Schumpeterian Macroeconomic Production Function for Open Economies

Abstract

The macroeconomic production function is a traditional key element of modern macroeconomics, as is the more recent knowledge production function which explains knowledge/patents by certain input factors such as research, foreign direct investment, or international technology spillovers. This study is a major contribution to innovation, trade, FDI, and growth analysis, namely in the form of a combination of an empirically relevant knowledge production function for open economies—with both trade and inward FDI as well as outward foreign direct investment plus research input—with a macro production function. Plugging the open economy knowledge production function into a standard macroeconomic production function yields important new insights for many fields: the estimation of the production potential in an open economy, growth decomposition analysis in the context of economic globalization, and the demand for labor as well as long-run international output interdependency of big countries, and this includes a view at the asymmetric case of a simple two country world in which one country is at full employment while the other is facing underutilized capacities. Finally, there are crucial implications for the analysis of broad regional integration schemes such as TTIP or TPP and a more realistic and comprehensive empirical analysis.

Appendix

5.1.1 A.1 Optimal Choice of the Size of the R&D Sector

In the above equation (V) in logs, one can replace z′L′ by z″L since z″Y = z′L′y (recall that z″Y is the output of the R&D sector), and z″Y/y = z′L′ and z″Y/y = z″L so that z″L = z′L′ and hence ln(z′L′) can be replaced by ln(z″L). The research share z″ in output that maximizes Y thus can be derived (or one maximizes y with respect to z″). Taking the derivative dlnY/dz″ and setting it equal to zero gives the necessary condition (while assuming: V′ + V″ < 1)

$$ d \ln Y/dz^{{\prime\prime} }=\left(-1/\left(1-z^{{\prime\prime}}\right)\right)\Big(1/\left(1-\left({V}^{\prime }+{V}^{{\prime\prime}}\right)\left(1-\beta \right)\right)+\left(1-\beta \right){V}^{{\prime\prime} }\ \left(1/z^{{\prime\prime}}\right)/\left(1-\left({V}^{\prime }+{V}^{{\prime\prime}}\right)\left(1-\beta \right)\right)=0 $$

(5.38)

$$ -1/\left(1-{\mathrm{z}}^{{\prime\prime}}\right)+\left(1-\mathrm{\ss}\right){\mathrm{V}}^{{\prime\prime} }/{\mathrm{z}}^{{\prime\prime} }=0 $$

(5.39)

$$ z^{{\prime\prime} }/\left(1-z^{{\prime\prime}}\right)=\left(1-\beta \right){V}^{{\prime\prime} } $$

(5.40)

$$ 1/z^{{\prime\prime} }=1+1/\left(\left(1-\beta \right){v}^{{\prime\prime}}\right) $$

(5.41)

$$ z^{{\prime\prime} }=1/\Big(1+\left(1/\left(\left(1-\beta \right){v}^{{\prime\prime}}\right)\right) $$

(5.42)

Assume 1/((1 − β)v″) is close to zero.

$$ \ln z^{{\prime\prime}}\approx 1/\left(\left(1-\beta \right){v}^{{\prime\prime}}\right) $$

(5.43)

Note: dlnz″/dβ < 0; dlnz″/dv″ > 0.

For a maximum, the second derivative should be negative and it is given by the expression

$$ {d}^2 \ln Y/{dz{\prime\prime}}^2=\left[\left(-1/{\left(1-z{\prime\prime} \right)}^2\right)\left(1/\left(1-\left({V}^{\prime }+{V}^{{\prime\prime}}\right)\left(1-\beta \right)\right)\right)\right]\hbox{--} \left\{\left(1-\beta \right){V}^{{\prime\prime}}\left(1/z^{{\prime\prime} }{}^2\right)/\left(1-\left({V}^{\prime }+{V}^{{\prime\prime}}\right)\left(1-\beta \right)\right)\right\}<0 $$

(5.44)

This equation is fulfilled if {…} > […].

$$ \left\{\dots \right\}/\left[\dots \right]>1 $$

(5.45)

$$ \Big(1/{\left(1-z{\prime\prime} \right)}^2/\left(\left(\left(1-\beta \right){v}^{{\prime\prime}}\right)/z^{{\prime\prime} }{}^2\right)<1 $$

(5.46)

$$ 1/\left(\left(\left(1-2z^{{\prime\prime} }+z^{{\prime\prime} }{}^2\right)/z^{{\prime\prime} }{}^2\right)\left(1-\beta \right){v}^{{\prime\prime}}\right)<1 $$

(5.47)

$$ 1/\left(1/z^{{\prime\prime} }{}^2-2/z^{{\prime\prime} }+1\right)>\left(1-\beta \right){v}^{{\prime\prime} } $$

(5.48)

$$ -1/z{{\prime\prime}}^2+2/{z}^{\prime }>-\beta + \ln {v}^{{\prime\prime} } $$

(5.49)

$$ f\left({z}^{\prime}\right)>-\beta + \ln {v}^{{\prime\prime} } $$

(5.50)

An alternative approach could be to consider an endogenous growth model based on the Schumpeterian macroeconomic production function and then one considers the steady state situation and maximizes steady state per capita consumption through optimal choice of z″. Governments eager to obtain the maximum golden rule consumption per capita will have to consider the profit maximization condition of the R&D sector and on this basis should allocate an adequate subsidy rate to the R&D sector. An extended approach would then additionally include the government budget constraint G + f′Y = τY if one assumes that there is no government debt (f′ is the subsidy ratio that should reflect the difference between the social rate of return on innovation and the private rate of return on innovation and G is government consumption—with G/Y:= γ to be considered the relevant exogenous variable). This then leads to an optimum tax analysis where τ = f′ + γ.

5.1.2 A.2 Schumpeterian CES Function

The knowledge production function is given by

$$ A={\left(xY*/Y\right)}^H{j}^{V^{\prime }}{y}^{H+{V}^{\prime }+{V}^{{\prime\prime} }}{\left({z}^{\prime }{L}^{\prime}\right)}^{V^{{\prime\prime} }}{\left(\alpha *\beta /r\right)}^V{\left(\alpha \beta */r*\right)}^{V*} $$

(5.51)

The CES production function—compared to the Cobb–Douglas function, it is better suitable for analyzing income distribution issues—is given by

$$ Y=\lambda {\left[\left(1-\varOmega \right){(AL)}^{-{v}^{{\prime\prime} }}+\varOmega {K}^{-{v}^{{\prime\prime} }}\right]}^{-1/{v}^{{\prime\prime} }} $$

(5.52)

(Ω > 0; 0 < Ω < 1; ∞ ≥ v″ ≥ 1; v″ ≠ 0, elasticity of substitution σ″ = 1/(1+v″); λ > 0)

Inserting (i) in (ii) gives:

$$ \begin{array}{ll}\hfill & Y\\ {}& =\lambda {\left[\left(1-\varOmega \right){\left({\left(xY*/Y\right)}^H{j}^{V^{\prime }}{y}^{H+{V}^{\prime }+{V}^{{\prime\prime} }}{\left({z}^{\prime }{L}^{\prime}\right)}^{V^{{\prime\prime} }}{\left(\alpha *\beta /r\right)}^V{\left(\alpha \beta */r*\right)}^{V*}L\right)}^{-{v}^{{\prime\prime} }}+\varOmega {K}^{-{v}^{{\prime\prime} }}\right]}^{-1/{v}^{{\prime\prime} }}\hfill \end{array} $$

(5.53)

$$ \begin{array}{ll}\hfill & {Y}^{-{v}^{{\prime\prime} }}\\ {}& ={\lambda}^{-{v}^{{\prime\prime} }}\left[\left(1-\varOmega \right){\left({\left(xY*/Y\right)}^H{j}^{V^{\prime }}{y}^{H+{V}^{\prime }+{V}^{{\prime\prime} }}{\left({z}^{\prime }{L}^{\prime}\right)}^{V^{{\prime\prime} }}{\left(\alpha *\beta /r\right)}^V{\left(\alpha \beta */r*\right)}^{V*}L\right)}^{-{v}^{{\prime\prime} }}+\varOmega {K}^{-{v}^{{\prime\prime} }}\right]\hfill \end{array} $$

(5.54)

We can solve in a meaningful way for Y if one assumes that v″ = V′+V″:

$$ \begin{array}{ll}\hfill & {Y}^{-2{v}^{{\prime\prime} }}\\ {}& ={\lambda}^{-{v}^{{\prime\prime} }}\left[\left(1-\varOmega \right){\left({\left(xY*\right)}^H{j}^{V^{\prime }}{\left({z}^{\prime }{L}^{\prime}\right)}^{V^{{\prime\prime} }}{\left(\alpha *\beta /r\right)}^V{\left(\alpha \beta */r*\right)}^{V*}{L}^{1\_H-{V}^{\prime }-{V}^{{\prime\prime} }}\right)}^{-{v}^{{\prime\prime} }}+\varOmega {\left(K/Y\right)}^{-{v}^{{\prime\prime} }}\right]\hfill \end{array} $$

(5.55)

$$ \begin{array}{ll}\hfill & Y\\ {}& ={\lambda}^{0.5}{\left[\left(1-\varOmega \right){\left({\left(xY*\right)}^H{j}^{V^{\prime }}{\left({z}^{\prime }{L}^{\prime}\right)}^{V^{{\prime\prime} }}{\left(\alpha *\beta /r\right)}^V{\left(\alpha \beta */r*\right)}^{V*}{L}^{1-H-{V}^{\prime }-{V}^{{\prime\prime} }}\right)}^{-{v}^{{\prime\prime} }}+\varOmega {\left(K/Y\right)}^{-{v}^{{\prime\prime} }}\right]}^{-1/2{v}^{{\prime\prime} }}\hfill \end{array} $$

(5.56)

Dividing (iv) by ΩK^−v″ gives:

$$ \begin{array}{ll}\hfill & \left(Y/\left(\varOmega K\right)\right)\\ {}& ={\lambda}^{-{v}^{{\prime\prime} }}\left[\left(1-\varOmega \right){\left({\left(xY*/Y\right)}^H{j}^{V^{\prime }}{y}^{H+{V}^{\prime }+{V}^{{\prime\prime} }}{\left({z}^{\prime }{L}^{\prime}\right)}^{V^{{\prime\prime} }}{\left(\alpha *\beta /r\right)}^V{\left(\alpha \beta */r*\right)}^{V*}L\right)}^{-{v}^{{\prime\prime} }}/{\left(\varOmega K/Y\right)}^{\hbox{--} {v}^{{\prime\prime} }}+1\right]\hfill \end{array} $$

(5.57)

$$ \mathrm{Define}\kern0.5em {z}^{\prime }:{=}^{``}\left[\left(1-\varOmega \right){\left({\left(xY*/Y\right)}^H{j}^{V^{\prime }}{y}^{H+{V}^{\prime }+{V}^{{\prime\prime} }}{\left({z}^{\prime }{L}^{\prime}\right)}^{V^{{\prime\prime} }}{\left(\alpha *\beta /r\right)}^V{\left(\alpha \beta */r*\right)}^{V*}L\right)}^{-{v}^{{\prime\prime} }}/{\left(\varOmega K/Y\right)}^{\hbox{--} {v}^{{\prime\prime} }}+1\right] $$

(5.58)

Hence, taking logs and using the approximation ln(1 + Z ^′) ≈ Z ^′—for Z′ close to zero—we can use the approximation:

$$ -3{v}^{{\prime\prime} } \ln (Y)- \ln \Big(\left(\varOmega \right)-{v}^{{\prime\prime} } \ln (K)=-{v}^{{\prime\prime} } \ln \left(\lambda \right)+{Z}^{\prime } $$

(5.59)

$$ \ln (Y)=\left\{- \ln \right(\left(\varOmega \right)-{v}^{{\prime\prime} } \ln (K)+{v}^{{\prime\prime} } \ln \left(\lambda \right)-{Z}^{\prime}\Big\}/3{v}^{{\prime\prime} } $$

(5.60)