Externalities and Public Goods

What factors cause market failure?

• Externalities, where the utility of one individual depends on the consumption and actions of other agents

• The pricing of public goods in private markets doesn’t result in efficient allocations

• Some firms become large enough that they have sufficient market power to become price-setters rather than price-takers

• If information is incomplete trade might be impeded resulting in inefficiencies

The tragedy of the commons

• This is a common example of how externalities cause market failure

• Take a common good, lie a pasture for grazing which many farmers use; if farmers aren’t conscientious about their actions, they might overconsume leading to depletion of the public good; in the same way, their utility depends on the actions of other farmers

How might externalities be positive or negative?

• Externalities are exogenous factors which affect an agent, or the side-effects of transactions between two parties which affects a third uninvolved party

• They can be positive and negative; positive things are those like education, vaccinations, the appearance of a neighbours house, negative externalities are things like smoking, pollution, and chewing loudly

  • This distinction helps us to know what is over or under-produced and create policy to address this

Negative externalities graph

• The equilibrium at Q is one where the good is overproduced because it doesn’t account for the overall social cost; it only cares about the private/monetary cost

• This means there is a deadweight loss, which can be addressed by things like Pigouvian taxes

How can markets be made more efficient in the face of negative externalities?

• The main way to restore efficiency is to internalise the externality; account for the extra social cost the good has when selling it

  • Thus, one should impose a tax or subsidy on it dependent on whether it is negative or positive

• It could also be interpreted as a missing market; pollution causes damage to air, but there is no market for clean air so it is difficult to value it

What do Pigouvian taxes do?

• The idea is to include the extra social cost a good has (negative externality) in the price of the good when selling it

• This adjusts the Private Marginal Cost so that it is equal to the Private Marginal Benefit at the social optimum

• The tax is set equal to the Marginal Damage which is the difference between the Marginal Social Cost and the Private Marginal Cost

Pigouvian Tax diagram

How to calculate a Pigouvian tax

• Find the social optimum quantity

• Find the demand and supply prices at this quantity

• To find the optimal tax, compare the demand price with the supply price at the social optimum outcome: subtract the supply price from the demand price

How to determine the socially optimal quantity?

• Set demand and supply equal, X_d = A - P / α, and X_s = B + P / β and solve for price P = (βA - βα) / (β + α), the price without externality

• Plug this into supply/demand to get the equilibrium quantity: X_d = (A + B) / (β + α)

• This quantity is wrong because it doesn’t account for externality cost, C_E(x) = (δx)² = δ²x²

• Social optimum is where MSB = MSC, so we need to find both curves

  • MSC is comprised of MPC and MEC. To find MEC, differentiate total cost, = 2δ²x

  • The MPC is the inverse of the supply function, MPC = βx - B

  • MSC = MPC = βx - B + 2δ²x

• Assuming no positive externalities, MPB = MSB = A - αx

• Set MSB = MSC for optimal quantity, x* = (A + B) / (β + δ² + α)

How do we calculate the demand and supply price at the social optimum? Then calculate the optimal tax.

• To find the demand price set the socially optimal quantity equal to the demand function and solve to find P_d = (A(β + 2δ²) - αβ) / (β + 2δ² + α)

  • This is the price consumers must face such that they’ll want to buy the socially optimum quantity

• To find supply price, set the socially optimal quantity equal to the supply function to get P_s = (βA - B(α + 2δ²)) / (β + 2δ² + α)

• To find the optimal tax, subtract the supply price form the demand price to obtain, T* = (2δ²(A + B)) / (β + 2δ² + α)

  • This tax is the vertical gap between the demand curve and the MPC curve at the social optimum

What is Coase Theorem?

• This is another solution to the externality problem which involves creating a new market to help refine the value of the negative impact (ie, introduce property rights to the element which impacts utility)

• In such a case, polluters would be forced to pay an extra cost for externalities

• Coase suggests that this procedure wouldn’t distort allocation efficiency in the market so long as there are no transaction costs and preferences are quasilinear

What are public goods?

• They possess the properties non rivalry (one agent consuming the good doesn’t prevent another from consuming it) and non-excludability (agents can’t be excluded from benefitting from the good)

• They are usually provided in the same way to all consumers (so that everyone receives a similar amount)

  • Properties of these goods aren’t clear-cut so there are intermediaries between private goods and public goods

What does private provision of public goods refer to?

• If agents choose their contribution to a public good privately, this is a game where we contribute, and then based on total contributions we get to enjoy the good

• Agents who benefit less from the good are more likely to contribute less and thus free-ride, which can lead to the under-provision of public goods in private markets

What is government provision?

• Refers to the case in which, due to underprovision in the private market, the government provides the main public goods, like public defence, or education

• It might be difficult for a government to determine how much public good to provide, as they don’t know each persons benefit

  • Thus, they should charge different prices to agents, proportional to their willingness to pay according to the Lindahl equilibrium

How can a government gauge social preferences?

• The government might do a survey or vote; ask individuals to rank alternatives and use a rule to aggregate individual preferences into one social choice

  • Unfortunately, results aren’t method-independent and are victim to Arrow’s impossibility theorem

• The government could also construct social preferences by aggregating individual utility functions

  • We could derive a utilitarian social welfare from a weighted sum or a Rawlsian social welfare function by picking the minimum utility each agent gets and then choose the option which maximises the chosen social welfare function