Many tractable TANK models treat redistribution as a contemporaneous wedge. I show that this view is incomplete once wage contracts are type-specific. In a tractable Two-Agent New Keynesian model, each household type adjusts its nominal wage relative to its own previous wage. This own-lag contract makes the cross-type wage gap a payoff-relevant distributional state variable. The wage gap follows a second-order expectational law of motion and feeds back into aggregate demand through consumption dispersion. Inflation stabilization or contemporaneous profit-wedge neutralization therefore generally fails to restore the corresponding representative-agent allocation. Under the maintained commitment benchmark, RANK-equivalent stabilization from period $t=1$ onward requires history-dependent transfers that respond to inherited wage dispersion, not only current profits. In the benchmark calibration, wage rigidity raises the peak output response to a transfer shock by a factor of 3.27, from $2.49\times 10^{-4}$ to $8.14\times 10^{-4}$.
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