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ANNEX I Construction of Relative Rental Rates of Capital
With geometric depreciation and perfect foresight the rental price of a capital good Kj in period t is given by:
where pij(t) denotes the investment price of capital good j in period t, r(t) is the economy-wide nominal rate of return between periods t and t – 1, and δj is the depreciation rate of capital good j. The above arbitrage equation says that the return on investing PIJ in capital good j, which is the rental rate plus any valuation changes minus depreciation, is equal to the opportunity cost of investing the same amount in an asset with known return r.
Investment series for each type of capital good in current and constant prices were used to obtain investment prices. The price series were smoothed by taking four period averages. Depreciation rates are taken from Young (1992).
Under constant returns to scale, the following equality holds:
where the left-hand side is the share of aggregate capital income in GDP and the right-hand side is the sum of the rental incomes of the five types of capital goods. This equality can be exploited to calculate the rental rates of the different capital goods. This is done by varying r(t) until the sum of rental payments to individual capital goods is equal to the share of aggregate capital income in GDP.
ANNEX II A Two-sector Model with Physical and Human Capital
The model in this appendix closely follows those developed by Uzawa (1965) and Lucas (1988). It has two sectors and two factors of production, human capital H and physical capital K. Sector 1 uses both inputs to produce consumption goods and capital goods, which have the same relative price:
where Y1 is output of sector 1,
where Y2 is output of sector 2 and B2 is a constant. Human capital employed in both sectors cannot exceed the available stock: H ≥ H1 + H2. GDP is the sum of output in sectors 1 and 2: GDP = Y1+ pY2, with p the relative price of H.
A crucial assumption in the Lucas’ model is that H is produced with constant returns to scale (β = 1), which, as can be shown, yields a long-run equilibrium in which K, H, and GDP all grow at the same rate.6 When the economy is in this equilibrium it is said to be on its balanced growth path (BGP). Along this path K/H is constant. In the case with β < 1, a BGP does not exist and the economy converges to a steady state in which growth comes to a halt in the absence of exogenous increases in H, just as in the neoclassical growth model (Solow, 1956). The case with β > 1 is not very interesting as it implies accelerating growth over the long-run, which is normally not observed over long time horizons.
The model has transitional dynamics. Because it takes time to accumulate K and H, an economy with an initial endowment K0/H0 that is different from the long-run ratio K*/H* cannot jump instantaneously to its BGP or steady state. For K0/H0 < K*/H*, K/H will gradually increase while the economy moves toward its long-run equilibrium, vice versa for K0/H0 > K*/H*.
Now let H be defined as the product of years of schooling s, raw labor LR, and the state of knowledge in the economy A:
The above relation shows the sources of human capital accumulation. In the long-run, however, only LR and A can be sources of growth in human capital, because educational attainment is bounded by the maximum number of years a person can spend in school. The state of knowledge increases through research and development, scientific research, learning-by-doing, and adoption and adaptation of foreign technologies. Learning-by-doing can be the by-product of investment, which can extend beyond the firm that makes the investment. Foreign technologies can be introduced into an economy through trade and foreign direct investment and possibly other channels.
Dividing equation (A2.2) by H it can be rewritten as:
where (1–u) = H2/H
If TFP is interpreted as Aα, then the product of TFP1/α and quality adjusted labor is a measure of human capital similar to that in equation (A2.3). The Uzawa-Lucas model then predicts that K/(sLRTFP1/α) is constant in the long-run. In logarithms this is:
where L = sLR is quality adjusted labor and θ0 is a constant. TFP is assumed to be a function of learning-by-doing through capital accumulation and foreign knowledge imported through trade XM:
This paper, which was prepared by Harm Zebregs (APD), reports preliminary results for a forthcoming IMF working paper.
See Appendix I for details.
The measure of labor input could be refined further by taking into account actual hours worked. These data are available on a quarterly basis from 1980 onward, but they appear to suffer from measurement errors in the early 1980s.
This literature suggests that productivity rises with increases in specialization in the production process. Specialization is proxied by the number of intermediate inputs incorporated in a unit of final output. Trade raises productivity because it gives producers access to a larger variety of intermediate inputs.