Money and optimal inflation

Transcription

1 Advanced Monetary Theory and Policy EPOS 2012/13 Money and optimal inflation Giovanni Di Bartolomeo

2 Before starting to avoid confusion Net inflation p=(p-p(-1))/p(-1), e.g. p=.02 (2% inflation rate); gross inflation P=P/P(-1), e.g. P=1.02. Note that P+1=p. Net nominal interest rate i=.03 (a unit bond give an interest of.03 i.e. 30 basis points, 3%); gross nominal interest rate 1+i (a unit bound give back 1.03) Bound price (Q) and interest rate: Q=1/(1+i) Fisher equation (RR real interest rate) RR = Ep i or RR = p i Recall that the gross RR is 1/b, if b=0.97, gross RR is 1.03, then the net is 3%. Quarterly data b=0.992, quarterly gross RR is 1.01, then the net is 1% (1% times 4 quarters is 3%)

3 Today problem (Friedman) We consider an economy where Markets are competitive and prices are flexible (it follows that quantitative theory of money holds) Production function is Y = A N a Households take advantage in holding money, i.e. it increases their utility (MIU short-cut) A social planner (a policymakers who maximizes the social welfare function, i.e. household utility) set the growth rate of money (nominal interest rate, inflation) Our question is: Which is the optimal rate of inflation (money growth rate)?

4 Money in the utility function (MIU) model Assume that the representative household cares about consumption (C), leisure (L=1-N), and money (real money balances, i.e. m=m/p), the period utility is U= u(c, L, m) The idea is that households care about money, because it facilitates transactions (instrument of exchange) Household s (nominal) budget constraint is: PC + M + QB + = WN + M(-1) + B(-1) Note that C is the real consumption (thus PC is the nominal one), Q is the financial asset price, M and B are nominal money and financial asset held

5 MIU model Household budget constraint (nominal): PC + M + QB + = WN + M(-1) + B(-1) Define the financial wealth as A = M(-1)+B(-1), then PC + M +QM QM + QB + = WN + A and we can finally rewrite the budget as PC + (1 Q)M + QA(+1) + = WN + A No Ponzi condition Limt t A 0 Debt cannot be paid by new debt forever

6 Maximize Subject to Household s problem U= u(c, L=1 N, M/P) WN + A ( PC + (1 Q)M + QA(+1) ) = 0 with respect to C (consumption), N (labor supply), A(+1) (asset demand), M (money demand).

7 Maximize Subject to Household s problem U= u(c, L=1 N, M/P) WN + A ( PC + (1 Q)M + QA(+1) ) = 0 with respect to C (consumption), N (labor supply), A(+1) (asset demand), M (money demand). It follows 1. uc l P = 0 => l = uc/p (MU consumption) 2. ul + l W = 0 (Labor supply) 3. l Q + l(+1) = 0 (Euler equation) 4. um/p= l (1-Q) (Money demand) Recall that between the price and the interest rate of a perpetual asset there is an inverse relationship: Q =1/1+i

8 Household s problem Use (1) l = uc/p in (4), then um/p = uc/p (1 Q) Recall that between the price and the interest rate of a perpetual asset there is an inverse relationship: Q =1/1+i It follow i.e. um = uc (1 1/1+i) um/uc = i/1+i Which is an implicit money demand As exercise derive the money demand M= assuming U = log(c) + h log(1 N) + (M/P) 1+z /1+z

9 Ramsey problem The policymaker acts as a social planner Maximizes U= u(c, L, M/P) subject to Y = A N a. It follows the Lagrangian First order condition i.e. 1. ul=l aan a-1 2. um=0 L= u(c, 1 N, M/P) + l(y - A N a ) 1. ul/uc= MPL (recall competitive, efficient economy) 2. um=0

10 Optimal monetary policy The social planner aims to implement: um=0 As the money demand is: um/uc = i/1+i He needs to set i=0 (i.e., Q=1). Recall that the real interest rate is determined by real forces (gross RR = 1/b), always if the prices are flexible in the long run with price stickiness Moreover, the Fisher equation implies RR = p i 3% = p 0 Optimal inflation is 3% (Friedman rule), at least in the long run

11 Friedman rule: Intuition Money marginal costs and benefits 1. MC = 0 (producing money cost almost noting) 2. MB = i (money production gives the interest rate) Optimality always implies MC=MB It follows that it is optimal a zero nominal interest rate As the real interest rate is positive, negative inflation is optimal Note that (for Friedman) money is a sort of public good, private banking would imply an overproduction of money and inflation. A different view (private banking) stresses that different private money would be associated to different brand, private banks will not overprovide money because of competition