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We analyze the rise in the acceptability fiat money in a Kiyotaki–Wright economy by developing a method that can determine dynamic Nash equilibria for a class of search models with genuine heterogeneous agents. We also address open issues regarding the stability properties of pure strategy equilibria and the presence of multiple equilibria, numerical experiments illustrate the liquidity conditions that favor the transition from partial to full acceptance of fiat money, and the effects of inflationary shocks on production, liquidity, and trade.