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Respond to the question: Newbie How To Solve Cournot With Capaci?

11/19/2012 05:22 PM by Andy C; Newbie: How To Solve Cournot With Capacity Constraints

Previously we have written min cost models to meet a set demand at minimum cost with various constraints. We want to start looking at gaming outcomes, specifically cournot-nash.

I've looked at Cournot and understand the simple examples shown in texts/web.

q_i = qty supplied by i
Price = intercept - slope x (sum(i, q_i))
each supplier i has a linear cost c_i
profit_i = (price - cost)q_i

Equilibrium is when all supplier 1st order differentiation of profit function = 0. ie intercept - slope x sum(i, q_i) - c_i = 0. Makes sense.

What I want to do next is introduce capacity constraints: 0 <= q_i <= QMAX_i

What I have seen so far really threw me for a loop. Quite often I see "MCP" stuff with references to "KKT" conditions. I have tried to follow this through, but I am struggling to see how the equations shown were formed from the KKT. Its probably easy once you know how :)

For example, before the capacity constraints we had to solve:

intercept - slope x sum(i, q_i) - c_i = 0

Now what I see is things like:

0 >= intercept - slope x sum(i, q_i) - c_i _|_ 0 <= q_i <= QMAX_i
where _|_ = orthogonal, and no obj function any more.

So my questions are:

can someone assist me in understanding how to introduce constraints into my cournot model - starting with the capacity constraints shown above. Either through a discussion or via references to good online resources etc. I've tried quite a web search and am still a bit lost, so starting simple would be really helpful.

In the long run I would like to introduce some more constraints which apply to each supplier (eg minimum supply requirements) and to the "system" as a whole (eg transportation limits between nodes, pollution limits). I mention this just in case the capacity constraints are something special rather than a general form.

Feel free to email as well: collingwood25 at hotmail com

AndyC [Manage messages]