5. Externalities and Public Goods
5. Externalities and Public Goods
• Welfare properties of Walrasian Equilibria rely on the hidden assumption of private goods:
“the consumption of the good by one person has no effect on other people’s utility, but
prevents them from consuming that good”
• Externalities (or external effects) and Public Goods are the most relevant cases of market
failures because of the existence of non-private goods
– Externalities appear when the decisions of one agent (consumer or firm) has a direct effect
on the welfare of other agents (consumers or firms)
– Public Goods are goods such that the consumption by one agent does not prevent others
from consuming them as well
5. Externalities and Public Goods
Externalities
• Loud music at 3:00 am (negative consumption externality)
• Well kept garden (positive consumption externality)
• Paper factory upstream on a river (negative production externality)
• Bee farming next to a flower production field (positive production externality)
5. Externalities and Public Goods
Public Goods
Public Goods types
Non-Rivalrous
Rivalrous
Non-Excludable
(Pure) Public Goods
Common-Pool Resources
Excludable
Club Goods
Private Goods
• Highways
• National Defense
• Pollution
• TV signal
• Rural community commons
5. Externalities and Public Goods
5. Externalities and Public Goods
Externalities
$
'
Plan
Definition 5.1 An externality is present whenever the well-being of a consumer or the production possibilities of a firm are directly affected by the actions of another agent in the
economy
• Externalities
– The problem: Market inefficiency
– The solutions: Quotes, Taxes, Bargaining and Property Rights
– On missing markets
&
Direct externalities (the effect of a paper factory on a fishery downstream) are not to be mistaken
with pecuniary externalities (the effect of the price of paper on the costs of the fishery). While
the later are negotiated in the market, the former are not.
• Public Goods
– The problem: The inefficiency of private provision
– The solutions: Lindahl Equilibria
• Common-Pool Resources
5. Externalities and Public Goods
5.1 Bilateral Externalities
5. Externalities and Public Goods
5.1 Bilateral Externalities
5.1 Bilateral Externalities
From the point of view of individual 2, h represents an external effect of consumers 1’s action.
We therefore assume that
∂u2(x21, . . . , x2n, h)
6= 0
∂h
The indirect utility function is defined by
• I = {1, 2}
• n traded commodities (x1 is a numeraire)
• p (prices) are given. Consumers are price-takers and cannot affect prices
• At prices p, individual’s initial endowments generate a wealth of w
i
• The externality h is an action taken by individual 1 at 0 monetary cost
• Each individual has preferences over the commodities he consumes and over h
ui(xi1, . . . , xin, h)
vi(p, wi, h) = max ui(xi, h)
s.t. p · xi ≤ wi
%
5. Externalities and Public Goods
5.1 Bilateral Externalities
5. Externalities and Public Goods
5.1 Bilateral Externalities
Competitive Equilibrium Outcome
A usual simplifying assumption is that preferences are quasilinear with respect to the numeraire
commodity x1. By doing so, it can be proven that the demand functions of the n − 1 nonnumeraire commodities do not depend on w. Then, the indirect utility function takes the form
vi(p, wi, h) = φi(p, h) + wi
• Choice of h by individual 1 : Assuming that we are at a competitive equilibrium (in the n
commodities) at prices p, individual 1 will choose h to maximize vi . Hence, he will choose
the level h∗such that it satisfies the First Order Condition (assuming interior solution)
φ′1(h∗) = 0
Since we assume that prices p are constant, we may write
vi(wi, h) = φi(h) + wi
• Even though consumer 2’s utility depends on h, it cannot affect the choice of h. Herein lies
the problem
We further assume that φi is twice differentiable and that φ′′i (h) < 0
5. Externalities and Public Goods
5.1 Bilateral Externalities
5. Externalities and Public Goods
5.1 Bilateral Externalities
Efficient Outcome
Analysis
Clearly, the efficient level h∗∗ given by
In contrast, any Pareto efficient level of hmust maximize the joint surplus, that is, the sum of
the utility functions (because of the quasilinearity assumption)
and the competitive level h∗given by
max φi(h) + φ2(h)
The First Order Condition for an interior maximum is
φ′1(h∗∗)
+
φ′2(h∗∗)
φ′1(h∗∗) = −φ′2(h∗∗)
will never coincide unless
=0
That is, if h∗∗is a (interior) Pareto efficient solution, then
φ′1(h∗∗) = −φ′2(h∗∗)
(1)
which would mean NO externality
φ′1(h∗) = 0
φ′2(h∗∗) = 0
5. Externalities and Public Goods
5.1 Bilateral Externalities
5. Externalities and Public Goods
5.1 Bilateral Externalities
Negative Externality
• If φ′2(·) < 0 (negative externality) we have
−φ′2 (h)

φ′1(h∗∗ ) = −φ′2(h∗∗ ) > 0 
φ′1(·) decreasing
⇒ h∗ > h∗∗

′
∗
φ1(h ) = 0
• If φ′2(·) > 0 (positive externality) we have

φ′1(h∗∗ ) = −φ′2(h∗∗ ) < 0 
φ′1(·) decreasing
⇒ h∗ < h∗∗

φ′1(h∗) = 0
5. Externalities and Public Goods
h∗∗
5.1 Bilateral Externalities
5. Externalities and Public Goods
h∗
φ′1 (h) + φ′2 (h)
h
φ′1 (h)
5.1 Bilateral Externalities
Positive Externality
• The efficient outcome does not mean that the externality is eliminated (h∗∗ 6= 0)
φ′1 (h)
+
φ′2 (h)
• In the case of a negative externality, the marginal benefit to individual 1 equals the marginal
cost to individual 2
• In the case of a positive externality, it must be the case that the sum of the marginal benefit
of both individuals equals zero
φ′2 (h)
h∗
h∗∗
′
φ1 (h)
h
• The efficient level of a negative externality is greater that zero even in cases such as pollution,
noise, etc. It is difficult to “value” the benefits-costs of such externalities as, generally, those
that produce the negative externality (polluters) think that the optimal level of the externality
should be larger than those who suffer from the externality
5. Externalities and Public Goods
5.2. Classical Solutions
5. Externalities and Public Goods
5.2 Classical Solutions
Quotas
There are 3 classical approaches to solve the inefficiencies produced by externalities
• Quotas: The government imposes a constraint on the level of the activity producing the
externality
• Taxes: The government charges with a tax the activity producing the externality
• Bargaining: Provide conditions so that the individuals reach an optimal agreement by themselves on the level of the activity producing the externality
5. Externalities and Public Goods
5.2. Classical Solutions
5.2. Classical Solutions
• In the presence of a negative externality (h∗ > h∗∗) the government can simply pass a law
prohibiting levels beyond h∗∗
• In the presence of a positive externality (h∗ < h∗∗) the government can simply pass a law
requiring individual 1 to produce at least h∗∗
• Although this is a simple solution, it is difficult to implement. It requires the government to
enforce the quota and to monitor the producer, which can be difficult and costly
It would be better if the government could make some adjustments to the market so that it
worked properly
5. Externalities and Public Goods
5.2. Classical Solutions
Taxes
The government can impose a tax on the level of production, h, of the (negative) externality
Let th be a tax that individual 1 must pay per unit of h produced. His maximization problem,
then, becomes
max φ1(h) − th · h
whose First Order Condition is
φ′1(ht) = th
satisfies the First Order Condition 2
Thus, the tax
ht = h∗∗
th = −φ′2(h∗∗)
known as Pigouvian tax, implements the efficient outcome
We know (efficient solution 1) that
φ′1(h∗∗) + φ′2(h∗∗) = 0
Thus, by setting
(2)
we make sure that
th = −φ′2(h∗∗) > 0
5. Externalities and Public Goods
5.2. Classical Solutions
5. Externalities and Public Goods
5.2. Classical Solutions
−φ′2 (h)
• The optimal tax equals the marginal externality at the efficient level h∗∗
• It is equal to the amount that individual 2 would be willing to pay to slightly reduce h from
its efficient level h∗∗
• Due to tax cost beard by individual 1, he is required to internalize the externality that he
imposes on individual 2
−φ′2 (h∗∗ ) = th
h
∗∗
h
∗
• Tax must be imposed on the externality producing activity directly
h
φ′1 (h)
5. Externalities and Public Goods
• To implement quotas/taxes the government must have detailed knowledge of the benefits
and costs of the externality to the individuals
5.2. Classical Solutions
Bargaining and Enforceable Property Rights
• A decentralized possibility is to let the two individuals to negotiate a solution to the problem
by themselves
• The success of such procedure depends on the whether property rights are clearly assigned
– Does individual 1 has the right to produce h ? If so, how much ?
– Can individual 2 prevent individual 1 from producing h? If so, how much ?
• Coase Theorem: As long as property rights are clearly established, the two parties will negotiate in such a way that the optimal level of h is implemented
5. Externalities and Public Goods
5.2. Classical Solutions
Example
Suppose that individual 2 has the right (by law) to a smoke-free environment
• Individual 2 can prohibit individual 1 from smoking
• But individual 1 can “buy” from individual 2 the right to smoke a few cigarettes a day in
exchange of some money transfer
• The two individual can bargain both over the size of the transfer and over the number of
cigarettes that are allowed to smoke
5. Externalities and Public Goods
5.2. Classical Solutions
5. Externalities and Public Goods
5.2. Classical Solutions
Bargaining Solution
Bargaining Mechanism (Individual 2 rights)
Suppose that individual 2 has the right to an externality-free environment.
The bargaining procedure could work as follows:
1. Individual 2 offers individual 1 a take-it-or-leave-it contract specifying a payment T2 and an
activity level h2
2. Individual 1 decides on whether to accept such offer
(a) If the offer is accepted then the contract is implemented (h = h2)
(b) If the offer is rejected then individual 1 cannot produce any externality (h = 0)
For an offer (h, T ) to be accepted by individual 1, it must be the case that it produces higher
utility than the disagreement level h = 0.
That is, individual 1 will accept (h, T ) if and only if
φ1(h) − T ≥ φ1(0)
Given this (incentive compatible) constraint, individual 2 will choose (h2, T2) in order to solve
max
h.T
φ2(h) + T
s.t. φ1(h) − T ≥ φ1(0)
5. Externalities and Public Goods
5.2. Classical Solutions
Since individual 2 will prefer higher T , the constraint will be binding at the optimum. Hence,
the problem becomes
max φ1(h) + φ2(h) − φ1(0)
h
The First Order Condition is given by,
φ′1(h2) + φ′2(h2) = 0
5. Externalities and Public Goods
Hence,
• individual 2 will offer h2 = h∗∗ in exchange of a transfer of T2 = φ1(h∗∗) − φ1(0)
• Individual 1 will accept such offer (h2, T2)
Thus, the bargaining mechanism implements the socially efficient outcome
which is, precisely, the condition 1 that defines the Pareto efficient level of h.
5.2. Classical Solutions
5. Externalities and Public Goods
5.2. Classical Solutions
5. Externalities and Public Goods
5.2. Classical Solutions
Bargaining Mechanism (Individual 1 rights)
Bargaining Solution
Suppose now that individual 1 has the right to produce a level h of the externality as high as he
wants
The bargaining procedure could work as before:
As before, for an offer (h, T ) to be accepted by individual 1, it must be the case that it produces
higher utility than the disagreement level h = h∗.
That is, individual 1 will accept (h, T ) if and only if
1. Individual 2 offers individual 1 a take-it-or-leave-it contract specifying a payment T1 and an
activity level h1
φ1(h) − T ≥ φ1(h∗)
Given this (incentive compatible) constraint, individual 2 will choose (h2, T2) in order to solve
2. Individual 1 decides on whether to accept such offer
(a) If the offer is accepted then the contract is implemented (h = h1)
(b) If the offer is rejected then individual 1 can produce as much as he wants (h = h∗)
5. Externalities and Public Goods
5.2. Classical Solutions
Again, since individual 2 will prefer higher T , the constraint will be binding at the optimum.
Hence, the problem becomes
max φ1(h) + φ2(h) − φ1(h∗)
h
The First Order Condition is given by,
φ′1(h2) + φ′2(h2) = 0
which, again, is the condition 1 that defines the Pareto efficient level of h.
max
h.T
φ2(h) + T
s.t. φ1(h) − T ≥ φ1(h∗)
5. Externalities and Public Goods
5.2. Classical Solutions
Hence,
• individual 2 will offer h1 = h∗∗ in exchange of a transfer of T1 = φ1(h∗∗) − φ1(h∗)
• Individual 1 will accept such offer (h1, T1)
Thus, the bargaining mechanism implements, again, the socially efficient outcome
5. Externalities and Public Goods
5.2. Classical Solutions
5. Externalities and Public Goods
5.2. Classical Solutions
Externalities and missing markets
• The fact that, regardless of how property rights are allocated, bargaining leads to a Pareto
efficient (h = h∗∗) outcome is known as the “Coase Theorem”
• The way property rights are established has distributional consequences: The transfer is larger
in the case individual 2 has the property right than when individual 1 has. Notice, though,
that under the quasilinearity assumption wealth transfers have no effect on social welfare
• Well-defined property rights, and quasilinearity, are essential to the result
• The government does not need to have any information on the benefits-costs of the externality
to any of the individuals. It only has to ensure that property rights are clear and enforceable
The problem generated by the presence of externalities is often seen as a problem of the absence
of a market for the externality generating activity
Suppose that
• There is a market for activity h
• Individual 2 has the right to prevent all activity h, but can sell the right to produce 1 unit of
h at the unit price ph
In this case, individual 1 faces the problem
max φ2(h) + phh
h
5. Externalities and Public Goods
5.2. Classical Solutions
5. Externalities and Public Goods
5.2. Classical Solutions
The First Order Condition in this case is,
The First Order Condition is
φ′1(h) = ph
φ′2(h) = −ph
which implicitly defines a supply function h2(ph)
In turn, when deciding how many rights to purchase, individual 1 solves the problem
max φ1(h) − phh
which implicitly defines a demand function h1(ph)
The market-clearing condition establishes that h1(ph) = h2(ph), that is,
φ′1(h) = −φ′2(h)
h
This condition, once more, is the equation 1 determining the efficient outcome
The existence of this market, therefore, induces an equilibrium level of the externality generating
activity equal to its social optimum h∗
5. Externalities and Public Goods
5. Externalities and Public Goods
5.3 Public Goods
5.3 Public Goods
Public Goods types
Definition 5.2 A Public Good is a commodity such that its consumption by one individual
does not preclude consumption by other individuals as well
• It should be noted that the definition refers to the same unit of the commodity
• The name public refers to the particular characteristic of the commodity that it is available to
the everybody at the same time, not that it is provided by a public organism (public school)
or community (public television) or that it is free (public domain or public licenses)
Non-Rivalrous
Rivalrous
Non-Excludable
Pure Public Goods (Police)
Common-Pool Resources
Excludable
Club Goods (Satellite TV)
Private Goods (Apples)
• A Public Bad can usually be redefined as a Public Good
– Pollution → Clean Air
– Noise → Silence
5. Externalities and Public Goods
5.3 Public Goods
5. Externalities and Public Goods
5.3 Public Goods
Pure Public Goods
As before, we assume that preferences are quasilinear with respect to the numeraire commodity
x1. Then, the indirect utility function takes the form
• I = {1, 2, . . . , I}
• n traded commodities (x1 is a numeraire)
• p (prices) are given. Consumers are price-takers and cannot affect prices. Decisions on the
Public Good cannot affect prices of the traded commodities.
• At prices p, individual’s initial endowments generate a wealth of wi
• The quantity of the Public Good is denoted by x.
• Each individual has preferences over the commodities he consumes and over x
ui(xi1, . . . , xin, x)
vi (p, wi, x) = φi(p, x) + wi
Since we assume that prices p are constant, we may write
vi(wi, x) = φi(x) + wi
We further assume that φi is twice differentiable with φ′′i (x) < 0, and that φ′i(x) > 0 in the
case of a public good (negative if it is a public bad)
On the supply side, the cost of supplying q units of the public good is c(q), where c′(q) >
0 and c′′(q) > 0
5. Externalities and Public Goods
5.3 Public Goods
5. Externalities and Public Goods
5.3 Public Goods
Efficient Outcome
A Pareto efficient level of provision of the public good must maximize the aggregate surplus,
max
q
I
X
c′ (q)
φi(q) − c(q)
i=1
I
X
i=1
whose corresponding First Order Condition is given by,
I
X
φ′i (q)
φ′i(q ∗∗) − c′(q ∗∗) = 0
i=1
q ∗∗
Thus, at the efficient level, the total marginal utility is equal to the marginal cost
5. Externalities and Public Goods
5.3 Public Goods
q
5. Externalities and Public Goods
5.3 Public Goods
Private Provision of a Public Good
The firm maximizes profits
Suppose now that the public good is to be provided by a profit-maximizer private company that
sells “1 unit” of the public good at the unit price p.
At this price p, each consumer decides how much of the public good to buy, xi
Consumer are utility-maximizers and thus the problem
max φi(xi +
xi
X
xj ) − pxi
j6=i
has, for each consumer, the First Order Condition
φ′i(x∗i + x∗−i ) = p
so that
max pq − c(q)
p = c′(q ∗)
The equilibrium condition requires that supply equals demand, that is,
x∗ =
I
X
x∗i = q ∗
i=1
Thus, for any individual that purchases a positive amount of the public good it must be the case
that
φ′i(x∗) = c′(x∗)
5. Externalities and Public Goods
5.3 Public Goods
5. Externalities and Public Goods
5.3 Public Goods
Analysis
Notice that for all individuals, even for those that do not contribute to (do not purchase) the
public good, we have that φ′i(x∗) > 0
Suppose that individuals 1 through K do not contribute while individuals K + 1 through I do
. Then,
I
X
φ′i(x∗)
=
K
X
=
K
X
I
X
φ′i(x∗) > c′(x∗)
i=1
φ′i(x∗)
whenever a positive amount of the public good is supplied, which does not coincide with the
Pareto efficient level
I
X
φ′i(x∗∗) = c′(x∗∗ )
φ′i(x∗) + (I − (K + 1))c′(x∗) > c′(x∗)
Thus, when people make voluntary contributions the market will provide too little of the public
good
φ′i(x∗)
i=1
i=1
so,
+
I
X
i=1
i=K+1
i=1
5. Externalities and Public Goods
5.3 Public Goods
5. Externalities and Public Goods
5.3 Public Goods
Explanation: The Free-Rider Problem
• Given x∗, the purchase of one more unit of the public good by consumer i involves
′
c (q)
I
X
i=1
– a private cost of p
– a private benefit of φ′i(x∗) P
– but also a public benefit of j6=i φ′j (x∗)
φ′i (q)
• Nevertheless, when deciding on xi individual i considers only the private benefit against the
private cost
• For the efficient solution, the society values the social benefit against the social cost (which
is equal to the private cost as p = c′(q))
q∗
q ∗∗
q
5. Externalities and Public Goods
5.3 Public Goods
5. Externalities and Public Goods
Example
5.4 Lindahl Equilibrium
Suppose that individuals can be ordered according to their marginal utilities
There are several approaches to the solution of the insufficient provision of a public good in the
market
φ′1(x) < φ′2(x) < · · · < φ′I (x) for all x
• The government could impose (by law) to each individual his contribution
In this case, the individual First Order Condition
• The government could take over the provision of the public good
φ′i(x∗) = φ′i(x∗i + x∗−i) = p
can hold for at most one individual, call it m. All i 6= m will choose x∗i = 0.
This implies that, ∀i 6= m, φ′i(0 + x∗−i ) < p = φ′m(x∗m + 0). Thus, m must be individual I
since φ′I (x) > φ′i(x) for all i 6= I
5. Externalities and Public Goods
5.4 Lindahl Equilibrium
For instance, the government can set a per unit subsidy to each individual equal to
si =
X
φ′i(x∗∗ )
(3)
j6=i
• The government could tax/subsidize the provision of public good in such a way that provides
incentives so that private benefits/cost are in line with public benefits/costs
5. Externalities and Public Goods
5.4 Lindahl Equilibrium
Substituting the values of the subsidy in 3 and adding the market-clearing equilibrium condition
we have,
X
φ′i(xi + x∗∗
φ′j (x∗∗) = p = c′(x)
−i ) +
j6=i
∗∗
which is satisfies when x = x , so that
Then, the individual i’s problem becomes (assuming all other individual choose
max φi(xi + x∗∗
−i ) + sixi − pxi
The corresponding First Order Condition is, in this case,
φ′i(xi + x∗∗
−i ) + si = p
x∗∗
j )
I
X
φ′i(x∗∗) = c′(x∗∗ )
i=1
Thus, the efficient level is implemented (but might be quite expensive !!)
5. Externalities and Public Goods
5.4 Lindahl Equilibrium
5. Externalities and Public Goods
5.4 Lindahl Equilibrium
Lindahl Equilibrium
Suppose that the public good can be unbundled into I private goods xi (i = 1 . . . , I) , where
each good can be interpreted as the “individual i’s enjoyment of the public good”. Each of these
goods will have its own price pi, known as the Lindahl price.
Given an equilibrium price, the individual’s problem is
On the supply side, when the firm produces one single unit of the public good x, it is producing
one unit of each of the personalized good x1, . . . , xI . Hence, for each unit of x the firm produces
it earns p1 + p2 + · · · + pI . Thus, the firm’s problem becomes
max φi(xi) − pixi
max
I
X
pix − c(x)
i=1
which has a First Order Condition
and the corresponding First Order Condition
φ′i(xi ) = pi for each i
I
X
pi = c′(x)
i=1
5. Externalities and Public Goods
5.4 Lindahl Equilibrium
5. Externalities and Public Goods
5.5 Common-Pool Resources
Combining the optimality conditions of the consumers and the firm we get
I
X
φi(x) = c′(x)
i=1
which, again, is satisfied at the efficient level x = x∗∗. Thus, the Lindahl Equilibrium implements
the socially efficient level of public good, yet
• Individuals act as price takers of its own personalized good
• For the equilibrium to work, individuals must believe that if they do not purchase any of the
public good, then they will not be able to consume any of it
• It could be a good model for Club goods
A common-pool resource is a kind of good whose consumption is rivalrous and non-excludable.
Examples are:
• local fishing grounds
• common grazing land
• irrigation systems
• common wood forests
The typical problem in these cases is the overuse of the resource
5. Externalities and Public Goods
5.5 Common-pool resources
5. Externalities and Public Goods
Example: A Local Fishery Ground
Pareto efficient number of boats
• There are I fishers, each owning ki fishing boats
• k=
PI
i=1 ki
5.5 Common-pool resources
The Pareto efficient solution is found from the problem
denotes the total number of operating fishing boats
max f (k) − c(k)
• All boats are equal, they can catch the same number of fishes
• f (k) represents the total number of fishes caught by k boats (f ′ > 0, f ′′ < 0, f (0) = 0)
′
whose corresponding First Order Condition is
f ′(k ∗∗) = c′(k ∗∗)
′′
• Fishing boats are produced at a cost c(k) (c > 0, c > 0)
(4)
• The price of fish is normalized to 1
• The quantity used of the public good is tied (by f ) to the number of boats k
5. Externalities and Public Goods
5.5 Common-pool resources
5. Externalities and Public Goods
5.5 Common-pool resources
The Market equilibrium number of boats
Market-clearing conditions imply that
Supply If p is the market-clearing price of a fishing boat, the boat producers must solve
f ′(k ∗)
max pk − c(k)
whose solution is given by
′
p = c (k)
(5)
Demand Each fisher solves the problem
max
ki
ki
f (k) − pki
ki + k−i
the First Order Condition is s
f ′(k)
ki f (k) k−i
+
=p
k
k k
(6)
∗
ki∗ f (k ∗) k−i
+
= c′(k ∗)
k∗
k∗ k∗
(7)
By symmetry, at equilibrium we must have ki∗ = kj∗for all i, j. Thus, condition 7 may be
rewritten as
1 f (k ∗) I − 1
f ′(k ∗) +
= c′(k ∗)
(8)
I
k∗
I
5. Externalities and Public Goods
5.5 Common-pool resources
5. Externalities and Public Goods
5.5 Common-pool resources
Altogether we have,
• Notice that the left-hand side of condition 8 is a convex combination of the marginal product,
f ′(k), and the average product, f (k)
k
• Since f is concave, we know that
f ′(k) <
• Thus,
f (k)
k
• At the Pareto efficient level k ∗∗we have
f ′(k ∗∗) = c′(k ∗∗)
• At the market equilibrium level k ∗we have
f ′(k ∗) < c′(k ∗)
∗
1 f (k ) I − 1
> f ′(k ∗)
c′(k ∗) = f ′(k ∗) +
I
k∗
I
• Since f ′ is decreasing and c′is increasing, we have k ∗ > k ∗∗
That is, the Market overuses the fishery
5. Externalities and Public Goods
5.5 Common-pool resources
5. Externalities and Public Goods
5.5 Common-pool resources
Solutions
c′ (k)
• Quotes
• Taxes
• Government control
f ′ (k)
k ∗∗
k∗
k
• Community Norms, promoting cooperation, punishment ... (Game theoretical machinery !)