Tatonnement beyond gross substitutes? Gradient descent to the rescue

We identify a class of economies for which tatonnement is equivalent to gradient descent. This is the class of economies for which there is a convex potential function whose gradient is always equal to the negative of the excess demand. Among other consequences, we show that a discrete version of tatonnement converges to the equilibrium for the following economies of complementary goods. i. Fisher economies in which all buyers have complementary CES utilities, with a linear rate of convergence. (In Fisher economies all agents are either buyers or sellers of non-numeraire goods, but not both.) This shows that tatonnement converges for the entire range of Fisher economies when buyers have complementary CES utilities, in contrast to prior work, which could analyze only the substitutes range, together with a small portion of the complementary range. ii. Fisher economies in which all buyers have Leontief utilities, with an O(1/t) rate of convergence.