General motivation behind the augmented Solow model

Transcription

1 General motivation behind the augmented Solow model Empirical analysis suggests that the elasticity of output Y with respect to capital implied by the Solow model (α 0.3) is too low to reconcile the model with empirical facts. For instance, the speed of convergence predicted by the model is far higher than observed in the data.

2 The model

3 Remark: according to MRW the technology to produce H and K are identical. Both forms of capital result directly from investment of final output in the accumulation of stocks. H s Y H Net investment in education h h Net investment in machinery k k K s Y K

4 Steady state in Solow: s k*( t) ng s y*( t) ng s y*( t) A( t) ng 1

5 s y*( t) A( t) ng 1 log y*( t) log A( t) log sk [ n g ] 1 1 elasticity / of / y / relative / to / sk 1

6 Steady state in Augmented Solow: 1 h 1 k y*( t) A( t) s s ng ng log y*( t) log A( t) log s k log s h [ n g ] elasticity / of / y / relative / to/ sk 1 elasticity / of / y / relative / to/ sh 1 Remark: the augmented Solow model is a true generalization of Solow 1956: by assuming β = 0, the last term vanishes and we are back in the Solow model.

7 Countries differ in terms of technology A j (t) saving rates s k, j and s h, j, and population growth rates n j.

8 Technology: A j = ε j A where ε is an exogenous technology shock This means initial technology explanatory variables s k and s h. A j is assumed uncorrelated to the exogenous

9 Focus on a world in which convergence to steady state has already taken place..

10 Solow versus H Augmented Solow the estimated equation for the Solow model is : log y j *( t) const. log s [ n g ] 1 1 k, j j t, j constant = loga + gt is not country specific country specific technology shocks are captured by the error term

11 Implied α

12 Findings on Solow As expected, the coefficients of ln(sk) and ln (n + g + ): have opposite sign their absolute size is in the same order of magnitude The main problem is that the implied α is far too large

13 the estimated equation for the H augmented Solow model is : log y *( t) j C log s log s [ n g ] C = constant k, j h, j j j, t

14 Implied α Implied β Observations

15 Findings on H augmented Solow the implied α is now consistent with the evidence α 0.3

16

17 estimate of β is too large check predictions following from estimates of α and β against microeconometric evidence of earning and marginal productivity effects of education: If markets are competitive as assumed in the model, the marginal productivity effects of education should be reflected by the earnings of educated and noneducated workers. This provides a way of estimating the size of β from microeconomic evidence This check suggests that the MRW estimate of β is too large!

18 E. Prescott (1998) A calibration analysis by Edward Prescott 1998, is based on the augmented Solow model, and concludes: 1. On the assumption that efficiency A is uniform across countries, the neoclassical model with physical and human capital cannot explain per capita income differences as large as those registered between the richest and poorest countries. 2. What we need, it is a theory explaining why the level A of total factor productivity (TFP) differs across countries.

19 Prescott s intangible capital Human capital is but one form of intangible capital which is largely unmeasured and missing in government statistics. Intangible capital includes not only school training to population in working age, but also on the job training, firm specific learning by doing, organization capital, and various forms of unmeasured R&D investment. Unmeasured investments I in official statistics imply that there is also unmeasured output Y, because Y = C + I. The unconventional part of Prescott s calibration exercise is addressed at dealing with this problem, but we skip this for simplicity.

20 Prescott s argument ln yj* = costant + (1 α β) 1 [αlnskj + βlnshj (α + β)ln(n + g +)] +εj (18) According to IMF estimates physical capital investment as a share of GDP is about 20% for rich and poor countries after This implies that, countries do not differ much in their investment ratios s k. To preserve the share s h in a plausible range for rich and poor countries, the burden of the explanation of per capita income differences falls on a high elasticity of output with respect to intangible capital H, that is a high β.

21 Suppose y R = 10 y P and country R and P differ in h (human capital) endowment, but are otherwise identical. In particular, A R = A P, and physical capital stocks per unit of efficiency are identical. y = Ak α h β y R / y P = 10 = Ak R α h R β / Ak P α h P β = (h R / h P ) β (h R / h P ) = 10 1 / β α β MPH R / MPH P = β k R h 1 α β R / βk P h 1 (β 1) P = (h R / h P ) MPH R / MPH P = 10 (β 1) / β (1 β) / β MPH P / MPH R = 10

22 (1 β) / β MPH P / MPH R = 10 Huge differences in rates of return to human capital are implied by observed differences in per capita income between Rich and Poor, unless β takes implausibly high values. Rates of return would be too high in poor countries relative to rich countries, and inconsistent with the facts suggesting that human capital does not flow from rich to poor, but the other way.

23 Prescott s Conclusion: The burden of explaining per capita income differences must partly fall on understanding why the constant in equation (18) is NOT uniform across countries, that is, what is needed, is a theory of TFP. Prescott holds to the basic neoclassical assumption that technical knowledge is transferable across countries at low cost: International differences in total factor productivity must be explained through institutionally based differences in work practices, not in useful knowledge. These differences affect the level of A

24 Why doesn t capital [and human capital] flow from rich to poor countries? (R. Lucas, 1990) Unlike in MRW, countries are not islands Financial capital and human capital flow across countries. If we assume that financial and human capital flow where returns are higher, we should expect a tendency towards the equalization of the rates on returns on both forms of capital. Neoclassical model with financial flows assumes perfect mobility of financial capital equalization of rates of returns is instantaneous Lucas 1990 reply draws upon his 1988 human capital model. It is a two sector AK model

25 Lucas 1988 assumes: Final output Y produced by human and physical capital (no raw labour) no exogenous technological progress human capital accumulation equation different from MRW 1 t t (1) Y t K t uhl t h t (1 u t ) h t (2) A and are exogenous constants expressing total factor productivities in the output and h sector, respectively. h is per capita human capital u is fraction of time spent by h in final output sector L is population growing at the exponential rate n

26 Divide equation (1) by L, and define y = Y/L, k = K/L (capital per worker) y = k α (uh) (1 α) (3) h t (1 u t ) h t endogenous growth is related to the linear structure of system (3), (3 ) (3 ) In particular g h * = θ(1 u*) Since human capital production is linear with respect to h input, the steady state growth rate g* is determined by the steady state fraction of time (1 u*) spent accumulating h. u* is a endogenous decision variable, which depends on preferences!!

27 y = k α (uh) (1 α) g y = αg k + (1 α)(g u + g h ) (4) in steady state: g y * = g k *, g u * = 0, u(t) = u* = constant (5) (1 α) g y * = (1 α) g h * g y * = g h * g y * = g h * g h * = θ(1 u*) In a optimizing framework, u(t) and u* depend on preferences: Other things equal, lower u(t) implies: higher future human capital stock h(t + dt) lower per capita output y(t) now, and lower per capita consumption c(t) at given propensity to consume c(t) / y(t)

28 1 Lucas (1) y t kt uh t t 1 t y t k t A Solow now compare output per worker in USA and India, y USA_1980s / y India_1980s 15 The Solow model with A t uniform in USA and India implies huge differences in the rates of return to k. If A t is uniform, we can normalize A t = 1. yt k t k t y 1/ t The gross rate of return to K is r = αk α 1 (α 1)/α = αy r r USA India = y y USA India ( 1)/ 58

29 Now consider the Lucas model Dividing equation (1) by the stock H Y = uhl of human capital in Y sector Output per efficient worker in Y sector is: y = k α where y = Y / uhl output per efficient worker k = K / uhl capital per efficient worker k = y 1/α the marginal product of capital is: r = αk (α 1) = α y (α 1)/α (6)

30 To estimate output per efficient worker Lucas has to resort to 1959 estimates by Ann Krueger: After taking into account education attainments, the ratio of output per efficient worker in 1959 USA and India is y USA / y India y uh y uh USA India 3. Notice that if we consider output per capita instead of output per efficient worker, the ratio would be far higher, because average education attainment in 1959 USA is far higher than in 1959 India With = 0.4 (average of USA and India capital shares) this gives: r r USA India = [ y USA / y India ] (α 1)/α = y uh y uh USA India ( 1)/ 1/5 Lucas observes that the ratio is still too large to be consistent with the evidence on capital flows.

31 For this reason, he adds an externality in equation (1): 1 t t (1) Y t K t uhl t h 1 y t k t uh t t h so that output per efficient worker is now: y = Ak α h γ the reason for the externality is that private human capital accumulation augments the socially available useful knowledge, which increases the productivity of private factors in the output sector. The productivity of one worker increases if the average education attainment and productivity of the workers she is working with is higher. (6 ) From Denison s USA data concerning , Lucas estimates the external effect A 10% increase in the average quality of those with whom I work increases my productivity by 3.6%.

32 the net MPK and interest rate is now r = αk (α 1) h γ = α y (α 1)/α h γ /α (7) Lucas concludes: On the crucial assumption that socially available useful knowledge is country specific, in the sense that there are not knowledge spillovers across countries, the estimated exactly eliminates the [capital] return differential in a 1959 India U.S. comparison.

33 Lucas 1990 indication: To explain per capita income differences between the rich countries and those that, unlike modern India, have failed to enter the mechanism of modern economic growth, we must break free of the straight jacket of the convex neoclassical model without externalities. Problem: As Lucas admits, the weakness in this argument is that it does not provide an explanation of why the poorest nations are unable to exploit useful knowledge created outside.