Economic Environment

The economy consists of identical infinitely lived agents, and population grows exogenously at rate n. Agents have preferences only over consumption, and choose to allocate their time endowment in three types of activities: consumption-good production, R&D effort, and human capital attainment.

Our model economy is characterized by the following three equations: First, at period t, output (Y_t) is produced using labor (L_Yt) and physical capital (K_t) according to the following aggregate Cobb-Douglas technology:

where h_t represents the effectiveness of average human capital level on labor, α is the share of capital, ξ is a technology externality, and A_t is the economy's technical level.

Second, the R&D equation that determines technological progress is given by

where L_At is the portion of labor employed in the R&D sector at time t, A*_t is the worldwide stock of existing technology that grows exogenously at rate

, ϕ is an externality due to the stock of existing technology, and λ captures the existence of decreasing returns to R&D effort. The above R&D equation is the one proposed by Jones (1995, 2002) plus a catch-up term (A*_t/A_t)^ψ, where ψ is a technology-gap parameter. The catch-up term is also consistent with the “relative backwardness” hypothesis of Findlay (1978) that the rate of technological progress in a relatively backward country is an increasing function of the gap between its own level of technology and that of the advanced country.

Third, we have the schooling equation that determines the way by which human capital accumulates. The human capital technology follows Bils and Klenow (2000), who suggest that the Mincerian specification of human capital is the appropriate way to incorporate years of schooling in the aggregate production function. Following their approach, human capital per capita is given by

where f(S_t)=ηS^β_t, η>0, β>0; and S_t is the labor force average years of schooling at time t. The derivative f′(S_t) represents the return to schooling estimated in a Mincerian wage regression: an additional year of schooling raises a worker's efficiency by f′(S_t).

We assume that, each period, agents allocate time to human capital formation only after output production has taken place.

From equation (4), we can write

which in turn implies

It is important to notice that the above human capital technology differs from the one employed in Jones (2002). In particular, Jones assumes that education investment fully depreciates each period or, in other words, that human capital does not accumulate. However, the optimal allocation to investment in education declines along the adjustment path and therefore without human capital accumulation, human capital index is larger in lower-income nations; clearly a counterfactual result. Our equation (6) does not suffer from this counterfactual result because the value of S rises with the income level, as the international evidence suggests.

Social Planner's Problem

Let C_t be the amount of aggregate consumption at date t. A central planner would choose the sequences {C_t, S_t, A_t, K_t, L_Yt, L_At, L_Ht}^∞_t=0 so as to maximize the lifetime utility of the representative consumer subject to the feasibility constraints of the economy, and the initial values L₀, K₀, S₀, and A₀. The problem is stated as follows:

subject to

where θ is the inverse of the intertemporal elasticity of substitution, ρ is the discount factor, and δ is the depreciation rate of physical capital. Equation (9) is a feasibility constraint as well as the law of motion of the stock of physical capital; it states that, at the aggregate level, domestic output must equal consumption plus physical capital investment, I_t. Equation (12) is the labor constraint; the labor force—that is, the number of people employed in the output and the R&D sectors—plus the number of people in school must be equal to population.

Solving this dynamic optimization problem obtains the Euler equations that characterize the optimal allocation of labor in human capital investment, in R&D investment, and in consumption/physical capital investment, respectively, as follows:

At the optimum, the planner must be indifferent between investing one additional unit of labor in schooling, R&D, and final output production. The LHS of equations (15) and (16) represent the return from allocating one additional unit of labor to output production. The RHS of equation (15) is the discounted marginal return to schooling, taking into account labor growth. The RHS term in brackets arises because human capital determines the effectiveness of labor employed in output production as well as in R&D. The RHS of equation (16) is the return to R&D investment. An additional unit of R&D labor generates [λ(A_t+1−A_t)]/L_At new ideas for new types of producer durables. Every new design increases next period's output by ξY_t+1/A_t+1 and R&D production by dA_t+2/dA_t+1 times [(1−α)Y_t+1]/L_Y,t+1{[λ(A_t+2−A_t+1)]/L_A,t+1}⁻¹, where the term [(1−α)Y_t+1]/L_Y,t+1{[λ(A_t+2−A_t+1)]/L_A,t+1}⁻¹ gives the value of one additional design that equalizes labor wages across sectors. Euler equation (17) is standard and states that the planner is indifferent between consuming one additional unit of output today and converting it into capital (thus consuming the proceeds tomorrow).

Steady-State Growth

We now derive the model's balanced-growth path. Solving for the interior solution, equation (12) implies that in order for the labor allocations to grow at constant rates, L_Ht, L_Yt, and L_At must all increase at the same rate as L_t. This means that the ratio L_Ht/L_t is invariant along the balanced-growth path. Hence, equation (11) implies that, at steady state (SS),

is constant and is given by

where

. Equation (18) shows that along the balanced-growth path the economy invests in human capital just to provide new generations with the steady-state level of schooling.

Let lowercase letters denote per capita variables, and g_x=G_x−1 denote the growth rate of x. The aggregate production function, given by equation (8), combined with the steady-state condition g_Y,ss=g_K,ss delivers the gross growth rate of output as a function of the gross growth rate of technology as

Since G_A,ss is a constant, it follows from equation (2) that

Equation (20) shows the relationship between the technology frontier growth rate and the technology growth rate of the model economy. Since

, it is easy to show that there is a unique point at which

Given the nature of the experiments that we want to carry out, we focus on the special case in which all countries grow at the same rate in steady state. That is, we assume that

is given by expression (21), and therefore so is G_A,ss.

Alternatively, we could assume that the technology leader shifts the world technological frontier outward according to equation (2), which now reduces to

where A*_t/A_t=1 as imitation is not possible at the frontier; and * denotes the value that variables take in the leading country. In such case

as in Jones (1995, 2002). Assuming that n=n*, and substituting G*_A into equation (20) delivers equation (21).

Consistent with Jones (1995, 2002) our balanced-growth path is free of scale effects. The reason why our model's long-run growth is equivalent to that of Jones even in the presence of a schooling sector, is that at steady state the mean years of education, S_t, reaches a constant level

.

Transition Dynamics

The aggregate production function, equation (8), suggests that we normalize variables by the term

. We then rewrite consumption, physical capital, and output as

,

, and

), respectively. Using equation (15) gives

where u_Yt and u_At are the labor shares in R&D and final-output production at time t, respectively. From the R&D equation (2), we get

where T=A*_t/A_t; and υ=μ(A*_t)^ϕ−1L^λ_t, which is a constant.

Finally, from equation (17) we get

The system that determines the dynamic equilibrium normalized allocations is formed by the conditions associated with three control and three state variables as follows:

Control Variables:

1. Euler equation for labor share in schooling, u_Ht: Equation (23)
2. Euler equation for labor share in R&D, u_At: Equation (25)
3. Euler equation for consumption,
   : Equation (26)

Subject to the constraint u_Yt=1−u_At−u_Ht.

State Variables:

1. Law of motion of human capital, S_t: Equation (6)
2. Law of motion of technology, A_t: Equation (24)
3. Law of motion of physical capital

where

and

Calibration

Because relative values of the cross-country data to which we compare the predictions of the model are taken with respect to U.S. levels, we calibrate the model parameters using, when possible, U.S. data as the steady-state outcome. Table 1 presents the parameter values used to carry out the simulations. We choose a value of 0.06 for the depreciation rate (δ), a value of 0.96 for the discount factor (ρ), and 0.36 for the capital share of output (α), which are standard in the literature. To assign values to per capita income growth rate in steady state (g_y,ss), and to population growth rate (n), we follow Jones (2002). In particular, we set g_y,ss equal to 2%, the average growth of output per hour worked between 1950 and 1993 in the United States, and n equal to 1.2%, the average growth rate of the labor force in the G-5 countries (France, West Germany, Japan, the United Kingdom, and the United States) during the period 1950–1993. The reason for using the average growth rate of labor in the G-5 rather than any other group of countries (or, for that matter, the whole sample) is that the main role of population growth rate in the model is to move the world technology frontier in steady state, and clearly the majority of world research effort is conducted in the G5 countries.¹¹

It is not clear what the steady-state value of the average educational attainment ought to be, given that mean years of schooling has been increasing over the past decades in most developed countries. We choose to set

to 12.5, to match the 1993 U.S. figure reported by Jones (2002). From equations (18), (23), and (26), it can be easily shown that the values of

, n, and G_y,ss imply an interest rate [given by the RHS of (26)] at steady state of 7.9%, which is well within U.S. observations.

where Mincer_i=f′(S_i) is the estimated Mincerian coefficient for country i; a and b equal ln(ηβ) and (β−1), respectively; and ε_i is a disturbance term. We obtain η=0.69 and β=0.43, which are very close to estimates by Bills and Klenow (2000).

Finally, we calibrate the R&D technology parameters. As a benchmark case, we set λ=0.5, and using equation (21) we recover the value of ϕ= 0.95.¹⁴

Transition Dynamics Predictions

Unlike steady-state regressions, in this section we assume that all countries belong to the same transitional path, approaching a unique, common balanced-growth path. Specifically, we perform two experiments as follows: First, we simulate the dynamics of a representative economy and study how well its adjustment path represents the cross-country data on key variables such as the state variables, interest rates, investment rates, and output growth rates. Second, we propose an exercise similar in spirit to that of MRW, which tries to assess how much of the cross-country output variation can be explained by transition dynamics.¹⁷

To carry out the first experiment, we need to estimate the policy rules that take state variables from given initial values to the steady state. Doing so requires the following two conditions: (a) given that the further away we move from the balanced-growth path, the lower the accuracy degree of the numerical approximation, we choose the initial values so that the numerical approximation provides a maximum-error measure of about 2% (see Table 2); (b) we start the adjustment paths inside the cloud of cross-country observations that compose our comprehensive sample.¹⁸

The goal of the first experiment is to see how well the transition dynamics of the model can explain important stylized facts such as cross-country dispersion of growth rates, cross-country dispersion of saving/investment rates, and cross-country equality of real interest rates. Figure 1 depicts cross-sectional data, along with off-steady-state predictions for physical capital, average years of schooling, TFP, interest rates, investment rates, and relative output growth. It is evident that the plotted data show wide cross-country dispersion in all variables, except for real interest rates, which are quite uniform across nations above the 25th percentile. State variables (K, S, A) generally increase with the relative level of output, and investment and growth rates generally depict weak inverted-U shapes, starting low and achieving their maximum values for middle-income nations.²¹

Adjustment paths for the non-oil sample: Benchmark case. Note: RGDPW is relative GDP per worker.

With fixed initial and final values of the state variables, the question is how well the transition path follows the data cloud in between. If we look at the charts in panels A, C, and E of Figure 1, the primary finding is that the simulated dynamics seem to fit well across the state-variable observations. These charts illustrate a number of other points worth noting. First, notice that a larger degree of relative backwardness (i.e., a larger value of ψ) induces faster technology catch-up, and slower human capital accumulation, making the adjustment paths better fit the data. Second, the simulated physical and human capital levels tend to diverge with respect to the rich countries' data. This is the result of calibrating the steady state to U.S. data. The two variables' divergent processes, however, offset each other and, as a result, the technology path captures well the observations.

Finally, let us pay attention to the charts in Panels B, D, and F of Figure 1. Below the 30th percentile in panel F, output-growth predictions are too large; thus, our model overpredicts output growth at early stages of development. Above the 30th percentile, however, predictions fall across the cloud of observations; thus, our model does much better in predicting output growth at later stages of development. In addition, panels B and D show that predictions capture the uniformity of the real interest rates (return to capital) above the 25th percentile, and are consistent with investment rates above the 35th percentile. As is the case with output growth, our model is weaker in replicating interest and investment rates at early development.

Our first experiment has shown that transition dynamics capture fairly well the cross-country equality of the real interest rates, and generate investment rates that are plausible, even though lower investment ratios at early levels of development would better capture the dispersion of the data. The main explanation for this lower dispersion of the predictions is probably a larger degree of market imperfections and distortions in less developed countries, something that the model cannot explain, and that can be perceived as a source of differences in steady states or in convergence speeds along the transition. The predicted output growth rates, on the other hand, are clearly impossible for countries with relative real GDP below the 30th, but reasonable for more developed nations. For the sake of comparability between transition dynamics predictions and steady-state regression predictions, it is important to mention that steady-state predictions are not very successful in accounting for the observed cross-country income growth dispersion either. Steady-state growth regressions of the MRW type need to make use of transitional factors to be able to minimally fit the data.

Can Transition Dynamics Explain the Cross-Country Output Data?

We now turn attention to the main issue of the paper: How well can the transition dynamics of the model explain cross-country income dispersion? More specifically, our second experiment tries to assess quantitatively how well the transition dynamics fit the output-per-worker data. This is important because fitting the cross-country income data is where steady-state regressions achieve their great success and therefore such an experiment is well motivated. Since we want to compare the transition dynamics predictions of our model with those of steady-state income regressions, we need to construct a measure of fit for transition dynamics that can be compared with a measure of fit in level regressions (namely, the OLS R²).

Taking logs in the Cobb-Douglas representation of the aggregate production function, and substituting inputs for their balanced-growth values, we end up with a standard in the literature steady-state econometric equation

where

is the estimate of output per worker;

and

represent estimates of k and S, respectively, derived from steady-state conditions using investment rates;

are estimated coefficients; and ε is a random disturbance term. Evidently, for the underlying model to be consistent with the data, estimated coefficients must be plausible according to the weight assigned by the national accounts to the different inputs. To each combined{\pagebreak} value

the regression assigns a predicted output level in log-scale, and all of the predicted output levels are in turn translated into a measure of fit (the OLS R²).

Following an equivalent procedure, we first calculate for each country the combined value e^f(S)(1−α)[K/(L_A+L_Y)]^α implied by the data, imposing the calibrated parameter values. Notice that this extendedstate variable represents the per worker human capital term (i.e., e^f(S)(1−α)), and the per worker physical capital term {i.e., [K/(L_A+L_Y)]^α}, as specified in the production function given by equation (8). Second, to each country's value of the combined state variable, we assign the output per worker level Y/(L_A+L_Y) predicted by the transition path.²²

Because the simulated adjustment path is a discrete set of pairs

we use interpolation methods to generate the predicted output level.

As mentioned previously, to generate the adjustment-path simulation, we employ initial values for the relative per worker physical capital stock, and the average educational attainment of 5.4% and 2.7 years, respectively. It works out that these two initial values imply a minimum value of the relative extended state variable of 18.9%. The sample that we employ to compute the measure of fit must then consist of those 51 nations that provide values of the extended state variable above 18.9%.²³

Figure 2 displays the actual output data (plot), and the predicted output data for the two values of ψ (continuous lines). To assess the fit of the adjustment paths, we employ the following statistic, which is equivalent to the OLS R²:

where

and x_j are the predicted and actual values of variable x for country j, respectively; and N is the number of countries included in the sample. Our variable x must be the natural log of relative GDP per worker to make the pseudo-R² comparable to the R² reported in steady-state regressions.

Adjustment-path predictions of GDP per worker for 51-nation sample: Benchmark case. Note: GDPW and KW denote GDP per worker and physical capital per worker, respectively.

For the adjustment-path predictions expressed in natural logs, Table 3 reports estimates of the pseudo-R². As it is shown, the transition path can explain up to 76% of the variation of relative output per worker in both the 51 non-oil and the 21 OECD samples. These numbers compare pretty well with the R² obtained by steady-state regressions. For example, MRW report a maximum R² of 78 for their non-oil sample, and 28% for the OECD group. Nonneman and Vanhoudt (1996), in turn, obtain an R² of 78% for OECD nations. These numbers are just a bit above the ones delivered by the transition predictions.

How can one interpret our results in the context of the existing empirical literature? Our results imply that the transition dynamics of an R&D model with endogenous human capital can explain the cross-country output variation as well as the more popular steady-state regressions can. Our findings do not discredit in any way the common steady-state regression exercises. They do, however, provide evidence that transition dynamics may be at least as important as steady states in explaining income differences.

Data sets and computer programs

The data and programs used in this paper are available by the authors upon request.

• Income (GDP) and its components (Source: PWT 5.6). Cross-country GDP per worker and real investment shares are taken from the Penn World Tables (PWT), Version 5.6, as described by Summer and Heston (1991). This data set is available on line at: http://datacentre.chass.utoronto.ca/pwt/index.html.
• Physical capital stocks (Source: STARS, PWT 5.6, and perpetual inventory approach). For the non-oil cross-country sample, we follow the perpetual inventory approach. The capital stock is calculated by summing investment from its earliest available year (1960 or before) to 1986 with the depreciation rate set at 6%. The initial capital stock is determined by the initial investment rate, divided by the depreciation rate plus the growth rate of investment during the subsequent 10 years. In the calibration of the parameter ψ, the Japanese physical capital stock in 1960 and South Korean physical capital in 1963 are obtained by deflating the 1965 PWT data (which unfortunately do not extend to 1960), using growth rates implied by the STARS physical capital data.
• Labor force (Source: PWT 5.6). The cross-country data set on the labor force is also taken from the Penn World Table, Version 5.6.
• Education [Source: STARS (World Bank)]. Annual data on educational attainment are the sum of the average number of years of primary, secondary, and tertiary education in labor force. These series were constructed from enrollment data using the perpetual inventory method, and they were adjusted for mortality, dropout rates, and grade repetition. For a detailed discussion on the sources and methodology used to build this data set, see Nehru et al. (1995).
• Return to capital. Annual data on return to capital (r_t) is calculated as r_t=α(Y/K).

Countries in the comprehensive sample

Our comprehensive sample includes the 79 countries from the Mankiw et al. (1992) non-oil sample for which annual data on income, raw labor, human capital, and investment rates were available for every year of the MRW sample period, 1960–1985. The table below provides a list of these nations along with the 1960–1985 average value of relevant variables for each country. An asterisk denotes the 51 nations included in the sample used to carry out the second experiment.