Abstract
Based on the framework for R&D heterogeneity-based endogenous ESTC in industry introduced in Chap. 3, this chapter is to develop a mathematical model of the same theme by specifying the utility function for representative households and the production technology and economic conditions of representative manufacturers, and to compare the outcomes of the model using competitive equilibrium analysis and centrally planned equilibrium analysis.
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Notes
- 1.
As Wen and Zhaoxi (2012) do not mention the ESTC R&D sector, I assume that the reference parameters of this sector are identical with those of the intermediate R&D sectors.
- 2.
To highlight the importance of the elasticity of substitution between capital and energy, I disregard the form of (KEL) here. In fact, if both sides of the (KEL) function are divided by \(L_Y\), then the form of (KEL) is similar to Eq. (4.83). In addition, as the present book focuses on ESTC, it does not discuss the specific forms of capital quality in depth, but represents them with capital-augmenting technological change.
- 3.
If \(\sigma_{K,L} = 1\) in Eq. (4.82), then \(\sigma_{K,E}\) is equal to \(\sigma_{KL,E}\).
- 4.
Because \(\sigma_{K,E} > 0\), if \(\sigma_{K,E} + g_{Q_K }^{ss} < 1\), then \(0 < \sigma_{K,E} < 1\) and \(0 < g_{Q_K }^{ss} < 1\), or \(\sigma_{K,E} > 1\) and \(g_{Q_K }^{ss} < 0\). Suppose \(\sigma_{K,E} + g_{Q_K }^{ss} \equiv x < 1\), as shown in Eq. (4.103), substituting \(x\) will turn Eq. (4.97) into \(g_{Q_E }^{ss} = \left( {\frac{{1 - \theta_{EY}^{ss} }}{{\theta_{EY}^{ss} }} + 2\rho } \right){g_{Q_K }^{ss} }^2 - 2\rho x\). If \(0 < \sigma_{K,E} < 1\) and \(0 < g_{Q_K }^{ss} < 1\), that means capital quality and ESTC are mutually complementary, so the equilibrium rate of ESTC will not be great. In addition, as \(\sigma_{K,E}\) is also the elasticity of substitution between capital and energy, substituting Eq. (4.104) into (4.85) will lead to \(g_E^{ss} \sim \frac{{1 - \sigma_{K,E} }}{{\sigma_{K,E} }}\frac{{1 - \theta_{EY}^{ss} }}{{\theta_{EY}^{ss} }}g_{Q_K }^{ss}\). Therefore, if \(0 < \sigma_{K,E} < 1\) and \(0 < g_{Q_K }^{ss} < 1\), then \(g_E^{ss} < 0\). Because \(g_K^{ss} = g_Y^{ss} > 0\), \(g_{Q_E }^{ss} < 0\). When \(g_{Q_K }^{ss} \to 1\), \(g_{Q_E }^{ss}\) reaches its maximum value \(\frac{{1 - \theta_{EY}^{ss} }}{{\theta_{EY}^{ss} }} + 2\rho\). If \(\sigma_{K,E} > 1\) and \(g_{Q_K }^{ss} < 0\), that means capital quality and ESTC are mutually substitutable, so the equilibrium rate of ESTC and capital quality change in opposite directions. In addition, as \(\sigma_{K,E}\) is also the elasticity of substitution between capital and energy, if \(\sigma_{K,E} > 1\) and \(g_{Q_K }^{ss} < 0\), then \(g_E^{ss} < 0\). As the capital stock in steady state will be larger than energy, \(g_{Q_E }^{ss} < 0\). When \(g_{Q_K }^{ss} \to - \sigma_{K,E}\), \(g_{Q_E }^{ss}\) reaches its maximum value \(\left( {\frac{{1 - \theta_{EY}^{ss} }}{{\theta_{EY}^{ss} }} + 2\rho } \right){\sigma_{K,E} }^2\).
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He, X. (2023). The Mathematical Model of R&D Heterogeneity-Based Endogenous ESTC in Industry. In: The Endogenous Energy-Saving Technological Change in China's Industrial Sector . Springer, Singapore. https://doi.org/10.1007/978-981-19-7485-4_4
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