Externalities
• In large markets where each individual agent has no control over aggregate
outcomes, each agent might not take the full consequences of their actions
into account:
• ... driving a car or flying leads to noise and air pollution
• ... buying a puppy from a puppy mill instead of adopting supports puppy
mills
• ... eating cod means cod are closer to being endangered
• ... killing mosquitos to combat zika is killing millions of bees
• ... those bees provide benefits to flower farms by pollinating them essentially for free
• ... people who live near flower farms get less beautiful views and smells
• When one agent’s consumption or production of a good affects the payoffs
of another agent, it is an externality
Pollution externalities
• We’ll start with a simple, classic problem: externalities from pollution
• The goal is to (1) precisely describe where the inefficiencies in the market
arise, and (2) look at optimal taxes and quotas, the classic solutions
Pollution externalities
Here’s a simple model: There is a representative household that takes price
and pollution X as given, and maximizes its utility subject to its budget constraint:
max v(q) − pq − X + w,
q
and a representative firm that takes price as given and maximizes profits,
max pq − C(q),
q
and aggregate pollution is
X(q) = xq.
What is the price-taking/perfectly competitive equilibrium?
1
Pollution externalities
The household’s demand is determined by
v 0 (q ∗ ) = p∗ ,
the firm’s supply is determined by
p∗ = C 0 (q ∗ ),
and pollution is
xq ∗ .
So in equilibrium, v 0 (q ∗ ) = C 0 (q ∗ ), and the market equates the private marginal
benefit of consumption to the household with the marginal cost of the firm.
Pollution externalities
But a utilitarian social planner would sum household welfare and firm profits,
yielding
max {v(q) − pq − xq} + {pq − C(q)},
q |
{z
} |
{z
}
Household utility
Firm profits
or
max v(q) − C(q) − xq.
q
The social planner would then pick a q satisfying
v 0 (q o ) − c0 (q o ) − x = 0,
so the socially optimal quantity is less than the perfectly competitive quantity:
the market is inefficient.
Pollution externalities
With externalities, markets are no longer efficient:
2
P
S =c ' (q)
Equilibrium
Efficient
D=v ' (q)
v ' (q)− x
q
Pollution externalities
• The key is that the private marginal benefit to the household is
v 0 (q)
while the social marginal benefit is
v 0 (q) − x.
• This is why many problems feel intractable: your incentives are to ignore
your influence on the outcome, since you can’t do much to change it, but
you know that “giving in” and doing something that indirectly harms
others is the root of the problem in the first place.
Pollution externalities
• Taxes reduce behaviors at the margin, but what is the optimal tax?
• Let the household solve
max v(q) − (p + t)q + w − X
q
where t is the tax. This gives the FONC’s for the household and firm,
v 0 (q ∗ ) − p − t = 0,
3
p − c0 (q ∗ ) = 0.
If we combine and compare with the social planner’s FONC, we see at the
social optimum,
v 0 (q o ) − c0 (q o ) − x = 0
while the market picks
v 0 (q ∗ ) − c0 (q ∗ ) − t = 0.
• Equate the tax, t to x, then implements the social optimum.
Pollution externalities
• What if we tax firms instead?
• Let the household and firm solve
max v(q) − pq + w − X,
q
max(p − t)q − C(q)
q
where t is the tax. This gives the FONC’s for the household and firm,
v 0 (q ∗ ) − p = 0,
p − t − c0 (q ∗ ) = 0.
If we combine and compare with the social planner’s FONC, we see at the
social optimum,
v 0 (q o ) − c0 (q o ) − x = 0
while the market picks
v 0 (q ∗ ) − c0 (q ∗ ) − t = 0.
• So charge the firm x for each additional unit it produces to get the optimal
outcome.
Pollution externalities
• This sounds nice, but is harder than it looks.
• Suppose the pollution costs to society are x2 q 2 , v(q) = b log(q), and C(q) =
c 2
2 q . What is the optimal tax? How does the tax vary with x, b, and c?
•
r
∗
q =
b
,
c+x
√
x b
t =√
c+x
∗
• For a more complicated, real world example, reflect on how hard it would
be to do this perfectly.
4
Pollution externalities
Pros:
• The intuition is simple: internalize the externality that people are creating
• A simple tax scheme can implement the optimal outcome
Cons:
• Requires knowing the marginal social cost of pollution, at the optimal
quantity: this is really hard to guess
• Measuring the cost of pollution is difficult, especially in health/life-years
terms
• Getting such a policy passed is difficult
• Monitoring and enforcing policies can be very difficult (VW diesels)
Pollution Quotas
• There is a big debate about whether pollution policy (carbon emissions)
should be done through taxes (prices) or quotas (quantities)
• Suppose, instead, that the government was selecting quotas q̄ instead of
setting taxes t: how would it do it?
• Maximizing social surplus means:
max v(q) − c(q) −→ v 0 (q e ) − c0 (q e ) = 0
q
which is what we got from the taxes above: for every tax policy there’s
an identical quota policy, and vice versa
• Why might firms prefer quotas to taxes?
Cap and Trade
• Sometimes people advocate capping the total amount of pollution society
allows, and then letting firms trade permits to pollute
• They often argue that in a large market, firms will negotiate and come to
the socially optimal outcome
• What does it take to get that result in a simple model?
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Cap and Trade
• Suppose there are a huge number of firms j, each with a cost hj of producing a single unit, and all firms can produce only one unit
• Firms take the price of a permit, s as given, and the price in the market
of the good, p as given. Then they decide whether to sell the permit or
keep it and produce.
• If a firm j wants to sell, it must be the case that
p − hj ≤ s,
since otherwise it would be profitable to produce. If a firm k wants to
buy, it must not have a permit and
p − hk > s.
• But now, notice that a price of a permit is zero in some efficient pricetaking equilibrium: if the price of permits is zero, firms for whom p−hj < 0
are indifferent between selling and not, so they are willing to sell at zero.
Then firms with p − hk > 0 buy and produce.
Cap and Trade
Possible problems:
• Dynamics and Investment: firms might hold onto permits rather than
trade because the permits have option value, meaning they might be useful
in the future
• Speculation: firms might hold onto permits rather than trade because they
expect the price to go up in the future, giving them a better deal
• Power in the Market for Permits: firms might hold onto permits rather
than trade because it constraints rival firms, making the firms who hoard
permits more profitable
Beyond taxes and quotas
• We want to switch to thinking about how to give agents incentives to pick
socially efficient actions in general, not just in simple pollution models
• We also have other concerns: we want incentive systems to balance the
budget, or not disproportionately negatively affect certain people
• This leads to two ideas: Lindahl pricing, and the Coase theorem
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All of design economics is about externalities
We’re going to use arguments similar to those above to argue two things:
• Lindahl pricing: Under complete information, you can always give incentives for agents to take efficient actions, even when there are externalities
and market power
• Coase theorem: Under complete information and with no transactions
costs, there is always a way to implement the efficient outcome when
agents can each “opt out” if it is in their best interests
All of design economics is about externalities
• Suppose each agent i chooses xi , but the utility of everyone in the economy
depends on the choices of everyone else, so that i’s utility of outcome
x = (x1 , x2 , ..., xN ) is
ui (x1 , x2 , ..., xi , ..., xN ).
• We can impose a tax on agent i, so that he solves
max ui (x1 , x2 , ..., xi , ..., xN ) − ti (x1 , x2 , ..., xi , ..., xN )
xi
• But we’re interested in making society as well off as possible. The efficient
xe solves
N
X
max
ui (x1 , x2 , ..., xi , ..., xN )
x1 ,x2 ,...,xN
i=1
{z
|
Social surplus
}
• How do we incentivize each agent i to pick xei , so we maximize welfare?
All of design economics is about externalities
• Well, charge agent i a tax ti (x1 , x2 , ..., xi , ..., xN ) that depends on everyone’s choices, so i’s payoff is
max ui (x) − ti (x).
xi
• If we don’t have money from outside, like a government grant, and can’t
keep money paid to us by the agents, we need to ensure the sum of the
taxes is zero. We say the taxes are budget balanced if
I
X
ti (x) = 0.
i=1
Our question is, “Are there taxes that give everyone an incentive to pick
the socially efficient action, and sum to zero so the budget is balanced?”
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All of design economics is about externalities
Definition 1. The Lindahl prices or Lindahl taxes are
ti (xi ) =
hi
|{z}
A payment that doesn’t depend on i’s choice
X
−
uj (xe1 , xe2 , ..., xi , ..., xeN )
j6=i
|
{z
}
Utility of all other agents 6= i if they behave optimally
The idea is to add the welfare of the other agents onto agent i’s payoff minus
a constant, so that i’s motives are then to maximize social surplus.
All of design economics is about externalities
• At the Lindahl prices, i’s payoff becomes
max
xi
N
X
uj (xe1 , ..., xi , ..., xeN ) −
j=1
|
hi
|{z}
A payment that doesn’t depend on i’s choice
{z
}
Social surplus
• Notice that if everyone else is doing the optimal thing – xej – then i cannot
do better than choose xei , since that is what maximizes social surplus: this
incentivizes agents to take the efficient action
All of design economics is about externalities
So recall our question:
• Is there a tax scheme that incentivizes everyone to take the efficient action,
and is budget balanced?
• If we sum these taxes, we get
I
X
i=1
hi −
N X
X
uj (xe1 , xe2 , ..., xi , ..., xeN ).
i=1 j6=i
If agent i picks xei , we then get tax revenue
I
X
i=1
hi − (N − 1)
N
X
uj (xe1 , xe2 , ..., xei , ..., xeN ).
j=1
(if you don’t see this, write out the sums for N = 2, N = 3, and so on.)
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Lindahl pricing
Theorem 2. Suppose each agent i’s utility function is ui (x). Then the efficient
outcome xe can be implemented by imposing a tax on each agent i equal to
X
ti (xi ) = hi −
uj (xe1 , xe2 , ..., xi , ..., xeN ).
j6=i
The tax scheme is budget-balanced if
N
X
N
ui (xe ) =
i=1
1 X
hi ,
N − 1 i=1
which can always be achieved by setting, for example,
N
hi = h =
N −1X
ui (xe ).
N i=1
Lindahl pricing
To do this, there are five steps:
• Step 1: maximize social surplus to get the efficient actions
• Step 2: design the Lindahl prices
• Step 3: verify the Lindahl prices implement efficient decision-making
• Step 4: investigate budget balance
Example 1: Pollution taxes
There’s a representative household that solves
max v(q) − pq + w − X(q),
q
a representative firm that solves
max pq − C(q),
q
Example 1: Pollution taxes
Step 1: Maximize social surplus to find the efficient actions. Social
surplus is
max v(q) + w − C(q) − X(q),
q
so the efficient outcome satisfies
v 0 (q e ) − C 0 (q e ) − X 0 (q e ) = 0.
We want to implement q e in the market.
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Example 1: Pollution taxes
Step 2: design Lindahl taxes. The Lindahl tax for the firm is
tf (q) = hf − (v(q) − pq − X(q) + w)
and the household’s tax is
th (q) = hh − (pq − C(q))
Example 1: Pollution taxes
Step 3: verify the taxes implement efficient decision-making. Then
the firm solves
max pq − C(q) − tf (q) = max v(q) + w − X(q) − C(q) + hf ,
q
q
and the solution satisfies
v 0 (q ∗ ) − C 0 (q ∗ ) − X 0 (q ∗ ) = 0,
just as in the socially optimal outcome. Similarly the household solves
max v(q) − pq + w − X(q) − th (q) = max v(q) + w − C(q) − X(q) − hh ,
q
q
which again yields the optimal outcome.
Example 1: Pollution taxes
• Step 4: investigate budget balance. The sum of the taxes is
tf (q) + th (q) = hf − (v(q) − X(q) + q) + hh − C(q)
• So any payments hf , hh that satisfy
hf + hh = v(q e ) − X(q e ) − C(q e ) + w
will balance the budget.
• For example, firms make zero profit if
hf = v(q e ) − X(q e ) − C(q e ) + w = −hh ,
and then households get all the gains from trade. Similarly, if
hh = v(q e ) − X(q e ) − C(q e ) + w = −hf ,
the firms keep all the gains from trade.
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Example 2: Regulating a monopolist with Lindahl taxes
The household, as usually, takes prices as given:
max v(q) − pq + w,
q
and the monopolist firm has total costs C(q). Step 1: maximize social
surplus to get the efficient actions. Social surplus is
max v(q) − C(q),
q
with FONC v 0 (q e ) − C 0 (q e ) = 0. How do we get the monopolist to do this?
Example 2: Regulating a monopolist with Lindahl taxes
• Step 2: design Lindahl taxes. Impose a tax on the firm tf (q) =
hf − (v(q) − p(q)q + w), so its problem is
max p(q)q − tf (q) − C(q) = v(q) + w − C(q) − hf
q
• Impose a tax on the household, th (q) = −(p(q)q − C(q)) + hh , so its
problem is
max v(q) − th (q) − pq + w = max v(q) + w − C(q) − hh
q
q
Example 2: Regulating a monopolist with Lindahl taxes
Step 3: verify the taxes implement the efficient outcome. Since
t(q) = −(v(q) − p(q)q + w) + hf ,
the monopolist then solves
max p(q)q − t(q) − C(q) = v(q) − C(q) + w − hf ,
q
which yields the FONC
v 0 (q ∗ ) − C 0 (q ∗ ) = 0,
and the monopolist behaves socially optimally. The same applies to the household.
Example 2: Regulating a monopolist with Lindahl taxes
• Step 4: investigate budget balance. We want the tax paid by the
household to equal the tax received by the firm so the side payments
balance out.
• This means that
th (q e ) + tf (q e ) = hh + hf − (v(q ∗ ) + w − C(q ∗ )) = 0
• So any pair of numbers hh + hf that equal social surplus will work
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Example 2: Regulating a monopolist with Lindahl taxes
• Option 1: hf = v(q ∗ ) − C(q ∗ ) + w = −hh . Tax the monopolist up to
social welfare and redistribute the tax revenue back to the consumers and
implement the perfectly competitive outcome
• Option 2: hh = v(q ∗ ) + w − C(q ∗ ) and hf = 0. This lets the monopolist
have all the gains from trade. Consumer surplus is zero, but the optimal
quantity is implemented.
• Option 3: hh + hf = (v(q ∗ ) + w − C(q ∗ ))/2. Households and firms split
the gains from trade.
Example 3: Educational subsidies
• Suppose there are i = 1, 2, ..., N households who are deciding how much
education to get. The more educated each member of the economy is, the
more productive everyone is overall.
• Household i’s payoff is its wage less the cost of education:
wi (e1 , e2 , ..., ei , ..., eN ) − cei
• If household i chooses its education alone, taking the education levels of
the other households as given, it would pick
∂wi (e1 , ..., ei , ..., eN )
− c = 0.
∂ei
Example 3: Educational subsidies
• Social surplus, however, is
N
X
wi (e1 , ..., ei , ..., eN ) − c
i=1
N
X
ei ,
i=1
and (Step 1: maximize social surplus.) household i should be choosing
N
X ∂wj (e1 , ..., ei , ..., eN )
∂wi (e1 , ..., ei , ..., eN )
+
−
c
=0
|{z}
∂ei
∂ei
|
{z
} j6=i
|
{z
} Marginal cost
Private return to education
Externality to j 6= i
• (Step 2: design Lindahl taxes.) Then the Lindahl taxes are
X
ti (e) = −
(wj (e1 , ..., ei , ..., eN ) − cej ) + hi
j6=i
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Example 3: Educational subsidies
• (Step 3: verify taxes implement efficient decision-making.) Each
household i then solves
max wi (ee1 , ..., ei , ..., eiN ) − cei − ti (ei )
ei
= max
ei
N
X
wi (ee1 , ..., ei , ..., eeN ) − cej −hi ,
j=1
|
{z
Social surplus
}
so that i is really maximizing social surplus, and the efficient education
level is optimal.
Example 3: Educational subsidies
• Step 4: investigate budget balance. To get budget balance, then, we
need to pick hi ’s so that
N
X
N
(wi (ee ) − ceei ) =
i=1
1 X
hi
N − 1 i=1
(there are many options, depending on how you want to split the surplus
within society)
This is what optimal tuition assistance would look like.
The Coase theorem
• Lindahl’s theorem implies that the efficient outcome can be implemented
in a budget-balanced way
• But what if the agents can refuse to participate? Even the government
often can’t compel people to participate in a market like this
The Coase theorem
• Now, imagine that we add the constraint that each agent i’s payoff must
be weakly positive, so they are better participating in this scheme than
not:
ui (x) − ti (x) ≥ 0.
• If the above inequality is satisfied, we say that participation is individually
rational (e.g., you can’t pay the CEO a negative amount)
13
• If an agent i “opts out”, write that xi = ∅, and then the payoffs of the
other players are given by
uj (x1 , x2 , ..., xi = ∅, ...xN )
• When can we implement the efficient outcome when we have to respect
individual rationality?
The Coase theorem
• We’re going to use Lindahl pricing, but with a slight twist:
Definition 3. Let the tax for agent i be his contribution to social welfare:
X
ti (x) =
uj (x1 , ..., xi = ∅, ..., xN )
j6=i
{z
|
}
Surplus of the other agents when i refuses to participate
−
max
X
xj ,j6=i
|
uj (x)
(1)
j6=i
{z
}
Surplus of the other agents when i participates
• We’re now setting the tax equal to each agent’s contribution to overall
social welfare
The Coase Theorem
• Then agent i’s payoff is given by ui (x) − ti (xi ), and if the other agents
take the efficient action, i’s payoff is
max ui (xe1 , ...xi , ...xeN ) − ti (xi ) =
xi
max ui (xe1 , ...xi , ...xeN ) +
xi
xj ,j6=i
xi
|
N
X
uj (xe1 , ..., xi , ..., xeN )−
j6=i
− max
= max
X
X
uj (x1 , ..., xi = ∅, ..., xN )
j6=i
uj (xe1 , ..., xi , ..., xeN ) − max
xj ,j6=i
j=1
{z
Social surplus
}
Coase theorem
14
|
X
uj (x1 , ..., xi = ∅, ..., xN )
j6=i
{z
}
Surplus of the other agents if i opts out
• Since xei maximizes social welfare and i is solving
max
xi
|
N
X
uj (xe1 , ..., xi , ..., xeN ) − max
xj ,j6=i
j=1
{z
Social surplus given i’s decision
}
X
uj (x1 , ..., xi = ∅, ..., xN ),
j6=i
|
{z
}
Doesn’t depend on xi
so each agent i should pick the efficient action xei .
• This implements efficient decisions by the agents, and is individually rational as long as
max
x
I
X
uj (x) − max
xj ,j6=i
j=1
X
uj (x1 , ..., xi = ∅, ..., xN ) ≥ 0,
j6=i
which is an important inequality: it means that all of the agents contribute
a positive benefit to overall welfare.
The Coase theorem
• If, for all i = 1, 2, ..., N ,
max
x
|
I
X
uj (x) ≥ max
xj ,j6=i
j=1
{z
Social surplus
}
|
X
uj (x1 , ..., xi = ∅, ..., xN ),
j6=i
{z
Social surplus if i opts out
}
then each agent contributes something non-negative to social surplus by
participating. If this is true, say that all agents contribute to social surplus
The Coase theorem
Theorem 4. As long as all agents contribute to social surplus, transactions
costs are zero, and all parties are symmetrically informed, they can negotiate to
the optimal outcome regardless of the initial allocation of property rights.
Is there a budget balanced, individually rational tax scheme that implements
the efficient outcome when all agents contribute to social surplus? We can add
constants hi to the taxes yielding revenue


N 
N

X
X
X
1
ui (xe ) =
hi +
uj (xe1 , ..., xi = ∅, ..., xeN ) ,


N −1
i=1
i=1
j6=i
and we then pick the hi ’s much more carefully.
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The Coase theorem
• Step 1: maximize social surplus to find the efficient actions
• Step 2: design the Lindahl prices to reflect agents’ marginal contributions
to social welfare
• Step 3: verify the Lindahl prices implement the efficient actions
• Step 4: verify individual rationality
• (Step 5: investigate budget balance (if the question asks))
Example: fair division of profits at a firm
• There are two employees at a firm, i = 1, 2, and each has to choose to
work hard, e = 1, or opt out, e = 0
• If both agents work hard, the firm generates profits pi which they split
equally, and if either takes it easy and opt out, the firm generates zero
profits.
• The cost of working hard is c1 for agent 1 and c2 for agent 2, and the cost
of taking it easy is 0
Example: fair division of profits at a firm
• Step 1: maximize social surplus. Social surplus is
max
e1 =0,1;e2 =0,1
e1 e2 π − c1 e1 − c2 e2 .
So as long as π > 2c, it’s optimal for the agents to work.
• Step 2: design Lindahl taxes. The tax for agent 1 is
pi
t1 (e1 ) = −e1
− c2 + h1
2
and the tax for agent 2 is
t2 (e2 ) = −e2
pi
− c1
2
+ h2
• Step 3: verify the Lindahl prices implement efficient actions.
Agent 1’s payoff is
max e1 e2 (pi − c1 − c2 ) − h1 ,
e1 ∈0,1
so that e1 = 1 is optimal, and agent 2’s payoff is
max e1 e2 (pi − c1 − c2 ) − h2 ,
e2 ∈0,1
so e2 = 1 is optimal. These taxes implement efficient actions.
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Example: fair division of profits at a firm
• Step 4: verify individual rationality. Are each agent’s payoffs positive?
π
max e1
− c1 − t1 (e1 )
e1 =0,1
2
= max e1 (π − c1 − c2 ) − h1 = π − c1 − c2 − h1,
e1 =0,1
which is positive as long as h1 ≤ π − c1 − c2 . (A similar calculation applies
to agent 2.)
• Step 5: investigate budget balance. Budget balance will hold if
t1 (e1 ) + t2 (e2 ) = 0, or
t1 (1) + t2 (1) = − (π − c1 − c2 ) + h1 + h2 = 0.
Total profit π is greater than c1 + c2 , and each agent’s payoff is π − c1 −
c2 − h1 and π − c1 − c2 − h2 , respectively. Any h1 + h2 that sum to firm
profit and satisfy h1 ≤ π − c1 − c2 will work.
Example: fair division of profits at a firm
• For example, h = −(π − c1 − c2 )/2 to split evenly
• Or
c1
(π − c1 − c2 ),
c1 + c2
so the split is proportional to costs.
h1 = −
h2 = −
c2
(π − c1 − c2 )
c1 + c2
• Or give all the proceeds to agent 1, so h1 = −(π − c1 − c2 ) while h2 = 0
• So there are many ways to balance the budget
Transactions costs and missing markets
• My neighbor has a boxer named Rocky:
• Rocky often spends winter nights outside
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Transactions costs and missing markets
• There is no market where I can purchase the right to decide what happens
to Rocky on cold nights
• Options, in order of sane to crazy:
– Ask the neighbors to treat their dog better
– Try to buy the dog directly
– Trespass and give the dog a blanket
– Call the police
– Kidnap the dog
– Watch all night, waiting for the dog to keel over and then intercede
• These options are all costly to me in non-pecuniary ways, so I end up
taking the craziest and most costly action
• Inefficiencies arise when there are missing markets and transactions costs,
since then we can’t bargain efficiently
Why isn’t Lindahl/Coase used more broadly?
• Well, it was: FCC lotteries
• Net transfers might be small while gross transfers are enormous
• “Individual rationality” as in the CEO example
• Corruption/captured regulators
But the fatal flaw of classical design is...
Information
... that the market designer does not know ui (x) for each agent i.
• Firms wish to overstate their costs of pollution abatement, households
want to overstate their benefits of environmental goods
• Students and universities wish to overstate the benefits of education
• CEOs wish to overstate the costs of their effort
• Borrowers wish to overstate their credit-worthiness
• Insurance buyers wish to overstate their health
• Sellers wish to overstate the quality of their goods
How do we incorporate the idea of “information” and “agency” into markets?
18