Appendix
1.1 Proof of Proposition 1
Consider two lifetime-equal histories \(\left\{ T,n,0\right\} \) and \(\left\{ T,0,m\right\} \). In history \(\left\{ T,n,0\right\} \), total welfare for all individuals born at \(t\ge 0\) can be written as:
$$\begin{aligned}&nN_{-1}T\alpha +nN_{-1}\left[ \frac{Q}{N_{-(T-1)}+ \cdots +N_{-1}(1+n)}\right] ^{\sigma } \\&\quad +\,nN_{-1}\left[ \frac{Q}{N_{-(T-2)}+ \cdots +N_{-1}(1+n+n^{2})}\right] ^{\sigma } \\&\quad + \cdots +nN_{-1}\left[ \frac{Q}{N_{-1}\left( n+n^{2}+ \cdots +n^{T+1}\right) } \right] ^{\sigma } \\&\quad +\,n^{2}N_{-1}T\alpha +n^{2}N_{-1}\left[ \frac{Q}{ N_{-(T-2)}+ \cdots +N_{-1}(1+n+n^{2})}\right] ^{\sigma } \\&\quad +\,n^{2}N_{-1}\left[ \frac{Q}{N_{-(T-3)}+ \cdots +N_{-1}\left( 1+n+n^{2}+n^{3}\right) }\right] ^{\sigma } \\&\quad + \cdots +n^{2}N_{-1}\left[ \frac{Q}{N_{-1}\left( n^{2}+n^{3}+ \cdots +n^{T+2}\right) }\right] ^{\sigma } \\&\quad + \cdots \end{aligned}$$
This can be simplified to:
$$\begin{aligned}&\sum \limits _{t=1}^{\infty }n^{t}N_{-1}T\alpha +\sum \limits _{t=1}^{\infty }n^{t}N_{-1}\left[ \sum \limits _{s=t}^{t+T-1}\left( \frac{Q}{ \sum \nolimits _{z=0}^{T-1}N_{-(T-s-z)}}\right) ^{\sigma }\right] \\&\quad =\sum \limits _{t=1}^{\infty }n^{t}N_{-1}T\alpha +\sum \limits _{t=1}^{\infty }n^{t}N_{-1}\left[ \sum \limits _{s=t}^{t+T-1} \left( \frac{Q}{\sum \nolimits _{z=0}^{T-s-1}N_{-(T-s-z)}+\left( N_{-1}\right) \sum _{r=1}^{s}n^{r}}\right) ^{\sigma }\right] \end{aligned}$$
Let us suppose, for the sake of presentation, that T is an even number. Then, in history \(\left\{ T,0,m\right\} \), total welfare can be rewritten as:
$$\begin{aligned}&mN_{-2}T\alpha +mN_{-2}\left[ \frac{Q}{N_{-(T-1)}+ \cdots +mN_{-2}}\right] ^{\sigma }\\&\quad +\,mN_{-2}\left[ \frac{Q}{N_{-(T-2)}+ \cdots +mN_{-2}+mN_{-1}}\right] ^{\sigma } \\&\quad +\,mN_{-2}\left[ \frac{Q}{N_{-(T-3)}+ \cdots +N_{-2}\left( m+m^{2}\right) +mN_{-1} }\right] ^{\sigma } \\&\quad + \cdots +mN_{-2}\left[ \frac{Q}{N_{-2}\left( m+m^{2}+ \cdots +m^{T/2}\right) +N_{-1}\left( m+m^{2}+ \cdots +m^{T/2}\right) }\right] ^{\sigma } \\&\quad +\,mN_{-1}T\alpha +mN_{-1}\left[ \frac{Q}{N_{-(T-2)}+ \cdots +mN_{-1}}\right] ^{\sigma }\\&\quad +\,mN_{-1}\left[ \frac{Q}{N_{-(T-3)}+ \cdots +N_{-2}\left( m+m^{2}\right) +mN_{-1}}\right] ^{\sigma } \\&\quad +\,mN_{-1}\left[ \frac{Q}{N_{-(T-4)}+ \cdots +N_{-2}\left( m+m^{2}\right) +N_{-1}\left( m+m^{2}\right) }\right] ^{\sigma } \\&\quad + \cdots +mN_{-1}\left[ \frac{Q}{N_{-2}\left( m^{2}+m^{3}+ \cdots +m^{T/2}\right) + \cdots +N_{-1}\left( m+m^{2}+ \cdots +m^{(T/2)+1}\right) }\right] ^{\sigma } \\&\quad +\,m^{2}N_{-2}T\alpha +m^{2}N_{-2}\left[ \frac{Q}{N_{-(T-3)}+ \cdots +N_{-2} \left( m+m^{2}\right) +mN_{-1}}\right] ^{\sigma } \\&\quad +\,m^{2}N_{-2}\left[ \frac{Q}{N_{-(T-4)}+ \cdots +N_{-2}\left( m+m^{2}\right) +N_{-1}\left( m+m^{2}\right) }\right] ^{\sigma } \\&\quad +\,m^{2}N_{-2}\left[ \frac{Q}{N_{-(T-5)}+ \cdots +N_{-2}\left( m+m^{2}+m^{3}\right) +N_{-1}\left( m+m^{2}\right) }\right] ^{\sigma } \\&\quad + \cdots +m^{2}N_{-2}\left[ \frac{Q}{N_{-2}\left( m^{2}+ \cdots +m^{(T/2)+1}\right) +N_{-1}\left( m^{2}+ \cdots +m^{(T/2)+1}\right) }\right] ^{\sigma } \\&\quad + \cdots \end{aligned}$$
This expression can be rewritten as:
$$\begin{aligned}&\sum \limits _{t=1}^{\infty }m^{t}N_{-2}T\alpha +\sum \limits _{t=1}^{\infty }m^{t}N_{-1}T\alpha \\&\quad +\,\sum \limits _{t=1}^{\infty }m^{t}N_{-2}\left[ \sum \limits _{s=t}^{t+T-1} \left( \frac{Q}{\sum \nolimits _{z=0}^{T-1}N_{-(T-s-z)}}\right) ^{\sigma } \right] \\&\quad +\sum \limits _{t=1}^{\infty }m^{t}N_{-1}\left[ \sum \limits _{s=t}^{t+T-1}\left( \frac{Q}{\sum \nolimits _{z=0}^{T-1}N_{-(T-s-z)}} \right) ^{\sigma }\right] \\&\quad =\sum \limits _{t=1}^{\infty }m^{t}N_{-2}T\alpha +\sum \limits _{t=1}^{\infty }m^{t}N_{-1}T\alpha +\sum \limits _{t=1}^{\infty }m^{t}N_{-2}\left[ \sum \limits _{s=t}^{t+T-1}\left( \hat{q}_{s}\right) ^{\sigma }\right] \\&\quad +\sum \limits _{t=1}^{\infty }m^{t}N_{-1}\left[ \sum \limits _{s=t}^{t+T-1}\left( \check{q}_{s}\right) ^{\sigma }\right] \end{aligned}$$
where
$$\begin{aligned} \hat{q}_{s}\equiv & {} Q/ \left[ \sum \nolimits _{z=0}^{T-s-1}N_{-(T-s-z)}+\sum \nolimits _{r=1,3,5, \ldots }^{s}\left( N_{-2}\right) m^{(r+1)/2}\right. \\&\qquad \left. +\sum \nolimits _{z=2,4,6, \ldots }^{s}\left( N_{-1}\right) m^{z/2}\right] \\ \check{q}_{s}\equiv & {} Q/ \left[ \sum \nolimits _{z=0}^{T-s-1}N_{-(T-s-z+1)}+mN_{-1}+\sum \nolimits _{r=1,3,5, \ldots }^{s}\left( N_{-2}\right) m^{(r+1)/2}\right. \\&\qquad \left. +\sum \nolimits _{z=2,4,6, \ldots }^{s}\left( N_{-1}\right) m^{z/2} \right] \end{aligned}$$
When the two histories are lifetime equal, we have: \(m=n\frac{N_{-1}}{ N_{-1}+N_{-2}(1-n)}\). Total well-being in the second history can now be written as:
$$\begin{aligned}= & {} \sum \limits _{t=1}^{\infty }\left( \frac{nN_{-1}}{N_{-1}+N_{-2}(1-n)} \right) ^{t}N_{-2}T\alpha +\sum \limits _{t=1}^{\infty }\left( \frac{nN_{-1}}{ N_{-1}+N_{-2}(1-n)}\right) ^{t}N_{-1}T\alpha \\&+\,\sum \limits _{t=1}^{\infty }\left( \frac{nN_{-1}}{N_{-1}+N_{-2}(1-n)} \right) ^{t}N_{-2}\left[ \sum \limits _{s=t}^{t+T-1}\hat{q}_{s}^{\sigma } \right] \\&+\,\sum \limits _{t=1}^{\infty }\left( \frac{nN_{-1}}{N_{-1}+N_{-2}(1-n) }\right) ^{t}N_{-1}\left[ \sum \limits _{s=t}^{t+T-1}\check{q}_{s}^{\sigma } \right] \end{aligned}$$
Hence social welfare is larger in history \(\left\{ T,n,0\right\} \) than in history \(\left\{ T,0,m\right\} \) if and only if:
$$\begin{aligned}&\sum \limits _{t=1}^{\infty }n^{t}N_{-1}T\alpha +\sum \limits _{t=1}^{\infty }n^{t}N_{-1}\left[ \sum \limits _{s=t}^{t+T-1}\left( \frac{Q}{ \sum \nolimits _{z=0}^{T-s-1}N_{-(T-s-z)}+\left( N_{-1}\right) \sum _{r=1}^{s}n^{r}}\right) ^{\sigma }\right] \\\gtrless & {} \\&\sum \limits _{t=1}^{\infty }\left( \frac{nN_{-1}}{N_{-1}+N_{-2}(1-n)} \right) ^{t}N_{-2}T\alpha +\sum \limits _{t=1}^{\infty }\left( \frac{nN_{-1}}{ N_{-1}+N_{-2}(1-n)}\right) ^{t}N_{-1}T\alpha \\&+\,\sum \limits _{t=1}^{\infty }\left( \frac{nN_{-1}}{N_{-1}+N_{-2}(1-n)} \right) ^{t}N_{-2}\left[ \sum \limits _{s=t}^{t+T-1}\hat{q}_{s}^{\sigma } \right] \\&+\,\sum \limits _{t=1}^{\infty }\left( \frac{nN_{-1}}{N_{-1}+N_{-2}(1-n) }\right) ^{t}N_{-1}\left[ \sum \limits _{s=t}^{t+T-1}\check{q}_{s}^{\sigma } \right] \end{aligned}$$
That expression can be simplified as follows. Note that \(\sum \nolimits _{t=1}^{\infty }n^{t}N_{-1}T\alpha =N_{-1}T\alpha \left( \frac{1}{1-n} -1\right) =\frac{nN_{-1}T\alpha }{1-n}\) and that \(\sum \nolimits _{t=1}^{\infty }\left( \frac{nN_{-1}}{N_{-1}+N_{-2}(1-n)}\right) ^{t}T\alpha \left( N_{-2}+N_{-1}\right) =T\alpha \left( N_{-2}+N_{-1}\right) \sum \nolimits _{t=1}^{\infty }\left( \frac{nN_{-1}}{N_{-1}+N_{-2}(1-n)}\right) ^{t}=\frac{T\alpha nN_{-1}}{(1-n)}\). Hence we have:
$$\begin{aligned}&\sum \limits _{t=1}^{\infty }n^{t}N_{-1}T\alpha =\sum \limits _{t=1}^{\infty }\left( \frac{nN_{-1}}{N_{-1}+N_{-2}(1-n)}\right) ^{t}N_{-2}T\alpha \\&+\,\sum \limits _{t=1}^{\infty }\left( \frac{nN_{-1}}{N_{-1}+N_{-2}(1-n)} \right) ^{t}N_{-1}T\alpha \end{aligned}$$
and the condition thus becomes:
$$\begin{aligned}&\sum \limits _{t=1}^{\infty }n^{t}N_{-1}\left[ \sum \limits _{s=t}^{t+T-1} \left( \frac{Q}{\sum \nolimits _{z=0}^{T-s-1}N_{-(T-s-z)}+\left( N_{-1}\right) \sum _{r=1}^{s}n^{r}}\right) ^{\sigma }\right] \\\gtrless & {} \\&\sum \limits _{t=1}^{\infty }\left( \frac{nN_{-1}}{N_{-1}+N_{-2}(1-n)} \right) ^{t}N_{-2}\left[ \sum \limits _{s=t}^{t+T-1}\hat{q}_{s}^{\sigma } \right] \\&+\sum \limits _{t=1}^{\infty }\left( \frac{nN_{-1}}{N_{-1}+N_{-2}(1-n) }\right) ^{t}N_{-1}\left[ \sum \limits _{s=t}^{t+T-1}\check{q}_{s}^{\sigma } \right] \end{aligned}$$
1.2 Proof of Corollary 1
Take the special case where space does not matter: \(\sigma =0\). The condition becomes:
$$\begin{aligned}&\sum \limits _{t=1}^{\infty }n^{t}N_{-1}T\alpha +\sum \limits _{t=1}^{\infty }n^{t}N_{-1}T \\\gtrless & {} \\&\sum \limits _{t=1}^{\infty }\left( \frac{nN_{-1}}{N_{-1}+N_{-2}(1-n)} \right) ^{t}N_{-2}T\alpha +\sum \limits _{t=1}^{\infty }\left( \frac{nN_{-1}}{ N_{-1}+N_{-2}(1-n)}\right) ^{t}N_{-1}T\alpha \\&+\sum \limits _{t=1}^{\infty }\left( \frac{nN_{-1}}{N_{-1}+N_{-2}(1-n)} \right) ^{t}N_{-2}T+\sum \limits _{t=1}^{\infty }\left( \frac{nN_{-1}}{ N_{-1}+N_{-2}(1-n)}\right) ^{t}N_{-1}T \end{aligned}$$
That condition can be simplified as:
$$\begin{aligned}&\sum \limits _{t=1}^{\infty }n^{t}N_{-1}\gtrless \left( N_{-2}+N_{-1}\right) \sum \limits _{t=1}^{\infty }\left( \frac{nN_{-1}}{ N_{-1}+N_{-2}(1-n)}\right) ^{t} \\&\quad \iff N_{-1}\left( \frac{n}{1-n}\right) \gtrless \left( N_{-2}+N_{-1}\right) \left( \frac{nN_{-1}}{N_{-1}+N_{-2}(1-n)-nN_{-1}}\right) \\&\quad \iff \frac{1}{1-n}\gtrless \frac{N_{-2}+N_{-1}}{\left( N_{-1}+N_{-2}\right) \left( 1-n\right) } \end{aligned}$$
That condition is always valid. Hence, independently from initial conditions, if space congestion does not matter, histories \(\left\{ T,n,0\right\} \) and \(\left\{ T,0,m\right\} \) bring the same total welfare. This is not surprising, since these two lotteries are, by construction, lifetime-equal, meaning that these yield to exactly the same number of life periods, and, hence, in the absence of concern for congestion, this makes the two lotteries equally good.
1.3 Proof of Proposition 2
Consider first the case of average utilitarianism. Assuming that the two histories are lifetime equal, the total number of individuals born at \(t\ge 0\) in history \(\left\{ T,n,0\right\} \) and in history \(\left\{ T,0,m\right\} \) is: \(\frac{TnN_{-1}}{1-n}\). In the light of this, the average total welfare in history \(\left\{ T,n,0\right\} \) is:
$$\begin{aligned} \frac{\sum \limits _{t=1}^{\infty }n^{t}N_{-1}T\alpha +\sum \limits _{t=1}^{\infty }n^{t}N_{-1}\left[ \sum \nolimits _{s=t}^{t+T-1} \left( \frac{Q}{\sum \nolimits _{z=0}^{T-s-1}N_{-(T-s-z)}+\left( N_{-1}\right) \sum _{r=1}^{s}n^{r}}\right) ^{\sigma }\right] }{\frac{TnN_{-1}}{1-n}} \end{aligned}$$
whereas the average total welfare in history \(\left\{ T,0,m\right\} \) is:
$$\begin{aligned}&\frac{\sum \nolimits _{t=1}^{\infty }\left( \frac{nN_{-1}}{N_{-1}+N_{-2}(1-n)} \right) ^{t}N_{-2}T\alpha +\sum \nolimits _{t=1}^{\infty }\left( \frac{nN_{-1}}{ N_{-1}+N_{-2}(1-n)}\right) ^{t}N_{-1}T\alpha }{\frac{TnN_{-1}}{1-n}} \\&\quad +\sum \limits _{t=1}^{\infty }\frac{\left( \frac{nN_{-1}}{N_{-1}+N_{-2}(1-n) }\right) ^{t}N_{-2}}{\frac{TnN_{-1}}{1-n}}\left[ \sum \limits _{s=t}^{t+T-1} \hat{q}_{s}^{\sigma }\right] +\sum \limits _{t=1}^{\infty }\frac{\left( \frac{ nN_{-1}}{N_{-1}+N_{-2}(1-n)}\right) ^{t}N_{-1}}{\frac{TnN_{-1}}{1-n}}\\&\quad \times \left[ \sum \limits _{s=t}^{t+T-1}\check{q}_{s}^{\sigma }\right] \end{aligned}$$
Given that \(\frac{TnN_{-1}}{1-n}\) divides both the LHS and the RHS of that condition, it is straightforward to see that average welfare for individuals born at \(t\ge 0\) is larger under history \(\left\{ T,n,0\right\} \) than in history \(\left\{ T,0,m\right\} \) if and only if the same condition as under CU is satisfied. The same rationale can be used to show that CLU yields exactly the same ranking as CU as far as the comparison of lifetime equal histories is concerned.
1.4 Proof of Lemma 2
Let us compute the total long-run population size under the two histories \( \left\{ T,n,0\right\} \) and \(\left\{ T,0,m\right\} \). In general, the total population follows the dynamics:
$$\begin{aligned} L_{0}= & {} N_{-(T-1)}+N_{-(T-2)}+ \cdots +N_{-2}+N_{-1}+N_{0} \\ L_{1}= & {} N_{-(T-2)}+N_{-(T-3)}+ \cdots +N_{-1}+N_{0}+N_{1} \\ L_{2}= & {} N_{-(T-3)}+N_{-(T-4)}+ \cdots +N_{0}+N_{1}+N_{2} \\&\ldots \\ L_{T-2}= & {} N_{-1}+N_{0}+N_{1}+ \ldots +N_{T-2} \\ L_{T-1}= & {} N_{0}+N_{1}+N_{2}+ \cdots +N_{T-1} \\&\ldots \\ L_{t}= & {} N_{t-T+1}+ \cdots +N_{t-1}+N_{t} \end{aligned}$$
Under the history \(\left\{ T,1,0\right\} \), that evolution takes the form:
$$\begin{aligned} L_{0}= & {} N_{-(T-1)}+N_{-(T-2)}+ \cdots +N_{-2}+N_{-1}+N_{-1} \\ L_{1}= & {} N_{-(T-2)}+N_{-(T-3)}+ \cdots +3N_{-1} \\ L_{2}= & {} N_{-(T-3)}+N_{-(T-4)}+ \cdots +4N_{-1} \\&\ldots \\ L_{T-2}= & {} TN_{-1} \\ L_{t}= & {} TN_{-1}\quad \text { for }\,\,t\ge T-2 \end{aligned}$$
Under the history \(\left\{ T,0,1\right\} \), that evolution takes the form (we assume T is an even number):
$$\begin{aligned} L_{0}= & {} N_{-(T-1)}+N_{-(T-2)}+ \cdots +N_{-2}+N_{-1}+N_{-2} \\ L_{1}= & {} N_{-(T-2)}+N_{-(T-3)}+ \cdots +N_{-1}+N_{-2}+N_{-1} \\ L_{2}= & {} N_{-(T-3)}+N_{-(T-4)}+ \cdots +N_{-1}+N_{-2}+N_{-1}+N_{-2} \\&\ldots \\ L_{T-2}= & {} N_{-1}+N_{-2}+N_{-1}+ \cdots +N_{-2} \\ L_{T-1}= & {} N_{-2}+N_{-1}+N_{-2}+ \cdots +N_{-1} \\&\ldots \\ L_{t}= & {} \frac{T}{2}N_{-2}+\frac{T}{2}N_{-1}=\frac{TN_{-2}\left( 1+\frac{N_{-1}}{ N_{-2}}\right) }{2} \end{aligned}$$
Hence, the asymptotic population size under \(\left\{ T,1,0\right\} \) and \( \left\{ T,0,1\right\} \) are ranked according to:
$$\begin{aligned} TN_{-1}\gtrless \frac{TN_{-2}\left( 1+\frac{N_{-1}}{N_{-2}}\right) }{2}\iff N_{-1}\gtrless N_{-2} \end{aligned}$$
Hence, if \(N_{-1}>N_{-2}\), the asymptotic population under \(\left\{ T,1,0\right\} \) exceeds the one under \(\left\{ T,0,1\right\} \). If \( N_{-1}=N_{-2}\), the asymptotic population under \(\left\{ T,1,0\right\} \) equals the one under \(\left\{ T,0,1\right\} \). If , \(N_{-1}<N_{-2}\), the asymptotic population under \(\left\{ T,1,0\right\} \) is smaller than the one under \(\left\{ T,0,1\right\} \).
1.5 Proof of Proposition 3
Social welfare at the stationary equilibrium is equal to (abstracting from time indexes):
$$\begin{aligned} LTu(q)=LTu\left( \frac{Q}{L}\right) =LT\left( \frac{Q}{L}\right) ^{\sigma }+LT\alpha =L^{1-\sigma }TQ^{\sigma }+LT\alpha \end{aligned}$$
where L denotes the asymptotic population size, and q the asymptotic space per head.
Under history \(\left\{ T,1,0\right\} \), that formula becomes:
$$\begin{aligned} TN_{-1}Tu(q)= & {} T^{2}N_{-1}\left[ \alpha +\left( \frac{Q}{TN_{-1}}\right) ^{\sigma }\right] \\= & {} T^{2}N_{-1}\alpha +T^{2-\sigma }\left( N_{-1}\right) ^{1-\sigma }Q^{\sigma } \end{aligned}$$
Under the history \(\left\{ T,0,1\right\} \), that formula becomes:
$$\begin{aligned} LTu(q)= & {} \frac{TN_{-2}\left( 1+\frac{N_{-1}}{N_{-2}}\right) }{2}T\left[ \alpha +\left( \frac{Q}{\frac{TN_{-2}\left( 1+\frac{N_{-1}}{N_{-2}}\right) }{2}}\right) ^{\sigma }\right] \\= & {} \frac{T^{2}N_{-2}\left( 1+\frac{N_{-1}}{N_{-2}}\right) \alpha }{2}+\left( \frac{ TN_{-2}\left( 1+\frac{N_{-1}}{N_{-2}}\right) }{2}\right) ^{1-\sigma }TQ^{\sigma } \end{aligned}$$
Hence the ranking of the CU planner depends on:
$$\begin{aligned} T^{2}N_{-1}\alpha +T^{2-\sigma }\left( N_{-1}\right) ^{1-\sigma }Q^{\sigma }\gtrless \frac{T^{2}N_{-2}\left( 1+\frac{N_{-1}}{N_{-2}}\right) \alpha }{2}+\left( \frac{ TN_{-2}\left( 1+\frac{N_{-1}}{N_{-2}}\right) }{2}\right) ^{1-\sigma }TQ^{\sigma } \end{aligned}$$
That expression can be written as:
If \(N_{-1}=N_{-2}\), the LHS and RHS are equal to 0, so that indifference holds. If \(N_{-1}<N_{-2}\), and \(\alpha >0\), the LHS is negative, while the RHS is positive (given \(\sigma \le 1\)). Hence the history \(\left\{ T,0,1\right\} \) is better. If \(N_{-1}>N_{-2}\), and \(\alpha >0\), the LHS is positive, while the RHS is negative (given \(\sigma \le 1\)). Hence the history \(\left\{ T,1,0\right\} \) is better. If \(N_{-1}<N_{-2}\), and \(\alpha <0\), the LHS is positive, while the RHS is positive (given \(\sigma \le 1\)), so that the ranking depends on \(\alpha ( \frac{N_{-1}-N_{-2}}{2}) \gtrless ( \frac{Q}{T}) ^{\sigma }[ ( \frac{ (N_{-2}+N_{-1})}{2}) ^{1-\sigma }-( N_{-1}) ^{1-\sigma } ] \). If \(N_{-1}>N_{-2}\), and \(\alpha <0\), the LHS is negative, while the RHS is negative (given \(\sigma \le 1\)), the same indeterminacy prevails.
1.6 Proof of Proposition 4
Under \(\left\{ T,1,0\right\} \), average social welfare is:
$$\begin{aligned} Tu(q)=T\left( \frac{Q}{TN_{-1}}\right) ^{\sigma }+T\alpha \end{aligned}$$
Under \(\left\{ T,0,1\right\} \), L equals \(\frac{TN_{-2}(1+\frac{N_{-1}}{ N_{-2}})}{2}\), so that average social welfare is:
$$\begin{aligned} Tu(q)=T\left( \frac{Q}{\frac{TN_{-2}\left( 1+\frac{N_{-1}}{N_{-2}}\right) }{2}}\right) ^{\sigma }+T\alpha \end{aligned}$$
Hence the ranking between \(\left\{ T,1,0\right\} \) and \(\left\{ T,0,1\right\} \) depends on:
$$\begin{aligned} T\left( \frac{Q}{TN_{-1}}\right) ^{\sigma }+T\alpha \gtrless T\left( \frac{Q }{\frac{TN_{-2}\left( 1+\frac{N_{-1}}{N_{-2}}\right) }{2}}\right) ^{\sigma }+T\alpha \iff N_{-1}\lessgtr N_{-2} \end{aligned}$$
Simplifications yield: \(N_{-1}\lessgtr N_{-2}\). Hence if \(N_{-1}>N_{-2}\), \( \left\{ T,0,1\right\} \) is preferred over \(\left\{ T,1,0\right\} \). If \( N_{-1}<N_{-2}\), \(\left\{ T,1,0\right\} \) is preferred over \(\left\{ T,0,1\right\} \). Indifference holds under \(N_{-1}=N_{-2}\).
1.7 Proof of Proposition 5
Under history \(\left\{ T,1,0\right\} \), social welfare under CLU at the stationary equilibrium is equal to (abstracting from time indexes):
$$\begin{aligned} LT\left[ u(q)-\hat{u}\right]= & {} TN_{-1}T\left[ u(q)-\hat{u}\right] \\= & {} TN_{-1}T\left( \frac{Q}{TN_{-1}}\right) ^{\sigma }+TN_{-1}T( \alpha - \hat{u}) \\= & {} T^{2-\sigma }\left( N_{-1}\right) ^{1-\sigma }Q^{\sigma }+T^{2}N_{-1}( \alpha -\hat{u}) \end{aligned}$$
Under history \(\left\{ T,0,1\right\} \), that formula becomes:
$$\begin{aligned} LT\left[ u(q)-\hat{u}\right] =\left( \frac{N_{-2}\left( 1+\frac{N_{-1}}{N_{-2}}\right) }{2 }\right) ^{1-\sigma }T^{2-\sigma }Q^{\sigma }+\frac{N_{-2}\left( 1+\frac{N_{-1}}{ N_{-2}}\right) }{2}T^{2}( \alpha -\hat{u}) \end{aligned}$$
Hence the ranking of the CLU planner depends on:
$$\begin{aligned}&T^{2-\sigma }\left( N_{-1}\right) ^{1-\sigma }Q^{\sigma }+T^{2}N_{-1}( \alpha -\hat{u}) \gtrless \left( \frac{N_{-2}\left( 1+\frac{N_{-1}}{N_{-2}}\right) }{2}\right) ^{1-\sigma }T^{2-\sigma }Q^{\sigma }\\&\quad +\,\frac{N_{-2}\left( 1+\frac{N_{-1} }{N_{-2}}\right) }{2}T^{2}( \alpha -\hat{u}) \end{aligned}$$
After simplifications, that expression becomes:
$$\begin{aligned} ( \alpha -\hat{u}) \left[ \frac{N_{-1}-N_{-2}}{2}\right] \gtrless \left( \frac{Q}{T}\right) ^{\sigma }\left[ \left( \frac{N_{-2}+N_{-1}}{2} \right) ^{1-\sigma }-\left( N_{-1}\right) ^{1-\sigma }\right] \end{aligned}$$
If \(N_{-1}=N_{-2}\), the LHS and RHS are equal to 0, so that indifference holds. If \(N_{-1}<N_{-2}\), and \(\alpha -\hat{u}>0\), the LHS is negative, while the RHS is positive (given \(\sigma \le 1\)). Hence the history \( \left\{ T,0,1\right\} \) is better. If \(N_{-1}>N_{-2}\), and \(\alpha -\hat{u} >0 \), the LHS is positive, while the RHS is negative (given \(\sigma \le 1\) ). Hence the history \(\left\{ T,1,0\right\} \) is better. If \(N_{-1}<N_{-2}\), and \(\alpha -\hat{u}<0\), the LHS is positive, while the RHS is positive (given \(\sigma \le 1\)), so that the ranking depends on \(( \alpha -\hat{ u}) [ \frac{N_{-1}-N_{-2}}{2}] \gtrless ( \frac{Q}{T} ) ^{\sigma }[( \frac{N_{-2}+N_{-1}}{2}) ^{1-\sigma }-\left( N_{-1}\right) ^{1-\sigma }] \). If \(N_{-1}>N_{-2}\), and \( \alpha -\hat{u}<0\), the LHS is negative, while the RHS is negative (given \( \sigma \le 1\)), the same indeterminacy prevails.
1.8 Proof of Proposition 6
Under history \(\left\{ T,n,0\right\} \), the cumulated social welfare for individuals born at \(t\ge 0\) is:
$$\begin{aligned}&nN_{-1}T\alpha +nN_{-1}\left[ \frac{Q}{N_{-(T-1)}+ \cdots +nN_{-1}}\right] ^{\sigma }+nN_{-1}\left[ \frac{Q}{N_{-(T-2)}+ \cdots +n^{2}N_{-1}}\right] ^{\sigma } \\&\quad + \cdots +nN_{-1}\left[ \frac{Q}{nN_{-1}+n^{2}N_{-1}+ \cdots +n^{T+1}N_{-1}}\right] ^{\sigma }\\&\quad +\,nN_{-1}( ((T-1)n)^{\delta }+((T-2)n^{2})^{\delta }+ \cdots +(n^{T-1}) ^{\delta }) \\&\quad +\,n^{2}N_{-1}T\alpha +n^{2}N_{-1}\left[ \frac{Q}{ N_{-(T-2)}+ \cdots +nN_{-1}+n^{2}N_{-1}}\right] ^{\sigma }\\&\quad +\,n^{2}N_{-1}\left[ \frac{Q}{N_{-(T-3)}+ \cdots +n^{3}N_{-1}}\right] ^{\sigma } \\&\quad + \cdots +n^{2}N_{-1}\left[ \frac{Q}{n^{2}N_{-1}+n^{3}N_{-1}+ \cdots +n^{T+2}N_{-1}} \right] ^{\sigma }\\&\quad +\,n^{2}N_{-1}\left( ((T-1)n)^{\delta }+((T-2)n^{2})^{\delta }+ \cdots +(n^{T-1}) ^{\delta }\right) \\&\quad +\,n^{3}N_{-1}T\alpha +n^{3}N_{-1}\left[ \frac{Q}{ N_{-(T-3)}+ \cdots +n^{2}N_{-1}+n^{3}N_{-1}}\right] ^{\sigma }\\&\quad +\,n^{3}N_{-1}\left[ \frac{Q}{N_{-(T-4)}+ \cdots +n^{4}N_{-1}}\right] ^{\sigma } \\&\quad + \cdots +n^{3}N_{-1}\left[ \frac{Q}{n^{3}N_{-1}+n^{4}N_{-1}+ \cdots +n^{T+3}N_{-1}} \right] ^{\sigma }\\&\quad +\,n^{3}N_{-1}\left( ((T-1)n)^{\delta }+((T-2)n^{2})^{\delta }+ \cdots +(n^{T-1}) ^{\delta }\right) \\&\quad + \cdots \end{aligned}$$
This can be rewritten as:
$$\begin{aligned}&\frac{nN_{-1}T\alpha }{1-n}+\sum \limits _{t=1}^{\infty }n^{t}N_{-1}\left[ \sum \limits _{s=t}^{t+T-1}\left( \frac{Q}{\sum \nolimits _{z=0}^{T-s-1}N_{-(T-s-z)}+\left( N_{-1}\right) \sum _{r=1}^{s}n^{r}} \right) ^{\sigma }\right] \\&\quad +\frac{nN_{-1}}{1-n}( ((T-1)n)^{\delta }+((T-2)n^{2})^{\delta }+ \cdots +((T-T+1)n^{T-1}) ^{\delta }) \end{aligned}$$
Under history \(\left\{ T,0,m\right\} \), the cumulated social welfare for individuals born at \(t\ge 0\) is:
$$\begin{aligned}&mN_{-2}T\alpha +mN_{-2}\left[ \frac{Q}{N_{-(T-1)}+ \cdots +N_{-1}+mN_{-2}} \right] ^{\sigma } \\&\quad +\,mN_{-2}\left[ \frac{Q}{N_{-(T-2)}+ \cdots +mN_{-2}+mN_{-1}}\right] ^{\sigma } \\&\quad +\,mN_{-2}\left[ \frac{Q}{N_{-(T-3)}+ \cdots +N_{-2}\left( m+m^{2}\right) +mN_{-1} }\right] ^{\sigma } \\&\quad + \cdots +mN_{-2}\left[ \frac{Q}{N_{-2}\left( m+m^{2}+ \cdots +m^{T/2}\right) +N_{-1}\left( m+m^{2}+ \cdots +m^{T/2}\right) }\right] ^{\sigma } \\&\quad +\,mN_{-2}( ((T-2)m)^{\delta }+((T-4)m^{2})^{\delta }+ \cdots +((T-T+2)m^{T-2}) ^{\delta }) \\&\quad +\,mN_{-1}T\alpha +mN_{-1}\left[ \frac{Q}{N_{-(T-2)}+ \cdots +mN_{-2}+mN_{-1}} \right] ^{\sigma } \\&\quad +\,mN_{-1}\left[ \frac{Q}{N_{-(T-3)}+ \cdots +N_{-2}\left( m+m^{2}\right) +mN_{-1} }\right] ^{\sigma } \\&\quad +\,mN_{-1}\left[ \frac{Q}{N_{-(T-4)}+ \cdots +N_{-2}\left( m+m^{2}\right) +N_{-1}\left( m+m^{2}\right) }\right] ^{\sigma } \\&\quad + \cdots +mN_{-1}\left[ \frac{Q}{N_{-2}\left( m^{2}+m^{3}+ \cdots +m^{T/2}\right) + \cdots +N_{-1}\left( m+m^{2}+ \cdots +m^{(T/2)+1}\right) }\right] ^{\sigma } \\&\quad +\,mN_{-1}( ((T-2)m)^{\delta }+((T-4)m^{2})^{\delta }+ \cdots +((T-T+2)m^{T-2}) ^{\delta }) \\&\quad +\,m^{2}N_{-2}T\alpha +m^{2}N_{-2}\left[ \frac{Q}{N_{-(T-3)}+ \cdots +N_{-2} \left( m+m^{2}\right) +mN_{-1}}\right] ^{\sigma } \\&\quad +\,m^{2}N_{-2}\left[ \frac{Q}{N_{-(T-4)}+ \cdots +N_{-2}\left( m+m^{2}\right) +N_{-1}\left( m+m^{2}\right) }\right] ^{\sigma } \\&\quad +\,m^{2}N_{-2}\left[ \frac{Q}{N_{-(T-5)}+ \cdots +N_{-2}\left( m+m^{2}+m^{3}\right) +N_{-1}\left( m+m^{2}\right) }\right] ^{\sigma } \\&\quad + \cdots +m^{2}N_{-2}\left[ \frac{Q}{N_{-2}\left( m^{2}+m^{3}+ \cdots +m^{(T/2)+1}\right) +N_{-1}\left( m^{2}+m^{3}+ \cdots +m^{(T/2)+1}\right) }\right] ^{\sigma } \\&\quad +\,m^{2}N_{-2}( ((T-2)m)^{\delta }+((T-4)m^{2})^{\delta }+ \cdots +((T-T+2)m^{T-2}) ^{\delta }) \\&\quad +\cdots \end{aligned}$$
Substituting for m in the context of equal lifetime histories, that expression can be rewritten as:
$$\begin{aligned}&\sum \limits _{t=1}^{\infty }\left( \frac{nN_{-1}}{N_{-1}+N_{-2}(1-n)} \right) ^{t}N_{-2}T\alpha +\sum \limits _{t=1}^{\infty }\left( \frac{nN_{-1}}{ N_{-1}+N_{-2}(1-n)}\right) ^{t}N_{-1}T\alpha \\&\quad +\sum \limits _{t=1}^{\infty }\left( \frac{nN_{-1}}{N_{-1}+N_{-2}(1-n)} \right) ^{t}N_{-2}\left[ \sum \limits _{s=t}^{t+T-1}\hat{q}_{s}^{\sigma } \right] \\&\quad +\sum \limits _{t=1}^{\infty }\left( \frac{nN_{-1}}{N_{-1}+N_{-2}(1-n) }\right) ^{t}N_{-1}\left[ \sum \limits _{s=t}^{t+T-1}\check{q}_{s}^{\sigma } \right] \\&\quad +\sum _{t=1}^{\infty }N_{-2}\left( \frac{nN_{-1}}{N_{-1}+N_{-2}(1-n)} \right) ^{t}\left( \left( \frac{(T-2)nN_{-1}}{N_{-1}+N_{-2}(1-n)}\right) ^{\delta }\right. \\&\quad \left. + \cdots +\left( 2\left( \frac{nN_{-1}}{N_{-1}+N_{-2}(1-n)}\right) ^{T-2}\right) ^{\delta }\right) \\&\quad +\sum _{t=1}^{\infty }N_{-1}\left( \frac{nN_{-1}}{N_{-1}+N_{-2}(1-n)} \right) ^{t}\left( \left( \frac{(T-2)nN_{-1}}{N_{-1}+N_{-2}(1-n)}\right) ^{\delta }\right. \\&\quad \left. + \cdots +\left( 2\left( \frac{nN_{-1}}{N_{-1}+N_{-2}(1-n)}\right) ^{T-2}\right) ^{\delta }\right) \end{aligned}$$
That expression can be rewritten as:
$$\begin{aligned}&\frac{T\alpha nN_{-1}}{(1-n)} \\&\quad +\sum \limits _{t=1}^{\infty }\left( \frac{nN_{-1}}{N_{-1}+N_{-2}(1-n)} \right) ^{t}N_{-2}\left[ \sum \limits _{s=t}^{t+T-1}\hat{q}_{s}^{\sigma } \right] \\&\quad +\sum \limits _{t=1}^{\infty }\left( \frac{nN_{-1}}{N_{-1}+N_{-2}(1-n) }\right) ^{t}N_{-1}\left[ \sum \limits _{s=t}^{t+T-1}\check{q}_{s}^{\sigma } \right] \\&\quad +\left( \left( \frac{(T-2)nN_{-1}}{N_{-1}+N_{-2}(1-n)}\right) ^{\delta }+ \cdots +\left( (T-T+2)\left( \frac{nN_{-1}}{N_{-1}+N_{-2}(1-n)}\right) ^{T-2}\right) ^{\delta }\right) \\&\quad \times \frac{nN_{-1}}{(1-n)} \end{aligned}$$
Hence history \(\left\{ T,n,0\right\} \) has a larger or a lower social welfare than history \(\left\{ T,0,m\right\} \) if and only if:
$$\begin{aligned}&\frac{nN_{-1}T\alpha }{1-n}+\sum \limits _{t=1}^{\infty }n^{t}N_{-1}\left[ \sum \limits _{s=t}^{t+T-1}\left( \frac{Q}{\sum \nolimits _{z=0}^{T-s-1}N_{-(T-s-z)}+\left( N_{-1}\right) \sum _{r=1}^{s}n^{r}} \right) ^{\sigma }\right] \\&\quad +\,\frac{nN_{-1}}{1-n}( ((T-1)n)^{\delta }+((T-2)n^{2})^{\delta }+ \cdots +((T-T+1)n^{T-1}) ^{\delta }) \\&\quad \gtrless \frac{nN_{-1}T\alpha }{(1-n)}+\sum \limits _{t=1}^{\infty }\left( \frac{nN_{-1}}{N_{-1}+N_{-2}(1-n)}\right) ^{t}N_{-2}\left[ \sum \limits _{s=t}^{t+T-1}\hat{q}_{s}^{\sigma }\right] \\&\quad +\sum \limits _{t=1}^{ \infty }\left( \frac{nN_{-1}}{N_{-1}+N_{-2}(1-n)}\right) ^{t}N_{-1}\left[ \sum \limits _{s=t}^{t+T-1}\check{q}_{s}^{\sigma }\right] \\&\quad \times \left( \left( \frac{(T-2)nN_{-1}}{N_{-1}+N_{-2}(1-n)}\right) ^{\delta }+ \cdots +\left( (T-T+2)\left( \frac{nN_{-1}}{N_{-1}+N_{-2}(1-n)}\right) ^{T-2}\right) ^{\delta }\right) \\&\quad \times \frac{nN_{-1}}{(1-n)} \end{aligned}$$
That inequality can be rewritten as:
$$\begin{aligned}&\sum \limits _{t=1}^{\infty }n^{t}N_{-1}\left[ \sum \limits _{s=t}^{t+T-1} \left( \frac{Q}{\sum \nolimits _{z=0}^{T-s-1}N_{-(T-s-z)}+\left( N_{-1}\right) \sum _{r=1}^{s}n^{r}}\right) ^{\sigma }\right] \\&\quad +\,\frac{nN_{-1}}{1-n}( ((T-1)n)^{\delta }+((T-2)n^{2})^{\delta }+ \cdots +((T-T+1)n^{T-1}) ^{\delta }) \\&\quad \gtrless \sum \limits _{t=1}^{\infty }\left( \frac{nN_{-1}}{ N_{-1}+N_{-2}(1-n)}\right) ^{t}N_{-2}\left[ \sum \limits _{s=t}^{t+T-1}\hat{q} _{s}^{\sigma }\right] \\&\quad +\sum \limits _{t=1}^{\infty }\left( \frac{nN_{-1}}{ N_{-1}+N_{-2}(1-n)}\right) ^{t}N_{-1}\left[ \sum \limits _{s=t}^{t+T-1}\check{ q}_{s}^{\sigma }\right] \\&\quad +\left( \left( \frac{(T-2)nN_{-1}}{N_{-1}+N_{-2}(1-n)}\right) ^{\delta }+ \cdots +\left( (T-T+2)\left( \frac{nN_{-1}}{N_{-1}+N_{-2}(1-n)}\right) ^{T-2}\right) ^{\delta }\right) \\&\quad \times \frac{nN_{-1}}{1-n} \end{aligned}$$
1.9 Proof of Corollary 2
Fixing \(\sigma =0\) in the condition:
$$\begin{aligned}&\sum \limits _{t=1}^{\infty }n^{t}N_{-1}\left[ \sum \limits _{s=t}^{t+T-1} \left( \frac{Q}{\sum \nolimits _{z=0}^{T-s-1}N_{-(T-s-z)}+\left( N_{-1}\right) \sum _{r=1}^{s}n^{r}}\right) ^{\sigma }\right] \\&\quad +\frac{nN_{-1}( ((T-1)n)^{\delta }+((T-2)n^{2})^{\delta }+ \cdots +((T-T+1)n^{T-1}) ^{\delta })}{1-n} \\&\quad \gtrless \sum \limits _{t=1}^{\infty }\left( \frac{nN_{-1}}{ N_{-1}+N_{-2}(1-n)}\right) ^{t}N_{-2}\left[ \sum \limits _{s=t}^{t+T-1}\hat{q} _{s}^{\sigma }\right] \\&\quad +\sum \limits _{t=1}^{\infty }\left( \frac{nN_{-1}}{ N_{-1}+N_{-2}(1-n)}\right) ^{t}N_{-1}\left[ \sum \limits _{s=t}^{t+T-1}\check{ q}_{s}^{\sigma }\right] \\&\quad \times \left( \left( \frac{(T-2)nN_{-1}}{N_{-1}+N_{-2}(1-n)}\right) ^{\delta }+ \cdots +\left( (T-T+2)\left( \frac{nN_{-1}}{N_{-1}+N_{-2}(1-n)}\right) ^{T-2}\right) ^{\delta }\right) \\&\quad \times \frac{nN_{-1}}{1-n} \end{aligned}$$
yields:
$$\begin{aligned}&( ((T-1)n)^{\delta }+((T-2)n^{2})^{\delta }+ \cdots +((T-T+1)n^{T-1}) ^{\delta }) \\&\quad \gtrless \left( \frac{(T-2)nN_{-1}}{N_{-1}+N_{-2}(1-n)}\right) ^{\delta }+ \cdots +\left( (T-T+2)\left( \frac{nN_{-1}}{N_{-1}+N_{-2}(1-n)}\right) ^{T-2}\right) ^{\delta } \end{aligned}$$
The LHS is unambiguously larger than the RHS, since \(m<n\) and coexistence time is necessarily reduced by birth postponement. Hence the history \( \left\{ T,n,0\right\} \) is preferred over \(\left\{ T,0,m\right\} \).
1.10 Proof of Corollary 3
Under AU, the condition for preferring \(\left\{ T,n,0\right\} \) over \( \left\{ T,0,m\right\} \) becomes:the average total welfare in history \( \left\{ T,n,0\right\} \) is:
$$\begin{aligned}&\frac{\left[ \begin{array}{l} \sum \limits _{t=1}^{\infty }n^{t}N_{-1}T\alpha +\sum \limits _{t=1}^{\infty }n^{t}N_{-1}\left[ \sum \limits _{s=t}^{t+T-1}\left( \frac{Q}{ \sum \nolimits _{z=0}^{T-s-1}N_{-(T-s-z)}+\left( N_{-1}\right) \sum _{r=1}^{s}n^{r}}\right) ^{\sigma }\right] \\ +\frac{nN_{-1}( ((T-1)n)^{\delta }+ \cdots +((T-T+1)n^{T-1}) ^{\delta })}{1-n} \end{array} \right] }{\frac{TnN_{-1}}{1-n}} \\&\quad \gtrless \frac{\sum \nolimits _{t=1}^{\infty }\left( \frac{nN_{-1}}{ N_{-1}+N_{-2}(1-n)}\right) ^{t}N_{-2}T\alpha +\sum \nolimits _{t=1}^{\infty }\left( \frac{nN_{-1}}{N_{-1}+N_{-2}(1-n)}\right) ^{t}N_{-1}T\alpha }{\frac{ TnN_{-1}}{1-n}} \\&\quad +\frac{\left( \left( \frac{(T-2)nN_{-1}}{N_{-1}+N_{-2}(1-n)}\right) ^{\delta }+ \cdots +\left( (T-T+2)\left( \frac{nN_{-1}}{N_{-1}+N_{-2}(1-n)} \right) ^{T-2}\right) ^{\delta }\right) \frac{nN_{-1}}{(1-n)}}{\frac{TnN_{-1} }{1-n}} \\&\quad +\sum \limits _{t=1}^{\infty }\frac{\left( \frac{nN_{-1}}{N_{-1}+N_{-2}(1-n) }\right) ^{t}N_{-2}}{\frac{TnN_{-1}}{1-n}}\left[ \sum \limits _{s=t}^{t+T-1} \hat{q}_{s}^{\sigma }\right] +\sum \limits _{t=1}^{\infty }\frac{\left( \frac{ nN_{-1}}{N_{-1}+N_{-2}(1-n)}\right) ^{t}N_{-1}}{\frac{TnN_{-1}}{1-n}}\\&\quad \times \left[ \sum \limits _{s=t}^{t+T-1}\check{q}_{s}^{\sigma }\right] \end{aligned}$$
Simplifying by the total number of births leads to the same condition as in Proposition 6. The same kind of argument holds for CLU.
1.11 Proof of Proposition 7
Under history \(\left\{ T,1,0\right\} \), social welfare at the stationary equilibrium is equal to (abstracting from time indexes):
$$\begin{aligned} LTu(q)= & {} TN_{-1}Tu\left( \frac{Q}{L}\right) \\= & {} TN_{-1}T\left( \frac{Q}{L}\right) ^{\sigma }\\&+\,TN_{-1}T\alpha +TN_{-1}( ((T-1)n)^{\delta }+((T-2)n^{2})^{\delta }+ \cdots +(1n^{T-1}) ^{\delta }) \\= & {} \left( TN_{-1}\right) ^{1-\sigma }TQ^{\sigma } \\&+\,TN_{-1}T\alpha +TN_{-1}( ((T-1)n)^{\delta }+((T-2)n^{2})^{\delta }+ \cdots +(1n^{T-1}) ^{\delta }) \end{aligned}$$
Under the history \(\left\{ T,0,1\right\} \), that formula is:
$$\begin{aligned} LTu(q)= & {} \frac{TN_{-2}\left( 1+\frac{N_{-1}}{N_{-2}}\right) }{2}T\left[ \alpha +\left( \frac{Q}{\frac{TN_{-2}\left( 1+\frac{N_{-1}}{N_{-2}}\right) }{2}}\right) ^{\sigma }\right] \\&+\frac{TN_{-2}\left( 1+\frac{N_{-1}}{N_{-2}}\right) }{2}\left( \left( \frac{(T-2)nN_{-1} }{N_{-1}+N_{-2}(1-n)}\right) ^{\delta }\right. \\&\left. + \cdots +\left( 2\left( \frac{nN_{-1}}{ N_{-1}+N_{-2}(1-n)}\right) ^{T-2}\right) ^{\delta }\right) \\= & {} \frac{T^{2}N_{-2}\left( 1+\frac{N_{-1}}{N_{-2}}\right) \alpha }{2}+\left( \frac{ TN_{-2}\left( 1+\frac{N_{-1}}{N_{-2}}\right) }{2}\right) ^{1-\sigma }TQ^{\sigma } \\&+\frac{TN_{-2}\left( 1+\frac{N_{-1}}{N_{-2}}\right) }{2}\left( \left( \frac{(T-2)nN_{-1} }{N_{-1}+N_{-2}(1-n)}\right) ^{\delta }\right. \\&\left. + \cdots +\left( 2\left( \frac{nN_{-1}}{ N_{-1}+N_{-2}(1-n)}\right) ^{T-2}\right) ^{\delta }\right) \end{aligned}$$
Hence the ranking of the CU planner depends on:
$$\begin{aligned}&\left( TN_{-1}\right) ^{1-\sigma }TQ^{\sigma }+TN_{-1}T\alpha +TN_{-1}( ((T-1)n)^{\delta }+((T-2)n^{2})^{\delta }\\&\quad + \cdots +((T-T+1)n^{T-1}) ^{\delta }) \\&\quad \gtrless \frac{T^{2}N_{-2}\left( 1+\frac{N_{-1}}{N_{-2}}\right) \alpha }{2}+\left( \frac{TN_{-2}\left( 1+\frac{N_{-1}}{N_{-2}}\right) }{2}\right) ^{1-\sigma }TQ^{\sigma } \\&\quad +\frac{TN_{-2}\left( 1+\frac{N_{-1}}{N_{-2}}\right) }{2}\left( \left( \frac{(T-2)nN_{-1} }{N_{-1}+N_{-2}(1-n)}\right) ^{\delta }\right. \\&\quad \left. + \cdots +\left( 2\left( \frac{nN_{-1}}{ N_{-1}+N_{-2}(1-n)}\right) ^{T-2}\right) ^{\delta }\right) \end{aligned}$$
That expression can be written as:
$$\begin{aligned}&T^{2}\alpha \left( \frac{N_{-1}-N_{-2}}{2}\right) \\&\quad \gtrless Q^{\sigma }T^{2-\sigma }\left[ \left( \frac{(N_{-2}+N_{-1})}{2} \right) ^{1-\sigma }-\left( N_{-1}\right) ^{1-\sigma }\right] \\&\quad +\,\frac{T(N_{-2}+N_{-1})}{2}\left( \left( \frac{(T-2)nN_{-1}}{ N_{-1}+N_{-2}(1-n)}\right) ^{\delta }+ \cdots \right. \\&\quad \left. +\left( 2\left( \frac{nN_{-1}}{ N_{-1}+N_{-2}(1-n)}\right) ^{T-2}\right) ^{\delta }\right) \\&\quad -TN_{-1}( ((T-1)n)^{\delta }+((T-2)n^{2})^{\delta }+ \cdots +((T-T+1)n^{T-1}) ^{\delta }) \end{aligned}$$
If \(N_{-2}=N_{-1}\) and \(\alpha \ge 0\), the LHS is equal to zero, while the RHS is negative, so that history \(\left\{ T,1,0\right\} \) is better. If \( N_{-1}>N_{-2}\), the LHS is positive, while the RHS is, under \(\alpha \ge 0\) , negative, so that history \(\left\{ T,1,0\right\} \) is better. If \( N_{-1}<N_{-2}\) and \(\alpha \ge 0\), the LHS is negative, and the RHS is undetermined. Under \(\alpha <0\), the sign of the RHS is undetermined. Hence whether history \(\left\{ T,1,0\right\} \) is preferred depends on whether the above condition holds, which depends on how large \(\alpha \) is with respect to T.
1.12 Proof of Proposition 8
Average welfare in the long-run population is, under history \(\left\{ T,1,0\right\} \), to (abstracting from time indexes):
$$\begin{aligned} Tu(q)= & {} T\left( \frac{Q}{L}\right) ^{\sigma }+T\alpha +( ((T-1)n)^{\delta }+((T-2)n^{2})^{\delta }+ \cdots +(1n^{T-1}) ^{\delta }) \\= & {} \left( TN_{-1}\right) ^{-\sigma }TQ^{\sigma }+T\alpha +(T-1)^{\delta }+(T-2)^{\delta }+ \cdots +(1)^{\delta } \end{aligned}$$
whereas it is equal, under history \(\left\{ T,0,1\right\} \), to:
$$\begin{aligned} Tu(q)= & {} T\left( \frac{Q}{L}\right) ^{\sigma }+T\alpha +( ((T-2)m)^{\delta }+((T-4)m^{2})^{\delta }+ \cdots +(2m^{T-2}) ^{\delta }) \\= & {} \left( \frac{T(N_{-2}+N_{-1})}{2}\right) ^{-\sigma }TQ^{\sigma }+[ T\alpha +(T-2)^{\delta }+(T-4)^{\delta }+ \cdots +(2)^{\delta }] \end{aligned}$$
Hence the ranking of the AU planner depends on:
$$\begin{aligned}&\left( TN_{-1}\right) ^{-\sigma }TQ^{\sigma }+T\alpha +(T-1)^{\delta }+(T-2)^{\delta }+ \cdots +(1)^{\delta } \\&\quad \gtrless \left( \frac{T(N_{-2}+N_{-1})}{2}\right) ^{-\sigma }TQ^{\sigma }+ [ T\alpha +(T-2)^{\delta }+(T-4)^{\delta }+ \cdots +(2)^{\delta }] \end{aligned}$$
When \(N_{-2}=N_{-1}\), the first terms of the LHS and RHS are equal, so that the LHS exceeds the RHS. Hence history \(\left\{ T,1,0\right\} \) is better. If \(N_{-2}>N_{-1}\), the LHS is also larger than the RHS, so that history \( \left\{ T,1,0\right\} \) is better. However, when \(N_{-2}<N_{-1}\), the first term of the RHS exceeds the first term of the LHS, but the second term of the RHS is smaller than the second term of the LHS, so the ranking is ambiguous.
1.13 Proof of Proposition 9
Under history \(\left\{ T,1,0\right\} \), social welfare at the stationary equilibrium is equal to (abstracting from time indexes):
$$\begin{aligned} LTu(q)= & {} TN_{-1}T\left[ u\left( \frac{Q}{L}\right) -\hat{u}\right] \\= & {} TN_{-1}T\left( \frac{Q}{L}\right) ^{\sigma }+TN_{-1}T( \alpha -\hat{u }) \\&+\,TN_{-1}( ((T-1)n)^{\delta }+((T-2)n^{2})^{\delta }+ \cdots +(1n^{T-1}) ^{\delta }) \\= & {} \left( TN_{-1}\right) ^{1-\sigma }TQ^{\sigma }+TN_{-1}T(\alpha -\hat{u} )\\&+\,TN_{-1}( ((T-1)n)^{\delta }+((T-2)n^{2})^{\delta }+ \cdots +(1n^{T-1}) ^{\delta }) \end{aligned}$$
Under history \(\left\{ T,0,1\right\} \), that formula becomes:
$$\begin{aligned}&\frac{T^{2}N_{-2}\left( 1+\frac{N_{-1}}{N_{-2}}\right) }{2}\left( \alpha -\hat{u} \right) +\left( \frac{TN_{-2}\left( 1+\frac{N_{-1}}{N_{-2}}\right) }{2}\right) ^{1-\sigma }TQ^{\sigma } \\&+\,\frac{TN_{-2}\left( 1+\frac{N_{-1}}{N_{-2}}\right) }{2}\left( \left( \frac{(T-2)nN_{-1} }{N_{-1}+N_{-2}(1-n)}\right) ^{\delta }\right. \\&\left. + \cdots +\left( (T-T+2)\left( \frac{ nN_{-1}}{N_{-1}+N_{-2}(1-n)}\right) ^{T-2}\right) ^{\delta }\right) \end{aligned}$$
Hence the ranking of the CLU planner depends on:
$$\begin{aligned}&\left( TN_{-1}\right) ^{1-\sigma }TQ^{\sigma }+TN_{-1}T(\alpha -\hat{u} )+TN_{-1}( ((T-1)n)^{\delta }+((T-2)n^{2})^{\delta }+ \cdots \\&\qquad +(1n^{T-1}) ^{\delta }) \\&\quad \gtrless \frac{T^{2}N_{-2}\left( 1+\frac{N_{-1}}{N_{-2}}\right) }{2}\left( \alpha -\hat{ u}\right) +\left( \frac{TN_{-2}\left( 1+\frac{N_{-1}}{N_{-2}}\right) }{2}\right) ^{1-\sigma }TQ^{\sigma } \\&\quad +\,\frac{TN_{-2}\left( 1+\frac{N_{-1}}{N_{-2}}\right) }{2}\left( \left( \frac{(T-2)nN_{-1} }{N_{-1}+N_{-2}(1-n)}\right) ^{\delta }+ \cdots \right. \\&\quad \left. +\left( 2\left( \frac{nN_{-1}}{ N_{-1}+N_{-2}(1-n)}\right) ^{T-2}\right) ^{\delta }\right) \end{aligned}$$
That expression can be rewritten as:
$$\begin{aligned}&(\alpha -\hat{u})T^{2}\left( \frac{N_{-1}-N_{-2}}{2}\right) \\&\quad \gtrless TQ^{\sigma }\left[ \left( \frac{T(N_{-2}+N_{-1})}{2}\right) ^{1-\sigma }-\left( TN_{-1}\right) ^{1-\sigma }\right] \\&\quad +\,\frac{T(N_{-2}+N_{-1})}{2}\left( \left( \frac{(T-2)nN_{-1}}{ N_{-1}+N_{-2}(1-n)}\right) ^{\delta }+ \cdots \right. \\&\quad \left. +\left( 2\left( \frac{nN_{-1}}{ N_{-1}+N_{-2}(1-n)}\right) ^{T-2}\right) ^{\delta }\right) \\&\quad -\,TN_{-1}( ((T-1)n)^{\delta }+((T-2)n^{2})^{\delta }+ \cdots +((T-T+1)n^{T-1}) ^{\delta }) \end{aligned}$$
If \(N_{-2}=N_{-1}\) and \(\alpha -\hat{u}\ge 0\), the LHS is equal to zero, while the RHS is negative, so that history \(\left\{ T,1,0\right\} \) is better. If \(N_{-1}>N_{-2}\), the LHS is positive, while the RHS is, under \( \alpha -\hat{u}\ge 0\), negative, so that history \(\left\{ T,1,0\right\} \) is better. If \(N_{-1}<N_{-2}\) and \(\alpha -\hat{u}\ge 0\), the LHS is negative, and the RHS is undetermined. Under \(\alpha -\hat{u}<0\), the sign of the RHS is undetermined. Hence whether history \(\left\{ T,1,0\right\} \) is preferred depends on whether the above condition holds, which depends on how large \(\alpha -\hat{u}\) is with respect to T.
1.14 Proof of Proposition 10
Under classical utilitarianism, total welfare for individuals born at \(t\ge 0\) is, without the transition:
$$\begin{aligned} \sum \limits _{t=1}^{\infty }n^{t}N_{-1}T\alpha +\sum \limits _{t=1}^{\infty }n^{t}N_{-1}\left[ \sum \limits _{s=t}^{t+T-1}\left( \frac{Q}{ \sum \nolimits _{z=0}^{T-s-1}N_{-(T-s-z)}+\left( N_{-1}\right) \sum _{r=1}^{s}n^{r}}\right) ^{\sigma }\right] \end{aligned}$$
whereas, under the transition, it is:
$$\begin{aligned}&mN_{-1}\left( \alpha +\left( \frac{Q}{mN_{-1}+N_{-1}+N_{-2}+ \cdots +N_{-(T-2)}} \right) ^{\sigma }\right) \\&\quad +\,mN_{-1}\left( \alpha +\left( \frac{Q}{mN_{-1}+N_{-1}+N_{-2}+ \cdots +N_{-(T-3)} }\right) ^{\sigma }\right) \\&\quad +\,mN_{-1}\left( \alpha +\left( \frac{Q}{ m^{2}N_{-1}+mN_{-1}+N_{-1}+N_{-2}+ \cdots +N_{-(T-4)}}\right) ^{\sigma }\right) \\&\quad +\,mN_{-1}\left( \alpha +\left( \frac{Q}{ m^{2}N_{-1}+mN_{-1}+N_{-1}+N_{-2}+ \cdots +N_{-(T-5)}}\right) ^{\sigma }\right) \\&\quad + \cdots +mN_{-1}\left( \alpha +\left( \frac{Q}{ m^{T/2}N_{-1}+ \cdots +m^{2}N_{-1}+mN_{-1}}\right) ^{\sigma }\right) \\&\quad +\,m^{2}N_{-1}\left( \alpha +\left( \frac{Q}{ m^{2}N_{-1}+mN_{-1}+N_{-1}+N_{-2}+ \cdots +N_{-(T-4)}}\right) ^{\sigma }\right) \\&\quad +\,m^{2}N_{-1}\left( \alpha +\left( \frac{Q}{ m^{2}N_{-1}+mN_{-1}+N_{-1}+N_{-2}+ \cdots +N_{-(T-5)}}\right) ^{\sigma }\right) \\&\quad + \cdots +m^{2}N_{-1}\left( \alpha +\left( \frac{Q}{ m^{(T/2)+1}N_{-1}+ \cdots +m^{2}N_{-1}}\right) ^{\sigma }\right) \\&\quad +\cdots \end{aligned}$$
This can be rewritten as:
$$\begin{aligned}&\sum _{t=1}^{\infty }m^{t}N_{-1}T\alpha +\sum \limits _{t=1}^{\infty }m^{t}N_{-1}\\&\quad \times \left[ \sum \limits _{s=t}^{t+T-1}\left( \frac{Q}{ \sum \nolimits _{z=0}^{T-s-1}N_{-(T-s-z)}+\left( N_{-1}\right) \sum _{r=1,3,5,...}^{s}m^{(r+1)/2}}\right) ^{\sigma }\right] \end{aligned}$$
Hence the transition is socially desirable iff:
$$\begin{aligned}&\sum \limits _{t=1}^{\infty }n^{t}N_{-1}T\alpha +\sum \limits _{t=1}^{\infty }n^{t}N_{-1}\left[ \sum \limits _{s=t}^{t+T-1}\left( \frac{Q}{ \sum \nolimits _{z=0}^{T-s-1}N_{-(T-s-z)}+\left( N_{-1}\right) \sum _{r=1}^{s}n^{r}}\right) ^{\sigma }\right] \\&\quad \gtrless \\&\quad \sum _{t=1}^{\infty }m^{t}N_{-1}T\alpha \\&\quad +\sum \limits _{t=1}^{\infty }m^{t}N_{-1}\left[ \sum \limits _{s=t}^{t+T-1}\left( \frac{Q}{ \sum \nolimits _{z=0}^{T-s-1}N_{-(T-s-z)}+\left( N_{-1}\right) \sum _{r=1,3,5,\ldots }^{s}m^{(r+1)/2}}\right) ^{\sigma }\right] \end{aligned}$$
Substituting for \(n=m\) and simplifying yields:
$$\begin{aligned}&\sum \limits _{t=1}^{\infty }n^{t}N_{-1}\left[ \sum \limits _{s=t}^{t+T-1} \left( \frac{Q}{\sum \nolimits _{z=0}^{T-s-1}N_{-(T-s-z)}+\left( N_{-1}\right) \sum _{r=1}^{s}n^{r}}\right) ^{\sigma }\right] \\&\quad \gtrless \\&\quad \sum \limits _{t=1}^{\infty }n^{t}N_{-1}\left[ \sum \limits _{s=t}^{t+T-1} \left( \frac{Q}{\sum \nolimits _{z=0}^{T-s-1}N_{-(T-s-z)}+\left( N_{-1}\right) \sum _{r=1,3,5,\ldots }^{s}n^{(r+1)/2}}\right) ^{\sigma }\right] \end{aligned}$$
Given that coexisting generations are always less numerous after the transition, we have that the RHS necessarily exceeds the LHS, leading to a socially desirable transition. The same argument holds when considering AU and CLU.
1.15 Proof of Proposition 11
When coexistence concerns, the condition becomes:
$$\begin{aligned}&\sum \limits _{t=1}^{\infty }n^{t}N_{-1}T\alpha +\sum \limits _{t=1}^{\infty }n^{t}N_{-1}\left[ \sum \limits _{s=t}^{t+T-1}\left( \frac{Q}{ \sum \nolimits _{z=0}^{T-s-1}N_{-(T-s-z)}+\left( N_{-1}\right) \sum _{r=1}^{s}n^{r}}\right) ^{\sigma }\right] \\&\quad +\sum \limits _{t=1}^{\infty }n^{t}N_{-1}( ((T-1)n)^{\delta }+((T-2)n^{2})^{\delta }+ \cdots +((T-T+1)n^{T-1}) ^{\delta }) \\&\quad \gtrless \\&\quad \sum _{t=1}^{\infty }n^{t}N_{-1}T\alpha \\&\quad +\sum \limits _{t=1}^{\infty }n^{t}N_{-1}\left[ \sum \limits _{s=t}^{t+T-1}\left( \frac{Q}{ \sum \nolimits _{z=0}^{T-s-1}N_{-(T-s-z)}+\left( N_{-1}\right) \sum _{r=1,3,5,...}^{s}n^{(r+1)/2}}\right) ^{\sigma }\right] \\&\quad +\sum \limits _{t=1}^{\infty }n^{t}N_{-1}\left( \left( \frac{(T-2)nN_{-1}}{ N_{-1}+N_{-2}(1-n)}\right) ^{\delta }\right. \\&\quad \left. + \cdots +\left( (T-T+2)\left( \frac{nN_{-1} }{N_{-1}+N_{-2}(1-n)}\right) ^{T-2}\right) ^{\delta }\right) \end{aligned}$$
Simplifications yield:
$$\begin{aligned}&\sum \limits _{t=1}^{\infty }n^{t}N_{-1}\left[ \sum \limits _{s=t}^{t+T-1} \left( \frac{Q}{\sum \nolimits _{z=0}^{T-s-1}N_{-(T-s-z)}+\left( N_{-1}\right) \sum _{r=1}^{s}n^{r}}\right) ^{\sigma }\right] \\&\quad +\sum \limits _{t=1}^{\infty }n^{t}N_{-1}( ((T-1)n)^{\delta }+((T-2)n^{2})^{\delta }+ \cdots +((T-T+1)n^{T-1}) ^{\delta }) \\&\quad \gtrless \\&\quad \sum \limits _{t=1}^{\infty }n^{t}N_{-1}\left[ \sum \limits _{s=t}^{t+T-1} \left( \frac{Q}{\sum \nolimits _{z=0}^{T-s-1}N_{-(T-s-z)}+\left( N_{-1}\right) \sum _{r=1,3,5,\ldots }^{s}n^{(r+1)/2}}\right) ^{\sigma }\right] \\&\quad +\sum \limits _{t=1}^{\infty }n^{t}N_{-1}\left( \left( \frac{(T-2)nN_{-1}}{ N_{-1}+N_{-2}(1-n)}\right) ^{\delta }\right. \\&\quad \left. + \cdots +\left( (T-T+2)\left( \frac{nN_{-1} }{N_{-1}+N_{-2}(1-n)}\right) ^{T-2}\right) ^{\delta }\right) \end{aligned}$$
The transition towards later births is not necessarily good: it is still true that the first term of the LHS is lower than the first term of the RHS. But the second term of the LHS is larger than the second term of the RHS. Hence the comparison depends on the relative strength of congestion versus coexistence concerns. The same condition holds for AU and CLU.
