A particular ram: local deviation from the maximin path

The maximin program prescribes a constant utility path in regular problems, and is not consistent with a growth pattern. Rawls (Reference Rawls1971) acknowledged that economic growth may be necessary to reach a state of the economy in which material resources are sufficient to implement a ‘just society.’ He pointed out that the maximin principle must be modified to allow for economic growth. We propose local deviations from the maximin program that allow for growth. These deviations are based on a reduced utility that results in extra savings.

Formally, these deviations are constructed as follows. Denote the co-state variables of the capital stocks in a maximin problem by $\mu _{i}$, $i=1,\ldots ,n$. The maximin problem can be mathematically expressed as the maximization of the Hamiltonian $H(X,c)=\sum _{i=1}^{n}\mu _{i}\dot {X}_{i}$, subject to the constraint $U(X,c)\geq m(X)$ (Cairns and Long, Reference Cairns and Long2006). It is equivalent to maximizing the Lagrangean:

\begin{align*} L(c,X,\nu ) &= H(X,c)+\nu \left( U(X,c)-m(X)\right) \\ &= \sum_{i=1}^{n}\mu _{i}\dot{X}_{i}+\nu \left( U(X,c)-m(X)\right), \end{align*}

where $\nu$ is the dual variable of the equity constraint $U(X,c)\geq m(X)$. The term $\nu ( U(X,c)-m(X))$ satisfies the complementarity slackness condition, and is always equal to zero. Cairns and Long (Reference Cairns and Long2006, proposition 1) show that the co-state variables of a maximin problem are equal to the derivatives of the maximin value function with respect to the state variables: $\mu _{i}={\partial m}/{\partial X_{i}}$. Therefore, $M( X,c) =\sum _{i=1}^{n}{\partial m( X) }/{\partial X_{i}}\dot {X}_{i}=\sum _{i=1}^{n}\mu _{i}\dot {X}_{i}$, and the previous Lagrangean can be written as follows.

(6)\begin{equation} L(c,X,\nu )=M\left( X,c\right) +\nu \left( U(X,c)-m(X)\right). \end{equation}

Equation (6) yields a new interpretation of the maximin problem and its point-wise conditions: the maximin problem is tantamount to maximizing the net investment at maximin shadow values, subject to the constraint that consumption is no less than the maximin value.Footnote ⁴ We use this interpretation in characterizing the paths that deviate from maximin according to a parametric policy of consumption growth toward an eventual maximin path.

The maximin solution can be interpreted as follows. Along a regular maximin path, utility equals the highest level that is compatible with the satisfaction of the equity constraint (3). The consumption and investment decisions result in an egalitarian and efficient utility path. Along such a path, the Hamiltonian is nil (the Hartwick (Reference Hartwick1977) rule is satisfied), i.e., $H(X,c)=M(X,c) =\sum _{i=1}^{n}\mu _{i} \dot {X}_{i}=0$. Net investment is maximized but takes a value of zero. There is no room for growth.

If utility is lower than the maximin level, the resources freed up can be invested to increase the maximin value, and hence the sustainability capacity of the economy. We consider ‘deviations’ from the maximin problem, in the sense that the sustainability constraint is relaxed and current utility is made lower than the maximin level. We explore the case in which the ram maximizes maximin investment subject to a postulated time path of the utility levels $\bar U(t)$. This program is equivalent to maximizing the modified Lagrangean,

\[ \tilde{L}(c,X,\tilde{\nu})=M \left( X,c\right)+\tilde{\nu}\left( U(X,c)-\bar U(t)\right) \;. \]

That is to say, the program is to maximize $M ( X,c)$ subject to a modified constraint, $U(X,c)\geq \bar U(t)$, $t\geq 0$. The modified complementarity slackness condition is again equal to zero.

Definition 1 Instantaneous maximization of sustainability improvement. The ram is said to maximize sustainability improvement at current time if decisions $c$ maximize the increase of the maximin value subject to the given current utility:

(7)\begin{align} c \quad & \mathrm{maximizes} \quad M\left( X,c\right) =\sum_{i=1}^{n}F_{i}(X,c)\frac{ \partial m\left( X\right) }{\partial X_{i}} \\ &\mathrm{s.t.} \quad U(X,c)\geq \bar{U}(t). \nonumber \end{align}

The interpretation of this particular ram is that the current generation consumes less than the sustainable, maximin level and relaxes the limit to growth as much as possible, given its own target utility. This mechanism has the advantage of generating trajectories along which the maximin value increases over time so long as the utility is lower than the maximin value. This program is in line with our focus on sustainable growth.Footnote ⁵

Effect of current sacrifice on instantaneous sustainability improvement

We first examine the effect of current reduction of consumption with respect to the maximin level (steady-state consumption) on sustainability improvement.

In this simple fishery model, for a capital stock $S < S_{MSY}$, maximin improvement is given by

(10)\begin{equation} M(S,E)=\frac{dm(S)}{dt}=\frac{dm(S)}{dS}\dot{S}=(1-2S)(S(1-S)-SE)\;. \end{equation}

Equation (10) gives the sustainability improvement as a function of the fishing effort $E$ and current stock $S$. This equation can be interpreted as the product of two effects: a current stock effect $(1-2S)$, corresponding to the marginal productivity of the stock, and a sacrifice effect corresponding to the net investment $(S(1-S)-SE)$ due to the foregone consumption with respect to the maximin level. Along a maximin path, one would have no sacrifice, with the full consumption of the produced resource, and a stationary resource stock, i.e., $SE=S(1-S)$ and $\dot {S}=0$. The greater the sacrifice with respect to current maximin value, the larger the sustainability improvement. Also, the smaller the current resource stock, the larger the maximin improvement for a given sacrifice. At or above the MSY stock $S_{MSY}$, sacrifices have no effect.

Long-run recovery under constant effort

We now consider the long-run effect of a reduction of the fishing effort, and thus of consumption, with respect to the sustainable, maximin level. For illustrative purposes, we consider a very simple ram, with constant fishing effort.Footnote ⁸

Let a level of effort be chosen and remain constant at the level $E_0\in [0,1]$. Such a strategy could aim at increasing the available resource and sustainable consumption while maintaining an acceptable level of employment in the fishery. Consumption is given by $C(t)=E_0 S(t)$ and the dynamics of the exploited resource becomes

(11)\begin{equation} \dot{S}\left( t\right) =S(t)\left( 1-E_0-S(t)\right) \text{.} \end{equation}

Along this trajectory, the stock evolves as

(12)\begin{equation} S(t)=\frac{1-E_0}{1+\left({1-E_0}/{S_0}-1\right) e^{-(1-E_0)t}}\;. \end{equation}

The system tends toward a limit, $S_\infty =1-E_0$. For $E_0=1$, the stock is eventually exhausted.

The rule of constant effort completely determines the trajectory of the fishery. By equation (9), when $S< S_{MSY}$, the maximin level of effort is $E^{mm}( S) =1-S$. This level of effort maintains the stock at a stationary level that may correspond to a ‘poverty trap.’ In order to recover from a period of overfishing, society must harvest less than the maximin level $m( S) =S( 1-S)$ so that the stock can grow and the maximin value function can increase along the trajectory. There is no ‘free lunch’ for the future. Current effort must be less than $E^{mm}(S_{0})$, and consumption less than $C^{mm}=S_{0}( 1-S_{0})$.

Under a strategy of constant effort, with $E(t)=E_{0}<1-S_{0}=E^{mm}(S_{0})$, consumption increases with the stock size. Figure 2 depicts the following trajectories through time, beginning at the stock $S_{0}=0.1$:

• • The natural growth of the stock (without harvesting).Footnote ⁹

• • The growth of the resource stock with constant fishing effort $E_0=E^{MSY}=\frac 12$. The stock tends toward $S_{MSY}$. This trajectory is labeled ‘stock recovery.’

• • The trajectory of the maximin value along the trajectory for $E_0=\frac 12$. The maximin value increases toward the MSY level.

• • The consumption pattern, which increases as the stock increases and catches up to the maximin value. Consumption tends toward the MSY.

Figure 2. Evolution of the maximin value function along a constant effort trajectory $E(t)=E^{MSY}=\frac {1}{2}$ leading to maximum sustainable yield (MSY).

We stress that the recovery of the fishery (and thus the increase in consumption) is possible only because consumption is lower than the maximin level at all times. The long-run consumption depends on the reduction of the current consumption, the constant fishing effort being between the maximin value $E^{mm}(S_{0})=1-S_{0}$ and the MSY value $E^{MSY}=\frac {1}{2}$. A lower fishing effort, and hence current consumption, entails a higher long-run consumption.Footnote ¹⁰ Figure 3 presents the trajectories of maximin value and catches for three different recovery strategies (for three different effort levels) with, again, an initial fish stock $S_{0}=0.1$. For this stock, the initial maximin value is $m(S_{0})=S_{0}(1-S_{0})=0.1(1-0.1)=0.09$. The evolutions of the stock for the different scenarios are not represented on the figure.

• • The first strategy (trajectories denoted by $MV_{0.9}$ and $C_{0.9}$) corresponds to a constant fishing effort $E_0=E^{mm}=0.9$. At this effort level, the stock is in equilibrium at the initial value, i.e., $S_\infty =S_0=0.1$. The harvest is equal to the maximin value from the initial stock at all times, i.e., $C(t)=C^{\infty }=0.09$. A policy maintaining effort or employment at this level entrenches poverty.

• • The second strategy (trajectories denoted by $MV_{0.7}$ and $C_{0.7}$) corresponds to a constant fishing effort $E_0=0.7 < E^{mm}$. The fish stock increases asymptotically toward a limit, $S_\infty =1-E_0=0.3$. The harvest increases toward the maximin harvest for this limit, $C^{\infty } = S_\infty (1-S_\infty )=0.21$, which is lower than the MSY.

• • The third strategy (trajectories denoted by $MV_{0.5}$ and $C_{0.5}$) is that depicted in figure 2, with the fishing effort set constant at the MSY equilibrium effort, $E^{MSY}\!=0.5$. The maximin value increases asymptotically toward the MSY value and the harvest increases toward the MSY, which is $C^{\infty}\!=0.25$.

Figure 3. Sensitivity analysis with respect to the constant effort level, with values $E_{0}=E^{MSY}=\frac {1}{2}$ (growth toward the MSY), $E_{0}=0.7$ (intermediate case), and $E_{0}=E^{mm}(S_{0})=0.9$ (maximin path without growth).

There is a non-linear relationship between $C_{0}$ and $C_{\infty }$ which is determined by the chosen (constant) effort level. Recovery effort belongs to $[E^{MSY},E^{mm}(S_{0})]$. If the effort is small and equal to $E^{MSY}$, present consumption is low ($C_{0}=E^{MSY}S_{0}$) and the limiting consumption is high, at the MSY. If the effort is equal to $E^{mm}(S_{0})$, the stock remains at the initial level $S_{0}$, and the present and limiting consumption are equal. (There is no growth.) This is the maximin path. Intermediate cases are defined according to the relationship

(13)\begin{equation} C_{\infty }=\lim_{t\rightarrow \infty }E_{0}S\left( t\right) =E_{0}\left( 1-E_{0}\right) =\frac{C_{0}}{S_{0}}\left( 1-\frac{C_{0}}{S_{0}}\right) \text{ ,} \end{equation}

for $C_{0}\in \lbrack S_{0}/2,S_{0}]$, i.e., for $E_{0}\in \lbrack 1/2,1]$. The possibility frontier between present and long-run consumption is described by figure 4.

Figure 4. Trade-off between present consumption and long-run consumption in a fishery with constant effort and $S_{0}=0.1$.

Any pair $(C_{0},C_{\infty })$ that is achievable with constant effort $E_{0}\geq 1/2$ belongs to this frontier. For this family of ram, different normative criteria would prescribe different initial consumption and result in different long-run consumption. Several particular solutions are represented in figure 4, including the green golden rule (Chichilnisky et al., Reference Chichilnisky, Heal and Beltratti1995), that maximizes $C_{\infty }$, the myopic behavior from open access, maximizing $C_{0}$, and the maximin that accounts for the minimal consumption level and results in no growth, with $C_{0}=C_{\infty }$.

The results in this section emphasize that there is a trade-off between the current consumption and long-run sustainability, given the postulated growth pattern. The model is, however, too simple to study investment patterns. We thus turn to a two-sector model.

Effect of current sacrifice on instantaneous sustainability improvement

Here again, we start by considering the case in which consumption is reduced with respect to the maximin value, and how it improves sustainability.

In the DHS model, maximin improvement is given by

(17)\begin{align} M(K,S,c,r) &= \frac{dm(K,S)}{dt} =\frac{\partial m(K,S)}{\partial K}\dot{K}+ \frac{\partial m(K,S)}{\partial S}\dot{S} \nonumber\\ &= \frac{\partial m(K,S)}{\partial K}(K^{\alpha }r^{\beta }-c)-\frac{ \partial m(K,S)}{\partial S}r. \end{align}

Contrary to the single-stock problem of the fishery, a sacrifice does not necessarily entail sustainability improvement in the DHS model. For a level of consumption $c<m( S,K)$, the expression in equation (17) may be either positive or negative depending on the extraction level, and thus production and investment.

As explained previously, we consider a particular ram that is a deviation from the maximin path. This is for illustrative purposes only, and the same exercise could be done with another ram. The level of natural resource extraction that maximizes $M$ conditional on the consumption level is given by the following extraction rule $\hat {r}(K,S)$:Footnote ¹²

(18)\begin{equation} \hat{r}(K,S)=(\alpha -\beta )^{({1}/{1-\beta })}S^{({1}/{1-\beta })}K^{- ({1-\alpha }/{1-\beta })}\;. \end{equation}

This feedback rule is the same as the one along the maximin path. At any given state, maximizing sustainability improvement entails producing the same as for the maximin path at the current state, and investing any amount of capital freed up by a sacrifice of current consumption.Footnote ¹³

Under this strategy of maximizing sustainability improvement through the extraction rule $\hat {r}(K,S)$, net investment can be expressed as a function of the state variables and the consumption only:

\[ M(K,S,c,\hat r(K,S))=\frac{\partial m}{\partial K}(K^{\alpha} \hat r^{\beta}-c)-\frac{\partial m}{\partial S}\hat r \;. \]

This expression is linear in the current consumption (or symmetrically, current sacrifice), just as in the fishery model. It equals zero when the consumption equals the maximin level and net investment is nil, and is positive whenever the consumption is lower than the maximin level, corresponding to a positive net investment. The marginal effect of the sacrifice is proportional to the shadow value of the capital stock.

Trade-off between current consumption and long-run consumption for constant growth rate development paths

Sustainable growth in the DHS model corresponds to growth from an initial level of consumption $c_{0}$ lower than the maximin value $m( S_{0},K_{0})$. In this model, sustainable growth without limit is possible, for example when considering hyperbolic discounting (Pezzey, Reference Pezzey2004) or following a constant saving rate rule, which can correspond to some undiscounted utilitarian optima (Asheim and Buchholz, Reference Asheim and Buchholz2004; Asheim et al., Reference Asheim, Buchholz, Hartwick, Mitra and Withagen2007; D'Autume and Schubert, Reference D'Autume and Schubert2008). As our purpose is to illustrate cases in which growth induces a catching up of the maximin value in finite time, we consider that society initially pursues consumption growth at a constant rate $g>0$, a growth pattern that is not sustainable in the long run without technological progress (Stiglitz, Reference Stiglitz1974; Llavador et al., Reference Llavador, Roemer and Silvestre2011).Footnote ¹⁴ We assume that the consumption side of the economy is determined by the constant growth rate pattern, with consumption

(19)\begin{equation} c\left( t\right) =c_{0}e^{gt}\;. \end{equation}

To complete the ram, we assume that the current generation maximizes sustainability improvement, though the feedback extraction rule $\hat {r}(K,S)$.

The limit to growth. There is a limit to the time for which growth can be supported at rate $g$. If growth is pursued without considering the maximin value and its evolution, consumption overshoots the sustainable level at some time, and the economy ultimately collapses, as illustrated in figure 5. To avoid this unsustainable type of trajectory, the economy must switch at some time $T$ from the exponential growth path to a maximin path characterized by constant consumption $c_{\infty }\equiv m(S(T),K(T))$. In fact, the long-run level of consumption is endogenous, and is defined at the time at which consumption catches up with the maximin level.Footnote ¹⁵ Figure 6 illustrates two sustained-development paths starting from the same initial state and thus the same maximin value, but with different initial consumption and growth rates. The long-run sustained consumption is different. The first path corresponds to a sustainable version of the overshooting path of figure 5, in which the consumption pattern switches to constant consumption once the maximin level is reached. The other path has a lower initial consumption (higher sacrifice), a higher growth rate, a longer growth period and a larger long-run consumption level.

Figure 5. Exponential consumption and maximin value function, for $(K_{0},S_{0})=(10,100)$, $\alpha =2/3$ and $\beta =1/3$, $m(K_{0},S_{0})\approx 12.17$, initial consumption $C_{0}=0.9m(K_{0},S_{0}) \approx 10.95$ and growth rate $g=0.05$.

Figure 6. Exponential consumption and maximin value function, for $(K_{0},S_{0})=(10,100)$, $\alpha =2/3$ and $\beta =1/3$, $m(K_{0},S_{0})\approx 12.17$. Path 1 is plotted with $C_{0}=0.9m(K_{0},S_{0})\approx 10.95$, a growth rate $g=0.05$ and $C^{\infty }\approx 13.35$. Path 2 is plotted with $C_{0}=0.7m(K_{0},S_{0})\approx 8.52$, a growth rate $g=0.1$ and $C^{\infty }\approx 18.25$.

Growth at rate $g>0$ demands that $c_{0}<m( S_{0},K_{0})$. The larger the initial sacrifice, the more room there is for growth. In the example in figure 6, the maximin value is increased by about $50$ per cent with the path with the largest sacrifice (and by about $10$ per cent for the other path), in spite of a larger growth rate. The growth rate and the duration of the growth period are linked to the initial consumption and the long-run consumption. If two of the four are given, the two others can be derived. For any initial pair of stocks $( S_{0},K_{0})$ and the associated maximin value $m_{0}=m( S_{0},K_{0})$, it is possible at time $t=0$ to choose any pair

\[ (c_{0},g)\in \left\{ ]0,m_{0}[\,\times \,]0,\infty \lbrack \right\} \,\cup \,(m_{0},0)\;. \]

The path in which $( c_{0},g) =(m_{0},0)$ is the maximin path starting from the initial state. It has no growth. A path in which $( c_{0},g) \in \left \{ ]0,m_{0}[\,\times \,]0,\infty \lbrack \,\right \}$ (so that $c_{0}<m_{0}$ and $g>0$) has growth. However, growth at a constant rate cannot go on forever. There is an endogenous time $T(S_{0},K_{0},c_{0},g)$ at which consumption catches up to the dynamic maximin value, i.e., $c(T)=c_{0}e^{gT}=m[ S( T) ,K( T) ]$. From then on, growth is no longer sustainable, and the level of consumption must remain at the maximin level; i.e., for $t\geq T$, sustainability implies that $c( t) =m( S( T) ,K( T) )$.Footnote ¹⁶

Along any growth path, the ram that society implements imposes the long-run utility level. In our example of an exponential growth path, initial consumption, the rate of growth, and the very long-run consumption are interconnected. Figure 7 depicts the convex-concave correspondence from the initial pair $( S_{0},K_{0})$ to the attainable frontier, $\left \{ (c_{0},g,c_{\infty })\text { feasible from } ( S_{0},K_{0}) \right \}$. Growth is possible only if $c_{0} < m(S_{0},K_{0})$.Footnote ¹⁷ For a given growth rate, a lower level of initial consumption allows a higher long-run level. For a given initial consumption, a lower growth rate allows a higher long-run consumption (as the actual consumption catches the maximin level more slowly). Given the initial level of consumption $c_{0}$, there is a trade-off between the eventual maximin consumption that is sustained after some time $T$ (endogenous to the path followed) and the rate of growth that is sustained up to that level. A level of present consumption that is closer to the maximin value $m(S_{0},K_{0})$ entails a lower prospect for growth.

Figure 7. Links between initial consumption, growth rate, and long-run (sustained) consumption in the DHS model.