How ongoing structural change creates a double dividend: outdating of technologies and green growth

Abstract

In this paper we study how the ongoing transition to newer technologies, what we refer to as dynamic structural change, in a doubly-differentiated R&D-based economy can generate both higher economic growth and a slowdown of climate change, thus, creating a win-win situation known as double dividend. We first consider the unregulated decentralised economy and show that ongoing structural change can lead to lower environmental damages than the economy with a fixed structure. Next, we consider an emission tax differentiated across sectors which raises economic growth and reduces emissions in the economy. Our results indicate that promoting transition to newer technologies indeed may serve as a partial substitute to conventional abatement policy options.

Appendices

Appendix

Households derivations

The objective function of the household is

$$\begin{aligned} J^H=\int \limits _{0}^{\infty }{} \mathbf{e} ^{-\rho t}U(C) dt\;. \end{aligned}$$

(61)

with $U(C)=\ln C$ being the utility function from composite consumption C consisting of the continuum of products,

$$\begin{aligned} C= \left[ \int _{N_{min}}^{N_{max}} C_i^\frac{\varepsilon -1}{\varepsilon }di \right] ^\frac{\varepsilon }{\varepsilon -1}\;. \end{aligned}$$

(62)

with $1<\varepsilon <\infty$ being the elasticity of substitution between goods. All these goods are choice variables. The dynamic constraint is (86). We use the Lagrangian multiplier method to derive optimal $C_{i}$ (this is equivalent to Hamiltonian approach in our case).

The Lagrangian of the household is

$$\begin{aligned} L= \left[ \int _{N_{min}}^{N_{max}} C_i^\frac{\varepsilon -1}{\varepsilon }di \right] ^\frac{\varepsilon }{\varepsilon -1} - \lambda \left( \int _{N_{min}}^{N_{max}} P_i C_i di - ra + \dot{a} + L \right) \;. \end{aligned}$$

(63)

The first order condition for consumption good i is

$$\begin{aligned} C_i^{-\frac{1}{\varepsilon }} C^{\frac{1}{\varepsilon }} = \lambda P_i\;. \end{aligned}$$

(64)

Taking the F.O.C. for i and for j and substituting in yields

$$\begin{aligned} C_i = C_j \left( \frac{P_i}{P_j}\right) ^{-\varepsilon }\;. \end{aligned}$$

(65)

Substituting this back into the equation for expenditure, Equation (6) yields

$$\begin{aligned} C_j \left( \frac{1}{P_j}\right) ^{-\varepsilon } \int _{N_{min}}^{N_{max}} P_i^{1-\varepsilon }di = E\;, \end{aligned}$$

(66)

which can be rearranged to yield

$$\begin{aligned} C_i = E \frac{P_i^{-\varepsilon }}{\int _{N_{min}}^{N_{max}} P_j^{1-\varepsilon }dj}\;. \end{aligned}$$

(67)

and the standard Euler equation implies that the optimal growth rate for expenditure is given by

$$\begin{aligned} \frac{\dot{E}}{E} = r-\rho \;. \end{aligned}$$

(68)

Production sector derivations

Output of good i is given by (10):

$$\begin{aligned} Y_i=\frac{1}{1+T}A_{i}^{\alpha } L^{Y}_{i}\;. \end{aligned}$$

(69)

where $0<\alpha <1$ is the productivity of technology in production. The maximization problem of firm i is

$$\begin{aligned} \max _{P_{i}}\Pi _i = P_i Y_i - L^{Y}_i - \Psi \;, \end{aligned}$$

(70)

where $\Psi$ is a fixed operating cost (in labour units).

The only use for output of all goods i is consumption, so that $C_i=Y_i$. The only product used for investments is financial capital a and is excluded from this spectrum.

The output by an individual firm $Y_i$ equals to the consumption of that good $C_i$, so that we can insert Equation (67) into the profit function:

$$\begin{aligned} \Pi _i= & {} P_i Y_i - L^{Y}_i - \Psi \end{aligned}$$

(71)

$$\begin{aligned}= & {} P_i Y_i -Y_i A_i^{-\alpha }(1+T) - \Psi \end{aligned}$$

(72)

$$\begin{aligned}= & {} P_iE \frac{P_i^{-\varepsilon }}{\int _{N_{min}}^{N_{max}} P_j^{1-\varepsilon }dj} - E(1+T) \frac{P_i^{-\varepsilon }}{\int _{N_{min}}^{N_{max}} P_j^{1-\varepsilon }dj} A_i^{-\alpha } - \Psi \end{aligned}$$

(73)

The baseline model is obtained by letting $T=0$ above.

We use further the assumption of zero mass of each individual product in the price index, (9) which is usual when the continuum of goods is employed.

Maximizing profit with respect to the price under this non-atomic assumption yields

$$\begin{aligned} \frac{\partial \Pi _i}{\partial P_i} = \frac{E}{\int _{N_{min}}^{N_{max}} P_j^{1-\varepsilon }dj} (1-\varepsilon ) P_i^{-\varepsilon } -(1+T) \frac{E}{\int _{N_{min}}^{N_{max}} P_j^{1-\varepsilon }dj} P_i^{-\varepsilon -1} (-\varepsilon ) A_i^{-\alpha } = 0\;. \end{aligned}$$

(74)

The price is thus

$$\begin{aligned} P_i = \frac{\varepsilon }{\varepsilon -1} (1+T)A_i^{-\alpha }\;. \end{aligned}$$

(75)

All products out of the range $N_{max}-N_{min}$ have zero prices:

$$\begin{aligned} P_{i}={\left\{ \begin{array}{ll} 0,\,t<t_{max}(i), t_{max}(i): \Pi _i= 0, \dot{\Pi _i}>0;\\ (1+T)\frac{\varepsilon }{\varepsilon -1} A_i^{-\alpha },\, t_{max}(i)<t\le t_{min}(i),t_{min}(i): \Pi _i=0, \dot{\Pi _i}<0;\\ 0, t>t_{min}(i). \end{array}\right. } \end{aligned}$$

(76)

Labour employed in sector i is thus a sum of fixed costs and a function of the relative productivity of labour in sector i. Repeating arguments being made for the price formation we have piecewise-defined labour demand:

$$\begin{aligned} L^{D}(i)={\left\{ \begin{array}{ll} 0,\, t<t_{max}(i),t_{max}(i): \Pi _i= 0, \dot{\Pi _i}>0;\\ \frac{\varepsilon -1}{\varepsilon }E\frac{A_{i}^{-\alpha (1-\varepsilon )}}{\int \limits _{{N_{min}}}^{N_{max}}A_{j}^{-\alpha (1-\varepsilon )}dj}+\Psi ,\, t_{max}(i)<t\le t_{min}(i),t_{min}(i): \Pi _i=0, \dot{\Pi _i}<0;\\ 0, t>t_{min}(i). \end{array}\right. } \end{aligned}$$

(77)

where $\Psi$ is production-independent part of labour demand.

The patent for each new technology has a price

$$\begin{aligned}&p_{A}(i)\overset{def}{=}\int \limits _{t_{max}}^{t_{min}}\mathrm {e}^{-r(t-t_{max})}\Pi _{i}dt. \end{aligned}$$

(78)

which does not depend on time (only on integration limits). To see this, substitute for profits from (71,72,73), prices from (76), labour from (77). All these are functions of time. However once we integrate over the time period $t_{max}$ to $t_{min}$ the patent price becomes a function of $t_{max}(i), t_{min}(i)$ only and thus for each i it is a given quantity, albeit varying across technologies.

The markets clearing implies constant expenditures as long as $\mathcal {O}$ is constant:

$$\begin{aligned} E(t) &=& \int \limits _{N_{min}}^{N}P(i,t)C(i,t)di{\mathop {=}\limits ^{(76)}}\int \limits _{N_{min}}^{N_{max}}(1+T)\frac{\varepsilon }{\varepsilon -1} A_i^{-\alpha }Y(i,t)di{\mathop {=}\limits ^{(10)}}\nonumber \\ &= & {} \frac{\varepsilon }{\varepsilon -1}\int \limits _{N_{min}}^{N_{max}}L^{Y}(i,t)di= \frac{\varepsilon }{\varepsilon -1}(L-\mathcal {O}\Psi ). \end{aligned}$$

(79)

R&D derivations

The incentive for horizontal innovation is the potential profit from selling the (improved) technology to manufacturing firms. Thus, the value of horizontal R&D consists solely in expected future profits from vertical innovations:

$$\begin{aligned} V_{N}=\max _{u(\cdot )}\int \limits _{0}^{\infty }\mathrm {e}^{-rt}\left( \delta \pi ^{R}(i)|_{i=N} u(t)-\frac{1}{2}u^{2}(t)\right) dt \end{aligned}$$

(80)

Here profit of developing next technology $i=N$ is the value of vertical innovation into technology i, which is given by:

$$\begin{aligned} \pi ^{R}(i)|_{i=N}=p_{A}(N)-\frac{1}{2}\int \limits _{t_{0}(N)}^{t_{min}(N)}\mathrm {e}^{-r(t-t_{0})}g^{2}(N,t)dt. \end{aligned}$$

(81)

with g(N, t) being investments costs of developing technology N during the patent duration.

The HJB equation for the problem given by (80), (17) is:

$$\begin{aligned} rV=\max _{u(\cdot )}\left\{ \delta \pi ^{R}(i)|_{i=N} u(t)-\frac{1}{2}u^{2}(t)+\frac{\partial V}{\partial N}\delta u(t)\right\} . \end{aligned}$$

(82)

Taking F.O.C. we have

$$\begin{aligned} u^{*}=\delta \pi ^{R}(i)|_{i=N}+\delta \frac{\partial V}{\partial N}. \end{aligned}$$

(83)

Substituting back into the HJB equation, we find that it can be satisfied only for $V=const$, as long as $\pi ^{R}(i,t)|_{i=N}$ is constant.

This last has to be constant, since there is a free entry condition for vertical innovations: if some of the technologies yielded higher profits, all of the resources would go into the development of only those more profitable technologies. However, the investments are symmetric, thus, requiring constant and equal profits across technologies. Hence, we have

$$\begin{aligned} u^{*}=\delta \pi ^{R}(i)|_{i=N}=\delta \pi ^{R} \end{aligned}$$

(84)

yielding (28).

The profit of the single R&D firm developing technology i is:

$$\begin{aligned} \pi ^{R}(i)=p_{A}(i)-\frac{1}{2}\int \limits _{t_{0}}^{t_{min}}\mathrm {e}^{-r(t-t_{0})}g^{2}(i,t)dt. \end{aligned}$$

(85)

with investments going into increase of productivity:

$$\begin{aligned} \dot{A}(i,t)=\gamma g(i,t)-\beta A(i,t). \end{aligned}$$

(86)

where $\gamma$ is the efficiency of investments into productivity increase (equal for all sectors) and $\beta$ is the cost of supporting the productivity on the current level (this abstracts infrastructure and human capital, required for reproduction of the current level of technology).

The aggregate problem for vertical R&D reads:

$$\begin{aligned} V=\max _{g}\int \limits _{0}^{\infty }e^{-rt}dt\int \limits _{N_{min}(t)}^{N(t)}p_{A}(i)di-\int \limits _{0}^{\infty }e^{-rt}\int \limits _{N_{min}(t)}^{N(t)}\frac{1}{2}g^{2}(i,t)di dt;\end{aligned}$$

(87)

$$\begin{aligned} s.t.\end{aligned}$$

(88)

$$\begin{aligned} \forall i\in [N_{min},N]\subset \mathbb {R}_{+}: \dot{A}(i,t)=\gamma g(i,t)-\beta A(t)\end{aligned}$$

(89)

$$\begin{aligned} \int \limits _{N_{min}(t)}^{N(t)}g(i,t)di=a(t)-u{\mathop {=}\limits ^{def}}G(t). \end{aligned}$$

(90)

Implementing Maximum Principle approach, we derive optimal investments for each R&D firm as a function of shadow costs of investments, $\psi (i,t)$, this last being the function of price of the patent:

$$\begin{aligned}&\dot{\psi }(i,t)=r\psi (i,t)-\frac{\partial p_{A}(i)}{\partial A(i)},\nonumber \\&g^{*}(i,t)=\gamma \psi (i,t)-\frac{\int \limits _{N_{min}(t)}^{N(t)}\gamma \psi (i,t)di-G(t)}{N(t)-N_{min}(t)}. \end{aligned}$$

(91)

it may be demonstrated, that shadow costs of investments are the same across all existing technologies:

$$\begin{aligned} \psi ^{*}(i,t)=\psi ^{*}=\frac{\frac{\partial p_{A}(i)}{\partial A(i)}}{r+\beta } \end{aligned}$$

(92)

with $\frac{\partial p_{A}(i)}{\partial A(i)}=const$ by homogeneity of technologies (see Bondarev and Greiner (2019) for detailed proof). Combining (91) and (92) yields (35).

Dynamics of productivities themselves differ by the depreciation rate:

$$\begin{aligned} \dot{A}(i,t)=\frac{a(t)-u}{N(t)-N_{min}(t)}-\beta A(i,t). \end{aligned}$$

(93)

with u given by (28) and a(t) being the solution of (38).

Constant core derivation

The fact that the size of the economy is constant follows from definitions of $N_{min}$, $N_{max}$

$$\begin{aligned}&N_{min}(t):\, \frac{1}{\varepsilon -1}(L-\mathcal {O}\Psi )\frac{A_{N_{min}}^{\alpha (\varepsilon -1)}}{\int \limits _{{N_{min}}}^{N_{max}}A_{j}^{\alpha (\varepsilon -1)}dj}-\Psi =0; \end{aligned}$$

(94)

and

$$\begin{aligned}&N_{max}(t): \frac{1}{\varepsilon -1}(L-\mathcal {O}\Psi )\frac{A_{N_{max}}^{\alpha (\varepsilon -1)}}{\int \limits _{{N_{min}}}^{N_{max}}A_{j}^{\alpha (\varepsilon -1)}dj}-\Psi =0; \end{aligned}$$

(95)

Now observe that maximum profit for any sector i is reached at the point of $\dot{\Pi }(i)=0$. Then it follows, that growth of $N_{min}$ and $N_{max}$ is equal:

$$\begin{aligned} \dot{\Pi }(i)=0 \Leftrightarrow \left( \frac{\dot{A}(i,t)}{A(i,t)}-\int \limits _{N_{min}}^{N_{max}}\frac{\dot{A}(j,t)}{A(j,t)}dj\right) =\frac{\Psi }{\alpha L} (\dot{N}_{max}-\dot{N}_{min}) \end{aligned}$$

(96)

However, bracket in the lefthandside has to be equal to zero, since it equalizes growth rate of productivity of sector i and average growth rate of productivity in the economy. Since all the technologies are symmetric except for the time of their invention, it is straightforward to say that maximum profit for the given industry is reached at the point where its productivity grows at the average rate of the economy. Otherwise there will be still room for improvements of technology or the technology is already overdeveloped. From this it follows that $\dot{N}_{max}-\dot{N}_{min}=0$ yielding constant range of operational sectors in the economy, $\mathcal {O}=const$.

Growth rate of the benchmark economy

We start with the observation that (94) and (95) lead to

$$\begin{aligned} \forall i\in [N_{min},N_{max}]:\frac{A(i,t_{max}(i))^{\alpha (\varepsilon -1)}}{A(i,t_{min}(i))^{\alpha (\varepsilon -1)}}=\frac{{\int \limits _{N_{min}(t_{max}(i))}^{N_{max}(t_{max}(i))}A(j,t_{max}(i))}^{\alpha (\varepsilon -1)}dj}{{\int \limits _{N_{min}(t_{min}(i))}^{N_{max}(t_{min}(i))}A(j,t_{min}(i))}^{\alpha (\varepsilon -1)}dj}. \end{aligned}$$

(97)

Thus all technologies’ productivities grow at the same average speed during the time period of operational activity of the technology,

$$\begin{aligned} \forall i \in [N_{min},N_{max}], \forall t\in [t_{max}(i),t_{min}(i)]:\dot{A}=\frac{a-u}{N-N_{min}}. \end{aligned}$$

(98)

To obtain output growth rate consider aggregate output:

$$\begin{aligned} Y=\int \limits _{N_{min}}^{N_{max}}\left( \frac{A(i,t)^{\alpha \varepsilon }}{\int \limits _{N_{min}}^{N_{max}}A(j,t)^{\alpha (\varepsilon -1)}}\right) di=\frac{\int \limits _{N_{min}}^{N_{max}}A(i,t)^{\alpha \varepsilon }di}{\int \limits _{N_{min}}^{N_{max}}A(j,t)^{\alpha (\varepsilon -1)}dj} \end{aligned}$$

(99)

The direct calculation of output growth rates from (99) yields with the help of (98):

$$\begin{aligned}&\frac{\dot{Y}}{Y}=\frac{\dot{\left( \int \limits _{N_{min}}^{N_{max}}A(i,t)^{\alpha \varepsilon }di\right) }}{\int \limits _{N_{min}}^{N_{max}}A(i,t)^{\alpha \varepsilon }di}-\frac{\dot{\left( \int \limits _{N_{min}}^{N_{max}}A(j,t)^{\alpha (\varepsilon -1)}dj\right) }}{\int \limits _{N_{min}}^{N_{max}}A(j,t)^{\alpha (\varepsilon -1)}dj}{\mathop {=}\limits ^{\dot{N}_{min}=\dot{N}_{max}}}\alpha \frac{a-u}{N-N_{min}}\frac{N_{max}-N_{min}}{\bar{A}}\nonumber \\&= \alpha \frac{\dot{\bar{A}}}{\bar{A}}(N_{max}-N_{min})>0. \end{aligned}$$

(100)

Proof of Lemma 3

Total expenditures of households are unchanged for given $N_{min},N_{max}$:

$$\begin{aligned} E^{\tau }=\int _{N^{\tau }_{min}}^{N^{\tau }_{max}}P^{\tau }_{i}C^{\tau }_{i}di=\frac{\varepsilon }{\varepsilon -1}\int _{N^{\tau }_{min}}^{N^{\tau }_{max}}L^{Y}_{i}di \end{aligned}$$

(101)

Tax revenues coming to households are:

$$\begin{aligned} R{\mathop {=}\limits ^{def}}\int _{N^{\tau }_{min}}^{N^{\tau }_{max}}\frac{1}{i}P^{\tau }_{i}Y^{\tau }_{i}di{\mathop {=}\limits ^{(52),(53)}}\frac{\epsilon }{\epsilon -1}\int _{N^{\tau }_{min}}^{N^{\tau }_{max}}\frac{1}{i}L^{Y}_{i}di \end{aligned}$$

(102)

Rearranging terms we get

$$\begin{aligned} R-E^{\tau }=\frac{\epsilon }{\epsilon -1}\int _{N^{\tau }_{min}}^{N^{\tau }_{max}}\frac{1-i}{i}L^{Y}_{i}di \end{aligned}$$

(103)

Further using

$$\begin{aligned}&\int _{N^{\tau }_{min}}^{N^{\tau }_{max}}L^{\tau }_{i}di=\int _{N^{\tau }_{min}}^{N^{\tau }_{max}}\frac{i-1}{i}L^{Y}_{i}di=\nonumber \\&\frac{\epsilon }{\epsilon -1}\int _{N^{\tau }_{min}}^{N^{\tau }_{max}}\frac{i-1}{i}L^{Y}_{i}di-\frac{1}{\epsilon -1}\int _{N^{\tau }_{min}}^{N^{\tau }_{max}}\frac{i-1}{i}L^{Y}_{i}di \end{aligned}$$

(104)

and accounting for (52), (53) we get from (59)

$$\begin{aligned} \dot{a}^{\tau }=ra^{\tau }-\frac{1}{\epsilon -1}\int _{N^{\tau }_{min}}^{N^{\tau }_{max}}\frac{i-1}{i}L^{Y}_{i}di \end{aligned}$$

(105)

Observe now that both (103) and the second term of (105) are positive or negative simultaneously depending on whether $N_{max}\le 1$ (both positive) or $N_{min}\ge 1$ (both negative).

It remains to observe that since both $N_{min}, N_{max}$ are monotonic, these are exactly the only two regimes for assets dynamics.

Since the size of the economy increases in time, the actual expenditures change by

$$\begin{aligned} \Delta E= E^{\tau }-E=\frac{\varepsilon }{\varepsilon -1}\left( \int \limits _{N^{\tau }_{min}}^{N^{\tau }_{max}}L^{Y}_{i}-\int _{N_{min}}^{N_{max}}L^{Y}_{i}di\right) >0 \end{aligned}$$

(106)

because the size of the core increases and $N^{\tau }_{max}>N_{max},N^{\tau }_{min}>N_{min}$ holds.