Tragedy of the Commons
First definitions:
Modern examples of a common resource:
Critical characteristics of a commons
Externalities
Consumption  Production  
Current  
Future 
The tragedy of the commons and A Civil Action
In Woburn, MA Beatrice Foods and W R Grace both had plants that as a byproduct of their output produced toxic chemicals. Rather than pay to store, haul away and treat those chemicals it became their practice to pour those chemicals on the ground. While Grace and Beatrice avoided part of the cost of production it was a cost that was imposed on the community in the form of contamination of the local ground water. The toxic chemicals leached out of the ground and into the water system, from which the residents drew their drinking water.
A Model of the Overuse of the Commons
Two players, Beatrice and Grace, consume c_{B} and c_{G} units of the common resource, y, such that . If their combined consumption today exceeds y then c_{B} = c_{G} = y/2. Whatever they don't consume of y today is available for consumption tomorrow and is calculated as y  c_{B}  c_{G} . In the future each firm gets (y  c_{B}  c_{G} )/2 for its own consumption.
The utility from the consumption of the resource is given by ln(c), and looks like:
The graph is the same for both firms.
Beatrice's problem is to choose its utility maximizing consumption given Grace's choice. Beatrice''s utility from consumption in the two periods is
and she must choose her consumption so that
The first derivative with respect to Beatrice's consumption is
with a little algebra


We find Beatrice's best response in consumption to Grace's choice is
The graph of this looks like
The problem is symmetric so we have and the graph looks like
Putting Beatrice's and Grace's best response functions together we get the following
Solving the pair of best response functions for the two unknowns
If we solve the second equation for c_{B}, then we can set the two equal and solve for c_{G}, then substitute back to get c_{B}. After all of the manipulation we get
The firms consume 2/3 of the resource in the first period and 1/3 in the second period.
By contrast, let us see what happens when we maximize aggregate utility over the two periods. We'll define aggregate utility as the sum of the utility for the two firms over the two periods.
So now the problem is to
To find the maximum we take the derivatives with respect to c_{B} and c_{G}, set the results to zero, and solve for the two unknowns, c_{B} and c_{G}. The two derivatives are known as the 'first order conditions'. They are


If you solve this pair of equations for the two unknowns then
We now see that each firm consumes 1/4 of the resource in each period, whereas they each had previously consumed 1/3 of the resource in the first period. Why do they consume less? In the first phase when Beatrice and Grace choose independently, if Beatrice reduces consumption by 1 now then she will only get to consume half of that unit in the next period. So, her incentive is to not defer consumption. The problem is worse in a big population since the unit of consumption of the common resource you give up today is shared with everyone tomorrow.