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This paper uses an infinite-horizon model based on individual maximizing behavior to study whether explosive price-level paths unrelated to monetary growth--speculative hyperinflations--can be equilibrium paths under rational expectations. In a pure fiat money regime, speculative hyperinflations can be excluded only through severe restrictions on individual preferences; but when the government fractionally backs the currency by guaranteeing a minimal real redemption value for money, speculative hyperinflations are impossible, even if agents are not completely certain that they can redeem their money in any given period. The analysis also confirms that implosive price-level paths and divergent paths for capital asset prices are not equilibria under either monetary regime.