Journal article: We define continuous-time dynamics for exchange economies with fiat money. Traders have locally rational expectations, face a cash-in-advance constraint, and continuously adjust their short-run dominant strategy in a monetary strategic market game involving a double-auction with limit-price orders. Money has a positive value expect on optimal rest-points where it becomes a "veil " and trade vanishes. Typically, there is a piecewise globally unique trade-and-price curve both in real and in nominal variables. Money is not neutral, either in the short-run or long-run and a localized version of the quantity theory of money holds in the short-run. An optimal money growth rate is derived, which enables monetary trade curves to converge towards Pareto optimal rest-points. Below this growth rate, the economy enters a (sub-optimal) liquidity trap where monetary policy is ineffective ; above this threshold inflation rises. Finally, market liquidity, measured through the speed of real trades, can be linked to gains-to-trade, households' expectations, and the quantity of circulating money.