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We study optimal monetary policy in response to the cost-push uncertainty shock, which is a second-moment shock, in a textbook New Keynesian model. Following a cost-push uncertainty shock, optimal monetary policy faces a trade-off between output gap and inflation stabilization. This is because, even in the absence of first-moment cost-push shocks, cost-push uncertainty generates a time-varying gap between natural output and efficient output. These results contrast with those under a conventional productivity uncertainty shock, which leads to complete stabilization of the output gap and inflation.