Public goods and Externalities Flashcards
The goods we have studied so far have what properties?
Depletability
Excludability
What does Depletability mean and what was an example of this in our model, also define Non- depletability?
Depletability means = more for one is less for another ( edgeworth box, if someone had more books, it meant someone else had fewer books)
Non- depletability = more for one doesn’t mean less for another
What does excludability mean and what does non-excludability mean?
Excludability = private goods you can exclude ppl from consuming the good ( e..g cake)
Non - excludiability - public good, you cannot exclude ppl from consuming the good.
What are public goods?
These are non excludable and non rival/ non-depletable.
What is another way of thinking about non-rival/depletable?
for any level of production, the cost of providing it to a marginal (additional) individual is zero.
Catgeorize software code, free to air tv lentils and common land into Excludable vs Non-excludable and depletable vs non- depletable.
Categotrize ciagrattes, air pollution, virus site and congested non-toll roads ( these are bads) into Excludable vs non-excludable and Depletable vs Non-depletable.
What does the tragedy of commons mean ( termed by Russell Hardin? )
If a resource is held in common for use by all then ultimately that resource will be destroyed e.g air pollution, overfishing.
As we consume things that a largely in common resources and we take actions which affect how much common resource there is for others to share, what does this mean?
we need to find ways of living together more effectively.
What is a externality and what is a negative externality give example?
Costs or benefits that spillover to third parties, external to a market mechanism
A negative spillover effect tot third parties of a market transaction; social costs exceed private costs ( such as climate change)
We are going to look at a model of public good now suppose that there are N goods in goods in our core model but good 1 is special ( pure public good) with individuals enjoying equally their own consumption and that of all members of society.
What is the Utility function and what is G a function of, what is the utility function increasing in.
The amount of public good i enjoy depends on the whole set of private actions by all the people in the economy. ( this explains G)
Utility increases in F.
In the model what 2 extreme cases will we look at to find the equilibrium of whether the public good gets provided or not?
Perfect substitutes case ( sum of contributions -- refers to prisoners dilemma game) Perfect complements ( public good is a product of everyone decision, if one person does provide anything there isn't a public good - refers to assurance game
So continuing in building our model, we are going to assume a model of 2 individuals A and B, where are are 2 goods
Good 1 is public good and good 2 is private good
1) First of all write the budget constraint of the 2 individuals if p2=p1=1, and income is fixed.
2) show that good one is an element of 0 or 1 ( i.e you can choose to contribute to public good or not)
Now we are going to assume to make it easier to make a 2x2 matrix that the utility function is linear in both goods, but for the public good G there is a gamma in front showing how much i like the public good, what is gamma an element of?
Give a day to day example of our model of public good provision with relates to our model?
In our model of public good provision, what will it be based off, which we have learnt before ?
Game theory, nash equilibrium ( a set of strategies that are best responses to each other)
Now we are going to think generally how we would make a payoff matrix using the Utilities of each individual A and B. We need to plug in the budget constraint within the utility functions which is x1^a + x2^a = m^A and x1^b + x2^b = m^b into the utility function and this will give a payoff of amount of public good and provide good. Show this generally ( HINT write G as a function of individual A and B’s valuation of public good) then write the private good ( using the budget constraint)
So now we have come to the point where we can finally make a payoff matrix, we are now going to assume that the provision of public good in the payoff matrix is the sum of contributions( perfect substitutes case).
1) So write what G is in a formula
2) what are the 3 possibiities of G ( write what G is an element off)
3) Write Utility function of A and B
4) show that individuals choose the public good x1 as either to provide or not to provide.
We know the utility functions for perfect substitutes as attached now we actually want to import numbers in the matrix solving the utility function, so suppose income for the 2 individuals are the same m^A and m^B = 2 and gamma = 0.8 if i contribute and the other person doesn’t, but if both contribute gamme = 2gamma = 1.6 ( you give each other both 0.8 if both contribute). If one doesn’t contribute and the other doesn’t, the one that doesn’t gets a higher payoff as they free ride, so now show this in a matrix where individual A is the row player and individual b is the column player. What is the nash equilibrium.
Same principles as utility function.
What is the end key about why no public good is provided here, is there a pareto superior place?
Private benefit < Private cost < Social benefit.
If person A contributes to public good, person B chooses whether or not to contribute, he gets less private consumption if he contributes ( 0.8<1) so will be worse off. Thus the provision of public good is problematic as you don’t take into account social benefits
There is a pareto superior place where both contribute, but there is an incentive to deviate ( they can free ride and not pay cost.
So to summarise in a world of prisoners dilemma, it is what and what is always true.
Its hopeless to use private incentives to provide a public good, as you will not get ppl contributing as private benefit < private cost ( people act in self interest, and don’t care about social benefit.
KEY TO NOTICE THIS ALWAYS HOLDS IF GAMMA < THE COST.
What does the prisoners dilemma game lead to when consumers are lead to decisions by themselves over public good?
Market failure ( it is possible to make everyone better off with other choices, so First welfare theroem doesn’t hold. WHY? When there is interdependent consumption, individuals cant optimize over everything in utility function, the market doesn’t allow this, so there is not reason to think neighbour will do what i like( especially when they have to pay cost.of doing something i like)
Now lets see if conclusion yields differently in the assurance game ( perfect complements, this is where public goods are products of what everyone does, so G = ( x1^A x x1^B). So the public good is only provided if everyone contributes. Suppose that gamma = 1.2 and income for both is 2. Firstly what is significant about gamma and what is the payoff matrix, and nash equilibrium. EXPLAIN
If i know the other person will provide to the public good, i will too as it will give me 1.2 units of utility vs 1 unit of utility I lose by not having as much of good 2 to consume.
So we know in the insurance game outcome of both contributing is pareto superior than only 1 contributing, what does this mean?
We have to get everyone on board to contributing to the public good, player A needs assurance that player B will choose to contribute too.
Is there a free riding problem in the assurance game?
Yes because contributing to the public good is not worthwhile if the other party does not do so.
What are 2 solutions to the free rider problem?
1) Voluntary cooperation ( e.g. if flatmate playing loud music stopping you from studying you don’t need the govt to solve this problem, you find a way to work with housemate and solve this.)
2) Government intervention
Where does voluntary cooperation work the best to solve free rider problems?
It works best in small scale societies ( households and families), this wouldn’t work as well in prisioner dilmma and assurance games, because even if we get everyone to collaborate, there is still incentive to cheat.
So we assumed so far that self interest is what people care about, but what are the ways we can relax this?
Altruism - individuals care directly about each others payoffs
Reociporal preference - individuals care conditionally about the actions of others.
There is now a lot of experimental evidence, that fit with the framework we have studied, where individuals were given tokens, which they can contribute to public good or not, what was findings?
Evidence showed ppeople are not fully self interested but not fully pareto efficient, it varies culture to culture.
Now lets introduce the idea of negative reciprocity ( i can punish someone of the belief what they did is not the right action)
So 2 scenarios
A imposes a penalty size pie if x1^A = 1 and x1^B = 0
B imposes a penalty of size pie if x1^A = 0 and x1^B =1
Will this solve the free rider problem?
Show the new payoff matrix and with pie and what is the value of n which will mean x1A = x1B = 1
If pie is > 0.2 then there will be no incentive to deviate.
We are now going to model the coarse theorem. So suppose that individual A owns the public good, so they can decide if it’s provided or not. It will only be provided if B contributes, and A can in the contract offer to contribute too. Remember in PD the optimal level of G was what. And when will the level of public good that A sets = 0.
It was Pareto optimal in Prisonsers dilemma where 2 units were provided by each individual ( 1 each).
COASE THEOREM: suppose that B contributes ( and A as well so x1^b = 1 and x1^a = 1 to, what is the utility function of B if he signs the contract as opposed if he doesn’t, supposing that gamma = 0.8.
Is A better off?
Remember gamma =2 if A provides so 2.6> 2 ( this is if he just spends his income b)
Yes A is also better off compared to G = 0.
How will our utility function look like with a fine if you don’t contribute to public good from government?
So essentially you are getting taxed upon what you don’t contribute.
What tax will be large enough such that people will contribute the public good
Private benefit > private cost ( the benefit of providing will mean you don’t have to pay a cost too.
Suppose we focus on the case where the level of public good is the summation of tax collected, it collects taxes at income rate t and government chooses how much to spend on G, what is the budget constraint of the government?
m^i is individuals incomes
We are now going to model income taxation to fund public good ( We are trying to find the optimal tax rate) suppose the government is utilitarian( cares about sum of utlites and the government don’t want to set a tax rate such at that people tax evade but will the social welfare function be? Assuming it’s linear in utility?We
utility i get from public goods + consumption of good 2 = income x the remaining factor that is residual from tax, e.g. if tax rate is 40% the amount they spend on good 2 is 60% of income.
By subbing in the fact that the government have a government budget constraint what can be get from the social welfare function?
the amount the government spends on public good, is proportional to income tax, so M gamma is everyone’s benefit of public good( as a unit you spend on public good doesn’t only benefit you but benefits everyone else - loss to individual of private consumption as they have less private consumption
So suppose the government can tax people and spend it on the public good, when will it be welfare optimal to tax people to provide the public good? How do we find this out ?What do we find out and explain the equation?
We ask is utilitarian welfare increasing in the tax rate( if i tax people will i increase welfare? ). So we find partial deraritve with respect to t and find that it is increasing with t. We compare social benefit to private cost and MY(gamma) and if it is >1, then putting up tax rate to spend on public goods is a good thing. We keep doing this till t = maximum tax rate.
So explaining the derative function = we times the sum of income because if we put up the tax rate just a little bit, we will generate more revenue proportionate to aggregrate income.
How can we show the samuelson rule is a more conventional way, where utltiy is not linear?
So you have a utility function over public and private goods , the amount of private good, is what is left over after you paid your taxes.
The social welfare utilitarian function is just adding up all the utility functions, plugging in government budget constraint
We can do the same princple as before and find whether welfare is increasing with higher t with partial deratives.
What is the optimal amount of public good provided that maximises Utilarian SWF with the samuelson role?
So we keep increasing taxes, reducing peoples private consumption, transferring that to the state until we balance the benefits of that, against the costs of that
What is another way to think about the sameulson rule?
Think about it as a tradeoff between the relative value of public and private consumption ( MRS, you want more public consumption but it means less of private good, you would like to increase both, but you cant( like an indifference curve. we care about the sum of these MRS’s
Lets look at distubtion a bit more closely, suppose there are 2 ppl, and income A > b, SO A pays more than B, but gamma is still the same for everyone, we know id the absence of government there will be no public good provided, what is the is social welfare function if we have a preference of equality( 1-thy) and what is government budget constraint
What is important to note about the SWF?
Note everyone will be better off, some will be better off and some will be worse off.
With a preference of equality government can increase SW even if one agent loses out ( this is because the weights of better off individuals are less than the worse off
How will the utility function’s of A and B look like if we use the linear utility function from before? What is this a form off?
( note its kind of like a form of redistribution because your taking income away from a to fund b but also a is also redistubted back too )
What will the utility B for a and b if we raise the tax rate to fund public good?
What do we do?
Then what are 3 observations that are made when
Gamma A < 0.9
Gamma b>0.1 and
Gamma A + B > 1
Find the FOC of utility function with respect to t
Notice Gamma a might still be less than one, so you can add a and b and total utlitiy is higher.
Explain this
A and B’s utility are increasing in public good but not aggregrate utility, Suppose we start where A is richer than B, then by providing the public good, you are keeping average utility the same, moving down a 45% line where A’s fall in utility is offset by B’s gain. In terms of utilitarian function we don’t want that, but in our social welfare function we care about B’s gains rather than A;s loses.
Explain this?
For A THE VALUE OF public good is less than cost of providing, but for b its the opposite, now when we produce public good we increase average utility.
Explain this ?
We have a pareto improvement from providing the public good for a and b
PROBLEM SET 4
Question 2
Question 3
Question 4
Question 4b) (b) In what sense is welfare higher with everyone is vaccinated compared to a case where nobody is vaccinated?
