Silviase/jgov_v1.4_full_ann huggingface dataset:
Constant parameter $ \phi\in(0,1) $ represents the weight for local knowledge. The exogenous rate of quality improvement at the world frontier for any intermediate good is denoted by g>0: $ \bar{A}_{t}=(1+g)\bar{A}_{t-1} $ . Note that if $ A_{t-1}=\bar{A}_{t-1} $ and $ H_{t}^{R}(i)=\bar{H} $ , i.e., if the follower country has the same knowledge level and employs the same amount of skilled labor as the frontier countries, the rate of quality improvement is g.

### 2.3 Consumption

Both skilled and unskilled workers are risk neutral, and the discount rate of any worker is r. In period t, each “young” worker born in that period receives her wage and consumes in the middle of that period. She may save part of her wage and purchase shares of intermediate firms at the end of period t for future consumption. In the middle of period $ t + 1 $ , shares of each firm yield dividends, which is equal to production profits less costs of quality improvement. After that, “old” consumers sell all their shares to the “young” generation and consume the quantity of the sum of the dividends and the value of their shares.

Under these assumptions, the value of the total shares of intermediate firm i, $ v_{t}(i) $ , is determined by the arbitrage condition:

$$ (1+r)v_{t}(i)=v_{t+1}(i)+\pi_{t}(i)-w_{t}^{H}\big(H_{t}^{M}(i)+H_{t}^{R}(i)\big). $$

Since the producers’ solutions determine $ v_{t}(i) $ whereas $ v_{t}(i) $ does not affect the producers’ decisions, the equilibrium analysis in the next section focuses on the producers’ side of the model.

## 3 Equilibrium Analysis

### 3.1 Dynamic and Steady-State Equilibrium

The firm producing intermediate good i chooses levels of $ H^{M}(i) $ and $ H^{R}(i) $ to maximize its monopoly profits from production, given by equation (5), less the cost of wages paid to skilled workers as follows:

$$ \max_{H_{t}^{M}(i),H_{t}^{R}(i)}(1-\alpha)A_{t}(i)L-w_{t}^{H}\big(H_{t}^{M}(i)+H_{t}^{R}(i)\big), $$
where $ w_{t}^{H} $ is the wage rate of skilled labor at time t. The first-order conditions are

$$ \left(1-\alpha\right)\eta M^{\prime}\big(H_{t}^{M}(i)\big)\bar{A}_{t-1}=w_{t}^{H}, $$

and

$$ \left(1-\alpha\right)\left(1+g\right)R^{\prime}\big(H_{t}^{R}(i)\big)\tilde{A}_{t-1}^{\phi}\bar{A}_{t-1}^{1-\phi}=w_{t}^{H}. $$

These conditions suggest that intermediate good sectors are symmetric, and hence $ H_{t}^{J}(i) = H_{t}^{J} \equiv \int_{0}^{1} H_{t}^{J}(i) di $ for J = M, R and $ A_{t}(i) = A_{t} $ . Thus, we obtain

$$ \frac{M_{t}^{\prime}}{R_{t}^{\prime}}=\frac{1+g}{\eta}\left(a_{t-1}-\frac{\eta}{1+g}M_{t-1}\right)^{\phi}. $$

where $ M_{t} \equiv M(H_{t}^{M}) $ , $ R_{t} \equiv R(H_{t}^{R}) $ , and $ X' $ indicates the first-order derivative of any function X. In addition, $ a_{t-1} \equiv A_{t-1}/\bar{A}_{t-1} $ , or $ a_{t-1} $ is the quality level of the follower country relative to the world frontier.

Equation (6) can be written as

$$ a_{t}=\frac{\eta}{1+g}M_{t}+R_{t}(a_{t-1}-\frac{\eta}{1+g}M_{t-1})^{\phi}. $$

Combining equations (10) and (11) yields

$$ a_{t}=\frac{\eta}{1+g}\left(M_{t}+\frac{M_{t}^{\prime}}{R_{t}^{\prime}}R_{t}\right). $$

Incorporating equation (12) and its first-order lag into equation (11) and the labor-market clearing condition $ H_{t}^{M} + H_{t}^{R} = H $ for any t yields

$$ \frac{M_{t}^{\prime}}{R_{t}^{\prime}}=\left(\frac{1+g}{\eta}\right)^{1-\phi}\left(\frac{M_{t-1}^{\prime}}{R_{t-1}^{\prime}}R_{t-1}\right)^{\phi}. $$

Quality improvement generates a non-negative profit if $ (1-\alpha)A_{t}L-w_{t}^{H}H\geq0 $ . Using equations (8), (9), and (12) to rewrite this yields

$$ L>\frac{H}{\frac{M_{t}}{M_{t}^{\prime}}+\frac{R_{t}}{R_{t}^{\prime}}}. $$

The fact that the maximum value of the right-hand side of this inequality is either $ \mu $ or $ \rho $ yields the following proposition.
Proposition 1. (Profitability of Quality Improvement) All firms producing intermediate goods engage in quality improvement in every period if $ L \geq \mu $ and $ L \geq \rho $ .

This proposition states that if the number of production workers, and hence the size of the market, is sufficiently large, quality improvement generates a non-negative profit. We assume that these conditions are met throughout this paper.

The equilibrium of the model can now be formally defined.

Definition 1. (Dynamic and Steady-State Equilibrium) Given $ a_{0} $ , $ H_{1}^{R} $ is determined by equation (10) for t = 1, and the dynamics of $ H_{t}^{R} $ in equilibrium for $ t \geq 2 $ are determined by equation (13). $ H_{t}^{R} $ and equation (12) yield the equilibrium $ a_{t} $ for $ t \geq 1 $ . The steady-state equilibrium is characterized by $ H^{R*} $ and $ a^{*} $ such that $ H^{R*} = H_{t}^{R} = H_{t-1}^{R} $ and $ a^{*} = a_{t} = a_{t-1} $ .

To graph $ H_{t}^{R} $ as a function of $ H_{t-1}^{R} $ and analyze the dynamics of $ H^{R} $ , we first consider the following equality satisfied in the steady state:

$$ q(H^{R*})\equiv\left(\frac{R^{\prime*}}{M^{\prime*}}\right)^{\frac{1-\phi}{\phi}}R^{*}=\left(\frac{\eta}{1+g}\right)^{\frac{1-\phi}{\phi}}, $$

where $ M^{*} \equiv M(H - H^{R*}) $ , $ R^{*} \equiv R(H^{R*}) $ , and $ ^{\prime*} $ indicates the first-order derivative of these functions evaluated at the steady state.

The shape of the function q, and hence the characteristics of the equilibrium, depends on the parameter values. The left-side figures of Figure 1 illustrate this dependence. First, if $ \phi + \rho < 1 $ , $ q(H^{R*}) $ for $ 0 \leq H^{R*} \leq H $ is monotonically decreasing between positive and negative infinity. Hence, equation (14) has a unique solution. Second, if $ \phi + \rho > 1 $ and $ H > \theta $ , where

$$ \theta=[\rho-(1-\phi)\mu]\left[\frac{\bar{H}^{\rho+\phi-1}+\frac{\mu}{\rho}\frac{\eta}{1+g}^{1-\phi}}{(\rho+\phi-1)^{\rho+\phi-1}\{(1-\mu)(1-\phi)\}^{(1-\mu)(1-\phi)}}\right]^{\frac{1}{\rho-(1-\phi)\mu}}, $$

the function q is increasing in $ H^{R*} $ for $ H^{R*} < \frac{\rho-(1-\phi)}{\rho-(1-\phi)\mu} H \equiv \hat{H} $ and decreasing for $ H^{R*} > \hat{H} $ . Since the maximum of q, $ q(\hat{H}) $ , is greater than $ \left( \frac{\eta}{1+g} \right)^{\frac{1-\phi}{\phi}} $ ,
equation (14) has two solutions. Third, if $ \phi + \rho > 1 $ and $ H < \theta $ , no solution to equation (14) exists for $ 0 \leq H^{R*} \leq H $ .

In any of the three cases, equation (13) implies $ \frac{dH_{t}^{R}}{dH_{t-1}^{R}} \geq 0 $ . The equilibrium $ H_{t}^{R} $ equals H when $ H_{t-1}^{R} = H $ , and it is zero when $ H_{t-1}^{R} = 0 $ . In addition, equation (13) implies

$$ \lim_{H_{t-1}^{R}\rightarrow0}H_{t}^{R\prime}(H_{t-1}^{R})=\begin{cases}\infty&if\quad\phi+\rho<1\\ 0&if\quad\phi+\rho>1.\end{cases} $$

By combining these results, we can graph $ H_{t}^{R} $ as a function of $ H_{t-1}^{R} $ for each case, as illustrated in the right-side figures of Figure 1.

These figures, in which $ H_{S}^{R*} $ indicates a positive stable steady-state level of $ H^{R} $ , while $ H_{U}^{R*} $ indicates an unstable steady state, suggest the following proposition.

### Proposition 2. (Three Patterns of Equilibrium)

(i) If $ \phi + \rho < 1 $ , there is a unique stable steady state in which $ H^{R*} $ is positive.

(ii) If $ \phi + \rho > 1 $ and $ H > \theta $ , there are two stable steady-state equilibria. If the initial level of $ H^{R} $ is sufficiently small, $ H^{R} $ converges to a steady state in which $ H^{R*} $ is zero. Otherwise, $ H^{R} $ converges to a steady state in which $ H^{R*} $ is positive.

(iii) If

, $ \phi i + \rho > 1 $ and $ H < \theta $ , there is a unique stable steady state in which $ H^{R*} $ is zero.

The key assumption that leads to the presence of multiple dynamic patterns and steady-state equilibria is a ‘biased’ dynamic externality that exists in technology adoption with local R&D but not in adoption without R&D. This externality results in a virtuous cycle in local R&D in some cases and a vicious cycle in others. If the externality is ‘unbiased,’ i.e., if the local knowledge expands through any type of adoption and $ \tilde{A}_{t-1}=A_{t-1} $ , there is only one stable steady-state equilibrium regardless of parameter values and initial conditions.
Let us consider the characteristics of each case in more detail. In case (i), the parametric condition, $ \phi + \rho < 1 $ , implies that the weight assigned to local knowledge in quality improvement through local R&D, $ \phi $ , is small, and hence the dynamic externality is relatively unimportant. As a result, there is a unique steady state whatever the initial quality level.

In case (ii), the dynamic externality is relatively important. Hence, when the initial quality level $ a_{0} $ , and hence the productivity of local R&D, is sufficiently high, is also high. The number of skilled workers in R&D activities is also large in the initial period. The subsequent knowledge improvement by the initial R&D leads to a virtuous cycle in R&D and convergence to a positive steady-state level of R&D. By contrast, if the follower country's quality level is initially low, gains from initial R&D are not sufficiently large for the country to enter the virtuous cycle.

In case (iii), the total amount of skilled labor H is sufficiently small. Hence, even when the current quality level is high, the quality improved by R&D, and hence the productivity of R&D, is not sufficiently high. Thus, whatever the initial conditions, the economy converges to the zero-R&D steady state.

### 3.2 Scale Effects

In equilibrium, equation (12) generates

$$ \frac{d a_{t}}{d H_{t}^{R}}=\frac{\eta}{1+g}\left(-\frac{M_{t}^{\prime\prime}}{M_{t}^{\prime}}-\frac{R_{t}^{\prime\prime}}{R_{t}^{\prime}}\right)\frac{M_{t}^{\prime}}{R_{t}^{\prime}}R_{t}>0, $$

where $ X'' $ denotes the second-order derivative of any function X. Using this and denoting a in the stable steady state in which $ H^{R} $ is positive by $ a_{S}^{*} $ yield

$$ \frac{\partial a_{S}^{*}}{\partial H}=\frac{\eta}{1+g}\frac{M_{S}^{\prime*}R_{S}^{\prime*}}{R_{S}^{*}}\frac{M_{S}^{\prime\prime*}R_{S}^{\prime*}+M_{S}^{\prime*}R_{S}^{\prime\prime*}+\frac{\phi}{1-\phi}\frac{\left(R_{S}^{\prime*}\right)^{2}M_{S}^{\prime*}}{R_{S}^{*}}+\frac{\phi}{1-\phi}M_{S}^{\prime\prime*}R_{S}^{\prime*}}{M_{S}^{\prime\prime*}R_{S}^{\prime*}+M_{S}^{\prime*}R_{S}^{\prime\prime*}+\frac{\phi}{1-\phi}\frac{\left(R_{S}^{\prime*}\right)^{2}M_{S}^{\prime*}}{R_{S}^{*}}}>0, $$

since

$$ M_{S}^{\prime\prime*}R_{S}^{\prime*}+M_{S}^{\prime*}R_{S}^{\prime\prime*}+\frac{\phi}{1-\phi}\frac{\left(R_{S}^{\prime*}\right)^{2}M_{S}^{\prime*}}{R_{S}^{*}}<0. $$

Thus, we obtain the following proposition.
Proposition 3. (Scale Effects in Levels) An increase in the number of skilled workers in the follower country raises the country’s quality in the steady state relative to its level in the frontier countries.

Diminishing marginal products of skilled labor in technology adoption should have resulted in these scale effects in levels, as Jones (1998) discusses.

### 3.3 Welfare

In this overlapping-generation model with two types of worker, it is not easy to define a social utility function for welfare analysis. Thus, we consider the welfare issue of whether subsidizing (or taxing) local R&D activities relating to inventive adaptation raises quality and output per capita in the steady state. $ ^{6} $

Denoting the rate of the subsidy to wages of skilled workers engaged in local R&D activities by $ \tau $ , we rewrite equation (12) as

$$ a_{t}=\frac{\eta}{1+g}\left(M_{t}+(1-\tau)\frac{M_{t}^{\prime}}{R_{t}^{\prime}}R_{t}\right). $$

Accordingly, the dynamics of $ H^{R} $ are given by

$$ \frac{M_{t}^{\prime}}{R_{t}^{\prime}}=\left(\frac{1+g}{\eta(1-\tau)}\right)^{1-\phi}\left(\frac{M_{t-1}^{\prime}}{R_{t-1}^{\prime}}R_{t-1}\right)^{\phi}, $$

and its steady-state value is implicitly given by

$$ q(H^{R*})=\left(\frac{\eta(1-\tau)}{1+g}\right)^{\frac{1-\phi}{\phi}}, $$

where the function q is defined by equation (14).

Figure 2 illustrates how Figure 1 should be modified in the case of an R&D subsidy. The left-side figures show graphically that the number of skilled workers engaged in R&D in the positive-R&D steady state, $ H_{S}^{R*} $ , is increasing in $ \tau $ in cases (i) and (ii) described above. In addition, $ H_{U}^{R*} $ in case (ii) is lower with the subsidy than without it. Therefore, although some follower countries in case (ii) converge to a zero-R&D steady state in the absence of intervention, those countries would converge to a positive-R&D steady state if an R&D subsidy were provided.
Case (iii) can be divided into two subcases when an R&D subsidy is provided. First, if $ (1-\tau)^{\frac{1-\phi}{\rho-(1-\phi)\mu}}\theta0&if\tau<\phi\\ =0&if\tau=\phi\\ <0&if\tau>\phi.\end{cases} $$

This and the fact that $ \frac{\partial a}{\partial H^{R}} > 0 $ if $ \tau < \phi $ derived from equation (16) lead to the following proposition.

Proposition 4. (Efficiency Gains from R&D Subsidies) Subsidizing skilled workers engaged in local R&D activities at the rate $ \phi $ maximizes output per capita in the steady state if the subsidy leads to a stable steady state in which there is a positive level of local R&D activities; i.e., in cases (i), (ii), and (iii).

The reason for the efficiency gains from an R&D subsidy is again the biased dynamic externality. Since private agents do not appropriate the future gains from current R&D, they underinvest in R&D activities in the decentralized equilibrium.

## 4 Empirical Analysis

This section tests the assumptions of the model described by equation (6) and its main prediction, presented in Proposition 2, that there are two patterns of equilibrium dynamics.

### 4.1 Data

For this empirical investigation, we need proxies for the variables in the model. First, technology adoption with minor adaptation is often conducted by MNEs.
without R&D activities in the host country, as we have already argued. Therefore, the number of skilled workers engaged in technology adoption with minor adaptation, $ H^{M} $ , is represented by FDI inflows as a percentage of GDP, and denoted by FDI. Strictly speaking, we should use FDI inflows that are not associated with R&D activities, but such data are not available for most countries. The data on FDI inflows are taken from UNCTAD (2003), and GDP data are from the World Bank (2003).

Second, the number of skilled workers engaged in technology adoption with inventive adaptation through local R&D activities, $ H^{R} $ , is represented by the R&D expenditures as a percentage of GDP $ (R\&D) $ . These data come from UNESCO (1999). $ ^{7} $

Third, the total number of skilled workers in the economy, H, is represented by the number of workers with a secondary-school degree or higher educational qualifications. This is constructed by multiplying the proportion of the population who are over 25 years of age and have a secondary-school degree from Barro and Lee (2001) by the number of workers from the Penn World Tables Mark 6.0, which updates Summers and Heston (1991).

Finally, the follower country’s average quality relative to the frontier level, a, is represented by the level of TFP relative to that of the United States. Following Hall and Jones (1999), the TFP of each country at time t is defined as

$$ T F P_{t}=\frac{Y_{t}}{K_{t}^{\bar{\alpha}}\left(W_{t}e^{\kappa\left(s_{t}\right)}\right)^{1-\bar{\alpha}}}, $$

where Y, K, and W are real GDP, the real capital stock, and the number of workers, respectively. Y and W are from the Penn World Tables Mark 6.0, while observations on K are obtained from data on real investment from 1950 (from the same source), assuming a five-percent depreciation rate. s is average years of schooling from Barro and Lee (2001). $ \kappa(s) $ indicates effective units of labor, in which we assume $ \kappa' = 0.134 $ for s < 4, $ \kappa' = 0.101 $ for $ 4 \leq s < 8 $ , and $ \kappa' = 0.068 $ for $ s \geq 8 $ , on the basis of micro evidence from Psacharopoulos
(1994). In addition, we assume $ \bar{\alpha}=1/3 $ .

We collected these data for those countries for which they are available for the period 1970–1999 to generate a panel data set covering three 10-year periods. The sample does not include the United States, which is assumed to be at the world technology frontier. FDI and R&D are averages per period, while values for H and a from the beginning of each period are used to represent initial conditions. The data for the last two periods, 1980–1989 and 1990–1999, are presented in the Appendix Table. Since the data for R&D expenditures are rarely available before 1980, we omit pre-1980 R&D. In addition, since R&D data are missing on some countries for some years after 1980, R&D may be based on only a few annual observations, rather than on all 10.

### 4.2 Testing the Assumptions of the Model

Denoting the rate of quality improvement by $ g_{t} \equiv \frac{A_{t}}{A_{t-1}} - 1 $ , we obtain from equation (6) that indicates the assumptions on quality improvement:

$$ \frac{\partial g_{t}}{\partial H_{t}^{M}}=\frac{\partial g_{t}}{\partial H_{t}^{R}}>0, $$

and

$$ \frac{\partial^{2}g_{t}}{\partial H_{t}^{M}\partial a_{t-1}}=(1-\phi)\frac{\partial^{2}g_{t}}{\partial H_{t}^{R}\partial a_{t-1}}<\frac{\partial^{2}g_{t}}{\partial H_{t}^{R}\partial a_{t-1}}<0. $$

Inequality (19) implies that the number of skilled workers engaged in technology adoption has a positive effect on the growth of average quality, whether adoption involves local R&D or FDI without R&D. In addition, inequality (20) implies that the effect diminishes as a increases or as the follower country catches up with the frontier countries. This is because less new technology is available to the follower country the larger is a. Furthermore, the effect of skilled labor on growth declines by more when technology adoption is without R&D than when adoption is with R&D. In other words, the effect of FDI without R&D on growth is greater than the effect of R&D when the follower country is far behind the knowledge frontier, and vice versa.

These implications of the assumptions can be tested by the following TFP-
growth regression:

$$ \begin{aligned}\ln TFP_{i,t+1}-\ln TFP_{it}=\beta_{0i}+\beta_{1}a_{it}+\beta_{2}\ln FDI_{it}+\beta_{3}(\ln FDI_{it}\times a_{it})\\+\beta_{4}\ln R\&D_{it}+\beta_{5}(\ln R\&D_{it}\times a_{it})+u_{it},\end{aligned} $$

where subscript i denotes country i, subscript t denotes the 10-year period starting from year t, and u is the error term. If the assumptions of the model are satisfied so that inequalities (19) and (20) hold, we expect $ \beta_{1} < 0 $ , $ \beta_{2} > 0 $ , $ \beta_{4} > 0 $ , and $ \beta_{3} < \beta_{5} < 0 $ .

The constant term, $ \beta_{0i} $ , is assumed to be country specific because of the geographic and linguistic proximity to the frontier countries, particularly the United States. Recent empirical papers such as Branstetter (2001), Conley and Ligon (2002), and Keller (2002), have found that geographic, economic, and linguistic distance significantly influence the degree of international knowledge diffusion. To eliminate $ \beta_{0i} $ , we take first differences to obtain the following estimable equation:

$$ \begin{aligned}\Delta(\ln TFP_{i,t+1}-\ln TFP_{it})=\beta_{1}\Delta a_{it}+\beta_{2}\Delta\ln FDI_{it}+\beta_{3}\Delta(\ln FDI_{it}\times a_{it})\\+\beta_{4}\Delta\ln R\&D_{it}+\beta_{5}\Delta(\ln R\&D_{it}\times a_{it})+\Delta u_{it},\end{aligned} $$

where $ \Delta x_{it} \equiv x_{it} - x_{i,t-1} $ for any variable x.

Major problems relating to the estimation of this equation are the potential endogeneity of FDI and R&D and autocorrelation in the error term. Hence, although we start with ordinary least squares (OLS), the optimal generalized method of moments (GMM) is also used to alleviate these potential problems. Since $ FDI_{it} $ and $ R\&D_{it} $ are determined by $ a_{i,t-1} $ , a suitable instrument for estimating equation (21) is $ \Delta a_{i,t-1} $ . Other suitable instruments are the lagged regressors. However, because of the limitation of the R&D data, we assume that the R&D variables are exogenous. Consequently, we instrument $ \Delta \ln FDI_{it} $ and $ \Delta (\ln FDI_{it} \times a_{it}) $ by $ \Delta a_{i,t-1} $ , $ \Delta \ln FDI_{i,t-1} $ and $ \Delta (\ln FDI_{i,t-1} \times a_{i,t-1}) $ . In addition, robust standard errors are used in all regressions because of potential heteroscedasticity and/or autocorrelation in the error terms.

The estimation results are reported in Table 1. The OLS results in the first column are consistent with the theoretical assumptions to some extent.
The estimated coefficients of $ \Delta a $ , $ \Delta \ln FDI $ , and $ \Delta (\ln FDI \times a) $ are negative, positive, and negative, respectively, and all are statistically significant at the five-percent level. Column (2) shows that the optimal GMM estimates are qualitatively the same, but the sizes of the latter two are substantially larger than the corresponding OLS estimates.

The coefficients of $ \Delta\ln R\&D $ , and $ \Delta(\ln R\&D\times a) $ are statistically insignificant for both OLS and GMM, although their signs are consistent with theoretical predictions. This is puzzling because Griffith et al. (2000) and Guellec and de la Potteire (2001) find a positive and significant effect of R&D. A possible explanation for this difference is that these studies focus on OECD countries whereas this paper includes all countries except for the United States.

Although the effect of R&D is unclear, we can still examine the relevance of inequality (19) by testing $ \beta_{3} < \beta_{5} $ . The F statistics for the null hypothesis that $ \beta_{3} \geq \beta_{5} $ against the alternative that $ \beta_{3} < \beta_{5} $ are 1.70 (in OLS) and 4.06 (in GMM). Thus, at the five-percent level, the null hypothesis is not rejected in OLS but is in GMM.

To test whether the R&D variables are endogenous, we apply the test for strict exogeneity suggested by Wooldridge (2002, p. 285). We re-estimate equation (21), using the level data on $ a_{it} $ , $ \ln R\&D_{it} $ , and $ \ln R\&D_{it} \times a_{it} $ as additional regressors and compute the F statistic for the null hypothesis that the coefficients of the additional variables are jointly zero. Given an F statistic of 0.55, we cannot reject strict exogeneity at the five-percent level. This justifies the GMM estimation without lagged first-differences of the R&D variables as instruments.

Moreover, we test the strict exogeneity of the FDI variables by using the same method. The F statistic is 2.34, and therefore, strict exogeneity of the FDI variables cannot be rejected at the five-percent level but can at the 10-percent level. Thus, it is not clear whether the FDI variables are exogenous or whether the OLS or GMM is favored.

As a robustness check, we use a limited sample that includes countries with a TFP level less than the level of the United States in 1980. Columns (3) and (4) of Table 1 report the OLS and GMM results, respectively, using the limited