Brown Growth, Green Growth, and the Efficiency of Urbanization

Abstract

We analyze the efficiency of urbanization patterns in a stylized dynamic model of urban growth with three sectors of production. Pollution, as a force that discourages agglomeration, is caused by domestic production. We show that cities are too large and too few in number in uncoordinated equilibrium if economic growth implies increasing pollution (‘brown growth’). If, however, production becomes cleaner over time (‘green growth’) the equilibrium urbanization path reaches the efficient urbanization path after finite time without need of a coordinating mechanism. We also generalize these results by taking other forms of congestion and urban land markets into account.

A Appendix

1.1 A.1 Characterizing the Equilibrium for Given $n_i$

The first-order condition for the problem to maximize $v_i$ with respect to $E_i$ is

$$\begin{aligned} \frac{d v_i}{dE_i}= & {} \left( (1-\mu _i)\,D_i(t)\,n_i^{(1+\psi _i)\mu _i}\,e_i^{\eta _i\,(1-\mu _i)+\mu _i}\,E_i^{-\mu _i}-p(t)\right) \,\frac{1}{n_i e_i}\,\left( 1-E_i\right) ^{\phi }\nonumber \\&-\left( D_i(t)\,n_i^{(1+\psi _i)\mu _i}\,e_i^{\eta _i\, (1-\mu _i)+\mu _i}\,E_i^{-\mu _i}-p(t)\right) \,\frac{\phi \,E_i}{n_i e_i}\,\left( 1-E_i\right) ^{\phi -1}=0.\qquad \end{aligned}$$

(A.1)

Rearranging leads to (9a). Note that (9a) has a unique solution, as the right-hand side is monotonically increasing in $E_i$ from zero at $E_i=0$ to infinity as $E_i\rightarrow 1/(1+\frac{\phi }{1-\mu _i})$.

The first-order condition for the problem to maximize $v_i$ with respect to $e_i$ is

$$\begin{aligned} \frac{d v_i}{de_i}= & {} \bigg ((1-\eta _i)\,(1-\mu _i)\,D_i(t) \,n_i^{(1+\psi _i)\mu _i}\,e_i^{\eta _i\,(1-\mu _i)+\mu _i} \,E_i^{-\mu _i}-p(t)\bigg )\\&\times \,\frac{-1}{e_i^{2}}\,\frac{E_i}{n_i}\,\left( 1 -E_i\right) ^{\phi }\geqq 0, \end{aligned}$$

with equality for an interior solution and strict inequality for the corner solution $e_i=e_i^{\max }$ (the lower corner solution $e_i=0$ is impossible, as then $c_i=0$). Using $E_i>0$, $E_i<1$, and $e_i>0$, this condition becomes (9b).

For an interior solution,

$$\begin{aligned} \frac{D_i(t)}{p(t)}\,n_i^{(1+\psi _i)\mu _i}\,e_i^{\mu _i+\eta _i\,(1-\mu _i)}&=\frac{E_i^{\mu _i}}{(1-\eta _i)\,(1-\mu _i)}. \end{aligned}$$

(A.2)

Combining this interior first-order condition for $e_i$ with the first-order condition for $E_i$,  (9a), we solve for $e_i$ and $E_i$ in the interior equilibrium as:

$$\begin{aligned} E_i =\frac{1}{1 +\frac{\phi }{1-\mu _i}\,\left( \frac{\mu _i+(1-\mu _i)\eta _i}{\eta _i}\right) } \equiv E_i^I \end{aligned}$$

(A.3)

and

$$\begin{aligned} e_i=\left( \frac{D_i(t)}{p(t)} \,n_i^{(1+\psi _i)\mu _i} \frac{(1-\eta _i)\,(1-\mu _i)}{(E_i^I)^{\mu _i}}\right) ^{-\frac{1}{\mu _i +(1-\mu _i)\,\eta _i}}, \end{aligned}$$

(A.4)

which substituted into the utility function implies

$$\begin{aligned} v_i(n_i)= & {} \left( n_i^{ -(\eta _i (1-\mu _i)-\mu _i \psi _i)} {p(t)^{-(1-\eta _i)(1-\mu _i)}} \,{D_i(t)} \right) ^{\frac{1}{\mu _i +(1-\mu _i)\,\eta _i}} \nonumber \\&\times (\xi _i-1)^{\xi _i - 1} \xi _i^{-\xi _i} (E_i^I)^{1-\mu _i \xi _i} (1-E_i^I)^\phi , \end{aligned}$$

(A.5)

where $\xi _i \equiv 1/(\mu _i+(1-\mu _i)\eta _i)$. Equation (A.5) is equation (10b). Equation (A.4) gives $e_i$ as a decreasing function of $n_i$, so that the interior solution can apply only if the expression is below $e_i^{\max }$. Hence, we find:

$$\begin{aligned} e_i = e_i^{\max } \Leftrightarrow n_i \le \left( \frac{p(t) }{D_i(t)} \frac{ \left[ E_i^I \right] ^{\mu _i} }{(e_i^{\max })^{\mu _i+(1-\mu _i)\,\eta _i}\,\left( 1- \mu _i)(1-\eta _i \right) }\right) ^{\frac{1}{(1+\psi _i)\mu _i}} \equiv n_i^I \end{aligned}$$

(A.6)

For a corner solution, the equilibrium level of $E_i$ is given as the solution of (9a) with $e_i=e_i^{\max }$. A similar argument as in Appendix A.2 shows that the optimal pollution level is increasing with $e_i^{\max }$.

1.2 A.2 Proof that Efficient Pollution Level is Non-decreasing with Population

If the optimal emission intensity is in the interior, ${\hat{e}}_i<e_i^{\max }$, the efficient level of pollution is independent of $n_i$ (cf. Appendix A.3). For the corner solution ${\hat{e}}_i=e_i^{\max }$, we take the total differential of equation (9a). This yields

$$\begin{aligned}&(D/p)\,(e_i^{\max })^{\eta _i\,(1-\mu _i)+\mu _i}\,(1+\psi _i)\, \mu _i\, n_i^{(1+\psi _i)\,\mu _i-1}\,dn_i\nonumber \\&\quad =\left( \frac{\mu _i\,{\hat{E}}_i^{\mu _i-1}}{(1-\mu _i)}\, \frac{1-(1+\phi )\,{\hat{E}}_i}{1-(1+\frac{\phi }{1-\mu _i})\,{\hat{E}}_i} +\frac{{\hat{E}}_i^{\mu _i}}{(1-\mu _i)}\,\frac{\phi \, \frac{\mu _i}{1-\mu _i}}{\left( 1-(1+\frac{\phi }{1-\mu _i}) \,{\hat{E}}_i\right) ^2} \right) \,dE_i\qquad \end{aligned}$$

(A.7)

For ${\hat{E}}_i<1/(1+\phi /(1-\mu _i))$, both the term in front of $dn_i$ on the left hand side and the term in front of $dE_i$ on the right hand side are strictly positive so that $dE_i/dn_i>0$. For $E_i=0$, we have $n_i=0$; for $E_i\rightarrow 1/(1+\phi /(1-\mu _i))$, we have $n_i\rightarrow \infty$ from condition (9a). Hence in the corner solution, $dE_i/dn_i>0$ for all positive values of $n_i$ and a fortiori for the relevant range $n_i \in (0,n_i^I)$. A similar argument leads to the conclusion that ${\hat{E}}_i$ increases if $D_i(t)/p(t)$ increases, while $n_i$ remains constant.

1.3 A.3 Proof of Result 1

By the envelope theorem, the total derivative of utility with respect to population equals the partial derivative of utility with respect to population. Hence,

$$\begin{aligned} \frac{dv_i(n_i)}{dn_i}= & {} \frac{\partial v_i(n_i)}{\partial n_i} = -\frac{1}{n_i^2}\,\left( D_i(t)\,n_i^{(1+\psi _i)\mu _i} \,e_i^{\eta _i\,(1-\mu _i)+\mu _i}\,E_i^{-\mu _i}-p(t)\right) \,\frac{E_i}{e_i}\left( 1-E_i\right) ^\phi \nonumber \\&+\frac{1}{n_i^2}\,\mu _i\,(1+\psi _i)\, D_i(t)\,n_i^{(1+\psi _i) \mu _i}\,e_i^{\eta _i\,(1-\mu _i)+\mu _i}\,\frac{E_i^{1-\mu _i}}{e_i} \,\left( 1-E_i\right) ^\phi \nonumber \\= & {} -\frac{1}{n_i^2}\,\left( (1-\mu _i\,(1+\psi _i))\,D_i(t) \,n_i^{(1+\psi _i)\mu _i}\,e_i^{\eta _i\,(1-\mu _i)+\mu _i} \,E_i^{-\mu _i}-p(t)\right) \,\frac{E_i}{e_i}\nonumber \\&\times \left( 1-E_i\right) ^\phi \nonumber \\= & {} \frac{1}{n_i}\left[ -1+\frac{\mu _i\,(1+\psi )\,D_i(t) \,n_i^{(1+\psi _i)\mu _i}\,e_i^{\eta _i\,(1-\mu _i)+\mu _i} \,E_i^{-\mu _i}}{D_i(t)\,n_i^{(1+\psi _i)\mu _i}\,e_i^{\eta _i \,(1-\mu _i)+\mu _i}\,E_i^{-\mu _{i}}-p(t)}\right] \,v_i(n_i) \end{aligned}$$

(A.8)

Thus, utility has an interior maximum, if $1-\mu _i\,(1+\psi _i)=1-\mu _i+\frac{1-\mu _i}{\sigma _i-1}=\frac{1-\mu _i}{\sigma _i-1}\,\left( \sigma _i-2\right) >0$, which is true by the above assumption $\sigma _i>2$. Otherwise, utility would be monotonically increasing with $n_i$ due to very strongly increasing returns to scale. The condition for an interior extremum is

$$\begin{aligned} (1-(1+\psi _i)\mu _i)\,\frac{D_i(t)}{p(t)}\,n_i^{(1+\psi _i) \mu _i}\,e_i^{\eta _i\,(1-\mu _i)+\mu _i} =E_i^{\mu _i} \end{aligned}$$

(A.9)

Note that an interior maximum must be at a population size where optimal emission intensity is at the corner ${\hat{e}}_i=e_i^{\max }$, as utility is montonically decreasing with $n_i$ for an interior solution of emission intensity as becomes clear from equation (10b) under our assumption $1<(\sigma -1)\eta \Leftrightarrow \mu \psi < (1-\mu )\eta$. Combining (9a) and (A.9), we find that the pollution level at the interior extremum of utility satisfies

$$\begin{aligned} \frac{1-\mu _i\,(1+\psi _i)}{1-\mu _i}\,\frac{1-(1+\phi ) \,E_i}{1-(1+\frac{\phi }{1-\mu _i})\,E_i}=1, \end{aligned}$$

(A.10)

which can be rearranged to obtain

$$\begin{aligned} E_i=\frac{\psi _i}{\phi +\psi _i+\phi \,\psi _i} \equiv E_i^{FB}. \end{aligned}$$

(A.11)

Plugging into (A.9), we obtain the result (11).

To show that (11) is a maximum of utility, we show that the elasticity of utility with respect to $n_i$ is monotonically declining from a positive to a negative value when $n_i$ increases from zero to infinity, so that utility has a single peak at $n_i=n_i^{\text {opt}}$. First we rewrite (A.8) as an elasticity condition

$$\begin{aligned} \frac{dv_i (n_i)}{dn_i} \frac{n_i}{v_i}= \frac{f_i^* - f_i}{f_i^*(f_i-1)} \end{aligned}$$

(A.12)

with

$$\begin{aligned} f_i \equiv (D_i/p)n_i^{(1+\psi _i)\mu _i}(e_i)^{\mu _i+(1-\mu _i)\eta _i} E_i^{-\mu _i} \end{aligned}$$

(A.13)

and

$$\begin{aligned} f_i^* \equiv (1-(1+\psi _i)\mu _i)^{-1} \end{aligned}$$

(A.14)

Thus, the elasticity declines monotonically with $f_i$. From (9a) we know that $f_i$ is non-decreasing in $E_i$ and from Appendix A.2 we know that $E_i$ is non-decreasing in $n_i$. Hence, the elasticity of utility with respect to $n_i$ is non-increasing in $n_i$. Equation (9a) shows that the lower bound for $f_i$ is $1/(1-\mu _i)$, which substituted in (A.12) gives that $\psi _i>0$ is the upper bound of the elasticity of utility with respect to $n_i$. Equation (9b) shows that for $n_i$ critically large, an interior solution implies, so that $f_i=1/(1-\eta _i)(1-\mu _i)$ and the elasticity of utility with respect to $n_i$ becomes the constant $-[(1-\mu _i)\eta _i - \psi _i \mu _i]/[\mu _i+(1-\mu _i)\eta _i]<0$, consistent with both (A.12) and (10b). Hence the elasticity is negative for all $n_i>n_i^{\text {opt}}$ so that $\lim \limits _{n_i\rightarrow \infty }v_i(n_i)=0$.

To determine utility for $n_i=0$, we use the lower bounds ($f_i=1/(1-\mu _i)$, $E_i=0$, $e_i=e_i^{\max }$, $n_i=0$) into (10a) to find

$$\begin{aligned} \lim \limits _{n_i\rightarrow 0}v_i(n_i) = \lim \limits _{n_i\rightarrow 0}\frac{p}{e_i^{\max }} \frac{\mu _i}{1-\mu _i} \frac{E_i}{n_i}, \end{aligned}$$

(A.15)

and use $f_i=1/(1-\mu _i)$ into (A.13) to find

$$\begin{aligned} \frac{E_i}{n_i}=\left( (1-\mu _i) \frac{D_i}{p} (e_i^{\max })^{\mu _i+(1-\mu _i)\eta _i} \right) ^{1/\mu _i}n_i^{\psi _i}. \end{aligned}$$

(A.16)

Combining the two displayed results we find $\lim \limits _{n_i\rightarrow 0}v_i(n_i)=0$. We state this finding in Result 1.

Similarly we find $\lim \limits _{n_i\rightarrow 0}v'_i(n_i) \propto \psi _i (E_i/n_i^2)\propto n_i^{\psi _i-1}$. This result implies that a sufficient condition for stability of the Malthusian urbanization pattern is $\psi _i<1$, so that $v'_i(0)$ is infinitely large and small new cities attract workers from large old cities.

1.4 A.4 Considering the Urban Spatial Structure

We consider the following extension where urban spatial structure is described by an Alonso-Muth-Mills monocentric city model (e.g. Fujita 1989), closely following Behrens et al. (2014, Appendix F). Production does not need space and is concentrated in the central business district (CBD) in the city center. Each worker commutes from his place of residence in a distance z from the CBD, causing transport costs $T(z)=\tau (t)\,\frac{1+\gamma }{\gamma }\,2^\gamma \,z^\gamma$, with $\tau (t),\gamma >0$, and $\gamma \le (1+\psi _i)\,\mu _i-1$. We allow the transport cost parameter $\tau (t)$ to change over time, capturing potential improvements in commuting technology. We consider a linear city with residences of unit length, such that the total extension of a city with $n_i$ inhabitants is $z^*=n_i/2$ to both sides of the CBD. In residential equilibrium, all households have the same income net of land rent r(z) and commuting costs T(z), i.e. $r(z)+T(z)$ is a constant. Normalizing opportunity costs of land to zero, $r(z^*)=0$, and thus $r(z)=T(z^*)-T(z)$. All differential rent is taxed and redistributed in a lumb-sum fashion to current urban citizens. Aggregate differential land rent is

$$\begin{aligned} A(n_i)=2\,\int _0^{z^*} r(z)\,dz=\tau (t)\,\frac{1+\gamma }{\gamma }\,2^{1+\gamma }\,\int _0^{z^*} \left( (z^*)^\gamma -z^\gamma \right) \,dz=\tau (t)\,n_i^{1+\gamma } \end{aligned}$$

(A.17)

Disposable income of an individual living at a distance z from the CBD thus is increased by a share $n_i$ in $A(n_i)$ and decreased by $r(z)+T(z)=T(z^*)$. Equation (8) is thus replaced by

$$\begin{aligned} c_i(t)=\;&\frac{1}{n_i}\,\left( D_i(t)\,n_i^{(1+\psi _i) \,\mu _i}\,e_i^{\eta _i\,(1-\mu _i)+\mu _i}\,\frac{E_i^{1 -\mu _i}}{e_i}-p(t)\,\frac{E_i}{e_i}-{\hat{T}}_i(n_i)\right) , \end{aligned}$$

(A.18)

where

$$\begin{aligned} {\hat{T}}_i(n_i)=A(n_i)-n_i\,T(z^*)=\tau (t) \,n_i^{1+\gamma }-n_i\,\tau (t)\,\frac{1+\gamma }{\gamma } \,2^\gamma \,\left( \frac{n_i}{2}\right) ^\gamma =\frac{\tau (t)}{\gamma } \,n_i^{1+\gamma } \end{aligned}$$

(A.19)

Within the model, the following variant of Result 1 holds:

Result 7

When pollution is at a locally optimal level, utility has a unique interior maximum at a population size determined as the solution of

$$\begin{aligned} (1-\mu _i\,(1+\psi _i))\,D_i(t)\,n_i^{(1+\psi _i)\,\mu _i} \,e_i^{\eta _i\,(1-\mu _i)+\mu _i}\,\frac{{\hat{E}}_i^{1-\mu _i}}{e_i}-p(t) \,\frac{{\hat{E}}_i}{e_i}-\tau (t)\,n_i^{1+\gamma }=0\qquad \quad \end{aligned}$$

(A.20)

Furthermore, utility $v_i(n_i)$ tends to zero for $n_i\rightarrow 0$ and $n_i\rightarrow \infty$.

The optimal population size decreases with the level of economic development $D_i(t)$, and increases with the relative import price, p(t).

Proof

The proof is structured as follws. We first characterize optimal pollution and its comparative statics. Then we derive the optimal population size and show that it is a maximum. Finally we derive the comparative static results of $n_i^{opt}$.

Optimal pollution is determined by the condition

$$\begin{aligned} \frac{dv_i}{dE_i}= & {} \left( (1-\mu _i)\,D_i(t)\,n_i^{(1+\psi _i)\,\mu _i} \,e_i^{\eta _i\,(1-\mu _i)+\mu _i}\,E_i^{-\mu _i}-p(t)\right) \,\frac{1}{e_i} \,\left( 1-E_i\right) ^{\phi }\\&-\left( D_i(t)\,n_i^{(1+\psi _i)\,\mu _i}\,e_i^{\eta _i\,(1-\mu _i) +\mu _i}\,\frac{E_i^{1-\mu _i}}{e_i}-p(t)\,\frac{E_i}{e_i} -{\hat{T}}_i(n_i)\right) \,\phi \,\left( 1-E_i\right) ^{\phi -1}=0. \end{aligned}$$

Rearranging leads to

$$\begin{aligned} \frac{D_i(t)}{p(t)}\,n_i^{(1+\psi _i)\,\mu _i}\,e_i^{\eta _i \,(1-\mu _i)+\mu _i}=\frac{{\hat{E}}_i^{\mu _i}}{1-\mu _i}\,\frac{1 -(1+\phi )\,{\hat{E}}_i-\frac{e_i}{p(t)}\,\phi \,{\hat{T}}_i(n_i)}{1 -\left( 1+\frac{\phi }{1-\mu _i}\right) \,{\hat{E}}_i}. \end{aligned}$$

(A.21)

The derivative of the last term on the right-hand side of this condition with respect to ${\hat{E}}_i$ is

$$\begin{aligned} \frac{d}{d{\hat{E}}_i} \frac{1-(1+\phi )\,{\hat{E}}_i -\frac{e_i}{p(t)}\,\phi \,{\hat{T}}_i(n_i)}{1-\left( 1+\frac{\phi }{1 -\mu _i}\right) \,{\hat{E}}_i}=\frac{\frac{\mu }{1-\mu }\,\phi -\left( 1+\frac{\phi }{1-\mu _i}\right) \,\frac{e_i}{p(t)} \,\phi \,{\hat{T}}_i(n_i)}{\left( 1-\left( 1+\frac{\phi }{1-\mu _i}\right) \,{\hat{E}}_i\right) ^2} \end{aligned}$$

(A.22)

A positive level of consumption requires (cf. Eq. A.18)

$$\begin{aligned} e_i\,{\hat{T}}_i(n_i)&<p(t)\,{\hat{E}}_i\,\left( \frac{D_i(t)}{p(t)} \,n_i^{(1+\psi _i)\,\mu _i}\,e_i^{\eta _i\,(1-\mu _i)+\mu _i} \,{\hat{E}}_i^{-\mu _i}-1\right) \nonumber \\&\mathop {=}\limits ^{(A.21)}{\hat{E}}_i \,\left( \frac{1}{1-\mu _i}\,\frac{1-(1+\phi )\,{\hat{E}}_i-\frac{e_i}{p(t)} \,\phi \,{\hat{T}}_i(n_i)}{1-\left( 1+\frac{\phi }{1-\mu _i}\right) \,{\hat{E}}_i}-1\right) \nonumber \\&\Leftrightarrow \frac{e_i}{p(t)}\,\phi \,{\hat{T}}_i(n_i) <\frac{\mu _i}{1-\mu _i}\,\phi \,{\hat{E}}_i. \end{aligned}$$

(A.23)

Using this result in (A.22), we find that the right hand side (RHS) of (A.21) is increasing with ${\hat{E}}_i$, as

$$\begin{aligned}&\frac{\mu }{1-\mu }\,\phi -\left( 1+\frac{\phi }{1-\mu _i}\right) \,\frac{e_i}{p(t)}\,\phi \,{\hat{T}}_i(n_i)\nonumber \\&\quad>\frac{\mu }{1-\mu }\,\phi \,\left( 1-\left( 1+\frac{\phi }{1-\mu _i}\right) \,{\hat{E}}_i\right) >0. \end{aligned}$$

(A.24)

Thus, we have the following comparative static results: $\partial {\hat{E}}_i/\partial n_i>0$—the partial of the left-hand side (LHS) of (A.21) with respect to $n_i$ is positive, while the partial of the RHS with respect to $n_i$ is negative; $\partial {\hat{E}}_i/\partial D_i>0$—the partial of the LHS with respect to $D_i$ is positive; $\partial {\hat{E}}_i/\partial p(t)<0$—the partial of the LHS with respect to p(t) is negative, while the partial of the RHS with respect to p(t) is positive; and $\partial {\hat{E}}_i/\partial \tau (t)>0$—the partial of the RHS with respect to $\tau (t)$ is negative.

The properties of $v_i(n_i)$ at the corners $n_i=0$ and $n_i=\infty$ follow according to exactly the same logic as for Result 1.

By the envelope theorem, the total derivative of utility with respect to population equals the partial derivative of utility with respect to population. Hence,

$$\begin{aligned} \frac{dv_i(n_i)}{dn_i}= & {} \frac{\partial v_i(n_i)}{\partial n_i}\\= & {} -\frac{1}{n_i^2}\,\left( D_i(t)\,n_i^{(1+\psi _i)\,\mu _i} \,e_i^{\eta _i\,(1-\mu _i)+\mu _i}\,\frac{{\hat{E}}_i^{1-\mu _i}}{e_i}-p(t) \,\frac{{\hat{E}}_i}{e_i}-{\hat{T}}_i(n_i)\right) \,\left( 1-{\hat{E}}_i\right) ^\phi \nonumber \\&+\frac{1}{n_i^2}\,\left( \mu _i\,(1+\psi _i)\,D_i(t)\,n_i^{(1+\psi _i) \,\mu _i}\,e_i^{\eta _i\,(1-\mu _i)+\mu _i}\,\frac{{\hat{E}}_i^{1-\mu _i}}{e_i} -n_i\,{\hat{T}}_i'(n_i)\right) \nonumber \\&\times \left( 1-{\hat{E}}_i\right) ^\phi \nonumber \\= & {} -\frac{\left( 1-{\hat{E}}_i\right) ^\phi }{n_i^2}\, \bigg ((1-\mu _i\,(1+\psi _i))\,D_i(t)\,n_i^{(1+\psi _i)\,\mu _i} \,e_i^{\eta _i\,(1-\mu _i)+\mu _i}\,\frac{{\hat{E}}_i^{1-\mu _i}}{e_i}\nonumber \\&-p(t)\,\frac{{\hat{E}}_i}{e_i}-{\hat{T}}_i(n_i)+n_ i\,{\hat{T}}_i'(n_i)\bigg ) \end{aligned}$$

Using $-{\hat{T}}_i(n_i)+n_i\,{\hat{T}}_i'(n_i)=\tau (t)\,n_i^{1+\gamma }$ from (A.19) and defining

$$\begin{aligned} \Psi (n_i)=(1-\mu _i\,(1+\psi _i))\,D_i(t)\,n_i^{(1+\psi _i) \,\mu _i}\,e_i^{\eta _i\,(1-\mu _i)+\mu _i}\,\frac{{\hat{E}}_i^{1 -\mu _i}}{e_i}-p(t)\,\frac{{\hat{E}}_i}{e_i}-\tau (t)\,n_i^{1+\gamma }, \end{aligned}$$

(A.25)

and with $n_i^{opt}$ to denote the optimal population size, the first-order condition for the optimal population size is $\Psi (n_i^{opt})=0$, as stated in (A.20). This is a maximum of utility, as second derivative is negative:

$$\begin{aligned} \frac{d^2v_i(n_i^{opt})}{dn_i^2}=-\frac{\left( 1-{\hat{E}}_i \right) ^\phi }{n_i^2}\,\Psi '(n_i^{opt})&<0. \end{aligned}$$

(A.26)

$\Psi '(n_i^{opt})>0$ is true, because in the derivative of $\Psi (n_i)$ with respect to $n_i$, evaluated at $n_i^{opt}$,

$$\begin{aligned} \Psi '(n_i^{opt})&=(1+\psi _i)\,\mu _i\, (1-\mu _i\,(1+\psi _i))\,D_i(t) \,n_i^{(1+\psi _i)\,\mu _i-1}\,e_i^{\eta _i\,(1-\mu _i)+\mu _i} \,\frac{{\hat{E}}_i^{1-\mu _i}}{e_i}\nonumber \\&\quad -(1+\gamma )\,\tau (t)\,n_i^{\gamma }\nonumber \\&\quad +\,\left( (1-\mu _i)\,(1-\mu _i\,(1+\psi _i))\,D_i(t)\,n_i^{(1+\psi _i) \,\mu _i}\,e_i^{\eta _i\,(1-\mu _i)+\mu _i}\,\frac{{\hat{E}}_i^{-\mu _i}}{e_i} -\frac{p(t)}{e_i}\right) \,\frac{\partial {\hat{E}}_i}{\partial n_i}, \end{aligned}$$

(A.27)

The expression in the second line is larger or equal to $(1+\psi _i)\,\mu _i\,p(t)\,\frac{{\hat{E}}_i}{e_i\,n_i^{opt}}>0$ (using the above assumption $(1+\psi _i)\,\mu _i\ge 1+\gamma$), and the expression in brackets in the last line is positive, as follows from the first-order condition for the optimal level of ${\hat{E}}_i$, and $\partial {\hat{E}}_i/\partial n_i>0$.

The comparative statics of $n_i^{opt}$ with respect to $D_i(t)$ is found by differentiating (A.20) with respect to $D_i(t)$

$$\begin{aligned}&\left( (1-\mu _i\,(1+\psi _i))\,(1-\mu _i)\,D_i(t)\,n_i^{(1+\psi _i) \,\mu _i}\,e_i^{\eta _i\,(1-\mu _i)}\,\left( \frac{{\hat{E}}_i}{e_i}\right) ^{-\mu _i} -p(t)\right) \,\frac{1}{e_i}\,\frac{\partial {\hat{E}}_i}{\partial D_i(t)}\nonumber \\&\quad +\,(1-\mu _i\,(1+\psi _i))\,n_i^{(1+\psi _i)\,\mu _i}\,e_i^{\eta _i \,(1-\mu _i)+\mu _i}\,\frac{{\hat{E}}_i^{1-\mu _i}}{e_i} +\Psi '(n_i^{opt})\,\frac{dn_i^{opt}}{d D_i(t)}=0 \end{aligned}$$

(A.28)

The expression in brackets in the first term is negative: Using (A.23) and (A.25), we have

$$\begin{aligned}&(1-\mu _i\,(1+\psi _i))\,(1-\mu _i)\,D_i(t)\,n_i^{(1+\psi _i)\,\mu _i} \,e_i^{\eta _i\,(1-\mu _i)+\mu _i}\,\frac{{\hat{E}}_i^{-\mu _i}}{e_i}-p(t)\\&\quad =(1-\mu _i)\,\left( p(t)+\frac{e_i\,\tau (t)\,n_i^{1+\gamma }}{{\hat{E}}_i}\right) -p(t)\\&\quad <(1-\mu _i)\,\left( p(t)+\frac{\frac{\mu _i}{1-\mu _i}\,p(t)\,{\hat{E}}_i}{{\hat{E}}_i}\right) -p(t)=0 \end{aligned}$$

We thus can use the following to get the sign of $\partial n_i^{opt}/\partial D_i(t)$,

$$\begin{aligned} \frac{\partial {\hat{E}}_i}{\partial D_i(t)}&=\frac{1}{D_i(t)}\,\frac{\frac{D_i(t)}{p(t)}\,n_i^{(1+\psi _i) \,\mu _i}\,e_i^{\eta _i\,(1-\mu _i)+\mu _i}}{\frac{\partial }{\partial {\hat{E}}_i}\left( \frac{{\hat{E}}_i^{\mu _i}}{1-\mu _i}\,\frac{1-(1+\phi ) \,{\hat{E}}_i-\frac{e_i}{p(t)}\,\phi \,{\hat{T}}_i(n_i)}{1-\left( 1+\frac{\phi }{1-\mu _i}\right) \,{\hat{E}}_i}\right) } <\frac{{\hat{E}}_i}{\mu _i\,D_i(t)} \end{aligned}$$

(A.29)

Thus the first two terms in (A.28) are positive, as

$$\begin{aligned}&\left( (1-\mu _i\,(1+\psi _i))\,(1-\mu _i)\,D_i(t)\,n_i^{(1+\psi _i)\,\mu _i} \,e_i^{\eta _i\,(1-\mu _i)}\,\left( \frac{{\hat{E}}_i}{e_i} \right) ^{-\mu _i}-p(t)\right) \,\frac{1}{e_i}\,\frac{\partial {\hat{E}}_i}{\partial D_i(t)}\nonumber \\&\qquad +\,(1-\mu _i\,(1+\psi _i))\,n_i^{(1+\psi _i)\,\mu _i}\,e_i^{\eta _i \,(1-\mu _i)+\mu _i}\,\frac{{\hat{E}}_i^{1-\mu _i}}{e_i}\nonumber \\&\quad =\left( (1-\mu _i)\,\frac{e_i\,\tau (t)\,n_i^{1+\gamma }}{{\hat{E}}_i} -\mu _i\,p(t)\right) \,\frac{1}{e_i}\,\frac{\partial {\hat{E}}_i}{\partial D_i(t)}+\frac{p(t)\,{\hat{E}}_i}{e_i\,D_i(t)} +\frac{\tau (t)\,n_i^{1+\gamma }}{D_i(t)}\nonumber \\&\quad>\left( (1-\mu _i)\,\frac{e_i\,\tau (t)\,n_i^{1+\gamma }}{{\hat{E}}_i} -\mu _i\,p(t)\right) \,\frac{{\hat{E}}_i}{e_i\,\mu _i\,D_i(t)} +\frac{p(t)\,{\hat{E}}_i}{e_i\,D_i(t)}+\frac{\tau (t) \,n_i^{1+\gamma }}{D_i(t)}\nonumber \\&\quad =\frac{\tau (t)\,n_i^{1+\gamma }}{\mu _i\,D_i(t)}>0 \end{aligned}$$

(A.30)

This implies that the optimal population size is decreasing with $D_i(t)$, as $\Psi '(n_i^{opt})>0$. In a similar fashion it can be shown that the optimal population size is increasing in p(t). $\square$