2.1. Basics

The output production functions for the home and foreign (indicated by a ‘*’) country, ignoring capital and labour, are (1) $Q=A,Q^{\ast }=A^{\ast }$(1)

The productivity functions are (2) $A=F\left(A_{-1},R,t\right)\mathrm{a}\mathrm{n}\mathrm{d}{A}^{\ast }=F^{\ast }\left(A_{-1}^{\ast },R^{\ast },t\right)$(2)

R, R* are (1, 4) vectors of R&D stock elements $R_j,R_j^{\ast },j=b,g$ where b and g indicate business and non-business or (semi-) government R&D. The function may be shifted by exogenous technical change indicated by time t. Analogous to the ideas of ‘learning by doing’ (Arrow 1962), ‘perspectives of experience’ (Boston Consulting Group 1968), and ‘learning from watching’ (King and Robson 1993) we consider R&D capital stocks as research experience indicators. To keep the model simple, we assume that spillovers stem from R&D, for example through patenting including the publication of the innovation, and not from productivity, which includes complex phenomena such as organization of the firms and other microeconomic and institutional issues, which are beyond the scope of this macroeconomic paper. R&D capital builds up according to the perpetual inventory method in the empirical literature (suppressing the index for current time), (3) $R_{j,t+1}=R_j+I_j-{\delta }_j{R}_j,j=b,g$(3)

Again, we assume a similar equation for a foreign country. Investment, $I_j$, includes the minimized costs for human capital, labs and low- or medium skill support of researchers. The advantage of this approach is that we do not have to model all of these factors separately.⁹ The last term represents depreciation in the learning indicator. The disadvantage is that we cannot distinguish the quantities and prices of factors in this simple version of the model. Doing so would lead us to larger models, which go beyond the limited number of variables typically included in vector-error-correction models to which we want to relate our model. A share s of output goes into R&D investment (4) $I_j=s_{j}Q,j=b,g$(4)

2.2. Dynamics and stability

With the production function (1) in the investment Equation (4), and the result in the dynamic equations in (3) we get (3′) $\begin{array}{rl}& R_{b,t+1}=R_b+s_{b}F\left(A_{t-1},R_b,R_g;R_b^{\ast },R_g^{\ast },t\right)-{\delta }_b{R}_b,\\ & R_{g,t+1}=R_g+s_{g}F\left(A_{t-1},R_b,R_g;R_b^{\ast },R_g^{\ast },t\right)-{\delta }_g{R}_g\end{array}\left(3^{\mathrm{\prime }}\right)$(3′)

Foreign R&D terms are exogenous for the home country by assumption. Subtracting current R-terms on both sides and dividing by them results in the growth rate version of these equations. Subtracting also $g\equiv \left(A_{+1}/A\right)-1$ yields (3′′) $\begin{array}{rl}& g_b-g\equiv \frac{R_{b,t+1}-R_b}{R_b}-g=s_{b}F\left(A_{-1},R_b,R_g;R_b^{\ast },R_g^{\ast },t\right)/R_b-{\delta }_b-g,\\ & g_g-g\equiv \frac{R_{g,t+1}-R_g}{R_g}-g_A=s_{g}F\left(A_{-1},R_b,R_g;R_b^{\ast },R_g^{\ast },t\right)/R_g-{\delta }_g-g\end{array}\left(3^{\mathrm{\prime }\mathrm{\prime }}\right)$(3′′)

Constant A/R_i ratios can ensure constant growth rates. This requires that A and R_i have the same growth rate in the long run, and that this is compatible with the specification of the production functions used for F below. We define $y_g$ ≡ R_g/$A$, $y_b$ ≡ R_b/ $A$. With these definitions, (3″) can be re-written as follows. (3′′′) ${\hat{y}}_b=\frac{s_b}{y_b}-{\delta }_b-g,{\hat{y}}_g=\frac{s_g}{y_g}-{\delta }_g-g\left(3^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}\right)$(3′′′)

Multiplying the first equation by y_b and the second by y_g we get (3^{iv}) ${\dot{y}}_b=s_b-\left({\delta }_b+g\right)y_b,{\dot{y}}_g=s_g-\left({\delta }_g+g\right)y_g$(3^{iv})

The steady-state solution for y_g and y_b is (5) $y_b^s=\frac{s_b}{\left({\delta }_b+g\right)},y_g^s=\frac{s_g}{\left({\delta }_g+g\right)}$(5)

For positive (negative) growth rates, both sides in (3^iv) are positive (negative) and therefore y-terms are increasing (decreasing) and thereby decrease (increase) the right-hand side. The process is therefore stable, going to the steady state solution (5) if a constant growth rate g exists (even if it varies in the adjustment process), which depends also on the production functions assumed below. As a result, R_g, $A$, and R_b have the same growth rate, if the production functions relating them do not contradict this partial stability analysis.

2.3. Extension to endogenous savings

We assume that governments’ search for an optimal mix of policy instruments – in practice including evaluation studies surveyed in Ziesemer (2019) – can be approximated by a utility maximizing choice of public R&D, and that firms try the same.¹⁰ The country’s problem at time t in finding the optimal savings ratios is to maximize utility from consumption subject to the dynamic R&D equations: $\begin{array}{rl}& Ma{x}_{s_{b,},s_g}\sum _{\tau =t}^T{\rho }^{\tau -t}U\left[\left(1-s_{b,\tau }-s_{g,\tau }\right)F\left(A_{\tau -1},R_{\tau },\tau \right)\right]\\ & \mathrm{s}.\mathrm{t}.R_{b,\tau +1}=R_{b,\tau }+s_{b,\tau }F\left(A_{\tau -1},R_{\tau },\tau \right)-{\delta }_b{R}_{b,\tau },\\ & R_{g,\tau +1}=R_{g,\tau }+s_{g,\tau }F\left(A_{\tau -1},R_{\tau },\tau \right)-{\delta }_g{R}_{g,\tau },\end{array}$given initial values for$A_{t-1}=A\left(t-1\right),R_{b,t}=R_b\left(t\right)$ and $R_{g,t}=R_g\left(t\right)$, and discount factor ρ < 1.

T is the time horizon, τ is the index of future time periods with beginning value t as the present period. The constraints (3′) include (1), (2) and (4) and use the definition $R_{\tau }=\left(R_{b,\tau },R_{g,\tau };R_{b,\tau }^{\ast },R_{g,\tau }^{\ast }\right)$. If T goes to infinity this is a standard dynamic optimization problem. If T-t = 2 we have a two period model. If T-t = 1 this is a myopic one-period model with no reason to invest unless we specify a salvage function.

We solve the dynamic R&D equations (3′) for the savings ratio and insert them into the objective function. The result with $g_{j,\tau }=(R_{j,\tau +1}-R_{j,\tau })/R_{j,\tau }$ is $Ma{x}_{R_{b,\tau +1,},R_{g,\tau +1}}\sum _{\tau =t}^T{\rho }^{\tau -t}U\left[F\left(A_{\tau -1},R_{\tau },\tau \right)-\left(g_{b,\tau }+{\delta }_b\right)R_{b,\tau }-\left(g_{g,\tau }+{\delta }_g\right)R_{g,\tau }\right]$given initial values for$A_{t-1}=A\left(t-1\right),R_{b,t}=R_b\left(t\right)$ and $R_{g,t}=R_g\left(t\right)$.

The first-order conditions for an interior solution are¹¹ (6a) $\rho U{{}^{\mathrm{\prime }}}_{\tau +1}\left[F_{2,\tau +1}-{\delta }_b\right]-U{}^{\mathrm{\prime }}=0\mathrm{o}\mathrm{r}{F}_{2,\tau +1}-{\delta }_b=U{}^{\mathrm{\prime }}/\left(U{{}^{\mathrm{\prime }}}_{\tau +1}\rho \right)$(6a) (6b) $\rho U{{}^{\mathrm{\prime }}}_{\tau +1}\left[F_{3,\tau +1}-{\delta }_g\right]-U{}^{\mathrm{\prime }}=0\mathrm{o}\mathrm{r}{F}_{3,\tau +1}-{\delta }_g=U{}^{\mathrm{\prime }}/\left(U{{}^{\mathrm{\prime }}}_{\tau +1}\rho \right)$(6b)

The suffix 2 or 3 denotes derivations with respect to the second or third argument in the function F, evaluated at period τ+1. As control variables are substituted, concavity of the objective function w.r.t. $R_{b,\tau +1,},R_{g,\tau +1}$ and their changes is a sufficient condition for optimality (Kamien and Schwarz 2001). As the function under the utility function is linear in the changes, even under linear utility concavity of F with respect to R-terms is sufficient; under strictly concave utility even mildly increasing returns of F could be allow for.

The net marginal products should be equal to each other,¹² and, as usual, marginal products should equal capital costs. For an iso-elastic specification of the utility function $U\left(c\right)=\frac{c^{1-\omega }}{1-\omega },\mathrm{w}\mathrm{e}\mathrm{g}\mathrm{e}\mathrm{t}\frac{U^{\mathrm{\prime }}}{\left({U^{\mathrm{\prime }}}_{\tau +1}\rho \right)}=\frac{c^{-\omega }}{c_{\tau +1}^{-\omega }\rho }={\frac{{\left(1+g_c\right)}^{\omega }}{\rho }}^{}$

It follows from this result that the right-hand sides of (6a) and (6b) are constant if the growth rate of consumption, $c=F\left(A_{\tau -1},R_{\tau },\tau \right)-\left(g_{b,\tau }+{\delta }_b\right)R_{b,\tau }-\left(g_{g,\tau }+{\delta }_g\right)R_{g,\tau }$, is constant. This is the case for constant and identical growth rates of R-terms and F (sufficient). Convergence of the integral to a constant value is necessary for a maximum to exist. It requires that the growth rate of discounted utility is negative, which is the case if the discount rate dominates the growth rate of utility in the long run,¹³ $log\rho +\left(1-\omega \right)g<0,or\left(1-\omega \right)g<-log\rho$

3.1. Steady-state definition and growth rate in the Cobb–Douglas case

A special case of the function (2) for A is the Cobb–Douglas function with $1>\alpha ,\text{β}>0;C\ge 1,$and unspecified signs for the exponents of the lagged dependent variable, the foreign variables and exogenous technical change, $\varphi ,\gamma ,\mu ,b\lessgtr 0$, (2′) $A=\left(A_{-1}{)}^{\varphi }{R}_b^{\alpha }{R}_g^{\text{β}}\right(R_b^{\ast }{)}^{\gamma }\left(R_g^{\ast }{)}^{\mu }C{e}^{bt}\right(2^{\mathrm{\prime }})$(2′)

C is a constant to adjust the value level of the right-hand side to the requirements of the data on the left-hand side. As Kaldor and Mirrlees (1962), we add exogenous growth to the endogenous arguments. The empirical literature interprets $\gamma ,\mu >0$as dominance of positive spillovers, and $\gamma ,\mu <0$as dominance of negative spillovers and competition effects from abroad (Luintel and Khan 2004). Of course, in principle,$\gamma$ and$\mu$ could have opposite signs, for example positive spillovers from public R&D and dominant competition effects from private foreign R&D,$\gamma ⟨0,\mu ⟩0$. Taking growth rates, and assuming equality of growth rates for A and domestic R-terms we get (2′′) $\hat{A}=\frac{\gamma \widehat{R_b^{\ast }}+\mu \widehat{R_g^{\ast }}+b}{1-\varphi -\alpha -\text{β}}\equiv g\left(2^{\mathrm{\prime }\mathrm{\prime }}\right)$(2′′)

This result shows semi-endogenous growth, as A is a function of endogenous variables, but the growth result has exogenous variables on its right-hand side. If b > (<) 0, it adds an element of (negative) exogenous growth. So far, we assume $1-\varphi -\alpha -\text{β}>0$; foreign R&D growth should be stronger than a negative trend in order to get a positive productivity growth rate. Park (1995), Eaton and Kortum (1997), NESTI (2017), and Ahmed and Bhatti (2020) emphasize this strong role of foreign R&D for small countries. Whereas semi-endogenous growth models normally have the population growth rate on the right-hand side, here foreign R&D has this role. By implication, growth stops if foreign R&D would stop growing in case$\gamma ,\mu >0$ and the exogenous growth rate b is zero.¹⁴ Even if b is positive, $\gamma ,\mu <0$would imply that foreign R&D can have a competition effect, for example making the more productive domestic sectors smaller and thereby make productivity growth negative. In this new perspective it is therefore possible that productivity growth has phases of being negative, not only in recessions via the degree of capacity utilization (see Penn World Tables). However, the strong role of population growth may come back in future research if capital and labour would be re-introduced and foreign variables get endogenous.

3.2. Endogenous savings ratio for the Cobb–Douglas function

With a Cobb–Douglas production function like (2″) we get $F_2=\alpha A/R_b$, and $F_3=\text{β}A/R_g$ from (6a) and (6b). Replacing the marginal products in (6a) and (6b) yields solutions of R&D-stock/productivity ratios as functions of the consumption growth rates or vice versa: (7a) $F_{2,\tau +1}=\alpha A/R_b=\frac{{\left(1+g_c\right)}^{\omega }}{\rho }+{\delta }_b\mathrm{o}\mathrm{r}\frac{R_b}{A}=\frac{\alpha }{{\displaystyle \frac{{\left(1+g_c\right)}^{\omega }}{\rho }+{\delta }_b}}$(7a) (7b) $F_{3,\tau +1}=\text{β}A/R_g=\frac{{\left(1+g_c\right)}^{\omega }}{\rho }+{\delta }_g\mathrm{o}\mathrm{r}\frac{R_g}{A}=\frac{\text{β}}{{\displaystyle \frac{{\left(1+g_c\right)}^{\omega }}{\rho }+{\delta }_g}}$(7b)

Using these stock ratios in (5) with the definition of $y_j\equiv R_j/A$ and yields endogenous steady-state savings ratios, with g from (2″): (8) $s_b=y_b^s\left({\delta }_b+g\right)=\frac{\alpha \left({\delta }_b+g\right)}{{\displaystyle \frac{{\left(1+g_c\right)}^{\omega }}{\rho }+{\delta }_b}},s_g=y_g^s\left({\delta }_g+g\right)=\frac{\text{β}\left({\delta }_g+g\right)}{{\displaystyle \frac{{\left(1+g_c\right)}^{\omega }}{\rho }+{\delta }_g}}$(8)

As $(1+g{)}^{\omega }/\rho >1$ for ω > 0, and all other expressions are below unity, the savings ratios are below unity. In a steady state with constant growth rates for consumption and productivity, this determines the savings ratios. This result is only valid for productivity production functions such as the Cobb–Douglas function for which (7a, b) determine the A/R ratios, and in the steady state for which consumption c has the same growth rate as A, $R_b$, and $R_g$, because these assumptions have been used in the derivation of (2″).

3.3. Existence and stability of a steady state with Cobb–Douglas productivity function

The Cobb–Douglas function above is one possibility that may allow for a stable steady state, and it delivers the growth rate of A. We define$R^{\ast }\equiv \left(R_b^{\ast }{)}^{\gamma }\right(R_g^{\ast }{)}^{\mu }{e}^{bt}$, $y_g\equiv R_g/A$, $y_b\equiv R_b/A$. From (3′′′), we then get ${\hat{y}}_b=s_b(A_{-1}{)}^{\varphi }{R}_b^{\alpha -1}{R}_g^{\text{β}}{R}^{\ast }C-{\delta }_b-g$ ${\hat{y}}_g=s_g(A_{-1}{)}^{\varphi }{R}_b^{\alpha }{R}_g^{\text{β}-1}{R}^{\ast }C-{\delta }_g-g$with g from (2″). Multiplying through by y-terms then delivers (3^iv) again. Existence and stability are ensured for y terms, given g. The left-hand side equal to zero requires that domestic R and A terms have the same growth rate. Existence and stability of a steady state then requires that the Cobb–Douglas terms are constant and go to the growth rate of R and A terms. The production function (2′), using the definition of R*, in terms of y expressions is $A^{1-\alpha -\text{β}}=(A_{-1}{)}^{\varphi }{y}_b^{\alpha }{y}_g^{\text{β}}{R}^{\ast }C$Taking growth rates yields (2′′′) $\hat{A}=\frac{1}{\left(1-\alpha -\text{β}\right)}(\varphi \widehat{A_{-1}}+\alpha \widehat{y_b}+\text{β}\widehat{y_g}+\widehat{R^{\ast })}(2^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }})$(2′′′)

According to (3^iv), y terms converge to constant values and therefore their growth rates go to zero.

If ${\displaystyle \frac{\varphi }{\left(1-\alpha -\text{β}\right)}<1,}$is a stable process going to (2^{iv}) $\hat{A}=\widehat{A_{-1}}=\frac{\widehat{R^{\ast }}}{1-\varphi -\alpha -\text{β}},$(2^{iv})

This is (2″) again. Therefore, this proves existence and partial stability for y terms and the growth rate of A without further restrictions on the parameters of the Cobb–Douglas function. The stability analysis is only partial because the feedback effects from the growth of A to (3^iv) and vice versa are not yet considered. A special case of this is fishing out $-\varphi =\alpha +\text{β}$ requiring a special negative value of $\varphi$, making the denominator of (2^iv) equal to unity. Below we will consider other functions.

For the optimization case, things are only slightly more complicated. In the optimization, we have substituted the savings ratios from the dynamic equations into the objective function $s_{b,\tau }=\frac{\left(g_{b,\tau }+{\delta }_b\right)R_{b,\tau }}{F\left(A_{\tau -1},R_{\tau },\tau \right)}=\left(g_{b,\tau }+{\delta }_b\right)y_{b,\tau }$ $s_{g,\tau }=\frac{\left(g_{g,\tau }+{\delta }_g\right)R_{g,\tau }}{F\left(A_{\tau -1},R_{\tau },\tau \right)}=\left(g_{g,\tau }+{\delta }_g\right)y_{g,\tau }$

This corresponds to (5) in terms of non-steady-state values. Replacing the savings ratios in (3^iv) yields (3^{v}a) ${\dot{y}}_{b,\tau }=\left(g_{b,\tau }+{\delta }_b\right)y_{b,\tau }-\left({\delta }_b+g\right)y_{b,\tau }=\left(g_{b,\tau }-g\right)y_{b,\tau },$(3^{v}a) (3^{v}b) ${\dot{y}}_{g,\tau }=\left(g_{g,\tau }+{\delta }_g\right)y_{g,\tau }-\left({\delta }_g+g\right)y_{g,\tau }=\left(g_{g,\tau }-g\right)y_{g,\tau }$(3^{v}b)

There is an additional effect on the savings ratios now. The optimality conditions (7a) and (7b) are hard to fulfil with equality, because A/R-terms are given at any moment in time because productivity and R&D stocks move only slowly. This would suggest that the growth rate of consumption adjusts. However, it cannot balance two equations. Whenever the marginal product of private or public R&D is above (below) its optimum value, this is the case because its R&D level is below (above) optimum and therefore its saving ratio should be higher (lower)¹⁵ than in the steady state. Therefore, temporarily higher (lower) growth rates of private or public R&D lead to a speed up of the respective R/A ratio. This is the message of (3^va, b). The growth rate difference is a policy reaction. The growth rate difference is set to zero if stability leads to a steady state where all growth rates are the same and the marginal productivity conditions hold.

3.4. A translation to the long-term relations of a VECM for the Cobb–Douglas function relations

The marginal productivity condition in Equation (7a) suggests logR_b = logA−log(F₂/α), and Equation (7b) suggests logR_g = logA−log(F₃/β), where the last terms are a constant according to (6a, b) if consumption growth has reached its steady-state value. Combining the last two equations by elimination of logA leads to logR_b = logR_g+log(F₃/β)-log(F₂/α). Two of these or the Cobb–Douglas production function, could be long-term relationships in a cointegrated VAR in line with theoretical modelling.¹⁶ This would lead to four possible cases with due adjustment of lags:

1. The limiting case of a vector-error-correction model with full rank and estimation in levels with foreign terms as (weakly) exogenous, if confirmed by tests.

2. In case of no cointegration, the same model would be re-written in first differences.

3. In case of one cointegrating equation, we would have the following long-term relation from the productivity production function as expected value: ${\left[\left(\varphi -1\right)\mathrm{l}\mathrm{o}\mathrm{g}\left(\frac{A}{1+g}\right)+\alpha log{R}_b+\text{β}log{R}_g+\gamma log{R}_b^{\ast }+\mu log{R}_g^{\ast }+bt+c-g\right]}_{-1}=0$

4. In case of two long-term relations, we would imagine having (with u and v as residuals) $E\left(u_{-1}\right)=0={\left[logA-log{R}_b-\log \left(\frac{F_2}{\alpha }\right)\right]}_{-1},$ $E\left(v_{-1}\right)=0={\left[log{R}_b-log{R}_g-log\left(\frac{F_3}{\text{β}}\right)+log\left(\frac{F_2}{\alpha }\right)\right]}_{-1}$

In (iv), public R&D drives private R&D according to the second equation, and, in turn, private R&D drives productivity according to the first equation. The marginal product terms could be replaced by the right-hand side of (7a) and (7b) if they hold with equality. Time trends do not appear in this case. Several terms here have unit coefficients for the case of a Cobb–Douglas function. This may seem unrealistic at first sight, but unit coefficients feature prominently in the long-term relations of textbook examples. The consumption-income-investment relation (Lütkepohl 2005) and the liquidity-interest relationships (Juselius 2006) also have unit coefficients, which estimations only obtain approximately in the more flexible VECM approaches. Pesaran (1997) points out that only under special cases of functional forms we can link the VECMs directly to theoretical models. VECM estimation in a log–log approach could test whether the relations are similar to those obtained here for Cobb–Douglas functions. As this turns out to be dis-appointing (see Appendix VECMs), it may be interesting to look at other production functions as we do in the next section.

We may also use, in addition, output production function (1) suggesting logQ = log (A) as a long-term relation. Alternatively, this function could be used in one way or other to replace log(A)-terms above. Long-term relations in VECMs have two-way causality. The marginal productivity conditions would suggest, that given A, R-terms are determined or explained. Conversely, the three variables determine each other and R-terms drive A according to the productions function requiring adjustments of R-terms again. In sum, in principal we have a theoretical underpinning for the VECM work of Soete et al. (2020a, 2020b) but the Cobb–Douglas functions are quantitatively unrealistic.

5.1 Conditions for a balanced growth path with Mukerji VES functions

Narrowing down the consideration to explore the possibility of balanced growth results of the growth model leads to the following. Taking differences of Equation (9a) and assuming that productivity and efficient private R&D, $e^{bt}{R}_b$, have the same growth rate, we get $g=d\mathrm{l}\mathrm{o}\mathrm{g}A=\frac{\left(1-\alpha \right)}{\left(1-ϵ\right)}d\mathrm{l}\mathrm{o}\mathrm{g}{R}_b-\frac{\alpha b}{\left(1-ϵ\right)}=d\mathrm{l}\mathrm{o}\mathrm{g}{R}_b+b$Solving this for the growth rate of private R&D yields $d\mathrm{l}\mathrm{o}\mathrm{g}{R}_b=\frac{b+{\displaystyle \frac{\alpha b}{\left(1-ϵ\right)}}}{{\displaystyle \frac{\left(1-\alpha \right)}{\left(1-ϵ\right)}-1}}=\frac{b\left(1-ϵ\right)+\alpha b}{ϵ-\alpha }=b\frac{1-ϵ+\alpha }{ϵ-\alpha }$Solving for the productivity growth rate yields (10a) $g=d\mathrm{l}\mathrm{o}\mathrm{g}A=d\mathrm{l}\mathrm{o}\mathrm{g}{R}_b+b=b\frac{\left(1-ϵ\right)+\alpha }{ϵ-\alpha }+b=b\frac{\left(1-ϵ\right)+\alpha }{ϵ-\alpha }+\frac{b\left(ϵ-\alpha \right)}{ϵ-\alpha }=\frac{b}{ϵ-\alpha }$(10a)

The interpretation is that we have semi-endogenous growth, ultimately driven by exogenous technical change if the assumption of balanced growth of productivity and efficient private R&D holds.

The second Equation, (9b), shows a relation between productivity and public R&D. Equal growth rates of productivity and efficient public R&D, $e^{xt}{R}_g$, correspondingly lead to (10b) $g=d\mathrm{l}\mathrm{o}\mathrm{g}A=\frac{x}{ϵ-\text{β}},d\mathrm{l}\mathrm{o}\mathrm{g}{R}_b=x\frac{\left(1-ϵ\right)+\text{β}}{ϵ-\text{β}}$(10b)

A balanced growth path for these variables therefore requires $b\left(ϵ-\text{β}\right)=x\left(ϵ-\alpha \right)$, which holds if β = α and b = x (sufficient). The sufficient conditions for a balanced growth path are a step back towards a CES function from the given VES function. Only if public and private R&D are equally productive in the production of A and have identical rates of technical progress can we have a steady state with identical growth rates for productivity and efficient R&D variables.

For a set of sufficient conditions for the existence of a steady state, suppose

1. for the foreign country there is a VES function like (2v) with variables and parameters having a ‘*’, there are relations symmetric to (10a, b), (11) $g^{\ast }=\frac{b^{\ast }}{{ϵ}^{\ast }-{\alpha }^{\ast }}=\frac{x^{\ast }}{{ϵ}^{\ast }-{\text{β}}^{\ast }},{\alpha }^{\ast }={\text{β}}^{\ast },b^{\ast }=x^{\ast }$(11)

2. $\alpha =\text{β}=\varphi =ϵ,\gamma =\mu$, and for the foreign country ${\alpha }^{\ast }={\text{β}}^{\ast }={ϵ}^{\ast }={\varphi }^{\ast },{\gamma }^{\ast }={\mu }^{\ast }$.

3. $ϵg=\gamma (g^{\ast }-b^{\ast })$, ${ϵ}^{\ast }{g}^{\ast }={\gamma }^{\ast }\left(g-b\right)$ or ${\displaystyle \frac{g}{g^{\ast }-b^{\ast }}=\gamma /ϵ}$, ${\displaystyle \frac{g^{\ast }}{g-b}=}$ ϵ*/γ*.

Property (ii) for the foreign country follows from the previous lines. Equal growth rates of all terms on the right-hand side of the VES function (2^v) require $\varphi g=\alpha g=\text{β}g=\gamma (g^{\ast }-b^{\ast })=\mu (g^{\ast }-b^{\ast })$From this equation, we find property (iii) using the β = α = φ. The left-hand side must have a corresponding growth rate²¹ and therefore balanced growth requires β = α = φ = ϵ, and correspondingly for the foreign country ${\text{β}}^{\ast }={\alpha }^{\ast }={ϵ}^{\ast }={\varphi }^{\ast }$. The second part of property (iii) follows from the equality of growth rates for foreign terms in (2^v). Property (iii) implies that these constraints for a balanced growth path are restrictive in relation to the VES function.

Property (iv) follows from the requirement of equal growth rates for domestic and foreign terms in (2^v). As all other parameters have already been used, assumption (iv) can be seen as putting a constraint on γ and γ* as necessary for balanced productivity growth of a two-country model. With these assumptions, arguments in the VES function (2^iv) have exponential growth functions with identical exponents multiplied to t, and we can re-write the production function, together with properties (i)-(iv) for the home country as (12) $A=e^{\alpha gt/ϵ}[\underset{\_}{B}{]}^{\frac{1}{ϵ}}C=e^{\gamma (g^{\ast }-b^{\ast })t/ϵ}[\underset{\_}{B}{]}^{\frac{1}{ϵ}}C$(12)

Underlining of B expresses that all arguments in B, multiplied to the growth rate term $e^{gt}$ or identical ones according to (iv) above, are fixed to a certain value when hypothetically arriving at the steady state.

5.2. Dynamics with VES function and balanced growth constraints

Finding steady state conditions can be slightly different from finding a balanced growth path because we need the whole model. Therefore, we do not use the results of the previous sub-sections in the current one unless explicitly mentioned. We define $y_b\equiv e^{bt}{R}_b/A,y_g\equiv e^{xt}{R}_g/A$. From (3′), where these definitions were used for b = x = 0, with Mukerji-VES function, we then get the accumulation dynamics of the VES growth model: (13) ${\hat{y}}_b=s_b[a{\left(A_{-1}\right)}^{\varphi }+h{e}^{\alpha bt}{R}_b^{\alpha }+\left(1-h\right)e^{\text{β}xt}{R}_g^{\text{β}}+d(R{}_b^{\ast }{)}^{\gamma }+f(R{}_g^{\ast }{)}^{\mu }{]}^{\frac{1}{ϵ}}C/R_b-{\delta }_b-(g-b)$(13) (14) ${\hat{y}}_g=s_g[a{\left(A_{-1}\right)}^{\varphi }+h{e}^{\alpha bt}{R}_b^{\alpha }+\left(1-h\right)e^{\text{β}xt}{R}_g^{\text{β}}+d(R{}_b^{\ast }{)}^{\gamma }+f(R{}_g^{\ast }{)}^{\mu }{]}^{\frac{1}{ϵ}}C/R_g-{\delta }_g-(g-x)$(14) Multiplying through by y-terms yields (15) ${\dot{y}}_b=\frac{s_b{\left[a{\left(A_{-1}\right)}^{\varphi }+h{e}^{\alpha bt}{R}_b^{\alpha }+\left(1-h\right)e^{\text{β}xt}{R}_g^{\text{β}}+d(R{}_b^{\ast }{)}^{\gamma }+f(R{}_g^{\ast }{)}^{\mu }\right]}^{\frac{1}{ϵ}}C}{A{e}^{-bt}}-({\delta }_b+g-b)y_b$(15) (16) ${\dot{y}}_g=\frac{s_g{\left[a{\left(A_{-1}\right)}^{\varphi }+h{e}^{\alpha bt}{R}_b^{\alpha }+\left(1-h\right)e^{\text{β}xt}{R}_g^{\text{β}}+d(R{}_b^{\ast }{)}^{\gamma }+f(R{}_g^{\ast }{)}^{\mu }\right]}^{\frac{1}{ϵ}}C}{A{e}^{-xt}}-({\delta }_g+g-x)y_g$(16)

In an appendix, we show that we get a steady state only for very special cases of parameter values. For the domestic dynamic equations this can be summarized as

either (i) b = x = 0, or (ii) ϵ = 0, or (iii) α = β = φ with ϵ = 1, or (iv) the CES case for domestic variables, α = β = φ = ϵ, in all four cases together with $b{\displaystyle \frac{1-ϵ+\alpha }{ϵ-\alpha }ϵ=\gamma (\frac{1-{ϵ}^{\ast }+{\alpha }^{\ast }}{{ϵ}^{\ast }-{\alpha }^{\ast }})b^{\ast }\mathrm{a}\mathrm{n}\mathrm{d}{b}^{\ast }=x^{\ast }\mathrm{a}\mathrm{n}\mathrm{d}\gamma =\mu .}$

From the dynamic equations of the foreign countries, we would get by symmetry the conditions

either (i) b* = x* = 0, or (ii) ϵ* = 0, or, or (iii) ϵ* = 1 with α* = β* = φ*, or (iv) the CES case for foreign variables, α* = β* = φ* = ϵ*, in each of the four case together with $b{\displaystyle \frac{1-ϵ+\alpha }{ϵ-\alpha }{\gamma }^{\ast }={ϵ}^{\ast }(\frac{1-{ϵ}^{\ast }+{\alpha }^{\ast }}{{ϵ}^{\ast }-{\alpha }^{\ast }})b^{\ast }\mathrm{a}\mathrm{n}\mathrm{d}b=x\mathrm{a}\mathrm{n}\mathrm{d}{\gamma }^{\ast }={\mu }^{\ast }.}$

This proves that the existence of a steady-state solution in the presence of a restricted Mukerji VES function and given savings ratios, optimal or not, is possible only under very restrictive conditions. Stability towards the stationary y terms for given g−b, g−x and g*−b*, g*−x* at the end of (15) and (16) also works analogous to the analyses above. If g−b, g−x or g*−b*, g*−x* would be higher than the fraction of their steady state levels, y-dot terms would be negative and slow down the growth of the efficient R/A ratios, which would reduce A growth. However, feedback effects would again be too complicated for qualitative analysis. Domestic and foreign growth rates can differ in the steady state of this model, but they must be proportional to each other. Otherwise, countries will have non-proportional growth rates and steady states do not exist, because the VES without restrictions contradicts the part of the system without production function.

In general, a VECM with a linear time trend or a growth model with VES functions are more flexible than a balanced growth or steady state model, and the long-run solution can have different growth rates for each variable. Balanced growth or steady states are unlikely to occur because they require very special parameter constellation. We provide some evidence for this below in Section 7. Without parameter restrictions, the theoretical model would consist of Equations (3′) and (2^v) with exogenous foreign variables.