Price versus quantity with hysteretic
regime shifts
1. Introduction
What type of policy instrument works best to regulate environmental problems? This question
has been asked many times in the last decades and the literature is vast that describes the
advantages and disadvantages of different tools (See e.g. Xepapadeas 1997[42]; Sterner and
Corria 2012[37] for overviews). Many studies have focused on the choice between price
instruments (taxes, landing fees) and quantity regulations (quotas, emission standards,
tradable emission permits). Under the usual convexity assumptions, these instruments are
equivalent when full information is available while in case of uncertainty, the optimal choice
between price and quantity regulation, depends on the structure of the problem (See e.g.
Weitzman 1974[41]; Pizer 1999[29] ; Hoel and Karp 2002[19]; and Hepburn 2006[18] for a
review).
Several early contributions study the role of non-convexities for the theory of competitive
markets and decentralized resource allocation (See e.g. Farrell 1959[14]; Starrett 1972[35];
Baumol and Oates 1975[3]; Guesnerie 1975[15]; Calsamiglia 1977[6]; and Mas-Colell 1987[26]
for an overview). Dasgupta (1982[13]) addresses the choice between price and quantity
instruments with non-convexities. Assuming uncertain costs and benefits, he finds that for
resources with threshold effects in the form of nonlinear benefits, a regulation would be
preferred to a tax. No contribution in the economic literature on choice of policy instruments
seems to address the effect of hysteresis, the property of some threshold effects to be sticky
and thus difficult although not impossible to reverse (Scheffer et al 2001)[39].
This paper aims to shed light on the question: Are price and quantity regulations equivalent in
a context of full information, threshold effects and hysteresis? This question is particularly
relevant when managing natural systems. Often these respond in a smooth way to changes in
for example the climate, nutrient loading, habitat fragmentation or biotic exploitation. Yet
substantial and dramatic changes can interrupt these smooth processes when certain
thresholds are crossed, leading to so-called regime shifts (Scheffer et al. 1993[34]; Van de
Koppel et al. 1997[38]; Nyström et al. 2000[28]; Scheffer et al. 2001[33]; Gunderson and
Pritchard 2002[16]; Carpenter 2003[7]; Rockström et al. 2009[32]; Biggs et al. 2012[4]).
1
Regime shifts in nature often manifest as large, abrupt, persistent changes in the mix of
ecosystem services that the system produces, with potentially significant impacts on human
well-being (MA 2005[23]; Stern 2006[36]; Crépin et al. 2012[8]). In addition regime shifts are
difficult and sometimes impossible to reverse (hysteresis). There are many documented
situations where human activities put pressure on natural systems, which may trigger a
regime shift thereby changing the ecosystem's production possibility and its usefulness for
human activities (Scheffer et al. 2001[33]; Biggs et al. 2012[4]; Regime Shifts DataBase 20141).
Hence regime shifts in social-ecological systems pose a fundamental scientific challenge with
crucial implications for policy and management (Dasgupa and Mäler 2003[12]; Brock and
Starrett 2003[5]; Crépin 2007[11]). Policy implications have been studied to some extent in an
irreversibility context (Krysiak 2008[22]; Baldursson and von der Fehr 2004[2]; Margolis and
Nævdal 2008[25]), and for systems with multiple basins of attraction (Mäler et al 2003[24];
Crépin et al 2011[9]; Kossioris et al 2011[20]; Crépin et al 2012[8]; Heijdra and Heijnen
2013[17]). Weitzman (1974, p.477)[41] pointed out in a note that "without convexity it may
not be possible to find a price, which will support certain output levels". Furthermore, with
non-convexities the existence of a supporting price system derived from Pontryagin's principle
is not sufficient for an optimum. Such a price system could support even the worst possible
outcome. (Arrow 2002.)[1]
Although the advantages and disadvantages of price incentives and quantity regulations have
been extensively discussed under convexity assumptions and uncertainty, it is unclear how
policy instruments function when hysteretic regime shifts can occur, even with full certainty.
Even if some price level could support output amounts of interest, we cannot be sure that the
preferred policy instrument with non-convexity would still be the same as without. With
potential multiple equilibria and regime shifts, social planners must decide which equilibrium
to enforce in the first place. The main hypothesis is that existing results in the literature
assuming convexity assumptions should hold within the basin of attraction of the desired
state, whereas they may not when a regime shift is needed to reach the desired state.
Scientific articles dealing with regulation of resources with regime shifts typically set up and
attempt to solve dynamic optimisation problems or dynamic games with non-convexities in
the constraint. With few exceptions (e.g. Polasky et al 2011)[31], such problems have no
1
www.regimeshifts.org. Accessed 2015-09-04.
2
closed form solution and their study requires in-depth bifurcation analysis combined with
computer simulations of the optimised system and careful parameter sensitivity analysis (e.g.
Wagener 2003[39], Mäler et al 2003[24], Crépin 2003[10], Crépin et al 2011[9], Kossioris et al
2011[21]). This paper makes an attempt to simplify the problem enough to enable analytical
results without losing essential characteristics of regime shift problems like hysteresis.
The paper is organised as follows: section 2 presents the analytical set up, section 3 sets up
price and quantity regulations for this model and compares welfare outcomes with both kinds
of regulations; section 4 provides a discussion of the results, their implications for policy and
concludes the paper.
2. Analytical set-up
To examine the choice of policy instrument to regulate resources that could undergo a regime
shifts, the paper at hand uses the Weitzman (1974)[41] framework as a starting point but
transforms it into a more dynamic setting with regime shift but no uncertainty. Weitzman
addressed the problem of a planner who wished to regulate a market that would not
spontaneously internalize all costs and benefits. He didn’t explicitly model the externality but
instead discussed how the Pareto optimum could be reached if the market for some reason
wouldn’t reach the long term equilibrium. Rather than building a model with detailed
resource dynamics and technological constraints that affect costs, this paper focuses on the
implications of regime shifts on resource users.
More specifically, consider a situation in which at some point t in time, a particular ecosystem
contributes to the production of a quantity qt of some consumer good. The commodity
produced in quantity qt at time t is valuable to consumers and its price pt is set in perfect
competition. In each period, consumers demand qtD  f  pt  . The production of the good
impacts the ecosystem so that a substantial level of production jeopardizes the ecosystem’s
capacity to provide the good in the future. If some threshold level of pressure is reached, the
ecosystem shifts to another regime, after a transition period during which thresholds are
trespassed but the system continues to behave as if they were not. Such a regime shift
implies that the ecosystem’s contribution to the production of the good changes. Producers
may have to adopt new technologies, replace some input factors, travel to alternative
harvesting grounds, etc. These changes are likely to affect costs. Assume that such change can
3
be modelled as a jump in cost at a threshold so that a low cost, CL  qt  , prevails in one
regime, and a high cost, CH (qt ) , in the other, with CL  qt   0 and CH  qt   0 . Let q denote
the threshold at which production switches from low cost to high cost and q  q the
threshold at which it switches from high cost to low cost. These two thresholds differ to
model hysteresis in the system, which implies that there is an asymmetry in entering and
leaving a particular regime. Once a particular regime is reached it becomes persistent and
leaving it requires crossing other boundaries than entering it. The dynamics that lead to a
regime shift build up during a transition period before the shift occurs, as occurs in many real
world regime shifts. (Scheffer et al. 2001[33]; Biggs et al. 2012[4]; Regime Shifts Database
2014).
The cost function depends not only on current quantity produced ( qt ) but also on the cost
function in the previous period, Ci  qt 1  , i H , L . Equation (1) defines the cost
function C  qt , Ci  qt 1   , i H , L , recursively. See Figure 1 for an illustration.

qt  q
CL  qt  if 

or qt   q, q  and Ci  qt 1   CL  qt 1 


C  qt , Ci  qt 1    
qt  q

C
q
if


 H t

or qt   q, q  and Ci  qt 1   CH  qt 1 


(1)
Even if the cost function seems to be ambivalent in the interval  q, q  , it cannot take both
values at the same time. At each point t in time only one value prevails, which is either
CL  qt  or CH  qt  for the cost function, yielding CL'  qt  or CH'  qt  for the marginal cost
function depending on past cost pattern Ct 1 and current stock size qt . More specifically if the
system starts in the low nutrient regime once the quantity has passed an upper bound q the
system shifts to the other regime. Recovering the initial low regime would require going down
all the way to q in order to shift back to the low nutrient regime (Figure 1).
To simplify the problem even further we assume that there are only three phases in time
represented by three specific points in time. If a regime shift occurs, these phases represent
initial conditions (period 0), a transition period when the system is under pressure at a
4
threshold ( q or q ), quick changes occur but the cost function is sticky and remains the same
as in period 0 (period 1), and the final period when the cost function has shifted (period 2).
The same regime (either H or L) could also prevail in all three periods, which happens when no
regime shift occurs.
Figure 1: Cost function with hysteresis
If low cost prevails at start but a regime shift is about to happen, during the transition phase
the system remains at low cost and produces q . When the transition phase is over the system


will have high cost with quantity in the interval q;  . In contrast, if the system was in high
cost to start with, during the transition phase the system remains at high cost and produces q .
When the transition phase is over the system will have low cost with quantity in the

interval 0, q . Producers take price as given and in the absence of regulation they choose
quantities to maximize   p, q  the discounted sum of profits. :
5
max   p, q  
qt
  p q  C  q , C  q 
t t
t0,1,2
t
i
t 1
1
1   
t
The vector of solutions to this problem called q S   qtS 
t0,1,2
characterises the producers’
supply function. Necessary conditions for a solution are   p, q S   0 and, for all t, equation
(2) holds. A sufficient condition is that   p, q S  is concave in q S .
pt 
C  qtS , Ci  qt 1  
qt

C  qtS1 , Ci  qt   dCi  qtS 
Ci
dqt
0
(2)
Since we deal with potentially non-marginal change and discontinuities the sufficiency
condition is not always satisfied. Hence we also need to calculate the maximum profit
obtained in the different cases and compare these to each other to identify the profit
maximizing solution for the whole problem. Note also that the way the problem is designed
implies that except when quantity is at one of the thresholds ( q if costs are low or q if costs
are high), no shift in cost will occur in the following period so equation (3) follows.
C  qtS1 , Ci  qt   dCi  qtS 
Ci
dqt
0
(3)
Beside the case when the price is always too low to provide a non-negative
profit, pt  CL  0  , t , resulting in no production, qtS  0 , there are four different cases with
positive profit depending on initial conditions, i.e. the nature of the initial regime and whether
or not the system undergoes a regime shift. These cases are summarized in table A1 in
appendix. Among them two interesting cases occur when there is a regime shift, which for
given initial conditions occurs when the profit generated from triggering a regime shifts is
larger than the profit generated from not doing so. We show in appendix that when initial
costs are low, a regime shift will occur if and only if condition (4) holds.
p1 

q  q 
CL q  CL  q1L 
L
1

p2  q2L  q2H   CL  q2L   CH  q2H 
 q  q  1   
L
1
(4)
However when initial costs are high, a regime shift will occur if and only if condition (5) holds.
6
p1 
 
q  q 
CH q  CH  q1H 
H
1

p2  q2L  q2H   CL  q2L   CH  q2H 
 q  q  1   
H
1
(5)
It is never optimal to trigger a regime shift to high cost if the optimal quantity remains below
q , while it may be optimal to trigger a regime shift to low cost even though the optimal
quantity remains above q . This is because an optimal solution requires minimizing costs,
hence striving for CL . However if costs are high to begin with it may be optimal to keep high
 
costs even if q2R  q; q if the cost of triggering a regime shift is higher than the cost of
remaining in a high cost regime.
Consider an instantaneous market adjustment mechanism so that at each point in time supply
and demand adjust to the market clearing quantity qˆt  qtS  qtD . Note that qˆt are functions
of pt , due to the way supply and demand are defined in the consumer and producer
problems. Hence relation (6) holds in equilibrium at time t , which also defines equilibrium
price pˆ t .
C  qˆt   f   qˆt   pˆ t
(6)
However this market solution does not account for possible externalities associated with
consumption or production of the good. Let B  qt  denote the total benefits to society at
time t , which consists of the consumer utility and the net value (benefits minus costs) of the
externality associated with the commodity. The planner aims to maximize welfare W  q 
defined as the discounted sum of net benefits from the good where net benefits are social
benefits B  qt  less social costs C  qt , Ci  qt 1   , equation(7). For the sake of simplicity, social
costs coincide with private costs. However since the social benefit differs from private utility,
the market outcome will typically not maximize social welfare expressed in equation(7).
t1

W  q    B  qt   C  qt , Ci  qt 1  
t0
7
 1 1


t
(7)
3. Regulations
The planner can choose to regulate quantity by ordering quantity qtR or price by ordering
price ptR at any point t in time. Given a set of initial conditions  p0 , q0 , Ci  q   the regulation
is put in place in period 1. Price and quantity adjust immediately but as in the unregulated
case in Section 2, it takes a transition period for the cost function to shift between high and
low level. Hence in period 0, initial conditions are the same in both regulatory systems since
nothing has happened yet. In period 1, the policy is put in place and price and quantities
adjust accordingly but quantities cannot cross the relevant thresholds. In period 2, the policy
has been in place for one period and changes can now transmit to the cost function if
necessary according to the rules(1), when applied to the desirable regulated quantity in each
policy.
3.1
Quantity regulation
With a quantity regulation, the regulator imposes the quantity to be exchanged on the
market. Given initial conditions  q0 , p0 , C0  q0 , p0   the social planner solves problem(8):
T

max W  q    B  qt   C  qt , Ci  qt 1  
q
0
 1 1


(8)
t
The optimal solution q R   qtR  fulfils necessary condition expressed in equation(9):
t ,
dB  qtR 
dqt

C  qtR , Ci  qt 1  
qt



C qtR1 , Ci  q R t  dCi  qtR 
Ci
dqt
0
(9)
The sufficient condition is that W  q R  is concave in q R . With these conditions the outcomes
in each period are the following, where Wt denotes the present value of the social welfare in
that particular period:

Wt  qtR   B  qtR   C  q R t , Ci  qt 1  
8
 1 1 
t
(10)
R
t
In period 1 and 2 q fulfils condition (9) with

C qtR1 , Ci  q R t 
Ci
  0 if no regime shift is
needed to reach the optimal solution.
If a regime shift to high cost must be triggered, q1R  q ; if a regime shift to low cost must be
triggered, q1R  q . In each of these cases, condition (9) cannot apply anymore because we deal
with non-marginal change and discontinuities.
Several cases materialise depending on the initial conditions and on how the optimum looks
like. Detailed results are summarized in Table A 2. For most conditions the quantity regulation
succeeds in triggering the optimal outcome however and exception materialises when the
high cost pattern prevails initially, CH  q0  , but the optimal situation would be to trigger a
regime shift to low cost, with the optimal quantity in period 2 remaining within the hysteresis
 
range: q2R  q; q ; C2  q2R   CL  q2R  . In this situation a one-step policy cannot trigger the
necessary regime shift, because it does not provide any incentives to decrease production
below q the threshold for triggering a shift to the low cost regime. In such condition, a twosteps policy as suggested by Heijdra and Heijnen (2013[17]) would be necessary to reach
optimum.
3.2
Price regulation
With a price regulation, the regulator imposes the market price of the commodity to reach the
desired outcome. A way to make society internalize the whole benefit of the commodity
produced including possible externalities would be to ordinate in each period
price ptR  B  qtR  . Consumers should then demand qtD  0 or qtD  f  ptR  . Let qˆt  ptR  denote
the quantity to be produced in period t if the regulated price ptR prevails. It turns out that the
outcome of a particular regulation aiming to require price ptR for all values of t depends on
how the initial condition ( q0 ) relates to the boundaries q and q as well as the initial level of
 

the cost function C0 .and on CL  q   CH q or CL  q   CH q . Table A 4 and Table A 5
provide detailed results. In many cases a price regulation ptR  B  qtR  would lead to the
desired outcome. This is the case for example when no regime shift is needed but also in most
cases when a regime shift is needed a price regulation will provide enough incentives to adjust
9
quantities to reach the optimal outcome. However two exceptions materialise for price
regulations, which both occur within the hysteresis zone. The outcome depends on whether

 
CL  q   CH q or CL  q   CH q .
     does not result in a clear quantity
 
If CL  q   CH q , ordering a price ptR  CL q , CH q
since the cost function is not defined in that region. The optimal outcome would be instead to


 
order the price ptR  CL q  B q . If instead CL  q   CH q ordering a price
 
ptR  CH q , CL q  would only lead to an optimal outcome under the condition that no


regime shift must be triggered. Indeed this price level does not provide incentives to adjust
the quantity produced enough to trigger a regime shift. Hence if low cost prevailed to start
with, this is a preferred cost level because in the interval of study you can provide the same
amount at lower cost and thus price incentive would lead to an optimal outcome. However if
initially costs are high the price regulation cannot trigger a regime shift, hence the outcome of
such policy would be suboptimal. In that situation, a two-steps policy as suggested by Heijdra
and Heijnen (2013[17]) would be necessary to reach optimum.
3.3
Price vs quantity regulation
To compare price and quantity regulation let’s calculate  the comparative advantage of
price over quantity regulation. This quantity is defined in equation(11), where qtR denotes the
regulated quantity at time t , ptR is the regulated price and qˆt  ptR  is the quantity that will be
produced on the market if the regulated price prevails.
 


   B qˆt  ptR   Ct qˆt  ptR 
T
0
 1   
t

  B  qtR   Ct  qtR 
T
1
0
 


 
 1 1 
 


 
 1 1 
 B qˆ1  p1R   C0 qˆ1  p1R   B  q1R   C0  q1R 
 B qˆ2  p2R   C2 qˆ2  p2R   B  q2R   C2  q2R 
 1 1 
t
(11)
2
In situations when no regime shift is needed to reach the desired outcome,


  0 because qˆt  ptR   qtR and C2 qˆ2  p2R   C2  q2R  . However there are some cases when


qˆt  ptR   qtR or when C2 qˆ2  p2R   C2  q2R  . These situations occur in the cases highlighted in
10
sections 3.1 and 3.2 when a regime shift needs to be triggered and the desirable quantity in
period 2 is in the hysteresis zone q2R   q; q  . Two cases can be identified:
1) The highest marginal low cost level is lower than the lowest marginal high cost
 
level, CL  q   CH q
In that situation, Table A 4 prevails for price regulation. Such a price regulation would reach


the optimal solution if set to ptR  CL q  B q to trigger a regime shift to or maintain low
costs and produce the maximum possible quantity at that cost. In contrast one shot quantity
regulation could only be optimal if the low cost already prevails because the produced
quantity would not decrease enough to trigger a regime shift from high cost to the low cost
regime.2 So if the high cost prevails the quantity regulation would lead to a loss compared to a
price regulation. The size of this loss is given by the comparative advantage of price over
quantity regulation in equation (12):

   B q  CH q  B  q1R   CH  q1R 

I

II
  
 


1
 1  C qR   C qR 
H
2
L
2
2
 1   
1   

III


(12)
Figure 2 illustrates the difference between both policies in each period in a market diagram
sketching the particular situation when p1  p2 so that q1R  q2R .3 Surfaces I, II and III
correspond to the all positive quantities in equation (12), which are relevant for comparing
both policies and determining the size of the gain  of using a price regulation rather than a
quantity regulation in that particular situation. In the figure, the gain from producing q at
high cost rather than q1R in the first period is positive because it consists of discounted sum of
two positive amounts I  II . In the second period the discounted amount III illustrates the
2
A successful quantity regulation would have to be designed in two steps. The first step would be a “cold turkey”
regulation commanding q to trigger a regime shift to low costs (see Heijdra and Heijnen 2013 for a more
 
  1 1
detailed example). That would yield social welfare B q  CH q
in period 1. The second step would
implement the optimal quantity in period 2 once the cost has shifted to low cost yielding social
1
welfare B  q2R   CL  q2R 
. Comparison between one-step price regulation and two-steps quantity
2
1   


regulation would then yield the exact same outcome except for possible implementation costs, which we do not
consider here.
3
Here we use the characteristics that k , B  k    B  q  dq and Ct  k    Ct  q  dq
11
k
k
0
0
foregone loss of producing quantity q2R at high costs, this is also always positive. In this
situation we know for sure that   0 and a price regulation dominate over a quantity
regulation. The following three main factors influence the size of the welfare gain of using a
price regulation rather than a quantity regulation if everything else remains constant:
1. A longer distance between each of the desired outcomes ( q1R and q2R ) and the lower
threshold of the hysteresis zone ( q ) increases the gain.
2. A steeper slope of the marginal cost and marginal benefit curves increases the gain.
3. A lower discount rate increases the gain.
Figure 2: Price vs quantity when C  q   C  q  and high costs prevail at initial conditions
L
H
2) The highest marginal low cost level is higher than the lowest marginal high cost
 
level, CL  q   CH q
 

In this situation, if p2R  CH q , CL  q  neither policy can reach the desired outcome if they
are implemented using one step only. Indeed a quantity regulation would reach the desired
quantity but at high cost instead of low cost, while the price regulation would reach
12
quantity qˆ2  p2R    CH 
1
p q
R
2
R
2
. The comparative advantage of a one-step price over a
one-step quantity regulation is given by (13).
 


 
  B qˆ1  p1R   CH qˆ1  p1R   B  q1R   CH  q1R 
 


 
 B qˆ2  p2R   CH qˆ2  p2R   B  q2R   CH  q2R 
 1 1 
 1   
1
(13)
2
Figure 3 helps compare one-step price and quantity regulation and two-steps regulations in
this particular set up. It pictures the outcome in period 2. The comparisons can be made
similarly in period 1.
Figure 3: Price vs quantity when C  q   C  q  and high costs prevail at initial conditions
L
H
Compared to an optimal outcome, where the optimal quantity q2R would be produced at
price p2R , a one step quantity regulation would lead to q2R the right quantity to be produced
but at a too high price, incurring the loss pictured by area IV in Figure 3 . In contrast a onestep price regulation would lead to producing too little of the good, only qˆ  q2R  but at the
13
right price p2R so the loss would be pictured by area V instead. Consequently the overall sign
of  depends on how area of type IV compares to area of type V in each period. In Figure 3,
ocular inspection easily reveals that IV > V implying that the price regulation dominates over
the quantity regulation in this particular case. However the opposite could be true too.
Further ocular inspection reveals that the following factors influence the size and sign of the
comparative advantage of using a price regulation rather than a quantity regulation if
everything else remains constant:
1. An upward shift in the marginal high cost function increase the loss generated by a
quantity regulation IV and also the loss generated by a price regulation V . The
relative size of these increases depends on the steepness of the marginal benefit and
marginal cost curves. A tilt to a steeper slope of the marginal high cost curve has a
similar effect but the impacts on IV are likely to be higher than the impacts on V .
2. Similarly a downward shift or a steeper slope of the marginal low cost curve increases
the loss generated by both regulation. However the negative impact on IV is always
higher than on V . So such changes tend to increase the comparative advantage of
price over quantity regulation.
3. A longer distance between each of the desired outcomes ( q1R or q2R ) and the lower
threshold of the hysteresis zone ( q ) increases the loss generated by a quantity
regulation IV and thus  , favouring a price regulation.
4. A steeper slope of the marginal benefit curve increases the loss generated by a price
regulation V and thus lowers  , favouring a quantity regulation.
5. A lower discount rate increases the gains of choosing the most efficient policy.
4. Discussion
This paper has made an attempt to address the price versus quantity discourse in a nonconvex environment where regime shifts can occur. In particular it extends the literature
when threshold effects result in endogenously driven jumps in costs, which are difficult but
not impossible to revert ̶ so-called hysteresis. The set up used built on Weitzman (1974) [41]
but was dynamic, identifying three periods of relevance: initial conditions, policy
implementation and regime shift when this was an applicable outcome of the policy.
14
In contrast to contributions using a related but much richer set-up (Crépin et al 2011[9],
Kossioris et al 2011[21]), this paper managed to find analytical solutions and provide analysis
of the problem using market diagrams.
The results confirm that within the basin of attraction of a desired state, price and quantity
are equivalent when full information is available and Weitzman’s results hold. Even when a
regime shift must be triggered, price and quantity regulation are equivalent in this set-up,
provided the final state lies outside of the zone of hysteresis. However Weitzman’s results
may not hold if the desired state lies within the zone of hysteresis and a regime shift is needed
to reach the desired state. In this situation we find that price regulation is preferable to
quantity regulation when a regime shift to low cost is required to reach the optimal state and
the maximum level of the low cost curve is lower than the minimum level of the high cost
curve. This is the case when the regime shift is substantial leading to a radical jump in the cost
structure.
In the alternative case when the maximum level of the low cost curve is higher than the
minimum level of the high cost curve, the result is ambiguous. However even in that situation
we are still able to say that a relatively steeper slope of the benefit curve would act in favour
of a quantity regulation while a longer distance between the desired outcome and the
threshold for going back to low cost would act in favour of a price regulation.
Hence these results indicate that policy makers should consider regime shifts when designing
chosing policy instruments and in some situations a unique constant price or quantity
regulation will not contribute to reach a social optimum. Kossioris et al (2011)[21] showed
that in such a context, a non-linear (n-exponential) tax scheme may increase efficiency and
come arbitrarily close to the social optimum situation. However they also recognize that such
tax scheme is rather complex and could be difficult to implement. Instead and in case an
undesired regime prevails, Heijdra and Heijnen (2013)[17] suggest a two steps policy. In the
first step the regulator would implement a strong price or quantity instrument under a limited
period to make sure to trigger the shift back to the desired regime. In a second step the
regulator would implement the optimal policy, just as if there was only one feasible regime.
The results in this paper indicate that this kind of policy may be needed particularly when the
undesired regime prevails and the desired state lies in the zone of hysteresis. In all other cases
a simple price or quantity regulation will actually trigger the optimal outcome.
15
A critical underlying assumption of this model is that a regime shift for example in some
ecosystem triggers a shift in cost for the production of a good derived directly from that
system. It may be more efficient to address the ecosystem dynamics directly rather than
regulating the good produced within the system. However this is no always possible and
economic activities like exploitation of ecosystem goods are often important drivers of
ecosystem dynamics. Regulators may also want to regulate such goods directly because their
production causes other kinds of market imperfections. This paper provides them with some
guidelines.
The approach in this paper takes stock of the idea that the important dynamics in the system
is the shift between regimes and the transition period, while the dynamics within each
particular regime is not the focus of this paper. Possible extensions of the model used here
could include more time steps allowing for dynamics also within each regime and in transition.
Hence we could consider shorter transition phases compared to the other phases, which is
likely to influence the results as the costs of being wrong in transition would tend to weigh
less than the cost of being wring after the transition since this period lasts longer.
Our set-up discards many important aspects. We compared price and quantity regulations
designed to reach the same desired quantity level but one may instead want to compare an
optimal quantity regulation with an optimal price regulation, which may not necessarily yield
the same quantity. Further, this paper considers only two different regimes however empirical
evidence show that some systems like coral reefs (Norström et al) may experience more than
two regimes. Brock and Starrett (2003) suggest some ways to generalise their results to more
than two stable regimes, which could be useful here as well. Hence, introducing more explicit
dynamics driving the regime shifts, for example in the form of a difference equation for the
regime itself, or the possibility to control the timing of a shift could also provide interesting
insights.
Lack of information, which was the focus of the Weitzman contribution would certainly matter
here as well and likely introduce more richness to the results. With uncertainty, catastrophic
regime shifts could motivate a preference towards quantity instruments because they are
more likely to stabilize the stock level rapidly (Newell and Pizer 2003[27]). However Weitzman
(2002)[40] also suggests that price regulation may be preferable to controlling a bad proxy of
a quantity variable (e.g. harvest instead of escapement stock in a fishery). The set-up used
16
also discards technical progress, which could be an alternative way to overcome production
externalities that trigger regime shifts, without having to lower the quantity produced.
Research should focus on finding ways to more systematically identify characteristics that play
a major role for the choice of policy instrument. The role of information and stimulation of
technical development related to this issue is the scope for interesting future research.
Empirical evaluations of existing policy instruments taking potential regime shifts into account
should be performed to verify the results in this article. Research should also focus on finding
good ways to identify and test potential policy instrument e.g. using experiments.
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20
Appendix
 
q0  0, q 
q0
q2
q2  0, q 
 
q0  q; q
q0  q; q
C0  q0   CL  q0 
C0  q0   CH  q0 
No regime shift occurs
Regime shift from high cost to low
cost
 NL  q   p0 q0  CL  q0 
 
q2  q; q
C2  q2   CL  q2 
 1 1

 1 1 
 p2 q2L  CL  q2L 
 
with q  CL
L
1
q2R   q,  
 RL  q   p0 q0  CH  q0 

 p1q1L  CL  q1L 
1
 p1 

 p q
2
 
and q  CL
L
2
 RH  q   p0 q0  CL  q0 
1
L
2 2
 p2 
  1 1

 
with q  CH 
H
2
1
 CL  q2L 
 
with q2L  CL
1
 1 1 
 1 1 
No regime shift occurs
 NH  q   p0 q0  CH  q0 

 1 1

 1 1 
 p2 q2H  CH  q2H 
 
2
with q1H  CH 
 p2 
2
 p2 
 p1q1H  CH  q1H 
 p1 q  CL q
 p2 q2H  CH  q2H 
   1 1
 p1 q  CH q
Regime shift from low cost to high
cost

q0   q,  
1
2
 
 p1  and q2H
 CH 
Table A 1: Profit from producing the good in all three periods depending on initial and final regime
When initial costs are low, a regime shift will occur if and only if
 RH  q    NL  q   0
 


 p1 q  q1L  CL q  CL  q1L 
 p1 

q  q 
CL q  CL  q1L 
L
1

 1 1   p  q
 q2L   CH  q2H   CL  q2L 
H
2
2
 1 1 
2
0
p2  q2L  q2H   CL  q2L   CH  q2H 
 q  q  1   
L
1
When initial costs are high, a regime shift will occur if and only if
 RL  q    NH  q   0
 

 
 p1 q  q1H  CH q  CH  q1H 
 p1 
21
 
q  q 
CH q  CH  q1H 
H
1

 1 1   p  q
2
L
2
 q2H   CL  q2L   CH  q2H 
p2  q2L  q2H   CL  q2L   CH  q2H 
 q  q  1   
H
1
 1 1 
2
0
1
 p2 
q0
q
R
2
qtR  0, q 
 
q0  0, q 
q0  q; q
C0  q0   CL  q0 
C0  q0   CH  q0 
No regime shift needed


 B q
R
2
  C  q 
L
W  q   B  q0   CH  q0 
 1 1
R
2
     1 1
1
  B  q   C  q 
1   
 B q  CH q
1
1   
2
R
2
 
L
R
2
C2  q2R   CL  q2R 
W  q   B  q0   CH  q0 

 1 1

 1 1 
 B  q1R   CH  q1R 
 B  q2R   CH  q2R 
One step policy triggers a desired
regime shift
W  q   B  q0   CL  q0 
  B  q   CL  q  

Table A 2: Quantity regulation implementing
 1   
q2R
W  q   B  q0   CH  q0 

 1 1

 1 1 
 B  q2R   CH  q2R 
1
2
No regime shift needed
 B  q1R   CH  q1R 
1
1 
 B  q2R   CH  q2R 
22
2
Suboptimal outcome, a two steps
policy is needed to trigger a desired
regime shift to low costs
q2R  q; q
q2R   q,  
q0   q,  
One step policy triggers a desired
regime shift
W  q   B  q0   CL  q0 
 B  q1R   CL  q1R 
 
q0  q; q
2
and corresponding welfare outcomes with hysteresis
2
Table A3 illustrates the necessary condition for a producer choosing a profit maximizing
quantity, qtS in a particular period t for different intervals of the regulated price and given
that a particular cost function prevails.
0; CL  0  
Price ptR
q 0
S
t
Quantity
C  0 ,  
 ;  
L
 
q  CL
S
t
1
 ; 
 
qtS  q if CL  q   CH q
 p  >0
R
t
or
qtS   C  
1
p
S
t
, Ci  qt 1  

  min CL  q  ;CH  q 
q0  q; q
 
q0  q; q
C0  q0   CL  q0 
C0  q0   CH  q0 
q0
p  B  qt 
0; CL  q  
q0  0, q 
 
 
  B  qˆ  p    C  qˆ  p
R
2
2
L
2
R
2
 1 1
 1  

2
L
 

q0   q,  
     1 1
1
  B  qˆ  p    C  qˆ  p   
1   
 B q  CH q
R
2
L
2
2
 
One step policy triggers a desired
regime shift
W  q   B  q0   CL  q0 
  B  q   CL  q  
 
 B qˆ2  p
R
2
   C  qˆ  p   
2
W  q   B  q0   CH  q0 
 
 1 1
 
1
  B  qˆ  p    C  qˆ  p   
1   
 B qˆ1  p1R   CH qˆ1  p1R 
1
1 
H
No regime shift needed
R
2
2
1
1   
R
2
H
2
2
Table A 4: Price regulation when CL  q   CH  q  and corresponding welfare outcomes with hysteresis
23
R
2
Suboptimal outcomes, qt  q but ptR  B  qt   B qt
H
CH q ; 

L
W  q   B  q0   CH  q0 
2
 C  q  ; C  q  
R
t
permanently or in transition
1

 p  >0
One step policy triggers a desired
regime shift to low cost, quantities
go down below q either
W  q   B  q0   CL  q0 
 B qˆ1  p1R   CL qˆ1  p1R 
H
 
No regime shift needed
1
 ;   maxC q  ;C q 
Table A3: Produced quantity as a function of regulated price;
R
t
 
qtS  CH 
R
2
2
q0
p
 
0; CH q

 
q0  0, q 
R
2
 
q0  q; q
q0  q; q
C0  q0   CL  q0 
C0  q0   CH  q0 
No regime shift needed, even in
the hysteresis interval it is
optimal to keep low costs.
W  q   B  q0   CL  q0 
 

 
  B  qˆ  p    C  qˆ  p   
1   
 B qˆ1  p1R   CL qˆ1  p1R 
R
2
2
L
2
R
2
One step policy triggers a desired
regime shift, for p2R  CL q
 
quantities go down below q in
transition and rise to qˆ2  p2R 
1
1 
1
W  q   B  q0   CH  q0 
2
     1 1
1
  B  qˆ  p    C  qˆ  p   
1   
 B q  CH q
2
 
CH q ; CL  q 

q0   q,  



R
2
L
2
R
2
2
A one step policy cannot trigger a
desired regime shift to low cost
W  q   B  q0   CH  q0 
 
 1 1
 
1
  B  qˆ  p    C  qˆ  p   
1   
 B qˆ1  p1R   CH qˆ1  p1R 
2
C  q  ; 
L
One step policy triggers a desired
regime shift
W  q   B  q0   CL  q0 
  B  q   CL  q  
 
 B qˆ2  p
Table A 5: Price regulation
24
R
2
   C  qˆ  p   
2
H
2
R
2
2
No regime shift needed
W  q   B  q0   CH  q0 
 
 1 1
 
1
  B  qˆ  p    C  qˆ  p   
1   
 B qˆ1  p1R   CH qˆ1  p1R 
1
1 
H
R
2
R
2
2
1
1   
R
2
H
2
R
2
2
p2R , when CL  q   CH  q  and corresponding welfare outcomes with hysteresis
2