# Introduction to Money

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1 Introduction to Money (3f)-P.1 How does money fit into modern macro models? - Money M = = nominal units issued by the government. Price level p. Purchasing power 1/p. - Consider discrete periods: Household hold money and interest-bearing assets: c t + a t+1 + M t+1 / p t = w t + (1+ r t )a t + M t / p t - Real rate of return on money holdings = p t / p t+1. - Money is an inferior store of value whenever p t / p t+1 < 1+ r t+1 - Equivalent: Positive interest rate on nominal bonds. => Two branches of monetary theory: a. Monetary models with a transactions motive for money - Accept that money is dominated by other assets in terms of return => Assume frictions in exchange processes as motive for holding money b. Monetary theory with money as a store of value: - Most popular: Overlapping-generations models with dynamic inefficiency How important is money for economic activity? - Even in models with motive for holding money, it may not matter economically. - Why should small frictions in exchange have large effects? => Theories why money is important. Most popular: Sticky prices.

2 (3f)-P.2 Motives for holding money: 1. Cash-in-advance constraint (Clower 1967, Lucas 1990) - Implicitly an imperfect information story: Impossible to verify credit at the points of purchase. Need government-issued money to certify purchasing power. - Value of consumption purchases limited by the money stock: p t c t M t - More elaborate versions: Constraint over subset of commodities; constraint over investment, etc. 2. Money in the utility function u = u(c t,m t / p t ) = u(c t,m t ) - Shortcut for money is useful. Classic Paper: Sidrauski (1967) 3. Transactions cost models: Money saves time c t H(M t / p t,s t ), where s t = shopping time <=> s t G(M t / p t,c t ), where G(m,H(m,s)) = s n t + s t =1 l t u = u(c t,l t ) = u(c t,1 n t G(m t,c t )) u (c t,n t,m t ) =>like money-in utility. 4. Search models: Money is preferred to barter exchange (e.g., Lagos & Wright 2005) - Idea: Model with decentralized trading, many differentiated goods. Each agent produces a subset, must buy all others. Individuals meet randomly. Barter requires double coincidence of wants. Holding money allows purchases from sellers who don t want the buyer s goods. => Money facilitates trade. Technical analysis complicated due to multiple goods & randomness of search.

3 (3f)-P.3 The Sidrauski Model -time model: Application of Optimal Control. max e ρt t=0 where m = M/p = real money balances. c + dk + 1 dm p = w + r k + χ u(c(t),m(t)) where χ(t) = series of lump-sum transfers from the government dm = 1 dm m p p dp where π(t) = 1 p => c + dk + dm => c + da dp = 1 p = inflation rate dm m π = w + r k π m + χ = w + r a (r +π ) m + χ - Insight: Opportunity cost of money = Nominal interest rate 0 - Transversality condition: lim T a(t ) e r(v)dv = 0 T

4 Hamiltonian: H(c,m,a,λ,t) = e ρt u(c,m)+ λ [w c + r a (r +π ) m + χ] (3f)-P.4 1. FOC for choice variables (c,m): e ρt u c (c,m) = λ and e ρt u m (c,m) = λ (r +π) => u m (c,m) u c (c,m) = r +π Marginal rate of substitution = Nominal interest rate - ˆ λ t = H a = r λ => 1 λ λ t = r λ = λ e ρt : Then u c (c,m) = ˆ 1ˆ ˆ λ λ t = 1ˆ λ λ => ˆ λ λ and t e ρt + ρ λˆ λ eρt = 1 λ t λ + ρ t = λ ˆ (r ρ) - Insight: Steady state has constant (c,m) => constant ˆ Real interest rate = Rate of time preference λ * = u c (c *,m * ) => r * = ρ [Model without growth, otherwise adjust]

5 Equilibrium: - Interest and wages derived from production: f ' (k) δ = r, f (k) k f ' (k) = w - Government transfer = seignorage: χ = 1 p dm. Taken as given by individuals. Seignorage = government revenue from issuing money (3f)-P.5 Monetary policy is defined by the growth rate of money supply: σ = M 1 dm - Note that χ = 1 p dm = M p σ = mσ 1. Constant capital-labor ratio k * : Implies r * = ρ = f ' (k * ) δ The real interest rate does not depend on monetary factors. 2. Constant real money holdings m: Otherwise MRS between c and m would vary, so c would vary. 1 dm = 1 dm 1 dp = σ π. So dm = 0 π * = σ * m M P Money growth rate determines the inflation rate. 3. Budget constraint: c = w+ r a (r +π) m + χ = w+ r (a m) π m +σ m In steady state: c * = w * + r * k * π * m +σ * m = w * + r * k *

6 Neutrality = Changes in M have no real effects Super-neutrality = Neither changes in M nor changes in σ have real effects (3f)-P.6 -neutral in steady state. σ m * u [σ m * from u m (c*,m * ) u c (c *,m * ) = r* +σ. u because u = u(c,m) is increasing in m] => Optimal policy is to set σ = r * r * +π * = 0 ( The Friedman rule ): Optimal supply policy means making money available at a zero opportunity cost. => Nominal interest rate = zero. Argument: Optimal to equate private and social cost. Printing money is essentially costless. - Note that the Friedman rule requires (a) deflation; (b) negative seignorage, which is implicitly financed by lump-sum taxes.

7 Learning Objectives on Money (3f)-P.7 - Know about alternative ways of introducing money into macro models. - Know why money does not matter much unless there are additional frictions. - Know Friedman s optimal quantity of money rule. The Sidrauski model as example of optimal control.

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