Introduction to Money


 Elinor Morgan
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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 interestbearing 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: Overlappinggenerations 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. Cashinadvance constraint (Clower 1967, Lucas 1990)  Implicitly an imperfect information story: Impossible to verify credit at the points of purchase. Need governmentissued 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 moneyin 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 lumpsum 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 capitallabor 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 Superneutrality = 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 lumpsum 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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