Economics Game Theory Questions

Question

1. Four stores have a problem with theft and security is a public good. Let S stand for the number of person-hours of security patrols per week. The marginal benefit of security patrols to each of the stores is given by the formula MB = 200 – 2Q. Patrols cost $25 per hour.
a) What is the socially efficient level of security?
b) If each store provided security independently, how much would each store provide?
c) Draw a diagram to illustrate your answers.
2. A paper mill discharges gunk that causes damage of $200 to the fishermen’s business. The pollution damage can be eliminated in one of two ways: the mill can install a filter at cost $300 or fishermen can install a water treatment plant at cost $100.
a) Identify the efficient outcome.
Assume the fishermen have property right to no Pollution (and the property right is protected
by compensatory damages)
b) Construct a game matrix to illustrate the interaction between the factory and the resident.
c) Suppose first that it is prohibitively expensive for the mill and fishermen to negotiate. What will the outcomes?
d) Suppose instead that the mill and fishermen can bargain costlessly. What will the outcomes be? Note: Apply the Nash bargaining solution where any gain from a bargaining is shared evenly between parties.

Answer 1

Hi Michelle,

 

I'm going to use * to mean multiplication throughout.

 

As far as problem 1 goes, security may be a public good, but I notice there's no external marginal social benefit in this problem.  That leaves us with the stores.  So each store is willing to pay 200-2*1 = 198 for the first hour, 200-2*2 = 196 for the second hour, and so forth.  So the question is, when are they not willing to pay anymore?  Well, if I've interpreted the question correctly, a single hour of patrol costs each of the four businesses $6.25 (since 6.25 * 4 = 25, the total cost of a single hour of patrol). 

 

What we have now is a straightforward algebra problem.  200-2*96 = 8, which means each business will happily pay 6.25 for the 96th hour.  However, 200-2*97 = 6.  Since each business is only willing to pay $6 for the 97th hour, they will balk at paying $6.25.  Ergo, 96 hours of patrol is the socially efficient solution.

. 

What about b?  If each store provided security independently, they would stop when 200-2*Q < 25.  Hopefully you're ok with figuring out which value of Q that is. 

 

I don't have graphing tools here, so I'll leave that for now.  Hopefully you have enough to go on.

 

Now, as to problem 2.  I find this a strange question, and I am confused by it.  Let me run through some answers, and hopefully something will work for you.

 

So, if the fishermen have a property right enforced by compensatory damages, and compensatory damages work they way that are intended, they have no incentive to spend any money, since their worst possible outcome is to break even. The paper mill, on the other hand, is going to lose money -- it's just a question of how much.  This is where my confusion arises.  If the gunk is a one-time event, payoff to the paper mill for just letting it go is -$200, whereas the payoff for installing a filter is -$300.  I think that's the intended interpretation.  So our game matrix is as follows:

 

 

                                         mill / Fishermen      

                             F. Install plant              F. Do nothing

 

 

m. install filter          -300 / -100                      -300 / 0

 

 

m. do nothing            0 / -100                          -200 / 0

     

In reality, though, paper mills discharge gunk regularly.  If this problem is really intended to say that the damage is $200 per week, let's say, then the payoff per week for doing nothing is -$200 / week, but the payoff for installing a filter is -$300 the first week and (close to) zero for every week thereafter.  Thus, if the game goes two weeks or more, the filter is the optimal move. 

 

If it's prohibitively expensive for the mill and fishermen to negotiate, both will do nothing, at a cost of $200 to the mill and a cost of $0 to the fishermen.

 

On the other hand, if the mill and fishermen can bargain, they can arrive at the strategy with the maximal sum of payoffs.  In this case, that is the strategy in the lower left-hand corner, in which the fishermen install the water filtration plan.  And how is the cost apportioned?  Well, there is a net $200 gain over the previous solution, which means the paper mill will be paying only $150, and the fisherman will wind up making $50 profit. 

 

I hope that helps.

 

Cheers,

Mike N.