Assuming for simplicity no technological progress orlabor force growth, diminishing returns implies that atsome point the amount of new capital produced is onlyjust enough to make up for the amount of existing capitallost due to depreciation. At this point, because of theassumptions of no technological progress or labor forcegrowth, we can see the economy ceases to grow.→Assuming non-zero rates of labor growth complicatesmatters somewhat, but the basic logic still applies – inthe short-run the rate of growth slows as diminishing re-turns take effect and the economy converges to a constant“steady-state” rate of growth (that is, no economic growthper-capita).→Including non-zero technological progress is very sim-ilar to the assumption of non-zero workforce growth, in

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2 2 MATHEMATICS OF THE MODEL

terms of “effective labor": a new steady state is reachedwith constant output per worker-hour required for a unitof output. However, in this case, per-capita output growsat the rate of technological progress in the “steady-state”(that is, the rate of productivity growth).

1.5 Variations in the effects of productivity

In the Solow-Swan model the unexplained change in thegrowth of output after accounting for the effect of capi-tal accumulation is called the Solow residual. This resid-ual measures the exogenous increase in total factor pro-ductivity (TFP) during a particular time period. The in-crease in TFP is often attributed entirely to technologicalprogress, but it also includes any permanent improvementin the efficiency with which factors of production arecombined over time. Implicitly TFP growth includes anypermanent productivity improvements that result fromimproved management practices in the private or publicsectors of the economy. Paradoxically, even though TFPgrowth is exogenous in the model, it cannot be observed,so it can only be estimated in conjunction with the simul-taneous estimate of the effect of capital accumulation ongrowth during a particular time period.The model can be reformulated in slightly different waysusing different productivity assumptions, or differentmeasurement metrics:

• Average Labor Productivity (ALP) is economic out-put per labor hour.

• Multifactor productivity (MFP) is output divided bya weighted average of capital and labor inputs. Theweights used are usually based on the aggregate in-put shares either factor earns. This ratio is oftenquoted as: 33% return to capital and 67% return tolabor (in Western nations).

In a growing economy, capital is accumulated faster thanpeople are born, so the denominator in the growth func-tion under the MFP calculation is growing faster thanin the ALP calculation. Hence, MFP growth is almostalways lower than ALP growth. (Therefore, measuringin ALP terms increases the apparent capital deepeningeffect.) MFP is measured by the "Solow residual", notALP.

2 Mathematics of the model

The textbook Solow–Swan model is set in continuous-time world with no government or international trade. Asingle good (output) is produced using two factors of pro-duction, labor ( L ) and capital (K )in an aggregate pro-duction function that satisfies the Inada conditions, whichimply that the elasticity of substitution must be asymptot-ically equal to one.[6][7]

Y (t) = K(t)α(A(t)L(t))1−α

where t denotes time, 0 < α < 1 is the elasticity ofoutput with respect to capital, and Y (t) represents totalproduction. A refers to labor-augmenting technology or“knowledge”, thusAL represents effective labor. All fac-tors of production are fully employed, and initial valuesA(0) ,K(0) , and L(0) are given. The number of work-ers, i.e. labor, as well as the level of technology growexogenously at rates n and g , respectively:

L(t) = L(0)ent

A(t) = A(0)egt

The number of effective units of labor, A(t)L(t) , there-fore grows at rate (n+ g) . Meanwhile, the stock of cap-ital depreciates over time at a constant rate δ . However,only a fraction of the output ( cY (t) with 0 < c < 1 ) isconsumed, leaving a saved share s = 1−c for investment:

K(t) = s · Y (t)− δ ·K(t)

where K is shorthand for dK(t)dt , the derivative with re-

spect to time. Derivative with respect to time means thatit is the change in capital stock—output that is neitherconsumed nor used to replace worn-out old capital goodsis net investment.Since the production function Y (K,AL) has constantreturns to scale, it can be written as output per effectiveunit of labour:[note 1]

y(t) =Y (t)

A(t)L(t)= k(t)α

The main interest of the model is the dynamics of capitalintensity k , the capital stock per unit of effective labour.Its behaviour over time is given by the key equation of theSolow–Swan model:[note 2]

k(t) = sk(t)α − (n+ g + δ)k(t)

The first term, sk(t)α = sy(t) , is the actual investmentper unit of effective labour: the fraction s of the outputper unit of effective labour y(t) that is saved and invested.The second term, (n + g + δ)k(t) , is the “break-eveninvestment”: the amount of investment that must be in-vested to prevent k from falling.[8]:16 The equation im-plies that k(t) converges to a steady-state value of k∗ ,defined by sk(t)α = (n+ g + δ)k(t) , at which there isneither an increase nor a decrease of capital intensity:

k∗ =

(s

n+ g + δ

) 11−α

3

at which the stock of capital K and effective labour ALare growing at rate (n+g) . By assumption of constant re-turns, output Y is also growing at that rate. In essence, theSolow–Swan model predicts that an economy will con-verge to a balanced-growth equilibrium, regardless of itsstarting point. In this situation, the growth of output perworker is determined solely by the rate of technologicalprogress.[8]:18

Since, by definition, K(t)Y (t) = k(t)1−α , at the equilibrium

k∗ we have

K(t)

Y (t)=

s

n+ g + δ

Therefore, at the equilibrium, the capital/output ratio de-pends only on savings, growth, and depreciation rates.This is the Solow-Swan model’s version of the Goldenrule savings rate.Sinceα < 1 , at any time t themarginal product of capitalK(t) in the Solow-Swan model is inversely related to thecapital/labor ratio.

MPK =∂Y

∂K= αA1−α/(K/L)1−α

If productivityA is the same across countries, then coun-tries with less capital per worker K/L have a highermarginal product, which would provide a higher returnon capital investment. As a consequence, the model pre-dicts that in a world of open market economies and globalfinancial capital, investment will flow from rich coun-tries to poor countries, until capital/worker K/L and in-come/worker Y /L equalize across countries.Since the marginal product of physical capital is nothigher in poor countries than in rich countries,[9] the im-plication is that productivity is lower in poor countries.The basic Solow model cannot explain why productivityis lower in these countries. Lucas suggested that lowerlevels of human capital in poor countries could explainthe lower productivity.[10]

If one equates the marginal product of capital ∂Y∂K with

the rate of return r (such approximation is often used inneoclassical economics), then, for our choice of the pro-duction function

α =K ∂Y

∂K

Y=

rK

Y

so that α is the fraction of income appropriated by cap-ital. Thus, Solow-Swan model assumes from the begin-ning that the labor-capital split of income remains con-stant.

3 Mankiw–Romer–Weil version ofmodel

3.1 Addition of Human Capital

N. Gregory Mankiw, David Romer, and David Weil cre-ated a human capital augmented version of the Solow-Swan model that can explain the failure of internationalinvestment to flow to poor countries.[11] In this model out-put and the marginal product of capital (K) are lower inpoor countries because they have less human capital thanrich countries.Similar to the textbook Solow–Swan model, the produc-tion function is of Cobb–Douglas type:

Y (t) = K(t)αH(t)β(A(t)L(t))1−α−β

whereH(t) is the stock of human capital, which depreci-ates at the same rate δ as physical capital. For simplicity,they assume the same function of accumulation for bothtypes of capital. Like in Solow–Swan, a fraction of theoutcome, sY (t) , is saved each period, but in this casesplit up and invested partly in physical and partly in hu-man capital, such that s = sK + sH . Therefore, thereare two fundamental dynamic equations in this model:

k = sKkαhβ − (n+ g + δ)k

h = sHkαhβ − (n+ g + δ)h

The balanced (or steady-state) equilibrium growth path isdetermined by k = h = 0 , which means sKkαhβ−(n+g+ δ)k = 0 and sHkαhβ − (n+ g+ δ)h = 0 . Solvingfor the steady-state level of k and h yields:

k∗ =

(s1−βK sβH

n+ g + δ

) 11−α−β

h∗ =

(sαKs1−α

H

n+ g + δ

) 11−α−β

In the steady state, y∗ = (k∗)α(h∗)β .

3.2 Econometric estimates

Klenow and Rodriguez-Clare cast doubt on the validityof the augmented model because Mankiw, Romer, andWeil´s estimates of β did not seem consistent with ac-cepted estimates of the effect of increases in schoolingon workers’ salaries. Though the estimated model ex-plained 78% of variation in income across countries, theestimates of β implied that human capital’s external ef-fects on national income are greater than its direct effecton workers’ salaries.[12]

4 6 NOTES

3.3 Accounting for External Effects

Theodore Breton provided an insight that reconciledthe large effect of human capital from schooling in theMankiw, Romer and Weil model with the smaller effectof schooling on workers’ salaries. He demonstrated thatthe mathematical properties of the model include signifi-cant external effects between the factors of production,because human capital and physical capital are multi-plicative factors of production.[13] The external effect ofhuman capital on the productivity of physical capital isevident in the marginal product of physical capital:

MPK =∂Y

∂K= αA1−α(H/L)β/(K/L)1−α

He showed that the large estimates of the effect of humancapital in cross-country estimates of the model are con-sistent with the smaller effect typically found on work-ers’ salaries when the external effects of human capitalon physical capital and labor are taken into account. Thisinsight significantly strengthens the case for the Mankiw,Romer, andWeil version of the Solow-Swanmodel. Mostanalyses criticizing this model fail to account for the ex-ternal effects of both types of capital inherent in themodel.[13]

3.4 Total Factor Productivity

The exogenous rate of TFP (Total Factor Productivity)growth in the Solow-Swan model is the residual after ac-counting for capital accumulation. The Mankiw, Romerand Weil model provides a lower estimate of the TFP(residual) than the basic Solow-Swan model because theaddition of human capital to the model enables capitalaccumulation to explain more of the variation in incomeacross countries. In the basic model the TFP residual in-cludes the effect of human capital because human capitalis not included as a factor of production.

4 Conditional convergence

The Solow-Swan model augmented with human capitalpredicts that the income levels of poor countries will tendto catch up with or converge towards the income lev-els of rich countries if the poor countries have similarsavings rates for both physical capital and human capi-tal as a share of output, a process known as conditionalconvergence. However, savings rates vary widely acrosscountries. In particular, since considerable financing con-straints exist for investment in schooling, savings rates forhuman capital are likely to vary as a function of culturaland ideological characteristics in each country. [14]

Since the 1950s, output/worker in rich and poor coun-tries generally has not converged, but those poor countries

that have greatly raised their savings rates have experi-enced the income convergence predicted by the Solow-Swan model. As an example, output/worker in Japan, acountry which was once relatively poor, has converged tothe level of the rich countries. Japan experienced highgrowth rates after it raised its savings rates in the 1950sand 1960s, and it has experienced slowing growth of out-put/worker since its savings rates stabilized around 1970,as predicted by the model.The per-capita income levels of the southern states ofthe United States have tended to converge to the lev-els in the Northern states. The observed convergence inthese states is also consistent with the conditional con-vergence concept. Whether absolute convergence be-tween countries or regions occurs depends on whetherthey have similar characteristics, such as:

• Education policy

• Institutional arrangements

• Free markets internally, and trade policy with othercountries.[15]

Additional evidence for conditional convergence comesfrom multivariate, cross-country regressions.[16]

If productivity growth were associated only with hightechnology then the introduction of information technol-ogy should have led to a noticeable productivity accelera-tion over the past twenty years; but it has not: see: Solowcomputer paradox. Instead world productivity appears tohave increased relatively steadily since the 19th century.Econometric analysis on Singapore and the other "EastAsian Tigers" has produced the surprising result that al-though output per worker has been rising, almost noneof their rapid growth had been due to rising per-capitaproductivity (they have a low "Solow residual").[5]

5 See also

• Economic growth

• Endogenous growth theory

• Golden rule savings rate

6 Notes[1] Step-by-step calculation: y(t) = Y (t)

A(t)L(t)=

K(t)α(A(t)L(t))1−α

A(t)L(t)= K(t)α

(A(t)L(t))α= k(t)α

[2] Step-by-step calculation: k(t) = K(t)A(t)L(t)

−K(t)

[A(t)L(t)]2[A(t)L(t) + L(t)A(t)] =

K(t)A(t)L(t)

− K(t)A(t)L(t)

L(t)L(t)

− K(t)A(t)L(t)

A(t)A(t)

. Since

5

K(t) = sY (t) − δK(t) , and L(t)L(t)

, A(t)A(t)

aren and g , respectively, the equation simplifies tok(t) = s Y (t)

A(t)L(t)−δ K(t)

A(t)L(t)−n K(t)

A(t)L(t)−g K(t)

A(t)L(t)=

sy(t) − δk(t) − nk(t) − gk(t) . As mentioned above,y(t) = k(t)α .

7 References[1] Acemoglu, Daron (2009). “The Solow Growth Model”.

Introduction to Modern Economic Growth. Princeton:Princeton University Press. pp. 26–76. ISBN 978-0-691-13292-1.

[2] Solow, Robert M. (1956). “A Contribution to the Theoryof Economic Growth”. Quarterly Journal of Economics70 (1): 65–94. doi:10.2307/1884513.

[3] Swan, Trevor W. (1956). “Economic Growth and Cap-ital Accumulation”. Economic Record 32 (2): 334–361.doi:10.1111/j.1475-4932.

[4] Solow, Robert M. (1957). “Technical Change and the Ag-gregate Production Function”. Review of Economics andStatistics 39 (3): 312–320. doi:10.2307/1926047.

[5] Haines, Joel D. (2006). “A Framework for Managingthe Sophistication of the Components of Technology forGlobal Competition”. Competitiveness Review 16 (2):106–121.

[6] Barelli, Paulo; Pessôa, Samuel de Abreu (2003). “Inadaconditions imply that production function must be asymp-totically Cobb–Douglas”. Economics Letters 81 (3): 361–363. doi:10.1016/S0165-1765(03)00218-0.

[7] Litina, Anastasia; Palivos, Theodore (2008). “Do Inadaconditions imply that production function must be asymp-totically Cobb–Douglas? A comment”. Economics Letters99 (3): 498–499. doi:10.1016/j.econlet.2007.09.035.

[8] Romer, David (2011). “The Solow Growth Model”.Advanced Macroeconomics (Fourth ed.). New York:McGraw-Hill. pp. 6–48. ISBN 978-0-07-351137-5.

[9] Caselli, F.; Feyrer, J. (2007). “The Marginal Product ofCapital”. The Quarterly Journal of Economics 122 (2):535. doi:10.1162/qjec.122.2.535.

[10] Lucas, Robert (1990). “Why doesn't Capital Flow fromRich to Poor Countries?". American Economic Review 80(2): 92–96

[11] Mankiw, N. Gregory; Romer, David; Weil, David N.(May 1992). “A Contribution to the Empirics of Eco-nomic Growth”. The Quarterly Journal of Economics 107(2): 407–437. doi:10.2307/2118477. JSTOR 2118477.

[12] Klenow, Peter J.; Rodriguez-Clare, Andres (January1997). “The Neoclassical Revival in Growth Economics:Has It Gone Too Far?". In Bernanke, Ben S.; Rotem-berg, Julio. NBERMacroeconomics Annual 1997, Volume12. National Bureau of Economic Research. pp. 73–114.ISBN 0-262-02435-7.

[13] Breton, T. R. (2013). “Were Mankiw, Romer, and WeilRight? A Reconciliation of the Micro and Macro Effectsof Schooling on Income”. Macroeconomic Dynamics 17(5): 1023–1054. doi:10.1017/S1365100511000824.

[14] Breton, T. R. (2013). “The role of educationin economic growth: Theory, history and cur-rent returns”. Educational Research 55 (2): 121.doi:10.1080/00131881.2013.801241.

[15] Barro, Robert J.; Sala-i-Martin, Xavier (2004). “GrowthModels with Exogenous Saving Rates”. Economic Growth(Second ed.). New York: McGraw-Hill. pp. 37–51.ISBN 0-262-02553-1.

[16] Barro, Robert J.; Sala-i-Martin, Xavier (2004). “GrowthModels with Exogenous Saving Rates”. Economic Growth(Second ed.). New York: McGraw-Hill. pp. 461–509.ISBN 0-262-02553-1.

8 Further reading

• Agénor, Pierre-Richard (2004). “Growth and Tech-nological Progress: The Solow–Swan Model”. TheEconomics of Adjustment and Growth (Second ed.).Cambridge: Harvard University Press. pp. 439–462. ISBN 0-674-01578-9.

• Barro, Robert J.; Sala-i-Martin, Xavier (2004).“Growth Models with Exogenous Saving Rates”.Economic Growth (Second ed.). New York:McGraw-Hill. pp. 23–84. ISBN 0-262-02553-1.

• Burmeister, Edwin; Dobell, A. Rodney (1970).“One-Sector Growth Models”. Mathematical The-ories of Economic Growth. New York: Macmillan.pp. 20–64.

• Dornbusch, Rüdiger; Fischer, Stanley; Startz,Richard (2004). “Growth Theory: The Neoclas-sical Model”. Macroeconomics (Ninth ed.). NewYork: McGraw-Hill Irwin. pp. 61–75. ISBN 0-07-282340-2.

• Farmer, Roger E. A. (1999). “Neoclassical GrowthTheory”. Macroeconomics (Second ed.). Cincin-nati: South-Western. pp. 333–355. ISBN 0-324-12058-3.

• Gandolfo, Giancarlo (1996). “The NeoclassicalGrowth Model”. Economic Dynamics (Third ed.).Berlin: Springer. pp. 175–189. ISBN 3-540-60988-1.

• Intriligator, Michael D. (1971). Mathematical Opti-malization and Economic Theory. Englewood Cliffs:Prentice-Hall. pp. 398–416. ISBN 0-13-561753-7.

6 9 EXTERNAL LINKS

9 External links• Solow Model Videos - 20+ videos walking throughderivation of the Solow Growth Model’s Conclu-sions

• Java applet where you can experiment with param-eters and learn about Solow model

• Solow Growth Model by Fiona Maclachlan, TheWolfram Demonstrations Project.

• A step-by-step explanation of how to understand theSolow Model

• Professor José-Víctor Ríos-Rull’s course at Univer-sity of Minnesota

• Professor Alex Tabarrok’s Solow Growth Modellecture at MRUniversity

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