An externality is something that enters into the utility (or production) function of one agent but is chosen by another agent.
Classic example is something that enters directly:
 right now PM from wildfires is keeping some people indoors
However an important distinction is made between those cases and situations where the actions by one agent affect another's utility through prices (a pecuniary externality)
 for example if demand for short term rentals on AirBnB drives up rents and makes housing less affordable for some residents
Meade (1952) considered the case of a beekeeper and nearby orchard.
 Bees pollinate the orchard, and the orchard nourishes the bees, shifting the production possibilities of both businesses outward
from Phaneuf and Requate
Setup

two individuals $i \in (1,2)$ receive utility from private goods $x$ and $z$ and disutility from emissions $E$

$x$ is a dirty good produced using labor $l$ and emissions
 $x = f(l_x,E)$, with $df(E)/dE>0$

clean good $z$ produced only using labor: $g(l_z)$

labor in economy constrained by time endowment $l$
[familiar FOCs below]
$$\begin{aligned} Max_{x,z,l,E} & \hspace{5pt} U_1(x_1,z_1,E) + \lambda_u [U_2(x_2,z_2,E)  \bar{u_2}] \\ & + \lambda_x[f(l_x,E) x_1  x_2] + \lambda_z[g(l_z) z_1  z_2] \\ & + \lambda_l[l  l_x  l_z] \end{aligned}$$
Marginal rate of substitution equal for both individuals
$$\frac{\partial U_1()/\partial x_1}{\partial U_1()/\partial z_1} = \frac{\lambda_x}{\lambda_z} = \frac{\partial U_2()/\partial x_2}{\partial U_2()/\partial z_2}$$
Marginal product of labor equal to the shadow value on labor constraint in both markets
$$\lambda_x \frac{\partial f()}{\partial l_x} = \lambda_l = \lambda_z \frac{\partial g()}{\partial l_z}$$
Exchange efficiency sets slope of production possibility curve equal to slope of each individual's indifference curve
$$\frac{\partial U_i()/\partial x_i}{\partial U_i()/\partial z_i} = \frac{\lambda_x}{\lambda_z} = \frac{\partial g()/\partial l_z}{\partial f()/\partial l_x}$$
Reducing $E$ increase utility for both people directly, but decreases their utility indirectly by reducing $x$ consumed.
 Net effect depends on preferences.
Taking the FOC wrt $E$, and rearranging yields
$$ \left[ \frac{\partial U_1()/\partial E}{\partial U_1()/\partial x_1} + \frac{\partial U_2()/\partial E}{\partial U_2()/\partial x_2} \right] = \frac{\partial f()}{\partial E}$$
Optimal $E$ equates the sum of each individuals marginal willingness to tradeoff $E$ for $x$ with the physical reduction in $x$ production induced by changing $E$
 Here $E$ is not just an externality but also a public good
 Define prices $p_z$ and $p_z$, wage $w$, and income $y_i$
 Firms and individuals act as price takers
Individuals max
$$Max_{x_i,z_i} \hspace{5pt} U_i(x_i,z_i,E) + \lambda_i[y_i  p_x x_i  p_z z_i]$$
Firms producing $x$ and $z$ max
$$Max_{l_x,E} \hspace{5pt} p_x f(l_x,E)  w l_x$$
$$Max_{l_z} \hspace{5pt} p_z g(l_z)  wl_z$$
While the efficiency in labor use and exchange conditions are met, the emission allocation is no longer efficient
$$p_x \frac{\partial f()}{\partial l_x } = w$$
$$p_x \frac{\partial f()}{\partial E } = 0$$
Firms face zero marginal cost of $E$ and use too much of it.
Pareto:
$$ \left[ \frac{\partial U_1()/\partial E}{\partial U_1()/\partial x_1} + \frac{\partial U_2()/\partial E}{\partial U_2()/\partial x_2} \right] = \frac{\partial f()}{\partial E}$$
Consider tax $\tau$, so
$$\pi = p_x f(l_x,E)  w l_x  \tau E$$
Pigouvian tax:
$$\tau = p_x \bigg[ \frac{\partial U_1()/\partial E}{\partial U_1()/\partial x_1} + \frac{\partial U_2()/\partial E}{\partial U_2()/\partial x_2} \bigg]$$
Example:
 Baker and a doctor share a wall.
 The baker installs a machine which causes vibrations, impairing the doctor's ability to operate.
Pigou:
Efficiency restored by taxing baker at dentists marginal harm from vibrations.
Colloquial conclusion:
 Doesn't matter who has the property rights, as long as their well defined, efficient outcome will obtain
 Well defined property rights
 No transaction costs
 No income / endowment effects
These conditions (mainly 2) rarely satisfied in the real world
 starting point for theory of the firm (Williamson)
More recently, MyersonSatterthwaite theorem casts further doubt on viability of property rights alone to efficiently adjudicate externalities
 No good way for two parties to trade a good when they each have secret valuations
Most environmental regulations are not Pigouvian taxes, but "command and control".
 Government sets emission limits or rates by facility.
 How does this compare to Pigouvian solution?
Household utility: $U_i(y_i,E)= y_i  D_i(E)$
Summing over $i$ yields the damage function: $D(E)=\sum_i D_i(E)$
Firms can abatement emissions at cost $C_j(e_j)$
 these include full opportunity costs (not just direct costs)
 $C_j(\hat{e_j}) = 0$ at unregulated level.
Social net benefits of pollution = $\sum_j C_j(e_j)  D(E)$
$$C'_j(e_j) = D'(E) \hspace{15pt} \forall j=1,..J$$
Therefore, the marginal cost of each polluter are also equal
$$C'_j(e_j) = C'_k(e_k) \hspace{15pt} \forall j,k$$
This is a necessary and sufficient condition for costminimization of any policy (such policies are "cost effective")
Pigovian taxes achieve this by construction:
$$C'_j(e_j) = \tau \hspace{15pt} \forall j=1,..J$$

If regulator knows all the $C_j$'s, could chose $e_j$ appropriately

In practice, such regulations never firm specific
 Cost effectiveness only achieved if all polluters are identical
More importantly, regulator doesn't actually know $C$
 tax cost effective by construction
 but command and control requires full information
Another option: Capandtrade
 Regulator issues a fixed amount of emission permits $\bar Q$
 Assume these are auctioned off, with clearing price $\rho$
Firm's minimize total compliance costs:
$$TC_j(e_j) = C_j(e_j)  \rho e_j$$
$$s.t. \sum_j e_j \leq \bar{Q}$$
Assuming firms are price takers,
$$C'_j(e_j) = \rho \hspace{15pt} \forall j=1,..J$$
In static world of full information (ie known MC and MD), taxes and allowances are equivalent
 Note: If permits aren't auctioned, Coase independence assumptions apply.
Weitzman (1974) considered efficiency when there is uncertainty about the damage or cost functions.
 Assume planner must set policy based on expected MB and MC, and then true functions materialize
 If costs are uncertain and slope of MB > MC, use quantity instrument
 If costs are uncertain and slope of MB < MC, use price instrument
Intuition: Gradient tells you how bad it is to be wrong in each direction.
 Surprising result: If only MB are uncertain, instruments are equivalent (equally bad)
Intuition: pollution chosen by firms setting $MC = \tau$ or $\rho$. If costs don't change, resulting marginal prices don't change, and expected $E$ obtained.
Stavins 1996 considers correlated uncertainty.
Seems likely that uncertainty in MB and MC are positively correlated.
If that's the case, quantity instrument becomes more appealing (can see this visually)
Most recent capandtrade policies have included a price bounds
 Politically ceiling helpful assuaging business community fears
 Floor also increasingly important for environmental stakeholders

Some work on details of cap and trade, particularly over long run

Lots of work over past decade tests core predictions:
 Are outcomes independent of allocation?
 Is C&T costeffective?

But more interesting is how these theoretically firstbest policies compare to what's done in practice.
 Many policies contain elements that are not well motivated (economically)
 Carve outs; Grandfathering;
 What the costs of these?
Probably the largest area of research involves estimating costs and benefits

If policy maker wanted to set the optimal tax, what should it be?

Bulk of these papers estimate health (or related) impacts

Then use benefit transfer to convert to $

More recently people have linked to intermediate outcomes of obvious policy interest
As opposed to benefit transfer to assign value.
We'll spend a week on the workhorse model in this area: hedonic property valuation.
 Great example of reduced form vs structural approach to EE
 Interesting overlap with behavioral
 polluting markets often characterized by imperfect competition
 multiple market failures > actual DWL unclear
 market power complicates firm response to simple models presented above
 dynamics
 innovation
 behavioral!
IO tools well suited to recover preferences and run counterfactuals
[hopefully your term paper, eventually]

Test a theory
 tough to find low hanging fruit
 i.e. independence of allocation

Measure costs and benefits
 identification
 something we don't know about (like fine PM)
 some important extension: long vs short run; avoidance  adaptation; behavioral
 climate change per se important (but crowded)

Evaluate a policy
 not enough unless you tie to theory/ some broader question
 compare second (third..) best to ideal policy
 "dumb" policies often provide nice experiments to test other hypotheses

Environmental settings are good for learning about economics more broadly
 Hunt's papers
 Energy markets have nice properties (undifferentiated, clear mechanisms, good data, lots of policy )