Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

Solow empirical predictions

Mario Tirelli

2018

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

Kaldor-Solow’s facts summary

Kaldor (1957) and Solow (1957) first highlighted some regularities.

1 Real output roughly grows at a constant rate, g .

2 Real capital has roughly the same constant rate of growth ofoutput, gK ≈ g .

3 The ratio of profits on capital Π/K and of the real interestrate r are both roughly constant.

4 Cross-country comparisons reveal a high variance of outputper-capita and growth rates (recall Acemoglou’s chp.1).

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

Kaldor-Solow’s facts summary

Kaldor (1957) and Solow (1957) first highlighted some regularities.

1 Real output roughly grows at a constant rate, g .

2 Real capital has roughly the same constant rate of growth ofoutput, gK ≈ g .

3 The ratio of profits on capital Π/K and of the real interestrate r are both roughly constant.

4 Cross-country comparisons reveal a high variance of outputper-capita and growth rates (recall Acemoglou’s chp.1).

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

Kaldor-Solow’s facts summary

Kaldor (1957) and Solow (1957) first highlighted some regularities.

1 Real output roughly grows at a constant rate, g .

2 Real capital has roughly the same constant rate of growth ofoutput, gK ≈ g .

3 The ratio of profits on capital Π/K and of the real interestrate r are both roughly constant.

4 Cross-country comparisons reveal a high variance of outputper-capita and growth rates (recall Acemoglou’s chp.1).

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

Kaldor-Solow’s facts summary

Kaldor (1957) and Solow (1957) first highlighted some regularities.

1 Real output roughly grows at a constant rate, g .

2 Real capital has roughly the same constant rate of growth ofoutput, gK ≈ g .

3 The ratio of profits on capital Π/K and of the real interestrate r are both roughly constant.

4 Cross-country comparisons reveal a high variance of outputper-capita and growth rates (recall Acemoglou’s chp.1).

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

Kaldor-Solow’s facts summary

Kaldor (1957) and Solow (1957) first highlighted some regularities.

1 Real output roughly grows at a constant rate, g .

2 Real capital has roughly the same constant rate of growth ofoutput, gK ≈ g .

3 The ratio of profits on capital Π/K and of the real interestrate r are both roughly constant.

4 Cross-country comparisons reveal a high variance of outputper-capita and growth rates (recall Acemoglou’s chp.1).

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

Main implication

The first 3 facts on growth imply that the economy has a balancedgrowth, with the main economic variables growing at the sameconstant rate g and r is roughly constant.

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

Solow’s successful quality predictions

Balance growth: variables such as Y ,C ,K , I tend to grow atthe same constant rate g and the ratio of K/Y is roughlyconstant (so is for other ratios). Fact 1-4 hold. Also, r , Π/Yare constant at balance growth.

g depends on demographics n and on the rate of technologicalprogress µ.Saving rate s has only temporary effect on growth: onlyon ss variables in levels and on,

k∗

y∗=

s

δ + g

Contrary to the empirical evidence, saving rates do notcovariate with the growth rate.Conditional convergence: two economies with the samefundamentals, except for k0, are s.t. the poorest grows faster.

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

Solow’s successful quality predictions

Balance growth: variables such as Y ,C ,K , I tend to grow atthe same constant rate g and the ratio of K/Y is roughlyconstant (so is for other ratios). Fact 1-4 hold. Also, r , Π/Yare constant at balance growth.g depends on demographics n and on the rate of technologicalprogress µ.

Saving rate s has only temporary effect on growth: onlyon ss variables in levels and on,

k∗

y∗=

s

δ + g

Contrary to the empirical evidence, saving rates do notcovariate with the growth rate.Conditional convergence: two economies with the samefundamentals, except for k0, are s.t. the poorest grows faster.

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

Solow’s successful quality predictions

Balance growth: variables such as Y ,C ,K , I tend to grow atthe same constant rate g and the ratio of K/Y is roughlyconstant (so is for other ratios). Fact 1-4 hold. Also, r , Π/Yare constant at balance growth.g depends on demographics n and on the rate of technologicalprogress µ.Saving rate s has only temporary effect on growth: onlyon ss variables in levels and on,

k∗

y∗=

s

δ + g

Contrary to the empirical evidence, saving rates do notcovariate with the growth rate.

Conditional convergence: two economies with the samefundamentals, except for k0, are s.t. the poorest grows faster.

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

Solow’s successful quality predictions

Balance growth: variables such as Y ,C ,K , I tend to grow atthe same constant rate g and the ratio of K/Y is roughlyconstant (so is for other ratios). Fact 1-4 hold. Also, r , Π/Yare constant at balance growth.g depends on demographics n and on the rate of technologicalprogress µ.Saving rate s has only temporary effect on growth: onlyon ss variables in levels and on,

k∗

y∗=

s

δ + g

Contrary to the empirical evidence, saving rates do notcovariate with the growth rate.Conditional convergence: two economies with the samefundamentals, except for k0, are s.t. the poorest grows faster.

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

Mankiw, Romer and Weil (QJE, 1992)

[..] the empirical predictions of the Solow model are, to afirst approximation, consistent with the evidence.

Examin-ing recently available data for a large set of countries, wefind that saving and population growth affect income inthe directions that Solow predicted. Moreover, more thanhalf of the cross-country variation in income per capita canbe explained by these two variables alone.[p. 407]

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

Mankiw, Romer and Weil (QJE, 1992)

[..] the empirical predictions of the Solow model are, to afirst approximation, consistent with the evidence. Examin-ing recently available data for a large set of countries, wefind that saving and population growth affect income inthe directions that Solow predicted.

Moreover, more thanhalf of the cross-country variation in income per capita canbe explained by these two variables alone.[p. 407]

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

Mankiw, Romer and Weil (QJE, 1992)

[..] the empirical predictions of the Solow model are, to afirst approximation, consistent with the evidence. Examin-ing recently available data for a large set of countries, wefind that saving and population growth affect income inthe directions that Solow predicted. Moreover, more thanhalf of the cross-country variation in income per capita canbe explained by these two variables alone.[p. 407]

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

Countries with higher income per capita tend to have highersaving rate s and lower population growth n. Yet,quantitatively, the effect of population growth and of thesaving rates are too large.

Let Yt = Kαt (AtNt)

1−α. Recall, g = η = n + µ and noticethat,

Yt

Nt= Atk

αt , kt ≡

Kt

AtNt

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

Countries with higher income per capita tend to have highersaving rate s and lower population growth n. Yet,quantitatively, the effect of population growth and of thesaving rates are too large.

Let Yt = Kαt (AtNt)

1−α. Recall, g = η = n + µ and noticethat,

Yt

Nt= Atk

αt , kt ≡

Kt

AtNt

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

Steady-state capital (in efficiency units) is,

k∗ =

(s

δ + n + µ

) 11−α

In per capita, the production function is,

Yt

Nt= At

(Kt

AtNt

)α= Atk

αt

Taking logs and evaluating at bal. growth, kt = k∗,

log

(Yt

Nt

)= logAt + α log k∗

= logA0 + t log(1 + µ) +α

1− αlog s − α

1− αlog(δ + n + µ)

Since, n and s vary across countries, k∗ varies too, explainingdifferent levels of y across countries.

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

Steady-state capital (in efficiency units) is,

k∗ =

(s

δ + n + µ

) 11−α

In per capita, the production function is,

Yt

Nt= At

(Kt

AtNt

)α= Atk

αt

Taking logs and evaluating at bal. growth, kt = k∗,

log

(Yt

Nt

)= logAt + α log k∗

= logA0 + t log(1 + µ) +α

1− αlog s − α

1− αlog(δ + n + µ)

Since, n and s vary across countries, k∗ varies too, explainingdifferent levels of y across countries.

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

Steady-state capital (in efficiency units) is,

k∗ =

(s

δ + n + µ

) 11−α

In per capita, the production function is,

Yt

Nt= At

(Kt

AtNt

)α= Atk

αt

Taking logs and evaluating at bal. growth, kt = k∗,

log

(Yt

Nt

)= logAt + α log k∗

= logA0 + t log(1 + µ) +α

1− αlog s − α

1− αlog(δ + n + µ)

Since, n and s vary across countries, k∗ varies too, explainingdifferent levels of y across countries.

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

The empirical model to be used in the cross-countryregression is, for each country j :

log

(Yj

Nj

)= a0 + b1 ln sj + b2 ln(δ + µ+ nj) + εj

where a0 = lnA0 is the intercept capturing, not onlytechnology (e.g. resource endowments, climate, institutions )in the base year 1960; Yj/Nj refers to 1985.

εj (also interpretable as the stoch. component oftechnological progress) assumed iid across countries and w/rregressors.

One can test or restrict coefficient to satisfy,

b1 = −b2 =α

1− α

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

The empirical model to be used in the cross-countryregression is, for each country j :

log

(Yj

Nj

)= a0 + b1 ln sj + b2 ln(δ + µ+ nj) + εj

where a0 = lnA0 is the intercept capturing, not onlytechnology (e.g. resource endowments, climate, institutions )in the base year 1960; Yj/Nj refers to 1985.

εj (also interpretable as the stoch. component oftechnological progress) assumed iid across countries and w/rregressors.

One can test or restrict coefficient to satisfy,

b1 = −b2 =α

1− α

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

Coefficients on investment rate (ln I/GDP = ln s) and n havepredicted signs and are significant in most samples.

Equality restriction on b can’t be rejected. Regressions explaina quite large fraction of the cross-country income-variation:adj-R2 = .59 in the intermediate sample.

This is in contrast with the common wisdom and previousempirical findings, which says that differences in A explainmost of the cross-country income variation in the Solowmodel.

Nonetheless, estimates reveal that the impacts of s and n aremuch larger than the theoretical model predicts:α = b1(1 + b1)−1 exceeds 1/2 > 1/3 = α = YK/Y .

If one had to constrain α = 1/3, the constrained regressionwould see the R2 drop from 0.59 to 0.28.

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

Coefficients on investment rate (ln I/GDP = ln s) and n havepredicted signs and are significant in most samples.

Equality restriction on b can’t be rejected. Regressions explaina quite large fraction of the cross-country income-variation:adj-R2 = .59 in the intermediate sample.

This is in contrast with the common wisdom and previousempirical findings, which says that differences in A explainmost of the cross-country income variation in the Solowmodel.

Nonetheless, estimates reveal that the impacts of s and n aremuch larger than the theoretical model predicts:α = b1(1 + b1)−1 exceeds 1/2 > 1/3 = α = YK/Y .

If one had to constrain α = 1/3, the constrained regressionwould see the R2 drop from 0.59 to 0.28.

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

Coefficients on investment rate (ln I/GDP = ln s) and n havepredicted signs and are significant in most samples.

Equality restriction on b can’t be rejected. Regressions explaina quite large fraction of the cross-country income-variation:adj-R2 = .59 in the intermediate sample.

This is in contrast with the common wisdom and previousempirical findings, which says that differences in A explainmost of the cross-country income variation in the Solowmodel.

Nonetheless, estimates reveal that the impacts of s and n aremuch larger than the theoretical model predicts:α = b1(1 + b1)−1 exceeds 1/2 > 1/3 = α = YK/Y .

If one had to constrain α = 1/3, the constrained regressionwould see the R2 drop from 0.59 to 0.28.

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

Coefficients on investment rate (ln I/GDP = ln s) and n havepredicted signs and are significant in most samples.

Equality restriction on b can’t be rejected. Regressions explaina quite large fraction of the cross-country income-variation:adj-R2 = .59 in the intermediate sample.

This is in contrast with the common wisdom and previousempirical findings, which says that differences in A explainmost of the cross-country income variation in the Solowmodel.

Nonetheless, estimates reveal that the impacts of s and n aremuch larger than the theoretical model predicts:α = b1(1 + b1)−1 exceeds 1/2 > 1/3 = α = YK/Y .

If one had to constrain α = 1/3, the constrained regressionwould see the R2 drop from 0.59 to 0.28.

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

Coefficients on investment rate (ln I/GDP = ln s) and n havepredicted signs and are significant in most samples.

Equality restriction on b can’t be rejected. Regressions explaina quite large fraction of the cross-country income-variation:adj-R2 = .59 in the intermediate sample.

This is in contrast with the common wisdom and previousempirical findings, which says that differences in A explainmost of the cross-country income variation in the Solowmodel.

Nonetheless, estimates reveal that the impacts of s and n aremuch larger than the theoretical model predicts:α = b1(1 + b1)−1 exceeds 1/2 > 1/3 = α = YK/Y .

If one had to constrain α = 1/3, the constrained regressionwould see the R2 drop from 0.59 to 0.28.

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

MRW, essentially, conclude that the effect of n and s are toolarge because the model –in its textbook version– is‘misspecified’.

They suggest to include a broader definition of ‘capital’, inparticular, to include human capital.

Human capital accumulation = activities such as worktraining, schooling and others (e.g. including health care),enhance labor productivity through a costly and timely‘investment’.

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

MRW, essentially, conclude that the effect of n and s are toolarge because the model –in its textbook version– is‘misspecified’.

They suggest to include a broader definition of ‘capital’, inparticular, to include human capital.

Human capital accumulation = activities such as worktraining, schooling and others (e.g. including health care),enhance labor productivity through a costly and timely‘investment’.

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

MRW, essentially, conclude that the effect of n and s are toolarge because the model –in its textbook version– is‘misspecified’.

They suggest to include a broader definition of ‘capital’, inparticular, to include human capital.

Human capital accumulation = activities such as worktraining, schooling and others (e.g. including health care),enhance labor productivity through a costly and timely‘investment’.

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

To grasp the concept,

Yt = AtKαt (htNt)

1−α

Think at ht as

labor skills, increasing with education (e.g. years and qualityof schooling) and work training, or as

labor quality, increasing with the health-care accessibility(e.g. the extension and quality of the public health-caresystem), or as

labor effort increasing with self-motivation and workenvironment.

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

Omitting human cap. biases estimates (s, n); 2 reasons

Yt = AtKαt (htNt)

1−α︸ ︷︷ ︸human capital H1−α

t

, ht = ht−1 + shYt

Nt

1 Fix sh. Higher s or lower n leads to higher Y /N and thisincreases human cap. by shY /N; this is an indirect effectboosting up Y /N. So, one can explain higher Y /N for a unit∆s with a lower α.

2 Human capital accumulation may be correlated with s and n:the population spends more time in education in countrieswith a better edu. system; this decision is costly, implyingthat i) part of the family income is spent in education–lowers– and ii) people enter in the labor force later in life –lower n.Not accounting for sh implies higher α.

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

Omitting human cap. biases estimates (s, n); 2 reasons

Yt = AtKαt (htNt)

1−α︸ ︷︷ ︸human capital H1−α

t

, ht = ht−1 + shYt

Nt

1 Fix sh. Higher s or lower n leads to higher Y /N and thisincreases human cap. by shY /N; this is an indirect effectboosting up Y /N. So, one can explain higher Y /N for a unit∆s with a lower α.

2 Human capital accumulation may be correlated with s and n:the population spends more time in education in countrieswith a better edu. system; this decision is costly, implyingthat i) part of the family income is spent in education–lowers– and ii) people enter in the labor force later in life –lower n.Not accounting for sh implies higher α.

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

Omitting human cap. biases estimates (s, n); 2 reasons

Yt = AtKαt (htNt)

1−α︸ ︷︷ ︸human capital H1−α

t

, ht = ht−1 + shYt

Nt

1 Fix sh. Higher s or lower n leads to higher Y /N and thisincreases human cap. by shY /N; this is an indirect effectboosting up Y /N. So, one can explain higher Y /N for a unit∆s with a lower α.

2 Human capital accumulation may be correlated with s and n:the population spends more time in education in countrieswith a better edu. system; this decision is costly, implyingthat i) part of the family income is spent in education–lowers– and ii) people enter in the labor force later in life –lower n.Not accounting for sh implies higher α.

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

The orthogonality hypothesis

MRW assume {εj} are iid . This cross-country orthogonality istoo strong, for 2 reasons.

1. human capital sh,j is often correlated with productivity Aj .If Aj differ across countries and we omit sh,j in the regr., {εj}will be correlated.

2. even including sh,j in the regr. there might be reversecausality: countries with high Aj invest more in human capital.

In both cases rhs variables are correlated to the residuals; OLSleads to upward bias estimates of b′s.

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

The orthogonality hypothesis

MRW assume {εj} are iid . This cross-country orthogonality istoo strong, for 2 reasons.

1. human capital sh,j is often correlated with productivity Aj .If Aj differ across countries and we omit sh,j in the regr., {εj}will be correlated.

2. even including sh,j in the regr. there might be reversecausality: countries with high Aj invest more in human capital.

In both cases rhs variables are correlated to the residuals; OLSleads to upward bias estimates of b′s.

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

The orthogonality hypothesis

MRW assume {εj} are iid . This cross-country orthogonality istoo strong, for 2 reasons.

1. human capital sh,j is often correlated with productivity Aj .If Aj differ across countries and we omit sh,j in the regr., {εj}will be correlated.

2. even including sh,j in the regr. there might be reversecausality: countries with high Aj invest more in human capital.

In both cases rhs variables are correlated to the residuals; OLSleads to upward bias estimates of b′s.

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

The orthogonality hypothesis

MRW assume {εj} are iid . This cross-country orthogonality istoo strong, for 2 reasons.

1. human capital sh,j is often correlated with productivity Aj .If Aj differ across countries and we omit sh,j in the regr., {εj}will be correlated.

2. even including sh,j in the regr. there might be reversecausality: countries with high Aj invest more in human capital.

In both cases rhs variables are correlated to the residuals; OLSleads to upward bias estimates of b′s.

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

Where are we?

Including a broader notion of capital (including human c.)allows the Solow model to better explain cross-countryper-capita income variations.

Cross-country variations in Y /N can be explained forreasonable values of s and n. In MRW, α ≈ 1/3 includingyears of school. However, the R2 drops considerably if thesample is restricted to OECD countries (differences intechnology matters more and results in correlated errors).

Does the Solow model explain growth? Not really!

gK explains at most 1/3; A/A seems to play a big role. Butthe latest is unexplained by the theory/model.

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

Where are we?

Including a broader notion of capital (including human c.)allows the Solow model to better explain cross-countryper-capita income variations.

Cross-country variations in Y /N can be explained forreasonable values of s and n. In MRW, α ≈ 1/3 includingyears of school. However, the R2 drops considerably if thesample is restricted to OECD countries (differences intechnology matters more and results in correlated errors).

Does the Solow model explain growth? Not really!

gK explains at most 1/3; A/A seems to play a big role. Butthe latest is unexplained by the theory/model.

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

Where are we?

Including a broader notion of capital (including human c.)allows the Solow model to better explain cross-countryper-capita income variations.

Cross-country variations in Y /N can be explained forreasonable values of s and n. In MRW, α ≈ 1/3 includingyears of school. However, the R2 drops considerably if thesample is restricted to OECD countries (differences intechnology matters more and results in correlated errors).

Does the Solow model explain growth?

Not really!

gK explains at most 1/3; A/A seems to play a big role. Butthe latest is unexplained by the theory/model.

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

Where are we?

Including a broader notion of capital (including human c.)allows the Solow model to better explain cross-countryper-capita income variations.

Cross-country variations in Y /N can be explained forreasonable values of s and n. In MRW, α ≈ 1/3 includingyears of school. However, the R2 drops considerably if thesample is restricted to OECD countries (differences intechnology matters more and results in correlated errors).

Does the Solow model explain growth? Not really!

gK explains at most 1/3; A/A seems to play a big role. Butthe latest is unexplained by the theory/model.

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

Solow’s analysis

Using US data 1901-1949, estimated that Y /N−growth wasexplained for the 87.5% by “productivity growth” and forthe 12.5% by capital accumulation.

Revising his analysis with more recent data one, respectively,finds 2/3 and 1/3.

The contribution of variations in labor input n is roughly zero(changes of the average hours of work, per-worker, has notrend).

Things are reversed if one looks at business cycle frequencies.

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

Solow’s analysis

Using US data 1901-1949, estimated that Y /N−growth wasexplained for the 87.5% by “productivity growth” and forthe 12.5% by capital accumulation.

Revising his analysis with more recent data one, respectively,finds 2/3 and 1/3.

The contribution of variations in labor input n is roughly zero(changes of the average hours of work, per-worker, has notrend).

Things are reversed if one looks at business cycle frequencies.

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

Solow’s analysis

Using US data 1901-1949, estimated that Y /N−growth wasexplained for the 87.5% by “productivity growth” and forthe 12.5% by capital accumulation.

Revising his analysis with more recent data one, respectively,finds 2/3 and 1/3.

The contribution of variations in labor input n is roughly zero(changes of the average hours of work, per-worker, has notrend).

Things are reversed if one looks at business cycle frequencies.

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

Solow’s analysis

Using US data 1901-1949, estimated that Y /N−growth wasexplained for the 87.5% by “productivity growth” and forthe 12.5% by capital accumulation.

Revising his analysis with more recent data one, respectively,finds 2/3 and 1/3.

The contribution of variations in labor input n is roughly zero(changes of the average hours of work, per-worker, has notrend).

Things are reversed if one looks at business cycle frequencies.

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

Let output be represented by,

Yt = F (Kt ,Nt ,At)

Take logs and compute instantaneous rate of change,

Y

Y=

1

Y

(FK K + FNN + FAA

)=

1

Y

(FK K

K

K+ FNN

N

N+ FAA

A

A

)

=

εK︷ ︸︸ ︷FK

K

Y

K

K+

εN︷ ︸︸ ︷FN

N

Y

N

N+ FA

A

Y

A

A

= εKK

K+ εN

N

N+ FA

A

Y

A

A︸ ︷︷ ︸u

= εKgK + εNgN + u

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

Solow’s “residuals”

Solow estimates the contribution of productivity as theresiduals u of,

g = γ1gK + γ2gN + u

He found that u was explaining most of the long run outputgrowth.

Postwar analysis shows that productivity growth contributesfor about 2/3 and capital per worker for about 1/3.

Labor input is roughly constant in the long-run (suggesting a”natural” rate of employment).

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

Solow’s “residuals”

Solow estimates the contribution of productivity as theresiduals u of,

g = γ1gK + γ2gN + u

He found that u was explaining most of the long run outputgrowth.

Postwar analysis shows that productivity growth contributesfor about 2/3 and capital per worker for about 1/3.

Labor input is roughly constant in the long-run (suggesting a”natural” rate of employment).

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

Solow’s “residuals”

Solow estimates the contribution of productivity as theresiduals u of,

g = γ1gK + γ2gN + u

He found that u was explaining most of the long run outputgrowth.

Postwar analysis shows that productivity growth contributesfor about 2/3 and capital per worker for about 1/3.

Labor input is roughly constant in the long-run (suggesting a”natural” rate of employment).

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

Solow’s “residuals”

Solow estimates the contribution of productivity as theresiduals u of,

g = γ1gK + γ2gN + u

He found that u was explaining most of the long run outputgrowth.

Postwar analysis shows that productivity growth contributesfor about 2/3 and capital per worker for about 1/3.

Labor input is roughly constant in the long-run (suggesting a”natural” rate of employment).

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

Conclusions

We need a theory and a model that can capture the mainstylized facts on growth and per capita income.

What did we learn on the directions to explore:1 Reason about the notion of capital. Introducing human capital

leads to an economy with more ‘stocks’ accumulating andtechnologies that may be CRTS or even IRTS in capital overall- How does this affect growth then?

2 Productivity growth is important but is a black box in thetheory (a ‘Solow residual’). We need a theory that explainsproductivity dynamics.

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

Conclusions

We need a theory and a model that can capture the mainstylized facts on growth and per capita income.

What did we learn on the directions to explore:

1 Reason about the notion of capital. Introducing human capitalleads to an economy with more ‘stocks’ accumulating andtechnologies that may be CRTS or even IRTS in capital overall- How does this affect growth then?

2 Productivity growth is important but is a black box in thetheory (a ‘Solow residual’). We need a theory that explainsproductivity dynamics.

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

Conclusions

We need a theory and a model that can capture the mainstylized facts on growth and per capita income.

What did we learn on the directions to explore:1 Reason about the notion of capital. Introducing human capital

leads to an economy with more ‘stocks’ accumulating andtechnologies that may be CRTS or even IRTS in capital overall- How does this affect growth then?

2 Productivity growth is important but is a black box in thetheory (a ‘Solow residual’). We need a theory that explainsproductivity dynamics.

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

Conclusions

We need a theory and a model that can capture the mainstylized facts on growth and per capita income.

What did we learn on the directions to explore:1 Reason about the notion of capital. Introducing human capital

leads to an economy with more ‘stocks’ accumulating andtechnologies that may be CRTS or even IRTS in capital overall- How does this affect growth then?

2 Productivity growth is important but is a black box in thetheory (a ‘Solow residual’). We need a theory that explainsproductivity dynamics.

Mario Tirelli Solow empirical predictions

Growth factsEmpirics 1: Solow’s qualitative predictions

Empirics 2: Solow’s quantitative pred. (cross-country income variations)Empirics 3: main sources of growth

Conclusions

Conclusions

We need a theory and a model that can capture the mainstylized facts on growth and per capita income.

What did we learn on the directions to explore:1 Reason about the notion of capital. Introducing human capital

leads to an economy with more ‘stocks’ accumulating andtechnologies that may be CRTS or even IRTS in capital overall- How does this affect growth then?

2 Productivity growth is important but is a black box in thetheory (a ‘Solow residual’). We need a theory that explainsproductivity dynamics.

Mario Tirelli Solow empirical predictions