Prof. Werner Antweiler, Ph.D.
Many economic models are focused on competition among
firms, each maximizing profits and interacting with each
other in a market place. Economic models in which firms
interact with each other in the production process (as
opposed to the market for goods or intermediates) are not
very common. However, they are useful for example in the
context of fisheries economics where effort from
one fishing boat affects the overall harvest and thus
the stock available to all other fishing boats. The model
below is taken from this context, presented in a UBC workshop
by eminent fisheries economist Martin Smith. I am distilling this
model for the benefit of my students who may find that this
model has relevance in non-fisheries applications as well.
Consider economic agents that produce a common
good with output and joint effort
where each agent receives a share of output equal
to the share of effort , and maximizes profit
and faces constant marginal cost .
The shape of the production function is crucial.
In fisheries economics the function is backward-bending
as increased effort eventually starts depleting the stock;
for example, with .
In other applications, expanding effort will typically face diminishing returns,
such as ; or more generally with . The first-order condition
for a profit maximum yields that optimal effort share is
where is the marginal
product and is the average product. The optimal
amount of effort thus depends on both marginal and average
product, which is a departure from the more common framework
where everything depends on the margin.
What kind of economic situations would correspond to such a model?
Consider producing a video game where a group of programmers gets
together and commit different amounts of time to the project, and
agree that they will split the proceeds in proportionate share
of their time contribution. Making the video game more and more complex
will eventually lead to diminishing returns. So this may be a situation
where joint production problem may arise. Each contributor is unable
to observe others' marginal cost, and thus there is a type of
Cournot equilibrium that emerges.
Let us continue solving the model. Introducing
as the average cost of effort, summing over all agents reveals that
The above equation shows how the marginal product and average
product are weighted, with the smaller weight going to
the marginal product when .
We also need to pin down demand. Assume demand is linear so
that
We also assume that output is diminishing
in joint effort so that . Then
. Further introduce the short-hand notation
. Now
and therefore
Finally, going back to the effort contribution,
The contribution share equation looks a lot like the
outcome of a standard Cournot model. But note that the demand
parameters or the scale parameter do not feature in the
share equation. It is just the number of participants and
the cost distribution that matters for determining relative
effort contributions. However, the demand and scale parameters matter
for determining overall effort.
The share equation has some interesting implications.
In order for the effort share to be less than one, it must hold that
So the cost advantage of any one contributor cannot be too large,
or this contributor will take up all of the effort.
This imposes an important constraint on the cost distribution.
Consider the case of and assume , so contributor
1 is less costly than contributor 2. Then for both two contribute
it must hold that : the higher-cost contributor
cannot be more than twice as costly as the lower-cost contributor.
Eventually this cost distribution will shrink. The last contributor
to join cannot have costs that are much above the previous average
; specifically, it must hold that
In essence, collaborators on a joint production problem need to have
relatively similar costs, or the joint production will not happen
and instead the lowest-cost provider produces all of the output.
What are the takeaways from the above model? Joint production
doesn't all that different from Cournot competition. After all,
there is a Nash equilibrium embedded in the model. The lowest-cost
contributors get a larger share of the overall pie, just like
low-cost producers get a larger market share in the conventional
Cournot model. Where the model gets interesting, perhaps, is when
the collaboration function is inverse-U shaped as in
fisheries economics. To read more on this topic, please read
Banzhaf, Liu, Smith, and Asche: Non-Parametric Tests of the Tragedy of the Commons, NBER Working Paper 26398.