This note came out of the first sessions of a PhD course in monetary economics. The opening question was what “fiat” money is. The word is Latin for “let it be done”; it has nothing to do with the car.

Fiat money is an object that is intrinsically worthless and inconvertible: it gives no utility, has no use in production, and cannot be redeemed for anything that does (Wallace, 1980). That definition is the source of the difficulty. In Arrow–Debreu general equilibrium, markets are complete and all trades for all dates and states clear at date 0 under a single budget constraint. An intermediate object has no function there, and its equilibrium price is zero.

Every monetary model therefore breaks an Arrow–Debreu assumption. Money-in-the-utility models put real balances $M_t/P_t$ in the utility function. Cash-in-advance models require cash for some purchases, $P_t c_t \le M_t$ (Lucas and Stokey, 1987). Search models remove the double coincidence of wants and record-keeping (Kiyotaki and Wright, 1989). The first two assume that money is useful; search models derive it. Wallace’s legal restrictions theory addresses a further puzzle: cash pays nothing while Treasury bills pay interest, so cash survives only because the government prevents bonds from circulating as money.

Asset pricing gives the same point in one line. Every traded asset satisfies the Euler equation $1 = E_t[M_{t+1} R_{t+1}]$, where $M_{t+1} = \beta\, u'(c_{t+1})/u'(c_t)$ is the stochastic discount factor. Money earns a nominal return of zero, so it is held only if it carries a liquidity premium $L_t$:

$$1 = L_t + E_t\big[M^{\text{nom}}_{t+1}\big], \qquad L_t = \frac{i_t}{1+i_t}.$$

The premium equals the interest forgone by holding cash instead of bonds (Benigno, Schilling and Uhlig, 2022).

The cleanest case is Samuelson’s (1958) overlapping generations model. Agents live two periods, receive an endowment $y$ when young and nothing when old, and the good is perishable. Without money the outcome is autarky: the old consume nothing, and the young cannot lend to people who die before repaying. Now give the initial old a stock $M$ of paper. The young sell goods for money and spend it when old, so

$$c^y_t + \frac{P_{t+1}}{P_t}\, c^o_{t+1} = y,$$

and money earns the real return $P_t/P_{t+1}$. In a steady state, real balances per young person $z$ solve $u'(y-z) = \beta\, u'(z)$, and the price level is $P = M/(Nz)$.

Three results follow. First, doubling $M$ doubles $P$ and changes nothing real, which is the quantity theory. Second, autarky remains an equilibrium: money has value only because each generation expects the next to accept it. Fiat money is a rational bubble, and the multiplicity of equilibria is a property of the model, not a flaw. Third, the monetary steady state Pareto-dominates autarky, so the First Welfare Theorem fails; the proof requires finitely many agents and goods (Shell, 1971). A monetary equilibrium exists only if the autarky interest rate is below the growth rate, the same $r < g$ condition used in current arguments about government debt.

The initial old receive goods for paper and are never asked to repay. The arrangement lasts as long as each new generation agrees to it.

sources

  • Samuelson, P. (1958). “An Exact Consumption-Loan Model of Interest with or without the Social Contrivance of Money.” Journal of Political Economy 66(6).
  • Wallace, N. (1980). “The Overlapping Generations Model of Fiat Money.” In Kareken and Wallace (eds.), Models of Monetary Economies, Federal Reserve Bank of Minneapolis.
  • Shell, K. (1971). “Notes on the Economics of Infinity.” Journal of Political Economy 79(5).
  • Lucas, R. and N. Stokey (1987). “Money and Interest in a Cash-in-Advance Economy.” Econometrica 55(3).
  • Kiyotaki, N. and R. Wright (1989). “On Money as a Medium of Exchange.” Journal of Political Economy 97(4).
  • Benigno, P., L. Schilling and H. Uhlig (2022). “Cryptocurrencies, Currency Competition, and the Impossible Trinity.” Journal of International Economics 136.