Accounting for Cross-Country Income Differences Francesco Caselli

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1 Accounting for Cross-Country Income Differences Francesco Caselli Applied Macroeconomics - ITAM

2 INTRODUCTION Motivation Why are some countries so much richer than others? Use Development Accounting (growth accounting at a point in time for a cross-section of countries rather than over time for a country) - two proximate determinants: 1 factors of production - physical and human capital. 2 efficiency - total factor productivity (TFP). How much of the cross-country income variance can be attributed to differences in factors of production, and how much to differences in the efficiency with which factors are used? OR What would the dispersion of incomes be if all countries had the same A?

3 INTRODUCTION Objective Income = F(Factors, Efficiency). The key steps in development accounting are: (1) choosing a functional form for F, and (2) accurately measuring Income and Factors. Efficiency is backed out as a residual. As for the Solow residual, this residual is a measure of our ignorance. Current consensus is that efficiency is at least as important as factors in explaining income differences. Objective of the paper is to try to chip away at this residual by improving on steps (1) and (2). looking at alternative functional forms, more sophisticated measurement of Income and Factors.

4 INTRODUCTION Road Map In order to cip away at the residual pursue the following avenues: 1 Robustness checks - choice of parameters in the basic model used in the literature, and measurement errors in output, labor, and years of schooling. 2 Improving the measurement of human capital - capturing quality differences, and health. 3 Improving the measurement of physical capital - variation in capital composition may result in unobserved quality differences in the capital stock. 4 Role of the sectorial composition of GDP - largest fraction of work force in poor countries is employed in agriculture. 5 Change the production function specification - CES generalization where cross-country efficiency differences are allowed to be non neutral.

5 INTRODUCTION Summary of Findings Among the listed avenues, the paper finds that the composition of physical capital and the functional form of the production function are the most promising.

6 MEASURE OF OUR IGNORANCE Basic Model GDP of a country: Per capita GDP of a country: Y = AK α (Lh) 1 α, y = Ak α (h) 1 α, where k = K/L, h is average human capital, L is number of workers, and K is physical capital stock.

7 MEASURE OF OUR IGNORANCE Data for Basic Model Penn World Tables version 6.1 GDP per capita (y) - real GDP per worker (RGDPWOK). Capital stock (K)- perpetual inventory method: K t = I t +(1 δ)k t 1, where Investment - I t = RGDPL POP KI, where RGDPL is real income per capita (Laspeyres method), POP is population, and KI is investment share in total income. Lastly, δ = Initial capital stock - K 0 = I 0 /(g+δ) from steady state expression of capital in Solow Model. I 0 is value of investment series in the first year it is available, and g is average geometric growth rate for investment series between first year with available data and L = (RGDPCH POP)/RGDPWOK, where RGDPCH is real GDP per capita (chain method). Barro and Lee (2001) data on average years of schooling Average human capital h = e φ(s), where s is average years of schooling in the population over 25 year old, and function φ(s) is piecewise linear. φ(s) = sif s 4, φ(s) = (s 4) if 4 < s 8, φ(s) = (s 4) (s 8) if 8 < s.

8 MEASURE OF OUR IGNORANCE Sample Size and Mesures of Income Dispersion A country has complete data at date t if it has an uninterrupted investment series between 1960 and t, and it has an observation for s in s is observed in the data every five years. Focus on 1996 as the most recent year that affords the largest sample. Focus on two measures of income dispersion: The ratio between the 90th (Canada) and the 10th percentile (Togo) of the income distribution - about 21. The log-variance - about 1.30.

9 MEASURE OF OUR IGNORANCE Measures of Success Factor-only model: y KH = k α (h) 1 α. Therefore y = Ay KH. Compare the (observed) variation in y KH to the (observed) variation in y. How successful is the factor-only model at explaining cross-country income differences? Two measures of success: success 1 success 2 = var[log(y KH)] var[log(y)] = y90 KH /y10 KH y 90 /y 10.,

10 MEASURE OF OUR IGNORANCE Performance of Basic Model By both measures of success the dispersion of y KH is much less than the dispersion of y. The factor only model does better the smaller and more homogeneous the sub-samples - above the median income, OECD, Europe and Americas.

11 ROBUSTNESS CHECKS Depreciation Rate A change in value of δ changes the relative weight of old and new investment in the perpetual inventory method used to compute capital stock. A higher rate of depreciation will increase the relative capital stock of countries that have experienced high investment rates towards the end of the sample period - poor countries. Change inδ does not change the explanatory power of the factor only model much.

12 ROBUSTNESS CHECKS Initial Capital Stock Capital stock series depends on I and K 0. The surviving portion of the guessed initial capital stock, K 0, as a fraction of the final estimate of the capital stock is given by (1 δ) t K 0 (1 δ) t K 0 + t i=0 (1 δ)i I t i. Longer time series of I reduces importance of guessed K 0. Restricting sample to 50 countries whose I series starts in 1950 does not change results much (success 1 = 0.39, and success 2 = 0.48). Suppose we trusted the estimate K 0 = I 0 /(g+δ) for U.S. (where date 0 is 1950). Then for any other country we could estimate K 0 from ( ) Y α ( ) 0 K0 L0 h α 0 =. Y US K US L US h US This also does not alter the results much (success 1 = 0.41, and success 2 = 0.34).

13 ROBUSTNESS CHECKS Returns to Schooling Function Decreasing returns to years of schooling in φ(s) dampens variation across countries in human capital, potentially increasing role of differences in A. Assume φ(s) = φ ss, and experiment with various values of the (constant) return to schoolingφ s - requires very high φ s to explain the data.

14 ROBUSTNESS CHECKS Average Years of Schooling Average years of schooling data is noisy. Use alternative data from De la Fuente and Doménech (2002) for 15 of 94 countries - success 1 = and succeess 2 = Not surprising because De la Fuente and Doménech (2002) show that the discrepancies between their measures and the ones in the literature are (1) stronger in first differences than in levels; and (2) stronger at the beginning of the sample than at the end. Data on years of schooling for 25 years or older is appropriate for rich countries with a large share of college graduates. Combine Barro and Lee (2001) data on years of schooling for 15 or older with data on over-25 to arrive at a composite measure of schooling - success 1 = and succeess 2 = Poor countries experienced much faster growth in schooling than rich countries. i.e. poor success ratios imply that the education gap is much smaller for the cohort less than 25 years of age.

15 ROBUSTNESS CHECKS Share of Capital Higher α implies that variation in k will have larger impact on variation in y. Because of Cobb-Douglas production function increasing the explanatory power of k through increases in α also means lowering the explanatory power of h. In the data there is more variation in k than in h across countries - higherαshould help.

16 QUALITY OF HUMAN CAPITAL Conceptualization Average human capital of country is now produced as: h = A h e φ(s), where A h captures quality of human capital quality, which may be different across countries. A h = p φp m φm k φk h hφh t. p is the teacher-pupil ratio, m is amount of teaching materials per student (textbooks, etc.), k h is amount of structures per student (classrooms, gyms, labs, etc.), and h t is the human capital of teachers. φ s will capture elaticity of human capital h w.r.t. to each of these inputs that affect quality.

17 QUALITY OF HUMAN CAPITAL Teacher s Human Capital Assume φ p = φ m = φ k = 0. In steady state h t = h. Thus, h = e φ(s)/(1 φ h), and y = Ak α e (1 α)φ(s) 1 φ h. Estimates ofφ h are very low or close to zero - Bils and Klwenow (2002), Hanushek (2004) and Altonji and Dunn (1996).

18 QUALITY OF HUMAN CAPITAL Other Determinants of Quality 1 Pupil-teacher rartio (p φp ): Lee and Barro (2001) report data on p in a cross-section of countries for various periods since The highest empirical estimate of φ p is 0.5 at which the success ratios are around Spending (m φm, k φ k h ): no direct data on materials - m, and structures - kh. From Lee and Barro (2001) use a measure of government spending per student in PPP dollars. Now, A h = Spending φsp. The rage of values for φ sp is zero to 0.2. For the upper bound model success ratios around 0.5. Bulk of this spending goes to teacher salaries, so variation in these data also reflect differences in the number and possibly the quality of teachers per student.

19 QUALITY OF HUMAN CAPITAL Test Scores Suppose A h = e φττ, where τ is the test score. Thus, higher test scores signal higher human capital. Model performance improves for very high increases in wages with test scores - unreasonable. Data on test scores provided by Lee and Barro (2001) for several countries on multiple tests (e.g.: math, science, and reading), and for multiple grades, at different dates. Two tests with greatest country coverage - 28 countries with overlapping test, input, and output data - are a math and a science test imparted to 13-year-old children between 1993 and Take the simple average of the two test scores as τ. studies of the relationship between test scores and wages tend to report Coefficients from regressions of log wages on absolute test scores can give us φ τ. φ τ 100 value from literature range from between 0.08 and 0.34 by Murnane on the lower side to between 0.55 and 1.02 by Neal and Johnson (1996) on the higher side.

20 QUALITY OF HUMAN CAPITAL Model Performance with Test Scores

21 QUALITY OF HUMAN CAPITAL Experience and Health 1 Experience (age schooling 6): Klenow and Rodriguez-Clare (1997) find that the net effect is negative: experience is actually higher in poor countries. Hence, correcting for experience lowers the explanatory power of the factor-only model. 2 Health: Large cross-country differences in nutrition and health status can cause substantial differences in energy and capacity for effort. Assume A h = e φamramr, where clearly φ amr < 0: a higher adult mortality rate (AMR) implies a less energetic workforce. Data on AMRs from WDI. Weil s preferred value for φ amr 100 is Conditional on this value explanatory power improves improves by almost one third - well above the 50 percent threshold. Important contribution. Alternative proxy for health status can be average birthweight generated by Behrman and Rosenzweig (2004). Thus, A h = e φ bwbw where where BW is Behrman and Rosenzweig s mean birthweight (in pounds). Value ofφ bw implied by Behrman and Rosenzweig s log wage regressions is 0.076, which implies a trivially small improvement in the success of the factor-only model.

22 QUALITY OF PHYSICAL CAPITAL Differences in Composition of Capital Caselli and Wilson (2004) find different types of equipment constitute widely varying fractions of the overall capital stock across countries. Wide variation in R&D spending: since equipment shares vary so much, so does the embodied-technology content of the aggregate capital stock. If the R&D content of equipment determines its quality, the quality of capital and not only its quantity may vary across countries. these differences are systematic - richer countries appear to employ high R&D capital much more than poor countries.

23 QUALITY OF PHYSICAL CAPITAL Conceptualization of Compositional Differences Do rich countries use higher quality equipment, and this higher quality accounts for some of the residual TFP variance? 1/γ P Y = B (x p) γ, γ < 1, p=1 x p = A p(h pl p) 1 α (K p) α. Capital is heterogeneous - P different types of capital. Assuming human capital per worker is constant across sector, i.e. h p = h, and that labor is free to flow across sectors so that the marginal product of labor is the same for all equipment types, we get P Y = K α (hl) 1 α B (A p) p=1 where K is market value of total capital stockξ p = K p/k. γ αγ 1 (1 α)γ(ξ p) 1 (1 α)γ 1 (1 α)γ γ.

24 QUALITY OF PHYSICAL CAPITAL Elasticity between Capital of Different Types Caselli and Wilson (2004) propose a regression-based approach to make inferences on the various A p s. Unfortunately, even with this knowledge it is virtually impossible to bound the amount of income variance that theξ p s can explain. This is because the last term, involving A ps and ξ ps, is very sensitive to the value of γ, and at the moment there is no reliable approach to the calibration of this elasticity. Ifγ is sufficiently low, i.e. capital types are sufficiently poor substitutes for one another, the quality of capital accounts for all of the unexplained component of income differences. However, whether such values are reasonable or not our current state of knowledge cannot say. Further research on the composition of capital is an important priority for development accounting.

25 QUALITY OF PHYSICAL CAPITAL Vintage Capital Model Solow s (1959) paper on vintage capital formalized the idea that technological progress is embodied in capital goods. The vintage-capital model could be described by Y t t = (h tl t) 1 α A t ikt i α, i=0 K t i = (1 δ) i Ĩ t 1. Ĩ t 1 is investment at time t 1 in terms of consumption good. Capital installed by sacrificing one unit of consumption at date s yields A s(1 δ) t s efficiency units of capital at time t, while capital installed by sacrificing one unit of consumption at time t yields A t efficiency units. If A s < A t the earlier sacrifice in consumption contributes less to output today not only because of physical depreciation, but also because of the older vintage.

26 QUALITY OF PHYSICAL CAPITAL Evidence on Vintage Capital Model According to Greenwood, Hercowitz and Krusell (1997) t = (h tl t) 1 α A t i K t i α, Y t i=0 K t i = (1 δ) i q t i Ĩ t 1. Now, one unit of consumption produces increasing amounts of capital of the same quality as against equal amounts of capital of increasing quality at different dates. PWT 6.1 investment data are data on I s = q s Ĩ s, and not on Ĩ s. Therefore, vintage effect - those that reduce relative price of investment goods - should already be accounted for. Regress Y t/(h tl t) on I t, (1 δ)i t 1, (1 δ) 2 I t 2, and experimented with 5, 10, and 20 lags of investment. If all vintage effects are adequately captured by investment-good prices in PWT 6.1 all the coefficients would take the same value. The vintage effects hypothesis would predict that coefficients on recent lags of investment would be systematically larger than those on older lags. The result was somewhat inconclusive, in that both hypotheses were rejected: all coefficients were not statistically the same, but neither they fell monotonically with the lag of investment.

27 SECTORAL DIFFERENCES IN TFP Variation in TFP at Sector Level Large differences in TFP could also be the result of variation in the weights in GDP of sectors with different sectorial-level productivity. Within narrowly defined industries, and even among countries at very similar levels of development, total factor productivity presents remarkable variation. Hence, industry-level studies suggest that aggregate TFP differences are not solely due to differences in the weights of high- and low-tfp sectors. Focus on an agriculture-nonagriculture split of GDP: in the poorest countries of the world virtually everyone works in agriculture, and in the richest virtually nobody does. poor countries have much lower labor productivity in agriculture; poor countries have somewhat lower labor productivity outside agriculture; and they have larger share of employment in the sector that Ű on average Ű is less productive.

28 SECTORAL DIFFERENCES IN TFP Model Performance with Sectoral Decomposition Factor-only model: y i KH = P AA A y i A,KH li A + P A A A yi A,KH li A. Take US to be the reference country. Thus, for the U.S. y A = P AA Ay A,KH, y A = P A A A y A,KH, Back out P A A A and P A A A and estimate y i KH for every other country with these US values. Result is 0.34 for success 1, and 0.32 for success 2. This means that, if all countries had the same industry-level TFPs as the US, but their observed allocation of measurable factors to agriculture and non-agriculture, the world distribution of income would be about one third as unequal as it actually is. For this sub-sample the corresponding success measures are 0.45 and 0.39 when the sectorial composition of GDP is not taken into account, i.e. taking account of differences in sectorial composition actually decreases model s performance. Dispersion in agricultural labor productivity is a critical source of dispersion in per-capita income. Almost all of the variation in agricultural income comes from differences in agricultural TFP.

29 NON-NEUTRAL DIFFERENCES IN TECHNOLOGY Non-neutrality of Technology We view differences in efficiency as factor neutral: some countries simply use all of their inputs more efficiently than others. More general view allows for the possibility that differences in technology show up as differences in the efficiency with which specific factors are used. Y = [α(a k K) σ +(1 α)(a h Lh) σ ] 1/σ,α (0, 1), σ < 1, ( ) 1/σ Sk y A k = α k, ( ) 1/σ Sh y A h = 1 α h, where S k and S h are the shares of physical and human capital in income, respectively.

30 NON-NEUTRAL DIFFERENCES IN TECHNOLOGY Intuition The output-capital ratio (y/k) is decreasing in income (y). The output-human capital ratio (y/h), instead, is increasing. If we assume that factor shares, S k and S h, are constant across countries, but we allow for non-neutrality in technology differences, we reach the startling conclusion that rich countries use human capital more efficiently than poor countries, but they use physical capital less efficiently. If factor shares vary systematically with per-worker income, then it becomes critical to know what is the elasticity of substitutionη = 1/(1 σ). Ifσ > 0 the two factors are good substitutes increase the usage of the most efficient factor. Hence, when σ > 0 demand will concentrate on the factor with high efficiency high share in income for this factor. Withσ > 0, when we observe a high income share for factor x we can infer that this factor is efficient. Similarly, Withσ < 0, when we observe a high income share for factor x we can infer that this factor is used inefficiently.

31 NON-NEUTRAL DIFFERENCES IN TECHNOLOGY Value ofη Literature has a very wide range of values. Try different values of η, and for each value back out A k and A h for US, and use these to compute y KH for every other country.