The tragedy of product homogeneity and knowledge non-spillovers: explaining the slow pace of energy technological progress

Abstract

There is a growing body of literature mentioning the slow progress of energy technological innovation as compared to information technology (IT), but the reasons why the energy sector is perplexed by slow innovation remain unexplained. Based on a variety-expanding endogenous technological change model, this paper investigates the economic mechanism that underlines the slow pace of energy technological progress. We show that in the market equilibrium the growth rate of energy technology variety is always lower than that of IT variety, this outcome stems from both the market fundamentals where the homogeneity of end-use energy goods is less likely to harness the pecuniary externality embedded in the love-for-variety household, and the technology fundamentals where capital-intensiveness of energy technology assets inhibits the non-pecuniary technological externality associated with knowledge spillovers. We further show that the social planner allocation can raise the rate of technological progress in both energy and IT sectors, but still fails to achieve an outcome in which technological progress in the energy sector can catch up with the IT sector. Finally, using efficiency-improving revenue-neutral policy interventions that subsidize energy sectors and tax IT sectors, the decentralized market equilibrium can achieve an outcome in which both energy and IT sectors have the same rate of technological progress.

Appendices

Appendix 1: Solving the household problem

Solving the household’s problem requires setting the current-value Hamilton as,

$$\begin{aligned} \mathrm{H}(C_T, C_E, A,\lambda )= & {} InC_T (t)+InC_E (t)+\lambda (t)\cdot [r(t)\cdot A(t)\\&+w(t)\cdot 2L-P_T (t)\cdot C_T (t)-C_E (t)]. \end{aligned}$$

The interior necessary conditions with respect to four endogenous variables are as follow,

$$\begin{aligned} C_E:&\mathrm{H}_{C_E } (C_E, C_T, A,\lambda )=C_E (t)^{-1}-\lambda (t)=0 \end{aligned}$$

(40)

$$\begin{aligned} C_T:&\mathrm{H}_{C_T } (C_E, C_T, A,\lambda )=C_T (t)^{-1}-\lambda (t)\cdot P_T (t)=0 \end{aligned}$$

(41)

$$\begin{aligned} A:&\mathrm{H}_A (C_E, C_T, A,\lambda )=\lambda (t)\cdot r(t)=\rho \cdot \lambda (t)-\dot{\lambda }(t) \end{aligned}$$

(42)

$$\begin{aligned} \lambda :&\mathrm{H}_\lambda (C_E, C_T, A,\lambda )=r(t)\cdot A(t)+w(t)\cdot 2L-P_T (t)\cdot C_T (t)-C_E (t)=\dot{A}(t)\nonumber \\ \end{aligned}$$

(43)

Differentiating (40)–(41) with respect to time obtain

$$\begin{aligned} \frac{\dot{C}_E (t)}{C_E (t)}=-\frac{\dot{\lambda }(t)}{\lambda (t)}, \frac{\dot{P}_T (t)}{P_T (t)}+\frac{\dot{C}_T (t)}{C_T (t)}=-\frac{\dot{\lambda }(t)}{\lambda (t)}. \end{aligned}$$

(44)

Substituting (42) into (44) captures the time path of consumption of energy and IT products

$$\begin{aligned} \frac{\dot{C}_E (t)}{C_E (t)}=-\frac{\dot{\lambda }(t)}{\lambda (t)}=r(t)-\rho , \frac{\dot{P}_T (t)}{P_T (t)}+\frac{\dot{C}_T (t)}{C_T (t)}=r(t)-\rho \end{aligned}$$

and the transversality condition

$$\begin{aligned} \mathop {\lim }\limits _{t\rightarrow +\infty } A(t)\cdot \exp \left[ {-\int _0^t {r(s)\cdot ds} } \right] =0 \end{aligned}$$

$\square$

Appendix 2: Proof of Proposition 1

Given that the instantaneous flow profit $\pi _E (i,t)$ is independent of primary intermediate energy input variety, the market value of primary energy firms is independent of primary energy variety and the HJB equation takes the form as $r(t)\cdot V_E (t)-\dot{V}_E (t)=\pi _E (t)=\varepsilon _E ^{-1}L$. Meanwhile, the FEC of energy R&D requires that $\eta \cdot V_E (t)=1$ for all time t when there is positive R&D for energy technological progress, implying that $\dot{V}_E (t)=0$ for all time t. Substituting it into the HJB equation yields the equilibrium market interest rate $r(t)=\eta L/\varepsilon _E$. Substituting $r(t)=\eta L/\varepsilon _E$ into (2) yields the equilibrium growth rate of household consumption of final energy goods,

$$\begin{aligned} g_{C_E } (t)\equiv \frac{\dot{C}_E (t)}{C_E (t)}=r(t)-\rho =\frac{\eta L}{\varepsilon _E }-\rho . \end{aligned}$$

(45)

Given that energy market clearing condition always holds $C_E (t)+X_E (t)+R_E (t)=Y_E (t)$, we get that final energy consumption should grow at the same rate as the output of energy goods. Furthermore, aggregating the whole set of varieties of primary energy intermediate inputs obtains the total outputs of end-use, secondary energy products

$$\begin{aligned} Y_E (t)=\frac{1}{1-a}\left[ {\int _0^{N_E (t)} {x_E (i,t)^{1-a}di} } \right] \cdot L^{a}=\frac{\varepsilon _E }{\varepsilon _E -1}\cdot L\cdot N_E (t), \end{aligned}$$

(46)

(46) implies that the equilibrium growth rate of secondary energy goods outputs is equal to that of primary energy technology variety,

$$\begin{aligned} g_{C_E } (t)=g_{Y_E } (t)=g_{N_E } (t)=\frac{\eta L}{\varepsilon _E }-\rho , \end{aligned}$$

where $g_{Y_E } (t)\equiv \dot{Y}_E (t)/Y_E (t)$, $g_{C_E } (t)\equiv \dot{C}_E (t)/C_E (t)$, $g_{N_E } (t)\equiv \dot{N}_E (t)/N_E (t)$ is the growth rate of end-use energy outputs, final energy consumption, and primary energy variety, respectively.

Turn to IT sectors, given that the instantaneous flow profit of IT firms producing each differentiate variety of IT consumer product is independent of product varieties $\pi _T (t)=\pi _T (j,t)$, the HJB equation (18) thus takes the form $r(t)\cdot V_T (t) -\dot{V}_T (t)=\pi _T (t)$, and rearranging obtains

$$\begin{aligned} V_T (t)=\frac{\pi _T (t)}{r(t)-g_{V_T } (t)}, \end{aligned}$$

(47)

where $g_{V_T } (t)\equiv \dot{V}_T (t)/V_T (t)$ is the growth rate of the market value of IT firms at time t. From the FEC of IT R&D (20), we have $g_{N_T } (t)+g_{V_T } (t)=g_w (t)$, where $g_{N_T } (t)$, $g_{V_T } (t)$, $g_w (t)$ is the growth rate of IT technology variety, market value, and wage rate at time t, respectively. Plugging it into (47) obtains

$$\begin{aligned} V_T (t)=\frac{\pi _T (t)}{r(t)-g_w (t)+g_{N_T } (t)}. \end{aligned}$$

(48)

Then using the instantaneous flow profit $\pi _T (t)=(\varepsilon -1)^{-1}\cdot w(t)\cdot L_{TP} (t)/N_T (t)$ and FEC of IT R&D (20) to substitute $\pi _T (t)$ and $V_T (t)$, (48) can be simplified as,

$$\begin{aligned} \eta \cdot L_{TP} (t)=(\varepsilon _T -1)\cdot [r(t)-g_w (t)+g_{N_T } (t)]. \end{aligned}$$

(49)

Plugging the IPF in IT sectors (19) and the growth rate of wage rate $g_w (t)=r(t)-\rho$ into (49), we derive the amount of workforce employed in IT sectors for output production and technology R&D,

$$\begin{aligned} L_{TP} (t)=\frac{(\varepsilon _T -1)\cdot (\rho +\eta L)}{\varepsilon _T \cdot \eta }, L_{TR} (t)=\frac{\eta L-(\varepsilon _T -1)\cdot \rho }{\varepsilon _T \cdot \eta }. \end{aligned}$$

(50)

From the IPF in IT sectors (19) we obtain the equilibrium growth rate of the number of IT variety

$$\begin{aligned} g_{N_T } (t)\equiv \frac{\dot{N}_T (t)}{N_T (t)}=\eta \cdot L_{TR} (t)=\frac{\eta L-(\varepsilon _T -1)\cdot \rho }{\varepsilon _T }. \end{aligned}$$

(51)

Based on the output of each individual variety of IT product given by $c_T (j,t)=L_{TP} (t)/N_T (t)$, the amount of aggregate consumption of IT product composite is determined by,

$$\begin{aligned} C_T (t)=\left[ {\int _0^{N_T (t)} {c_T (j,t)^{\frac{\varepsilon _T -1}{\varepsilon _T }}dj} } \right] ^{\frac{\varepsilon _T }{\varepsilon _T -1}}=N_T (t)^{\frac{\varepsilon _T }{\varepsilon _T -1}}\cdot c_T (t)=L_{TP} (t)\cdot N_T (t)^{\frac{1}{\varepsilon _T -1}}, \end{aligned}$$

(52)

and the equilibrium growth rate of consumption of IT product is thus given by

$$\begin{aligned} g_{C_T } \equiv \frac{\dot{C}_T }{C_T }=\frac{1}{\varepsilon _T -1}\cdot g_{N_T } =\frac{\eta L-(\varepsilon _T -1)\cdot \rho }{\varepsilon _T (\varepsilon _T -1)}. \end{aligned}$$

(53)

$\square$

Appendix 3: Proof of Proposition 2

The growth rate of energy technology and IT variety $g_{N_E} (t), g_{N_T } (t)$ is given by (22) and (24), and the difference in their growth rate is equal to

$$\begin{aligned} g_{N_T } (t)-g_{N_E } (t)=\frac{\eta L+\rho }{\varepsilon _T }-\rho -\left[ {\frac{\eta L}{\varepsilon _E }-\rho } \right] =\eta L\frac{\varepsilon _E -\varepsilon _T }{\varepsilon _T \varepsilon _E }+\frac{\rho }{\varepsilon _T } \end{aligned}$$

Consider that the differentiated variety in both energy and IT sector are gross substitutes, and the elasticity of substitution of primary energy input variety is sufficiently larger than that of IT product variety $\varepsilon _E >\varepsilon _T >1$, we have $g_{N_T} (t)>g_{N_E } (t)$. Moreover, the growth rate of consumption of energy and IT products $g_{C_E } (t), g_{C_T} (t)$ is given by (22) and (23), and the difference in their growth rate is equal to

$$\begin{aligned} g_{C_T } (t)-g_{C_E } (t)=\frac{\eta L}{\varepsilon _T (\varepsilon _T -1)}-\frac{\rho }{\varepsilon _T }-\frac{\eta L}{\varepsilon _E }+\rho =\eta L\cdot \frac{\varepsilon _E -\varepsilon _T (\varepsilon _T -1)}{\varepsilon _E \varepsilon _T (\varepsilon _T -1)}+\rho \cdot \frac{\varepsilon _T -1}{\varepsilon _T} \end{aligned}$$

Given that $\varepsilon _E >>\varepsilon _T >1$, we have $g_{C_T }(t)>g_{C_E } (t)$. $\square$

Appendix 4: Proof of Proposition 3

The current-value Hamilton is given by (we drop time to simplify notation),

$$\begin{aligned} \mathrm{H}(C_E, C_T, N_E, N_T )= & {} InC_E +InC_T +\lambda _E \cdot \eta \cdot [(1-\varepsilon _E ^{-1})^{-\varepsilon }\cdot \varepsilon _E ^{-1}\cdot L\cdot N_E -C_E ] \\&+\lambda _T (t)\cdot \eta \cdot N_T (t)\cdot \left[ {L-C_T ^{S}(t)\cdot N_T ^{S}(t)^{\frac{1}{1-\varepsilon _T }}} \right] , \end{aligned}$$

where $C_E, C_T$ are control variables, and $N_E, N_T$ are state variables. The necessary conditions for an interior solution are as follows,

$$\begin{aligned}&C_E : C_E ^{-1}-\lambda _E \cdot \eta =0, \end{aligned}$$

(54)

$$\begin{aligned}&C_T : C_T ^{-1}-\lambda _T \cdot \eta \cdot N_T \cdot N_T ^{\frac{1}{1-\varepsilon _T }}=C_T ^{-1}-\lambda _T \cdot \eta \cdot N_T ^{\frac{2-\varepsilon _T }{1-\varepsilon _T }}=0 \end{aligned}$$

(55)

$$\begin{aligned}&N_E : \lambda _E \cdot \eta \cdot \left( 1-\varepsilon _E ^{-1}\right) ^{-\varepsilon _E }\cdot \varepsilon _E ^{-1}\cdot L=\rho \cdot \lambda _E -\dot{\lambda }_E, \end{aligned}$$

(56)

$$\begin{aligned}&N_T : \lambda _T \cdot \eta \cdot \left( L-C_T \cdot N_T ^{\frac{1}{1-\varepsilon _T }}\right) -\lambda _T \cdot \eta \cdot N_T \cdot C_T \cdot \frac{1}{1-\varepsilon _T }\cdot N_T ^{\frac{\varepsilon _T }{1-\varepsilon _T }}=\rho \cdot \lambda _T -\dot{\lambda }_T.\nonumber \\ \end{aligned}$$

(57)

From (54) and (56), we obtain the growth rate of energy consumption in the social optimum

$$\begin{aligned} \frac{\dot{C}_E ^{S}(t)}{C_E ^{S}(t)}=\eta \cdot \varepsilon _E ^{-1}\cdot \left( 1-\varepsilon _E ^{-1}\right) ^{-\varepsilon _E }\cdot L-\rho . \end{aligned}$$

(58)

From energy market clearing condition, we obtain that the growth rate for energy consumption should be equal to the growth rate of energy technology variety,

$$\begin{aligned} \frac{\dot{N}_E ^{S}(t)}{N_E ^{S}(t)}=\frac{\dot{C}_E ^{S}(t)}{C_E ^{S}(t)}=\eta \cdot \varepsilon _E ^{-1}\cdot \left( 1-\varepsilon _E ^{-1}\right) ^{-\varepsilon _E }\cdot L-\rho . \end{aligned}$$

(59)

To solve for the growth rate of IT technology variety in the optimal growth path, differentiating (55) with respect to time t obtains

$$\begin{aligned} \frac{\dot{C}_T }{C_T }=-\frac{\dot{\lambda }_T }{\lambda _T }-\frac{2-\varepsilon _T }{1-\varepsilon _T }\cdot \frac{\dot{N}_T }{N_T }. \end{aligned}$$

(60)

From (57) we obtain

$$\begin{aligned}&\eta \left( L-C_T \cdot N_T ^{\frac{1}{1-\varepsilon _T }}\right) -\frac{\eta }{1-\varepsilon _T }\cdot C_T \cdot N_T ^{\frac{1}{1-\varepsilon _T }}=\rho -\frac{\dot{\lambda }_T }{\lambda _T } \Rightarrow -\frac{\dot{\lambda }_T }{\lambda _T }\nonumber \\&\quad =\eta \cdot L-\eta \cdot \frac{2-\varepsilon _T }{1-\varepsilon _T }\cdot C_T \cdot N_T ^{\frac{1}{1-\varepsilon _T }}-\rho \end{aligned}$$

(61)

From the innovation possibility frontier we obtain

$$\begin{aligned} \frac{\dot{N}_T }{N_T }=\eta \cdot \left[ L-C_T \cdot N_T ^{\frac{1}{1-\varepsilon _T }}\right] \Rightarrow -\frac{2-\varepsilon _T }{1-\varepsilon _T } \frac{\dot{N}_T }{N_T }=-\frac{2-\varepsilon _T }{1-\varepsilon _T }\eta \cdot \left[ {L-C_T \cdot N_T ^{\frac{1}{1-\varepsilon _T }}} \right] . \end{aligned}$$

(62)

Substituting (61)–(62) into (60) obtains the growth rate of IT product consumption in the social optimal growth path (superscript “S” corresponds to the social optimum.)

$$\begin{aligned} \frac{\dot{C}_T ^{S}(t)}{C_T ^{S}(t)}=\frac{\eta L}{\varepsilon _T -1}-\rho . \end{aligned}$$

(63)

Furthermore, based on the IT market clearing condition $C_T (t)=N_T (t)^{\frac{1}{\varepsilon _T -1}}\cdot L_{TP} (t)$, we derive the growth rate of IT technology variety in the social optimal growth path as

$$\begin{aligned} \frac{\dot{N}_T ^{S}(t)}{N_T ^{S}(t)}=(\varepsilon _T -1)\cdot \frac{\dot{C}_T ^{S}(t)}{C_T ^{S}(t)}=(\varepsilon _T -1)\cdot \left( {\frac{\eta L}{\varepsilon _T -1}-\rho } \right) =\eta L-(\varepsilon _T -1)\rho \end{aligned}$$

(64)

Substituting (64) into the IPF derive the amount of workforce allocated for R&D and production in IT sector,

$$\begin{aligned} L_{TR} ^{S}(t)=\frac{1}{\eta }\cdot \frac{\dot{N}_T ^{S}(t)}{N_T ^{S}(t)}=L-(\varepsilon _T -1)\eta ^{-1}\rho , L_{TP} ^{S}(t)=L-L_{TR} ^{S}(t)=(\varepsilon _T -1)\eta ^{-1}\rho \nonumber \\ \end{aligned}$$

(65)

$\square$

Appendix 5: Proof of Proposition 4

Comparing (24) with (28), we obtain that the socially optimal growth rate of technology variety in the energy sector is always greater than that in the market equilibrium. Meanwhile, comparing (23) with (29), we obtain that the socially optimal growth rate of technology variety in the IT sector is always greater than that in the market equilibrium. Furthermore, we compare the growth rates of energy technology variety with that of IT variety in the optimal growth path,

$$\begin{aligned} g_E ^{S}(t)=\eta \cdot L\cdot \varepsilon _E ^{-1}\cdot \left( 1-\varepsilon _E ^{-1}\right) ^{-\varepsilon _E }-\rho , g_T ^{S}(t)=\eta L-(\varepsilon _T -1)\rho , \end{aligned}$$

where $g_E ^{S}(t)\equiv \dot{N}_E ^{S}(t)/N_E ^{S}(t)$, $g_T ^{S}(t)\equiv \dot{N}_T ^{S}(t)/N_T ^{S}(t)$ denotes the optimal growth rate of energy and IT technology variety, respectively. The comparison thus boils down to

$$\begin{aligned} g_E ^{S}(t)-g_T ^{S}(t)=\eta \cdot L\cdot \left[ \varepsilon _E ^{-1}\cdot \left( 1-\varepsilon _E ^{-1}\right) ^{-\varepsilon _E }-1\right] +(\varepsilon _T -2)\cdot \rho , \end{aligned}$$

(66)

Given that the amount of workforce allocated to production in IT sector is less than the workforce endowment available in IT sector, $L_{TP} ^{S}(t)=(\varepsilon _T -1)\rho \eta ^{-1}<L$. Substituting it into (66) obtains

$$\begin{aligned}&\eta \cdot L\cdot \left[ \varepsilon _E ^{-1}\cdot \left( 1-\varepsilon _E ^{-1}\right) ^{-\varepsilon _E }-1\right] \nonumber \\&\quad +\left( \varepsilon _T -2\right) \cdot \rho >\rho \cdot \left[ (\varepsilon _T -1)\cdot \left[ \varepsilon _E ^{-1}\cdot \left( 1-\varepsilon _E ^{-1}\right) ^{-\varepsilon _E }-1\right] +(\varepsilon _T -2)\right] , \end{aligned}$$

(67)

whether the right-hand side of (67) is positive (social optimal growth rate of energy technology variety is larger than that of IT variety) boils down to

$$\begin{aligned} \varepsilon _E ^{-1}\cdot (1-\varepsilon _E ^{-1})^{-\varepsilon _E }-1>\frac{2-\varepsilon _T }{\varepsilon _T -1} \Rightarrow \left( {\frac{\varepsilon _E -1}{\varepsilon _E }} \right) ^{-\varepsilon _E }>\left( {\frac{\varepsilon _T -1}{\varepsilon _E }} \right) ^{-1}. \end{aligned}$$

(68)

Given that the elasticity of substitution between primary energy inputs $\varepsilon _E$ is sufficiently greater than the elasticity of substitution between IT consumer products $\varepsilon _T$, (68) does not necessarily hold, that is, the social planner solution can’t achieve an outcome in which energy technology variety grows faster than IT variety.