Appendices
A: Laissez-Faire Equilibrium
A complete derivation of the laissez-faire equilibrium is provided in the Online Appendix. In this section, we summarize the main equilibrium results. To solve the model, first we use the profit maximization of final goods producers Y to get the demand for intermediates \(Q_z\). Second, the profit maximization of intermediate goods producers generates a demand for machines, which leads to the price for machines, their optimal quantity and the profits that can be realized if a scientist successfully patents an innovation on a given machine. Third, we obtain factor demands for L, K and R and express intermediate products’ prices \(p_z\) as functions of factors’ prices. Lastly, combining all results from the three levels of optimization above, we obtain the equilibrium conditions as functions of innovative technologies A, as follows
$$\begin{aligned} (p_{eC}^*)^{\phi }= & {} \left( \frac{1}{\psi }\right) ^{\frac{\psi }{1-\psi }}(X_\psi ) \frac{\left[ \left( A_{n}^k\right) ^\psi \left( A_{n}^{(k-1)}\right) ^\psi \right] ^{\phi }}{\left[ A_{eC}^k\left( A_{n}^{(k-1)}\right) ^\psi \right] ^{\phi }\overset{\_}{K}^k + \left[ A_{eC}^{(k-1)}\left( A_{n}^k\right) ^\psi \right] ^{\phi }\overset{\_}{K}^{(k-1)}} \end{aligned}$$
(19)
$$\begin{aligned} (p_{eF}^*)^{\frac{1}{1-\gamma }}= & {} \left( \frac{1}{\beta }\right) ^{\beta } (X_\beta )^{1-\beta } \left( \frac{(A_n^S)^\beta }{A_{eF}^S}\right) ^{\frac{1}{1-\gamma }} \left( \frac{1}{\overset{\_}{R}^{S}}\right) ^{\alpha } \left( \frac{1}{\overset{\_}{K}^S}\right) ^{1-\alpha -\beta } \end{aligned}$$
(20)
where \(\phi \equiv \frac{1}{(1-\psi )(1-\gamma )}\), \(X_\psi \equiv \frac{1-\upsilon }{\upsilon + \psi (1-\upsilon )} \left( (A_n^N)^{\frac{1}{1-\gamma }}\overset{\_}{L}^{N} + (A_n^S)^{\frac{1}{1-\gamma }} \overset{\_}{L}^{S}\right)\) and \(X_\beta \equiv \frac{(1-\upsilon )}{\upsilon +\beta (1-\upsilon )} \left( (A_n^N)^{\frac{1}{1-\gamma }}\overset{\_}{L}^{N} + (A_n^S)^{\frac{1}{1-\gamma }} \overset{\_}{L}^{S}\right)\), and the price of non-energy goods is set as the numeraire, \(p_n\equiv 1\). If fossil fuels are being produced, the equilibrium factor demands are
$$\begin{aligned} L_{eC}^{S*}= & {} L_{eC}^{N*}=0 \;\;\;\;\;\;\;\; L_{n}^{N*}=\overset{\_}{L}^{N} \;\;\;\;\;\;\;\;L_{n}^{S*}= \overset{\_}{L}^{S} - L_{eF}^{S*} \end{aligned}$$
(21)
$$\begin{aligned} L_{eF}^{S*}= & {} min \;\; \left\{ \overset{\_}{L}^{S}\;\; ; \;\; \frac{\beta (1-\upsilon )}{\upsilon + \beta (1-\upsilon )} \frac{(A_n^N)^{\frac{1}{1-\gamma }}\overset{\_}{L}^{N} + (A_n^S)^{\frac{1}{1-\gamma }} \overset{\_}{L}^{S}}{{A_{n}^S}^\frac{1}{1-\gamma }} \right\} \end{aligned}$$
(22)
$$\begin{aligned} K_{eF}^{S*}= & {} \overset{\_}{K}^{S} \;\;\;\;\;\;\;\; K_{eC}^{S*}=K_{eC}^{N*}=0 \;\;\;\;\;\;\;\; R_{eF}^{S*}=\overset{\_}{R}^S \end{aligned}$$
(23)
and the optimal amount of fossil fuel intermediates produced is
$$\begin{aligned} Y^{*S}_{eF}=\frac{A_{eF}}{\left( A_n^{S}\right) ^\beta }\;\;\left( X_{\beta }\right) ^{\beta +\gamma (1-\beta )}\;\;\; \left[ \beta ^{\beta } \left( \overset{\_}{K}^{S}\right) ^{(1-\alpha -\beta )} \left( \overset{\_}{R}^S\right) ^{\alpha }\right] ^{1-\gamma } \end{aligned}$$
(24)
If instead fossil fuels are not being produced, the equilibrium values are
$$\begin{aligned} L_{eF}^{S*}= & {} 0 \;\;\;\;\;\;\;\; L_{n}^{k*}=\overset{\_}{L}^k-L_{eC}^{k *} \end{aligned}$$
(25)
$$\begin{aligned} L_{eC}^{k*}= & {} min \left\{ \overset{\_}{L}^k ; \frac{\psi (1-\upsilon )}{\upsilon + \psi (1-\upsilon )} \frac{\left( (A_n^k)^{\frac{1}{1-\gamma }}\overset{\_}{L}^{k} + (A_n^{k-1})^{\frac{1}{1-\gamma }} \overset{\_}{L}^{k-1}\right) \left( A_{eC}^k \left( A_n^{(k-1)}\right) ^\psi \right) ^{\phi } \overset{\_}{K}^k}{\left( A_n^k\right) ^{\frac{1}{1-\gamma }} \left( \left[ A_{eC}^k\left( A_{n}^{(k-1)}\right) ^\psi \right] ^{\phi }\overset{\_}{K}^k + \left[ A_{eC}^{(k-1)}\left( A_{n}^k\right) ^\psi \right] ^{\phi } \overset{\_}{K}^{(k-1)}\right) } \right\} \end{aligned}$$
(26)
$$\begin{aligned} K_{eF}^{S*}= & {} 0 \;\;\;\;\;\;\;\;K_{eC}^{k*}=\overset{\_}{K}_{eC}^{k} \;\;\;\;\;\;\;\; R_{eF}^{S*}= 0 \end{aligned}$$
(27)
B: Innovation Dynamics
The allocation of scientists in equilibrium depends on the expected profitability of working in a sector. Profits are zero if innovation is not successful. From the profit maximization of machine producers, we get
$$\begin{aligned} E(\pi ^k_{n, t})= & {} \gamma \left( 1-\gamma \right) L^k_{n, t}\left( p_{n, t}(1+\varphi ) A^k_{n,t-1}\right) ^\frac{1}{1-\gamma } + 0 \end{aligned}$$
(28)
$$\begin{aligned} E(\pi ^k_{eC, t})= & {} \gamma \left( 1-\gamma \right) (L^k_{eC, t})^{\psi }(K^k_{eC, t})^{1-\psi }\left( p_{eC, t}(1+\varphi ) A^k_{eC,t-1}\right) ^\frac{1}{1-\gamma } + 0 \end{aligned}$$
(29)
$$\begin{aligned} E(\pi ^S_{eF, t})= & {} \gamma \left( 1-\gamma \right) (L^S_{eF, t})^{\beta }(K^S_{eF, t})^{1-\alpha -\beta } (R^S_{t})^{\alpha } \left( p_{eF, t}(1+\varphi ) A^S_{eF,t-1}\right) ^\frac{1}{1-\gamma } + 0 \end{aligned}$$
(30)
Only Southern scientists can work in the fossil fuel sector. The ratio of expected profits in Eqs. (28) and (29) or (30) yields the ratios of Eqs. (13) and (14) in the main text. In equilibrium, they become
$$\begin{aligned} \frac{E(\pi _{n,t}^{k})}{E(\pi _{eC,t}^{k})}= & {} \frac{\left( \frac{\upsilon }{1-\upsilon }\right) f(k)_t-\psi f(k-1)_t +\left( \frac{\upsilon }{1-\upsilon }+\psi \right) \frac{g(k-1)_t}{g(k)_t}}{f(k-1)_t +f(k)_t}\left( \frac{1+\varphi (1-m_{n,t}^k)}{1+\varphi m_{n,t}^k}\right) ^{\frac{1}{1-\gamma }} \end{aligned}$$
(31)
$$\begin{aligned} \frac{E(\pi _{n,t}^{S})}{E(\pi ^S_{eF,t})}= & {} \frac{\left( \frac{\upsilon }{1-\upsilon }\right) f(S)_t-\beta f(N)_t}{f(N)_t +f(S)_t}\left( \frac{1+\varphi (1-m_{n,t}^S)}{1+\varphi m_{n,t}^S}\right) ^{\frac{1}{1-\gamma }} \end{aligned}$$
(32)
where \(f(k)_t\equiv [(1+\varphi m_{n,t}^k) A_{n,t-1}^k]^{\frac{1}{1-\gamma }} \overset{\_}{L}^{k}\) and \(g(k)_t\equiv \left[ (1+\varphi (1-m_{n,t}^k))A_{eC, t-1}^k\left( (1+\varphi m_{n,t}^{k-1})A_{n, t-1}^{k-1}\right) ^\psi \right] ^{\phi }\overset{\_}{K}^{k}\).
We then have three possible cases, depending on the effect of scientists on the ratio of profits.Footnote 23
-
1.
Profit ratio increasing in scientists—\(\partial \frac{E(\pi _n)}{E(\pi _e)}/ \partial m_n>0\). In this case, when the number of scientists in n rises, the profits of the manufacturing sector also rise relative to the energy sector. Thus, it must hold that
$$\begin{aligned} \frac{E(\pi _{n} (m_n=1))}{E(\pi _{e} ((m_n=1))}> \frac{E(\pi _{n} (m_n=0))}{E(\pi _{e} ((m_n=0))} \end{aligned}$$
(33)
The equilibrium allocations of scientists (such that none would want to move to a more profitable sector) are
-
(a)
If \(\frac{E(\pi _{n} (m_n=1))}{E(\pi _{e} ((m_n=1))}> \frac{E(\pi _{n} (m_n=0))}{E(\pi _{e} ((m_n=0))} > 1\)—the only equilibrium is \(m^*_n=1\), as manufacturing is always more profitable than energy.
-
(b)
If \(1> \frac{E(\pi _{n} (m_n=1))}{E(\pi _{e} ((m_n=1))} > \frac{E(\pi _{n} (m_n=0))}{ E(\pi _{e} ((m_n=0))}\)—the only equilibrium is \(m^*_n=0\), viceversa.
-
(c)
If \(\frac{E(\pi _{n} (m_n=1))}{E(\pi _{e} ((m_n=1))}> 1 > \frac{E(\pi _{n} (m_n=0))}{ E(\pi _{e} ((m_n=0))}\)—the equilibrium is either \(m^*_n=1\), \(m^*_n=0\), or an interior solution \(m^*_n \in (0,1)\) such that \([E(\pi _{n} (m_n=m_n^*))]/[E(\pi _{e} ((m_n=m_n^*))]=1\).
-
2.
Profit ratio decreasing in scientists—\(\partial \frac{E(\pi _n)}{E(\pi _e)}/ \partial m_n<0\). Now, the opposite of (33) must hold, so the options for equilibrium allocations are
-
(a)
If \(\frac{E(\pi _{n} (m_{n}=0))}{E(\pi _{e} ((m_{n}=0))} < 1\)—the only equilibrium is \(m_{n}^*=0\). It would not be profitable for scientist to move to manufacturing, because that would only lower the profit ratio.
-
(b)
If \(\frac{E(\pi _{n} (m_{n}=1))}{E(\pi _{e} ((m_{n}=1))} > 1\)—the only equilibrium is \(m_{n}^*=1\), opposite to the above case.
-
(c)
If neither of the two corner solutions apply, we have an interior solution such that \(\frac{E(\pi _{n} (m_n=m_n^*))}{E(\pi _{e} ((m_n=m_n^*))}=1\).
-
3.
Profit ratio independent of scientists—\(\partial \frac{E(\pi _n)}{E(\pi _e)}/ \partial m_n = 0\). If the ratio is independent of scientists, then
-
(a)
If \(E(\pi _n)< E(\pi _e)\), the only equilibrium is \(m_n^*=0\)
-
(b)
If \(E(\pi _n)> E(\pi _e)\), the only equilibrium is \(m_n^*=1\)
Applying the conditions above, we can find the equilibrium allocation of scientists, either with a corner solution if the conditions of point (1) or (2) are fulfilled, or finding an interior solution \(m^*\) if case (3) applies.Footnote 24 Rearranging the profit ratios in Eq. (32) with laissez-faire equilibrium values and the respective equilibrium scientists’ allocation, we develop the following Lemma.
Lemma 1
Under laissez-faire, it is an equilibrium for innovation in the South at time t to occur:
(1) only in the fossil fuel sector when \(E(\pi ^S_n(m_n=0))< E(\pi ^S_{eF}(m_n=0))\) , so that \(m_n^*=0\), if
$$\begin{aligned} \left[ \frac{A_{n,t-1}^S}{(1+\varphi m_{n,t}^N)A^N_{n,t-1}} \right] ^{\frac{1}{1-\gamma }}\frac{{\bar{L}}^S}{{\bar{L}}^N} < \frac{1+\beta (1+\varphi )^{\frac{1}{1-\gamma }}}{\frac{\upsilon }{1-\upsilon }(1-\varphi )^{\frac{1}{1-\gamma }-1}} \end{aligned}$$
(34)
(2) only in the manufacturing sector when \(E(\pi ^S_n(m_n=1))> E(\pi ^S_{eF}(m_n=1))\), so that \(m_n^*=1\), if
$$\begin{aligned} \left[ \frac{A_{n,t-1}^S}{(1+\varphi m_{n,t}^N) A^N_{n,t-1}}\right] ^{\frac{1}{1-\gamma }}\frac{{\bar{L}}^S}{{\bar{L}}^N} > \frac{1+\beta (1+\varphi )^{\frac{1}{\gamma -1}}}{\frac{\upsilon }{1-\upsilon }(1-\varphi )^{\frac{1}{\gamma -1}}-1} \end{aligned}$$
(35)
(3) in both sectors when \(E(\pi ^S_n({\tilde{m}}))=E(\pi ^S_{eF}({\tilde{m}}))\)(so when Eq. (32equals to 1), such that \(m_n^*={\tilde{m}}\).
Thus the following condition ensures that all Southern scientists work in the fossil fuel industry
$$\begin{aligned} \left[ \frac{A_{n,t-1}^S}{(1+\varphi )A^N_{n,t-1}} \right] ^{\frac{1}{1-\gamma }}\frac{{\bar{L}}^S}{{\bar{L}}^N} < \frac{1+\beta (1+\varphi )^{\frac{1}{1-\gamma }}}{\frac{\upsilon }{1-\upsilon }(1-\varphi )^{\frac{1}{1-\gamma }-1}} \end{aligned}$$
(36)
which corresponds to the initial condition at time zero imposed in Assumption 2.
If the fossil fuel sector is the most profitable innovation activity in the first period, as imposed in Assumption 2, then it necessarily remains the most profitable also in the next period if fossil fuels are still produced, because with this scientists’ allocation \(A_{n, t}^S=A_{n, t-1}^S\) and \(A_{n, t}^N\ge A_{n, t-1}^S\), and thus Eq. (36) keeps being valid.
C: Regularity Condition: Initial Prices
According to Assumption 1, in the first period under fossil fuels are cheaper than renewable resources, so \(\frac{p_{eF, t=1}^*}{p_{eC, t=1}^*}<1\). This requires that fossil fuel technologies have an initial advantage such that, rearranging the equilibrium prices (19) and (20), the following condition holds
$$\begin{aligned} \dfrac{(A_{eF, t=1}^{S}) ^\phi }{\Omega } > \dfrac{ \left( \dfrac{\left( A_{n,t=1}^S\right) ^{\beta }}{(A_{n,t=1}^N)^\psi } \; A_{eC,t=1}^N\right) ^\phi \overset{\_}{K}^N + \left( \dfrac{\left( A_{n,t=1}^S\right) ^{\beta }}{(A_{n,t=1}^S)^\psi }\; A_{eC,t=1}^S\right) ^\phi \overset{\_}{K}^S}{\left[ (A_{n,t=1}^N)^{\frac{1}{1-\gamma }}\overset{\_}{L}^{N} + (A_{n,t=1}^S)^{\frac{1}{1-\gamma }} \overset{\_}{L}^{S}\right] ^{\frac{\beta - \psi }{1-\psi }} } \;\; \forall m^k_1\in [0,1] \end{aligned}$$
(37)
where for brevity we write the A technologies at time \(t=1\), but each of them contains the initial technology \(A_{t=0}\) and the contemporaneous allocation of scientists, namely \(A_{z, t=1}^k= (1+\varphi m_{z,t=1}^k) A_{z, t=0}^k\). Omega is a time-invariant component such that \(\Omega ^{1-\psi } \; \equiv \frac{\left[ \psi (1-\upsilon )\right] ^\psi }{\left[ \beta (1-\upsilon )\right] ^\beta } \; \frac{\left[ \upsilon +\psi (1-\upsilon )\right] ^{1-\psi }}{\left[ \upsilon +\beta (1-\upsilon )\right] ^{1-\beta }} \frac{1}{{\left( \overset{\_}{K}^S\right) ^{1-\alpha -\beta } \left( \overset{\_}{R}^S\right) ^\alpha }}\). A sufficiently large endowment of fossil fuel deposits in the South, R, or a sufficiently advanced fossil fuel technology \(A_{eF, t=0}\) can by themselves ensure that this condition holds.
D: Proof of Proposition 1
Proof
For \(Y_{eF}\) to be strictly increasing in time, we must show that \(p_{eC, t} >p_{eF, t} \;\; \forall t\), so that fossil fuels are always preferred to renewable energy intermediates, and that \(Y_{eF, t+1} > Y_{eF,t} \; \forall t\), so that fossil fuels increase every period. For the prices, we know from Assumption 1 that in the first period \(p_{eC, t=1}/p_{eF, t=1}>1\). From Assumption 2, we also know that, if fossil fuels are the cheapest input, the equilibrium allocation of scientists is \(m_n^S=0\) and \(m_{eF}^S=1\). Hence, only fossil fuel technology \(A_{eF}^S\) grows in the South. In the North, since there is no active energy sector, all scientists are in manufacturing (\(m_n^N=1\)) and only the \(A_n^N\) technology grows over time. Then, the regularity condition in Eq. (37) must hold in the subsequent period: its left-hand side grows by \((1+\varphi )^\phi\)—the increase in \(A_{eF}^S\) with all scientists in that sector, while the right-hand side falls since \(A_n^N\) increases the denominator. All other factors are constant, so the inequality holds in the following period. Iterating this process over time, fossil fuels are always cheaper than renewable energy inputs.
For the second point, using the equilibrium for fossil fuel intermediates \(Y_{eF}^*\) from Eq. (24) we show that
$$\begin{aligned} Y_{eF, t+1}= & {} \frac{(1+\varphi m_{eF, t+1}^S)A_{eF, t}}{\left[ \left( 1+\varphi m_{n, t+1}^S\right) A_{n,t}^{S} \right] ^\beta }\;\left[ \left( \left( 1+\varphi \right) A_{n, t}^N\right) ^{\frac{1}{1-\gamma }}\overset{\_}{L}^{N} + \left( \left( 1+\varphi m_{n, t+1}^S\right) A_{n, t}^S\right) ^{\frac{1}{1-\gamma }} \overset{\_}{L}^{S}\right] ^{\beta +\gamma (1-\beta )} \nonumber \\&> Y_{eF, t}= \;\;\frac{A_{eF, t}}{\left( A_{n, t}^{S}\right) ^\beta }\;\;\left[ \left( (A_{n, t}^N)^{\frac{1}{1-\gamma }}\overset{\_}{L}^{N} + (A_{n, t}^S)^{\frac{1}{1-\gamma }} \overset{\_}{L}^{S}\right) \right] ^{\beta +\gamma (1-\beta )}\;\;\; \end{aligned}$$
(38)
All scientists in the North are in manufacturing, since the energy sector is non-existent. Simplifying
$$\begin{aligned} \begin{aligned}&\left( A_{n, t}^N\right) ^{\frac{1}{1-\gamma }}\overset{\_}{L}^{N}\left[ \left( \frac{1+\varphi m_{eF, t+1}^S}{\left( 1+\varphi m_{n, t+1}^S\right) ^\beta }\right) ^{\frac{1}{{\beta +\gamma (1-\beta )}}}\left( 1+\varphi \right) ^{\frac{1}{1-\gamma }} -1\right] +\\&\left( A_{n, t}^S\right) ^{\frac{1}{1-\gamma }} \overset{\_}{L}^{S} \left[ (1+\varphi m_{eF, t+1}^S)^{\frac{1}{{\beta +\gamma (1-\beta )}}} \left( 1+\varphi m_{n, t+1}^S\right) ^{\frac{1}{1-\gamma } - \frac{\beta }{\beta +\gamma (1-\beta )}} -1 \right] > 0 \end{aligned} \end{aligned}$$
(39)
We can show that both square brackets are greater than zero, so that the following conditions hold
$$\begin{aligned} \left( \frac{(1+\varphi m_{eF, t+1}^S)}{\left[ 1+\varphi m_{n, t+1}^S\right] ^\beta }\right) ^{\frac{1}{{\beta +\gamma (1-\beta )}}}\left( 1+\varphi \right) ^{\frac{1}{1-\gamma }} > 1 \end{aligned}$$
(40)
and
$$\begin{aligned} \left( 1+\varphi m_{eF, t+1}^S\right) ^{\frac{1}{{\beta +\gamma (1-\beta )}}} \left( 1+\varphi m_{n, t+1}^S\right) ^{\frac{\gamma }{(1-\gamma )(\beta +\gamma (1-\beta ))}} > 1 \end{aligned}$$
(41)
Since \(\varphi >0\) and \(1 \ge m_{eF}, m_n \ge 0\), the terms \((1+\varphi m_{eF}^S)\ge 1\) and \((1+\varphi m_{n}^S)\ge 1\). It is not possible for both terms to be equal to one at the same time, because \(m_{eF}^S+m_n^S=1\). Moreover, since \(1>\beta , \gamma >0\), then \(\beta +\gamma (1-\beta )>0\) and all exponents are positive. The first condition (D.22) is least likely to hold when the denominator has its largest value with \(m_{n}=1\)
$$\begin{aligned} \left( \frac{1}{1+\varphi }\right) ^{\frac{\beta }{\beta +\gamma (1-\beta )}}\left( 1+\varphi \right) ^{\frac{1}{1-\gamma }} > 1 \end{aligned}$$
(42)
which holds if \(\frac{1}{1-\gamma } > \frac{\beta }{\beta +\gamma (1-\beta )}\), which is indeed the case, since their difference is \(\frac{\gamma }{(1-\gamma )(\beta +\gamma (1-\beta ))} > 0\). For any lower value of \(m_{n}^S\) the denominator is smaller, so the above condition necessarily holds. Also the second condition 41 holds, since the two terms are both positive and cannot be simultaneously equal to one, as stated before. Thus, \(Y_{eF}^S\) always strictly increases in time. \(\square\)
E: Proof of Proposition 2
Proof
To prove Proposition 2 we must show that the global use of fossil fuels under laissez-faire inevitably causes an environmental catastrophe. From Proposition 1, we know that the production of fossil fuel inputs \(Y^S_{eF, t}\) is strictly increasing over time. The environment is a decreasing function of fossil fuel production, according to Eq. (12). Since the other parameters determining the quality of the global environment \(\Delta\), \(\xi\) and \({\overline{G}}\) are fixed and of finite magnitude, while \(Y^S_{eF}\) keeps growing boundlessly, even if we start with a small amount of fossil fuels’ use, necessarily there will be a time d when the production of fossil fuels surpasses the regenerative capacity of the environment to return to its initial pristine state
$$\begin{aligned} G_{d}=\left( 1+\Delta \right) {\overline{G}} -\xi Y^S_{eF,d-1} < {\overline{G}} \end{aligned}$$
(43)
After this point, the environment starts declining more and more every year. Iteratively substituting the values of the environment over time in the above equation, with ever growing values of fossil fuels, we would get to a generic time T when \(Y_{eF}\) makes \(G_T\le 0\), namely when
$$\begin{aligned} \left( 1+\Delta \right) ^{T-d+1} {\overline{G}} \le \xi \sum _{a=d-1}^{T-1} \left( 1+\Delta \right) ^{T-a-1} Y^S_{eF,a} \end{aligned}$$
(44)
At which point the economy will be in a disaster, as per Definition D.1. \(\square\)
F: Minimum Purchase of Deposits R
The North only needs to buy the extractive rights for the minimum amount of deposits R that forces a switch away from fossil fuel production, by making renewable energy inputs less costly than fossil fuels, \(p_{eF}> {\tilde{p}}_{eC}\). Rearranging the equilibrium prices from Eqs. (20) and (19) we get that, at a given point in time, the maximum amount of R that can be left to the South is
$$\begin{aligned} \left( {\tilde{R}}^S\right) ^{\alpha } \le \dfrac{\psi ^\psi \left( X_\beta \right) ^{1-\beta }}{\beta ^\beta \left( X_\psi \right) ^{1-\psi }} \left( \dfrac{(A_{n_{(eF)}}^S)^{\beta }}{\left( A_{n_{(eC)}}^N\right) ^\psi \left( A_{n_{ (eC)}}^{S}\right) ^\psi }\right) ^{\frac{1}{1-\gamma }} \dfrac{\left( \left[ A_{eC}^N\left( A_{n_{ (eC)}}^{S}\right) ^\psi \right] ^{\phi }\overset{\_}{K}^N + \left[ A_{eC}^{S}\left( A_{n_{ (eC)}}^N\right) ^\psi \right] ^{\phi } \overset{\_}{K}^{S}\right) ^{1-\psi }}{(A_{eF}^{S})^{\frac{1}{1-\gamma }}\left( \overset{\_}{K}^S\right) ^{1-\alpha -\beta }} \end{aligned}$$
(45)
In the notation, we distinguish among the \(A_{n_{(e)}}\) technologies arising in the equilibrium with fossil fuel energy \(A_{n_{(eF)}}\) from those in the renewable energy equilibrium \(A_{n_{(eC)}}\), since they need not be equal. Again, as in Sect. C, all the A technologies are at time t (time subscript omitted for brevity), but contain the contemporaneous allocation of scientists and past technologies, namely \(A_{z, t}^k= (1+\varphi m_{z,t}^k)A_{z,t-1}\), so this condition must apply in the context of the final scientists’ equilibrium. The deposits available for production in the South must be less than \({\tilde{R}}^S\) to ensure no production of fossil fuel energy occurs. Therefore, \(R_{min}={\overline{R}}^S-{\tilde{R}}^S\).
G: Calibration
For the calibration exercise we select values as close as possible to Hémous (2016). Initial values for our simulations are based on the 2003–2007 world economy (from the UNIDO database). A standard approach is to identify Annex I countries with the NorthFootnote 25 and non-Annex I countries with the SouthFootnote 26. The energy intensive sector is identified with chemical, petrochemical, non-ferrous metals, non-metallic minerals, and iron and steel, while the manufacturing sector is identified with all other sectors. For fossil fuel resources, we use the Statistical Review of World Energy 2013, taking coal production for non-Annex I countries (million tonnes of oil equivalent) across all years under consideration. The discount rate is, as in Nordhaus (2008), 0.0015. We rely on Hémous’ calibration for the initial values of environment and productivity. The polluting factor associated with the use of fossil fuel is equalized to the polluting factor of the South in Hémous, the most polluting country in his model. Table 1 presents the full list of parameters.
Table 1 Calibration parameters