At any time t, let Y t represent the aggregate output, K t denote the aggregate physical capital stock, H t indicate the aggregate human capital stock, L t signify the total number of employed workers (typically assumed to mirror an economy’s population level), and represent the constant level of total factor productivity (TFP). The conventional labor-augmented production technology adheres to the Cobb-Douglas form.

Y t = A K t α H t β [ ( 1 + x ) t L t ] 1 − ( α + β ) ϵ t P , (1)

In Equation (1), we expand upon the production function proposed by (Mankiw et al., 1992) . In addition to accounting for constant returns to scale, we introduce an exogenous Harrod-neutral rate of technological progress, denoted as x, along with multiplicative stochastic shocks, ϵ t P , intended to model potential short-run business cycles. The exogenous parameters α, β, and x are not constrained, as their empirical estimates contribute to understanding the dynamics of endogenous and/or exogenous growth within an economy. These parameter configurations will be discussed later. Through division of both sides of Equation (1), it can be reformulated in per-capita terms:

y t = A ( 1 + x ) [ 1 − ( α + β ) ] t k t α h t β ϵ t P . (2)

Here, the lowercase variables, y, k, and h, denote per-capita output, physical capital, and human capital, respectively. Detrending Equation (2) enables the representation of all variables in the context of per efficiency-unit of labor input in production.

y ^ t = A k ^ t α h ^ t β ϵ t P , where y ^ t = Y t ( 1 + x ) t L t ; k ^ t = K t ( 1 + x ) t L t ; h ^ t = H t ( 1 + x ) t L t . (3)

For the sake of mathematical simplification, we presume complete depreciation for both types of capital stock in each period. However, the trajectories of their dynamics and the characteristics of steady-state values remain unaffected even if the depreciation rates are less than 100%.¹ Hence, the investment flow at time t translates into the capital stock at time t + 1. Output, at each time point, is allocated to consumption, as well as the investment in physical capital and human capital. An endogenous variable s t signifies the fraction allocated to accumulate physical capital, with ( 1 − s t ) representing the fraction allocated to human capital. Assuming a log-utility function for consumption per efficiency unit of labor ( c ^ t ), the optimization problem for the household-worker’s lifetime utility is formulated by the value function as follows.

V ( k ^ 0 , h ^ 0 ) = max { c ^ t , s t , k ^ t + 1 , h ^ t + 1 } t = 0 ∞ E 0 ( ∑ t = 0 ∞ δ t ln c ^ t ) ; discount factor δ ∈ ( 0 , 1 ) , (4)

subject to

k ^ t + 1 = s t y ^ t − c ^ t = s t A k ^ t α h ^ t β ϵ t P − c ^ t ; (5)

h ^ t + 1 = ( 1 − s t ) y ^ t = ( 1 − s t ) A k ^ t α h ^ t β ϵ t P ; (6)

k ^ 0 , h ^ 0 are given; and (7)

c ^ t , k ^ t + 1 , h ^ t + 1 ≥ 0 , s t ∈ [ 0 , 1 ] , (8)

for all t ∈ ℕ .

To address the intertemporal utility maximization problem of the household-worker, considering constraints (5) to (8), we apply the (Bellman, 1957) principle of optimality, transforming Equation (4) into the subsequent limiting value function:

V ( k ^ t , h ^ t , ϵ t P ) = max { c ^ t , s t , k ^ t + 1 , h ^ t + 1 } t = 0 ∞ ∑ t = 0 ∞ ln c ^ t + δ E 0 [ V ( k ^ t + 1 , h ^ t + 1 , ϵ t + 1 P | k ^ t , h ^ t , ϵ t P ) ] , (9)

subject to constraints (5) to (8). A conjecture of the aforementioned value function assumes the following form:

V ( k ^ t , h ^ t , ϵ t P ) = B 0 + B 1 ln k ^ t + B 2 ln h ^ t + B 3 ln ϵ t P , (10)

where B₀, B₁, B₂, and B₃ represent combinations of parameters within the system, to be verified subsequently. The optimality conditions for maximizing the right-hand side of the combined Equations (9) and (10) involve obtaining the following two first-order derivatives with respect to c ^ t and s t :

1 c ^ t = δ B 1 k ^ t + 1 and (11)

k ^ t + 1 h ^ t + 1 = B 1 B 2 . (12)

Equation (11) delineates the household-worker’s optimal intertemporal allocation of resources between current consumption and future physical capital stock. Equation (12) implies a constant and time-invariant ratio between physical capital and human capital for optimality. By substituting Equations (11) and (12) into Equations (5) and (6), the resulting expressions are:

c ^ t = ( 1 1 + δ B 1 ) s t A k ^ t α h ^ t β ϵ t P , (13)

s t = 1 + δ B 1 1 + δ B 1 + δ B 2 , (14)

k ^ t + 1 = ( δ B 1 1 + δ B 1 ) s t A k ^ t α h ^ t β ϵ t P , and (15)

h ^ t + 1 = ( δ B 2 1 + δ B 1 ) s t A k ^ t α h ^ t β ϵ t P . (16)

Substituting Equations (13) to (16) into the right-hand side of the combined Equations (9) and (10) facilitates the verification of

B 1 = α 1 − δ ( α + β ) and B 2 = β 1 − δ ( α + β ) . (17)

B₁ and B₂ serve as crucial determinants in resolving the optimal sequence of { c ^ t , s t , k ^ t + 1 , h ^ t + 1 } t = 0 ∞ for the representative household-worker. Derived from B₁, Equation (13) reveals the optimal consumption trajectory as:

c ^ t = ( 1 − δ α − δ β ) A k ^ t α h ^ t β ϵ t P . (18)

Here, the marginal propensity to consume equates to one minus the summation of one-period discounted shares of physical capital and human capital within the production process, ( δ α + δ β ) . These discounted shares, δ α and δ β , correspondingly denote the optimal fractions of output allocated to physical capital and human capital investment. This relationship is evident when substituting B₁ and B₂ into Equations (15) and (16), as follows.

k ^ t + 1 = δ α A k ^ t α h ^ t β ϵ t P and (19)

h ^ t + 1 = δ β A k ^ t α h ^ t β ϵ t P . (20)

The optimal investment fraction, s t , is deduced from equations (6) and (20), or obtained by substituting B₁ and B₂ into Equation (14) as:

s t = 1 − δ β . (21)

Hence, the optimal s t equals one minus the one-period discounted share attributed to human capital in output. As β approaches zero, s t tends towards one, indicating that the household-worker lacks the motivation to invest in human capital. In this case, the model simplifies to a conventional Ramsey planning economy, where the only accumulable factor in the system is physical capital. Conversely, a greater share of human capital in output signifies that accumulating human capital proves more productive than focusing on physical capital accumulation.

The model described above encompasses four distinct scenarios within an economy, contingent upon the interplay of parameters, as illustrated in Table 1.

Assuming ϵ t P = 1 and 0 < α + β < 1 for scenarios (a) and (b), the steady-state values of k ^ * and h ^ * from Equations (19) and (20) are both unique and stable, expressed as:

k ^ * = ϕ 1 − β 1 − ( α + β ) and (22)

h ^ * = ( δ β A ) 1 1 − β ϕ α 1 − ( α + β ) , where ϕ = ( δ α A ) ( δ β A ) β 1 − β > 0 . (23)

In scenarios (a) and (b), when the reproducible factors, k ^ t and h ^ t , exhibit diminishing returns to scale, sustained growth in per-capita output occurs in the long run solely if the exogenous rate of technological progress, x, maintains a positive value. Conversely, scenarios (c) and (d), characterized by constant returns to reproducible factors in output ( α + β = 1 ) , render the steady-state values of k ^ * and h ^ * nonexistent. In such instances, the long-run growth rates denoted by γ for per-capita variables along the balanced growth path remain identical, perpetual and non-explosive:

γ y = γ k = γ h = γ c = x + δ A α α ( 1 − α ) 1 − α − 1 , where x ≥ 0 . (24)

Utilizing the earlier theoretical findings, it becomes feasible to simulate short-term and long-term transitions of the system’s variables. Equations (18) to (20) delineate the resolutions to the household-worker’s intertemporal utility maximization

issue. In conjunction with Equations (3) and (21), we can represent the precise log-linear equilibrium conditions in a state-space format as:

[ 1 ln k ^ t + 1 ln h ^ t + 1 ln ϵ t P ] = [ 1 0 0 0 ln ( δ α A ) α β 1 ln ( δ β A ) α β 1 0 0 0 0 ] [ 1 ln k ^ t ln h ^ t ln ϵ t P ] and (25)

[ ln c ^ t ln y ^ t ] = [ ln [ ( 1 − δ α − δ β ) A ] α β 1 ln A α β 1 ] [ 1 ln k ^ t ln h ^ t ln ϵ t P ] . (26)

To calibrate the system outlined above, all parameters are assumed to fall within reasonable ranges. Specifically, for scenarios (a) and (b) in Table 1, we set A = 2 , α = 0.3 , and β = 0.2 . With the household-worker’s subjective discount rate defined as δ = 0.95 , the optimal investment fraction is calculated as s t = ( 1 − δ β ) = 0.81 across all time periods. This implies that, at each point in time, the distribution of output should allocate 52.5% to consumption, 28.5% to physical capital accumulation, and 19% to human capital accumulation. Concurrently, we assume ϵ t P = 1 for a deterministic case, intending to introduce this shock later during our theoretical impulse-response analysis. Figure 1 illustrates the optimal trajectories of k ^ t , h ^ t , y ^ t , and c ^ t based on the provided initial values of k ^ 0 = 0.1 and h ^ 0 = 0.05 .

Continuing, to illustrate scenarios (c) and (d) as detailed in Table 1, we alter the share of human capital in output from β = 0.2 to β = 0.7 to ensure ( α + β ) = 1 . Under these conditions, the steady-state equilibrium depicted in Figure 1 becomes nonexistent, and the growth rate of the system, as derived from Equation (24), can be expressed as:

γ y = γ k = γ h = γ c = x + δ A α α ( 1 − α ) 1 − α − 1 = { 3.15 % for   x = 0   ( scenario ( c ) ) 5.15 % for   x = 0.02   ( scenario ( d ) )

Next, incorporating a stochastic component ϵ t P ~ i i d N ( 1 , σ P 2 ) into the model

enables us to conduct a theoretical impulse-response analysis spanning from the short to the long run. Scenario (b) serves as an illustrative example for this purpose. Assuming a one-percentage increase in TFP, the percentage responses of k ^ t , h ^ t , y ^ t , and c ^ t are depicted in Figure 2.

In Figure 2, y ^ t and c ^ t exhibit contemporaneous and positive responses to the one-percentage-point shock, whereas the responses of k ^ t and h ^ t occur with a lag of one period. Across all variables, the initial positive responses gradually diminish, settling within 6 to 7 time periods.

Furthermore, this positive standard shock introduces fluctuations in the long-run growth rate. In the case of scenario (b), assuming the exogenous rate of technological progress is x = 0.02 , it also represents the long-run rate of growth in per-capita output. Following a one-percentage increase in TFP at time T, Figure 3 illustrates that this growth rate contemporaneously rises from 2% to 3%, but at time T + 1, it decreases to 1.5%, aligning with the reduction observed in the positive response from y ^ t shown in Figure 2. The long-run growth rate of 2% is then restored after 6 to 7 time periods.

In the preceding subsection, although our simulation results appear reasonable, it is crucial to note that the system parameters, α, β, and x, should not be arbitrarily assumed but instead properly estimated. In our theoretical model, we provide theory-consistent cointegrating spaces for the per-capita variables y t , k t , and h t . By log-linearizing Equations (19), (20), and (3), and applying a lag operator, L, we obtain:

[ 1 − ( α + β ) L ] ( ln k t − x t ) = λ 1 + L ln ϵ t P ; (27)

[ 1 − ( α + β ) L ] ( ln h t − x t ) = λ 2 + L ln ϵ t P ; (28)

[ 1 − ( α + β ) L ] ( ln y t − x t ) = λ 3 + ln ϵ t P . (29)

Here λ 1 , λ 2 , and λ 3 represent constant combinations of parameters, and ln ϵ t P is an i.i.d. shock with a zero mean and constant variance.

From a time-series perspective, ln y t , ln k t , and ln h t are potentially I(1) variables, allowing for a maximum of two linearly independent cointegrating vectors. Assuming that α + β = 1 to ensure non-explosive and perpetual growth, Equations (27) to (29) result in the cointegrating space as:

ln y t − ln k t = κ 1 + ln ϵ t P and (30)

ln y t − ln h t = κ 2 + ln ϵ t P , where κ 1 and κ 2 are constant. (31)

If the existence of a cointegrating space with two cointegrating vectors is rejected, then at most one cointegrating equation can be found. By multiplying Equation (30) on both sides by α and Equation (31) on both sides by ( 1 − α ) , and then summing them up, we obtain:

ln y t − α ln k t − ( 1 − α ) ln h t = κ 3 + ln ϵ t P , where κ 3 is constant. (32)

If we relax the constraint that α + β = 1 , the unrestricted cointegrating equation can be derived from Equation (2), resulting in:

ln y t − α ln k t − β ln h t − [ 1 − ( α + β ) ] x t − ln A = ln ϵ t P . (33)

Equation (32) emerges as a constrained version of Equation (33) when assuming α + β = 1 . The validity of this hypothesis, in conjunction with the impact of the exogenous rate of technological progress, x, will be subjected to empirical testing in the subsequent section of this paper.

The data source for all variables in the theoretical model is directly available from Penn World Table (PWT) version 10.01, updated in early 2023, encompassing a substantial time span from 1950 to 2019 for the United States and most advanced countries. This dataset stands as the most current and comprehensive worldwide macroeconomic dataset constructed by (Feenstra et al., 2015) . Within this dataset, we observe real GDP denoted as Y t , total population represented as L t , real aggregate physical capital stock indicated by K t , and the calculated index for aggregate human capital stock, noted as H t .² Both real GDP and aggregate physical capital stock are quantified at constant 2017 national prices, expressed in million 2017 US dollars. The division of Y t , K t , and H t by the total population allows us to observe the upward trajectory of the logarithmic lowercase per-capita variables: ln y t , ln k t , and ln h t , depicted in Figure 4.

Throughout our sample period, both the physical capital stock per capita ( k t ) and real GDP per capita ( y t ) exhibited consistent growth, with average annual rates of 2.03% and 1.65%, respectively. In contrast, the expansion of human capital stock per capita, h t , experienced a distinct slowdown starting in 1980. Specifically, the average annual growth rate of h t stood at 1.24% from 1950 to 1980 but notably declined to 0.48% between 1981 and 2019. This trend concurs with the assertion made by (Carnevale & Rose, 2011) that the United States has

witnessed a deficiency in producing college-educated workers since 1980.

In general, following logarithmic transformation, most macroeconomic variables tend to exhibit stochastic trends in the long run. To assess the stationarity of ln y t , ln k t , and ln h t , Figure 4 suggests that both intercepts and time trends should be accounted for when conducting conventional unit root tests for these variables in their original levels. However, when examining first differences, considering only intercepts is adequate, unless a significant residual trending component persists, as observed in the case of Δ ln k t in our sample. Table 2 provides a summary of the outcomes derived from the Augmented Dickey Fuller (ADF) unit root tests (Said & Dickey, 1984) .

Table 2 indicates, at a 5% significance level, that ln y t , ln k t , and ln h t are all identified as nonstationary variables. This result suggests that their long-run cointegrating relationships can be examined through balanced regressions. While incorporating these integrated variables into an ad hoc single aggregate production function may introduce endogeneity bias, utilizing cointegration properties derived from a theory-consistent model helps avoid spurious regressions and simultaneity bias. To illustrate this, we conduct a Johansen cointegration test (Johansen, 1991) employing a vector autoregressive (VAR) model.

a. Lag lengths were optimally chosen using the Schwarz Information Criterion (SIC), with a maximum lag length of 10. The decision to include a constant and/or trend depends on their graphical representation and significance in the ADF test equations.

When multiple non-stationary variables share common stochastic trends, cointegration relationships can be observed through specific stationary and linear combinations, and this can be empirically tested using a VAR system encompassing these variables. In this process, it is imperative to select appropriate lag lengths by employing criteria such as the Schwarz Information Criterion, Akaike Information Criterion, and Hannan-Quinn Information Criterion. Subsequently, during the execution of Johansen cointegration tests, the VAR model with lag p, or VAR(p), undergoes transformation into a Vector Error Correction (VEC) model with lag p − 1, denoted as VEC (p − 1). The Johansen cointegration test outcomes within the VEC (1) model are summarized in Table 3.

According to Table 3, at a standard significance level of 5%, both the trace and maximum eigenvalue test statistics indicate the presence of two cointegration vectors (CVs) encompassing ln y t , ln k t , and ln h t , incorporating a time trend. However, only one CV is identified at a 1% significance level based on both test statistics. The prospect of one or two CVs aligns with the theoretical model’s predictions. Following the transformation of the VAR model into a VEC model, two cointegration equations are derived for , incorporating a time trend.

ln y t = 0.8599 + 0.7871 ∗ ∗ ∗ ( 0.1179 ) ln k t + 0.0086 ∗ ∗ ∗ ( 0.0022 ) t + ln ϵ ^ t P and (34)

ln h t = 15.9079 + 1.4411 ∗ ∗ ∗ ( 0.2538 ) ln k t + 0.0195 ∗ ∗ ∗ ( 0.0048 ) t + ln ϵ ^ t P , (35)

where standard errors are in parentheses, and ϵ ^ t P represents the regression residual. The significance level of 1% is denoted by ***. The first CV represents a conventional production function in the Solow–Swan model, where labor and physical capital serve as the sole factor inputs under constant returns to scale. The estimated significant share of physical capital at 0.7871 signifies the United States as a conventionally capital-intensive economy. Additionally, this economy demonstrates exogenous growth, with an annual technological progress rate of 0.86% as the Harrod-neutral rate. This rate is inferred from the estimated coefficient derived from Equation (34), illustrating the time trend’s impact.

Upon reducing the significance level to 1% in the Johansen cointegration tests,

a. Significance levels of 5% and 1% are denoted by ** and ***, respectively.

as indicated by both the trace and maximum eigenvalue test statistics in Table 3, the analysis identifies the presence of only one cointegration vector, characterized by Equation (36):

ln y t = 6.5657 + 0.2761 ∗ ∗ ( 0.1409 ) ln k t + 0.2645 ∗ ∗ ∗ ( 0.1016 ) ln h t + 0.0147 ∗ ∗ ∗ ( 0.0026 ) t + ln ϵ ^ t P , (36)

where standard errors are represented in parentheses, and the significance levels of 5% and 1% are denoted by ** and ***, respectively. The major estimates in Equation (36) exhibit statistical significance and fall within reasonable ranges.

Notably, the estimated shares of physical and human capital in the production process, denoted as α and β, respectively, approximate at 0.2761 and 0.2645, indicating their nearly equivalent magnitudes. This parity underscores the comparable importance of human capital stock alongside physical capital stock in shaping the trajectory of the US economy across a 70-year span. In addition, the estimated coefficient of the time trend suggests a consistent and exogenously driven technological growth rate of approximately 1.47% in the United States. This finding implies a stable trend indicating sustained external influences fostering technological advancements within the economy over the specified period.

Furthermore, drawing from the earlier theoretical section that derived the optimal investment share in human capital as δβ, assuming a discount rate of δ = 0.95 alongside the aforementioned estimated α and β, computations suggest a recommended allocation of approximately 25.13% and 26.23% of real GDP towards developing human capital and accumulating physical capital, respectively, to attain long-term economic optimality in the United States.

In our theoretical model, we categorize four scenarios in Table 1 based on the estimates of the coefficients. Table 4 provides a summary of the test results derived from formal log-likelihood ratio (LR) tests for the VEC model.

The first test evaluates the null hypothesis of constant returns to per-capita variables in Equation (33). With a reasonable level of significance, the rejection of α + β = 1 in favor of α + β < 1 could be justified, suggesting the potential presence of exogenous growth in the United States. The second test centers on the null hypothesis of a zero Harrod-neutral rate of technological progress ( x = 0 ), and this hypothesis is rejected as well. These two outcomes are jointly

a. The calculation of LR test statistics is based on the determinants of residual covariance of restricted and unrestricted VEC models.

scrutinized and supported in the third test, affirming that, over the past 70 years, the United States has experienced periods of pure exogenous growth. During this phase, significant contributions from both human and physical capital were observed, even though these capital stocks exhibited diminishing returns.

As our VEC model fulfills the stability condition owing to the stationarity of ln y t , ln k t , and ln h t after first differencing, it allows us to conduct Granger causality tests. These tests aid in evaluating the predictability of one variable concerning another. The outcomes of the block homogeneity Granger causality tests are summarized in Table 5.

In a VAR or VEC system, a variable is considered relatively endogenous if it is Granger-caused by other variables without exhibiting reverse predictability. The findings from Table 5 indicate that the lagged effects of Δ ln h t and Δ ln k t on Δ ln y t lack significance, as evidenced by the associated test statistics—2.4176, 3.6241, and 4.3135—all failing to surpass the respective critical values. However, Δ ln y t significantly Granger causes both Δ ln h t and Δ ln k t , supported by respective test statistics of 6.6776 and 15.3013, both surpassing the corresponding critical values. Moreover, Δ ln h t displays uni-directional Granger causality towards Δ ln k t , with a causation strength of 6.3031 from Δ ln h t to Δ ln k t , while the reverse causation from Δ ln k t to Δ ln h t is comparatively weaker at 2.5184. These Granger causality tests elucidate that real GDP per capita appears relatively exogenous compared to both physical capital per capita and human capital per capita. Among these variables, physical capital per capita emerges as the most endogenous within the system. These lagged effects provide insights into

a. The above Granger causality tests are performed at 5% level of significance.

the transmission channels among the variables. However, it is crucial to emphasize that these outcomes do not challenge our previous statements regarding the endogenous growth of per-capita output linked with per-capita physical and human capital inputs. The estimations primarily focus on establishing the long-term cointegration relationship among ln y t , ln k t , and ln h t , rather than on their short-term causal interactions.

These short-run Granger causal relationships enable us to conduct generalized impulse response analyses,³ considering the test results within our VEC system order { ln y t , ln h t , ln k t } . While the impulse response analysis comprises nine interactions among the three variables, we choose to present two meaningful and intuitive figures. Firstly, Figure 5 illustrates the responses of the two factor inputs, ln k t and ln h t , to each other’s shocks.

Figure 5 illustrates the dynamic relationship between two key components of economic growth: physical capital and human capital. The depicted responses highlight their reactions to sudden changes, or impulses, in each other.

The observed negative reaction of ln k t to a positive shock in ln h t suggests an interesting dynamic. When there is an unexpected increase in human capital, initially, there is a slight negative response in the increase of physical capital. This reaction might imply a shift in focus or resource allocation away from building more physical infrastructure immediately after a surge in human capital. It hints that in the short term, an emphasis on improving human skills or knowledge might temporarily reduce the immediate drive to invest in additional physical resources.

On the other hand, the positive response of ln h t to a positive shock in ln k t indicates a different dynamic. An unforeseen increase in physical capital seems

to stimulate the growth or enhancement of human skills or knowledge. This positive reaction suggests that improvements or expansions in physical infrastructure might create an environment conducive to the development of human capabilities. It implies that a well-equipped or technologically advanced infrastructure might foster an environment conducive to nurturing and improving human capital over time.

Secondly, a transient surge in output per capita, as depicted in Figure 6, gives rise to sustained elevations in the growth rates of both physical and human capital per capita.

Figure 6 portrays the aftermath of a sudden boost in output per capita on the growth rates of physical and human capital. It reveals an interesting pattern in their responses over time. Initially, when there is a positive jolt to output per capita, human capital experiences a significantly higher growth spurt compared to physical capital. This suggests that an immediate increase in productivity initially prompts a rapid surge in the growth of skills, education, or knowledge among the workforce, outpacing the initial growth in physical capital. However, as time progresses, the growth rate of physical capital catches up and begins to escalate steadily, indicating a more prolonged and continuous increase in the growth of infrastructure, machinery, or tangible assets. Meanwhile, the growth rate of human capital, after its initial rapid ascent, starts to diminish over time, ultimately plateauing or declining back towards normalcy. This pattern implies that while an initial productivity surge triggers a sharp rise in human capital growth, the growth rate of physical capital gradually accelerates and catches up, eventually surpassing or sustaining its growth, while the growth rate of human capital subsides or returns to baseline levels over extended periods.