  A prevalent theme in the economics and computation literature is to identify
natural price-adjustment processes by which sellers and buyers in a market can
discover equilibrium prices. An example of such a process is t\^atonnement, an
auction-like algorithm first proposed in 1874 by French economist Walras in
which sellers adjust prices based on the Marshallian demands of buyers. A dual
concept in consumer theory is a buyer's Hicksian demand. In this paper, we
identify the maximum of the absolute value of the elasticity of the Hicksian
demand, as an economic parameter sufficient to capture and explain a range of
convergent and non-convergent t\^atonnement behaviors in a broad class of
markets. In particular, we prove the convergence of t\^atonnement at a rate of
$O((1+\varepsilon^2)/T)$, in homothetic Fisher markets with bounded price
elasticity of Hicksian demand, i.e., Fisher markets in which consumers have
preferences represented by homogeneous utility functions and the price
elasticity of their Hicksian demand is bounded, where $\varepsilon \geq 0$ is
the maximum absolute value of the price elasticity of Hicksian demand across
all buyers. Our result not only generalizes known convergence results for CES
Fisher markets, but extends them to mixed nested CES markets and Fisher markets
with continuous, possibly non-concave, homogeneous utility functions. Our
convergence rate covers the full spectrum of nested CES utilities, including
Leontief and linear utilities, unifying previously existing disparate
convergence and non-convergence results. In particular, for $\varepsilon = 0$,
i.e., Leontief markets, we recover the best-known convergence rate of $O(1/T)$,
and as $\varepsilon \to \infty$, e.g., linear Fisher markets, we obtain
non-convergent behavior, as expected.
