Enormous parts of economic theory are devoted to “comparative statics”: if one variable changes (one chickpea firm’s cost structure decreases, demand for pineapples goes up, firms receive better information about the size of an oil field before bidding on rights to that field), does another variable increase or decrease (chickpea prices rise or fall, equilibrium supply of pineapples rises or falls, revenue for the oil field auctioneer rises or falls)?
There are a couple ways we can check comparative statics. The traditional way is using the implicit function theorem. Call t the variable you wish to change, and x the endogenous variable who comparative static you wish to check. If you have a model for which you can solve for x as a function of t, then finding the sign of dx/dt (or the nondifferential analogue) is straightforward. This is rare indeed in many common economic models; for instance, if x is supply of firm 1 in a Cournot duopoly, and t is the marginal cost of firm 2, and all I know is that demand satisfies certain properties, then since I haven’t even specified the functional form of demand, I will not be able to take an explicit derivative. We may still be in luck, however. Let f(x(t),t) be the function an agent is maximizing, which depends on x the agent can choose, and t which is exogenous. If the optimal choice x* lies in the interior of a convex strategy set, and f is concave and twice differentiable, then the first order approach applies, and hence at any optima, f'(x*,t)=0. By the chain rule, f'(x*,t)=0=(df/dx*)(dx*/dt)+(df/dt), or dx*/dt=-(df/dt)/(df/dx*).
So far, so good. But often, our maximand is neither differentiable nor concave, our strategy set is not convex, or our solution is not interior. What are we to do? Here, the monotone comparative statics approach (Athey and Milgrom are the early reference here) allows us to sign without many assumptions. We still don’t have many good results for games of strategic substitutes, however; in these types of games, my best response is decreasing in your action, so if a parameter change causes you to increase your action, I will want to decrease my action. Further, there are many games where player actions are neither strategic complements nor strategic substitutes. A good example is a patent race. If you increase your effort, I may best respond by increasing my effort (in order to “stay close”) or decreasing my effort (because I am now so far behind that I have no chance of inventing first).
Acemoglu and Jensen show that both types of games can be handled nicely if the games are “aggregative games” where my action depends only on the aggregate output of all firms rather than the individual actions of other firms. My choice of production in a Cournot oligopoly depends only on aggregate production by all firms, and my choice of effort in an independent Poisson patent race depends only on the cumulative hazard rate of invention. In such a case, if the game is one of strategic substitutes in the choice variable x, and the aggregate quantity is some increasing function of the sum of all x and t, then an increase in t leads to a decrease in equilibrium choices x, and entry by another player increases the aggregate quantity. If t(i) is an idiosyncratic exogenous variable that affects i’s choice, and only affects other players’ actions through the aggregate x, then an increase in t(i) increases x(i) and decreases x(j) for all other j. [When I say “increase” and “decrease” for the equilibrium quantities, when there are multiple equilibria, this means that the entire set of equilibria quantities shift up or down.]
For example, take a Cournot game, where my profits are s(i)*P(X+t)-c(i)(s(i),t(i)), where s(i) is my supply choice, P(X+t) is price given the total industry supply plus a demand shifter t, and c(i) is my cost function for producing s(i) given some idiosyncratic cost shifter t(i) such that s(i) and t(i) obey the single crossing property. Assume P is twice differentiable. Then taking the derivative of the profit function with respect to other players’ supply, we have that my supply is a strategic substitute with other’s supply iff P’+s(i)P”<0. Note that this condition does not depend on any assumption about the cost function c. Then using the theorem in the last paragraph, we have that, in a Cournot game with *any* cost function, a decrease in the demand curve decreases total industry supply, an additional entrant decreases equilibrium supply from existing firms, and a decrease in costs for one firm increases that firm's equilibrium output and decreases the output of all other firms. All we assumed to get this was the strategic substitutes property, twice-differentiability of P, and the single crossing property of the cost function in s and t. As with many comparative statics results, checking strategic substitutability and making sure the signs of s and t are correct for the interpretation of results is often easier than taking implicit derivatives even if your assumptions are such that the implicit approach could conceivably work.
If the strategic substitutes property does not hold, as in a patent race, then with compact, convex strategy sets, a payoff function pseudoconcave in own player strategies, a boundary condition, and a “local solvability” condition are sufficient to get the same results. Local solvability is roughly the condition which guarantees that the player’s own effect on the aggregate when she reallocates after a change in t or t(i) does not alter the result from the previous paragraph enough to flip the comparative static’s sign.
April 2011 Working Paper, as yet unpublished (IDEAS version).
A Fine Theorem 
Really like your blog, keep at it.