Notes

1) McCandless & Weber 1995 – Money authorities

2) Bullard and Keating 95 – inflation & output

3) Barro 1995, 96 – inflation & growth

4) Taylor 1996 – inflation & unemployment

1) Inflation and the growth rate of money (.92-.96) are highly correlated

- Support quantity theory

- Does not imply causality between growth rate of money and inflation that depends on monetary policy, McCandless & Weber 1995

2) Growth rate of money and nominal variables (Pn,t) are highly correlated

3) Growth rate of money and output are uncorrelated

- Support QT

- Barro 1995 finds that high inflation, there is a negative relation between inflation and the rate of growth of output.

4) Growth rate of money and unemployment are uncorrelated.

QT >> M(n,t) Vt = Pt Yt ; V, velocity of money is the average number times the stock of money is used to carry all transactions (proxied by GDP) in the economy in a given year, V is negatively correlated with the demand of money.

Classical economists

- V = constant

- Stock of nominal money did not affect real GDP.

- Inflation is just a monetary phenomenon

Empirical evidence

- V is not constant (affected by velocity, technology and uncertainty)

Fisher equation > (1+i) = (1+ r)(1+ inflation)

Increase in inflation leads to increase in nominal interest rate > are presumed by classical economist, r is not affected by inflation.

Standard macroeconomic models are not very helpful

1) Individual are assumed to care about their consumption and to decide about their consumption and savings plans subject to their budget constraint in order to maximize their well-being.

2) If we introduce money are a mean of storage for individuals, money and capital cannot coexist. (as Rk > 0, Rm < 0 due to inflation)

- Capital yields a gross rate of return of 1+r> 1. The net return is r > 0.

- Net return for money is (-INFLATIONt+1)/(1+INLATIONt+1) <0

MIU models >> U (c, M) ; M = M(n,t)/Pt

- Postulate that real money balances enter the utility function, so that the contemporaneous utility of the individual

o Real money balances enter in the utility function (what we can buy)

o Motivation for MIU models: as short-cut to the double coincidence of wants problem; higher real money balances save time in transaction. MIY models could be interpreted as having leisure in the utility function.

A money in the utility function model

1) Definite horizon

2) 3 agents

3) All individuals and firms are identical

The representation individual

1) Identical individuals with infinite lifetime & rational expectation

2) No pop. Growth, Nt normalized to one.

3) The utility is a function of consumption and real money balances. The welfare of individual is

T=0 LIM infinity, beta t* u Cct, mt); beta is the discount factor > {1,0} ; if beta =1 present and future weighted equally ; beta = 1 /(1+r) ; beta = 0, r approaches infinity, value more on today. (t+ >> i+ >> b-)

4) The demand for money will be always positive if

M=0 LIM, Um (ct, mt) = infinity for all c

To ensure a monetary equilibrium exist (in red, conditions for positive money demand for existence of a high monetary equilibrium)

M=infinity LIM, Um (ct, Mt) <= 0 for m>= avg m (monetary equilibrium), for all c

o Marginal utility becoming –ve for sufficiently high money balances.

Expenditure

- Consumption, Ct

- Assets: kt, mt, bt

Income

- Wt

- Assets income: capital rt= ^rt – d; bonds pay a nominal interest, it, money pay no nominal interest.

- Gov transfer, zt

The budget constraint of the individual

N >> wt + rt-1 * kt-1 + (1+it-1) b (n,t) /Pt + M (n,t)/Pt + z (n,t)/Pt = Ct + kt + b (n,t)/Pt + M(n,t)/Pt

R >> wt + rt-1 * kt-1 + (1+it-1) / (1+ INFt) b (n,t) + m (n,t)/ (1+INFt) + zt = Ct + kt + bt + Mt

The optimization problem of the individual

- Taking as given wt, ^rt-1 and It-1, the individual will choose mt, kt, ct, and bt to max its welfare.

T=0 LIM infinity, Beta t * u (ct, mt) subject to budget constraint, no ponzi game

- wt + rt-1 * kt-1 + (1+it-1) / (1+ INFt) b (n,t) + m (n,t)/ (1+INFt) + zt = Ct + kt + bt + Mt

T=infinity LIM, Beta t LAMDA t k t >= 0

T=infinity LIM, Beta t LAMDA t b t >= 0

T=infinity LIM, Beta t LAMDA t m t >= 0

LAMDA >> the multiplier associated to budget constraint.

Lagrangian:

L = T=0 LIM infinity, Beta t * u (ct, mt) +T=0 LIM infinity, Beta t LAMDA t [ wt + rt-1 * kt-1 + (1+it-1) / (1+ INFt) b (n,t) + m (n,t)/ (1+INFt) + zt - Ct - kt - bt – Mt]

dL/dct = 0 >> LAMDA t = Uc (Ct, Mt) >> marginal good of each variables

dL/dkt = 0 >> LAMDA t = Beta t LAMDA t+1 (1 + rt) = Beta t LAMDA t+1 (^rt + 1 – d)

dL/dbt = 0 >> LAMDA t = Beta t LAMDA t+1 (1+it)/ (1+INFt+1)

dL/dmt = 0 >> LAMDA t = Um (ct, mt) + Beta t LAMDAt+1 1/ (1+INFt+1) ; (1+INFt+1) >> inflation; Um (ct, mt) >> direct utility from money, liquity gain ; Beta t LAMDAt+1 1/ (1+INFt+1) >> store value of money, gain at t+1

Concavity ensures that these necessary conditions are also sufficient.

Combining eq:

1) (1+2) >> Uc (Ct, Mt) = Beta t Uc (Ct+1, Mt+1) (1 + rt) MC of 1 extra unit of capital (the marginal unit of forgone consumption) = MB (discounted utility of consuming the principal plus the interest in the next period)

2) (2+3) >> (1 + rt) = (1+it)/ (1+INFt+1) fisher equation, which is not arbitrage condition between bonds and capital

3) (1+4) >> Uc (ct, mt) = Um (ct, mt) + Beta t Uc (Ct+1, Mt+1) 1/ {(1+INFt+1)}

4) Um (ct, mt)/ Uc (ct, mt) = it / (1+it) = Rt ; to find the money demand, Rt is the opportunity cost of holding money

The Representative Firm

- Many identical & competitive firms

- FOP, Kt-1 (capital) and Lt (labour)

- Aggregate production function, Yt = f (Kt-1, Lt) with FK > 0, FL > 0, FKK > 0, FLL > 0, constant return to scale

F (LAMDA Kt-1, LAMDA Lt-1) = LAMDA F (Kt-1, Lt-1)

- Inada condition

K approaches 0 LIM, Fk (Kt-1, Lt-1) = infinity; K approaches infinity LIM, Fk (Kt-1, Lt-1) = 0

Production function is per capita (labor) terms, y = f(kt-1)

The opportunity problem of the firm

Taking as given wt and ^rt-1, the firm chooses capital Kt-1 and labour Lt to maximize its profits.

IIt = F (Kt-1, Lt) - ^rt-1 Kt-1 – wt Lt

First order,

dIIt/dKt-1 = 0 >> ^rt-1 = Fk (Kt-1, Lt)

dIIt/Lt = 0 >> wt = FL (Kt-1, Lt)

F (Kt-1, Lt) = Fk *(Kt-1) + FL* (Lt), at IIt = 0

In the per capita terms, ^rt-1 = F’ (Kt -1 ) and wt = F(kt-1) – F’(kt-1) kt-1

The money authority

- M (N, S, t) = M (N, S, t-1) + Z (N, t)

- The ratio of growth of money supply >> $t

- M (N, S, t) = (1 + $t) M (N, S, t-1)

- Mt = (1+St) / (1 + INF t) Mt-1

- Z (N, t) = $t M (N, t-1) or in the real term, zt = $t/ (1+ INFt) Mt-1

The Aggregate economy and market equilibrium condition

N >> Yt + rt-1 * kt-1 + (1+it-1) b (n,t) /Pt + M (n,t)/Pt + z (n,t)/Pt >= ct + kt + b (n,t)/Pt + M(n,t)/Pt

R >> f(kt-1) + rt-1 * kt-1 + (1+it-1) / (1+ INFt) b (n,t) + m (n,t)/ (1+INFt) + zt >= ct + kt + bt + mt

- Pt = Pt-1 (1+ INFt)

- All market clear

o Lt = Nt

o Kss = Kdd

o Bss = Bdd

o Mss = Mdd

The competitive equilibrium

Given the government policy {$t, zt} t=0 approaches infinity; competitive equilibrium {ct, kt, bt, mt} t=0 approaches infinity; a price sequences {rt-1, it-1, wt, INFt} t=0 approaches infinity, such that: (i) the individual maximizes his welfare subject to his budget constraint and the no-Ponzi game conditions; (ii) FOP are paid their marginal products; (iii) all market clear

1) wt + rt-1 * kt-1 + (1+it-1) / (1+ INFt) b (n,t) + m (n,t)/ (1+INFt) + zt = Ct + kt + bt + Mt

2) Uc (ct, mt) = Beta t * Uc (ct+1, mt+1) (1+rt)

3) (1+it) = (1+rt) (1+ INF t+1)

4) Uc (ct, mt) = Um (ct, mt) + Beta t * Uc (ct+1, mt+1) / (1/ INFt+1)

5) Yt = F(Kt-1, Lt), yt = F(kt-1)

6) ^rt -1= Fk(Kt-1), whre rt-1 = ^rt-1 - d

7) wt = F(kt-1) – F‘(kt-1) * kt-1

8) M(N, S, t) = M(N, S, t-1) + z(N, t) ; Mt = (1+ $t)/(1+INFt) Mt-1

9) Zt = $t/ (1+ INFt) Mt-1

10) Bt = 0

Transversality conditions:

T approaches infinity LIM, Beta t Uc(ct, mt) kt = 0

T approaches infinity LIM, Beta t Uc(ct, mt) bt = 0

T approaches infinity LIM, Beta t Uc(ct, mt) mt = 0

Long run: Steady State

A steady state equilibrium is an optional solution such that all variables of the economy are constant.

Neutrality, superneutrality and long run growth

- Neutrality of money means that changes in the quantity of nominal money have no effects on real variables but lead to a proportional change in the price level.

INF = $ >> satisfy neutrality

- Superneutrality of money means that change in the rate of growth of nominal money have no effect on real variables but lead the proportionate change in the rate of growth of the price level (inflation).

The long run effect of money and inflation

- Long run effect of inflation, Beta = 1 / (1+r); 1/Beta = (1+i)/(1+INF)

o (1+i) = 1/Beta * (1+INF) >> Nominal interest rate increase with inflation

o Opp. Cost of money, Rt = it/ (1+it), is positively affect by inflation

o Um(ct,mt)/ Uc(ct, mt) = Rt is affected by inflation, therefore inflation affects welfare

The Quantity Equation Revisited

U(c,m) = In c + @ In m, with 1 > @ > 0

(4ss) >> 1/c = @/m + Beta (1/c) / (1+ INF)

C = y- dk (1ss)

M= [ (1+INF)@ / (1+INF- BETA) ] c = [ (1+INF)@ / (1+INF- BETA) ] [y-dk]

1/V = [ (1+INF)@ / (1+INF- BETA) ]

Money an the long run growth

Yt = B * (1+g)^t * Lt; B> 0, g >0

ct = yt = B(1+g)^t

mt = [ (1+INFt+1) / (1+INFt+1- BETA/ (1-g)) ] @ct = [ (1+INF) / (1+INF- BETA) ]@ B(1+g)^t

M(N, t ) / M(N, t-1) = Pt / Pt-1 * (1+g)

(1+$) = (1+ INF) (1+g)