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The current framework of network utility maximization for rate allocation and its price-based algorithms assumes that each link provides a fixed-size transmission "pipe" and each user's utility is a function of transmission rate only. These assumptions break down in many practical systems, where, by adapting the physical layer channel coding or transmission diversity, different tradeoffs between rate and reliability can be achieved. In network utility maximization problems formulated in this paper, the utility for each user depends on both transmission rate and signal quality, with an intrinsic tradeoff between the two. Each link may also provide a higher (or lower) rate on the transmission "pipes" by allowing a higher (or lower) decoding error probability. Despite nonseparability and nonconvexity of these optimization problems, we propose new price-based distributed algorithms and prove their convergence to the globally optimal rate-reliability tradeoff under readily-verifiable sufficient conditions. We first consider networks in which the rate-reliability tradeoff is controlled by adapting channel code rates in each link's physical-layer error correction codes, and propose two distributed algorithms based on pricing, which respectively implement the "integrated" and "differentiated" policies of dynamic rate-reliability adjustment. In contrast to the classical price-based rate control algorithms, in our algorithms, each user provides an offered price for its own reliability to the network, while the network provides congestion prices to users. The proposed algorithms converge to a tradeoff point between rate and reliability, which we prove to be a globally optimal one for channel codes with sufficiently large coding length and utilities whose curvatures are sufficiently negative. Under these conditions, the proposed algorithms can thus generate the Pareto optimal tradeoff curves between rate and reliability for all the users. In addition, the distributed algorithms and convergence proofs are extended for wireless multiple-inpit-multiple-output multihop networks, in which diversity and multiplexing gains of each link are controlled to achieve the optimal rate-reliability tradeoff. Numerical examples confirm that there can be significant enhancement of the n- etwork utility by distributively trading-off rate and reliability, even when only some of the links can implement dynamic reliability.