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Classical monetary model

Giovanni Di Bartolomeo

giovanni.dibartolomeo@uniroma1.it

Sapienza University of Rome

Department of economics and law

Advanced Monetary Theory and Policy EPOS 2013/14

Assumptions

• Main assumptions

1. Perfect competition in goods and labor markets

2. Flexible prices and wages

3. No capital accumulation

4. No fiscal sector

5. Closed economy

• Assumptions 1 and 2 are crucial. Note that they imply

Pareto efficiency (First Fundamental Welfare Theorem)

• Assumptions 3-5 are simplifications. They can be

removed. Note that they also imply 𝐶𝑡 = 𝑌𝑡.

The household’s problem

• Representative household solves

𝑚𝑎𝑥𝐸0 𝑡𝑈(𝐶𝑡 , 𝑁𝑡)

𝑡=0

• Subject 𝑃𝑡𝐶𝑡 + 𝑄𝑡𝐵𝑡𝐵𝑡−1 + 𝑊𝑡𝑁𝑡 − 𝑇𝑡

• for t = 0, 1, 2 … plus solvency constraint (no-Ponzi

condition)

• Optimality conditions

−𝑈𝑛,𝑡

𝑈𝑐,𝑡=

𝑊𝑡

𝑃𝑡

𝑄𝑡 = 𝐸𝑡

𝑈𝑐,𝑡+1

𝑈𝑐,𝑡

𝑃𝑡

𝑃𝑡+1

Exercise (two-period economy)

• Representative household utility flow

𝑈 𝐶𝑡 + 𝑈(𝐶𝑡+1)

• Budget constraint in period 1 and 2: 𝑃𝑡𝐶𝑡 + 𝑄𝐵𝑡𝑊

𝑃𝑡+1𝐶𝑡+1𝐵𝑡

• Note that 𝑄 and 𝑊 are given

• Draw the inter-temporal budget constraint (solve the

budget constraints for 𝐵𝑡 and equate)

• Find the household's first order condition

• Find 𝐶𝑡 and 𝐶𝑡+1 assuming 𝑈 𝐶𝑘 , 𝑁𝑘 = 𝑙𝑛 𝐶𝑘

Just do it! Solve the household’s problem

• Lagrangean at time t:

L = 𝐸𝑡 𝑡 𝑈 𝐶𝑡+𝑖 , 𝑁𝑡+𝑖 + 𝑡+𝑖(𝐵𝑡−1+𝑖 + 𝑊𝑡+𝑖𝑁𝑡+𝑖

𝑖=0

− 𝑇𝑡+𝑖 − 𝑃𝑡+𝑖𝐶𝑡+𝑖 − 𝑄𝑡+𝑖𝐵𝑡+𝑖)

• i.e.

L = 𝐸𝑡 𝑈 𝐶𝑡 , 𝑁𝑡 + 𝑈 𝐶𝑡+1, 𝑁𝑡+1 + 2𝑈 𝐶𝑡+2, 𝑁𝑡+2 + … + 𝑡 𝐵𝑡−1 + 𝑊𝑡𝑁𝑡 − 𝑇𝑡 − 𝑃𝑡𝐶𝑡 − 𝑄𝑡𝐵𝑡

+ 𝑡+1(𝐵𝑡 + 𝑊𝑡+1𝑁𝑡+1 − 𝑇𝑡+1 − 𝑃𝑡+1𝐶𝑡+1

− 𝑄𝑡+1𝐵𝑡+1)+ 2𝑡+2 (𝐵𝑡+1 + 𝑊𝑡+2𝑁𝑡+2 − 𝑇𝑡+2 − 𝑃𝑡+2𝐶𝑡+2

− 𝑄𝑡+2𝐵𝑡+2) + … .

• Derive the above expression for 𝐶𝑡, 𝑁𝑡, 𝐵𝑡 and rearrange

Specification of utility

• Considering

𝑈 𝐶𝑡 , 𝑁𝑡 =𝐶𝑡

1−

1 − −

𝑁𝑡1+

1 +

• Implied optimality conditions:

𝐶𝑡𝑁𝑡

=𝑊𝑡

𝑃𝑡

𝑄𝑡 = 𝐸𝑡

𝐶𝑡+1

𝐶𝑡

−𝑃𝑡

𝑃𝑡+1

• Note 𝑡 =𝑃𝑡+1

𝑃𝑡 is the gross inflation rate.

Log-linear version

• Optimality conditions:

𝑐𝑡 + 𝑛𝑡 = 𝑤𝑡 − 𝑝𝑡

𝑐𝑡 = 𝐸𝑡 𝑐𝑡+1 −1

𝑖𝑡 − 𝐸𝑡 𝑡+1 −

• Where 𝑖𝑡 = −𝑙𝑜𝑔 𝑄𝑡 and = −𝑙𝑜𝑔 .

• Perfect foresight steady state (with zero growth):

𝑟 = 𝑖 − =

• Hence it implies a real rate

𝑖𝑡 = −𝑙𝑜𝑔 𝑄𝑡

Asset prices and interest rates

• A save asset A1 gives you a gross interest equal to 𝐼𝑁𝑇𝑡.

It means that investing X, you will obtain X times 𝐼𝑁𝑇𝑡

after one year

• Another safe asset A2 gives you 1$ after one year. How

much should it cost?

• Note that both assets are safe so their value should be

the same!

• The present value of 1$ today is 1

𝐼𝑁𝑇𝑡. It means that by

investing 1

𝐼𝑁𝑇𝑡 in A1, you can obtain 1$ after one year.

• Thus, by arbitrage the price 𝑄𝑡 of A2 should be 1

𝐼𝑁𝑇𝑡

• In the slide before, this explains 𝑖𝑡 = −𝑙𝑜𝑔 𝑄𝑡

The firm’s problem

• Representative firm with technology

𝑌𝑡 = 𝐴𝑡𝑁𝑡1−

• Profit maximization

max𝑃𝑡𝑌𝑡 − 𝑊𝑡𝑁𝑡

• Subject to the above firm technology, taking prices and

wages as given (perfect competition)

• Optimality condition 𝑊𝑡

𝑃𝑡= 1 − 𝐴𝑡𝑁𝑡

• In log-linear terms

𝑤𝑡 − 𝑝𝑡 = 𝑎𝑡 − 𝑛𝑡 + 𝑙𝑜𝑔 1 −

Equilibrium

• Good market clearing

𝑦𝑡 = 𝑐𝑡

• Labor market clearing

𝑐𝑡 + 𝑛𝑡 = 𝑎𝑡 − 𝑛𝑡 + 𝑙𝑜𝑔 1 −

• Asset market clearing

𝑦𝑡 = 𝐸𝑡 𝑦𝑡+1 −1

𝑖𝑡 − 𝐸𝑡 𝑡+1 −

• Aggregate production relationship

𝑦𝑡 = 𝑎𝑡 + 1 − 𝑛𝑡

• where 𝑎𝑡 is a stochastic process (e.g. 𝑎𝑡 = 𝑎𝑎𝑡−1 + 𝑡,

𝑡𝑁0,𝑎2)

• 4 equation for 5 unknowns!!!

Equilibrium

• Good market clearing

𝑦𝑡 = 𝑐𝑡

• Labor market clearing

𝑐𝑡 + 𝑛𝑡 = 𝑎𝑡 − 𝑛𝑡 + 𝑙𝑜𝑔 1 −

• Asset market clearing

𝑦𝑡 = 𝐸𝑡 𝑦𝑡+1 −1

𝑖𝑡 − 𝐸𝑡 𝑡+1 −

• Aggregate production relationship

𝑦𝑡 = 𝑎𝑡 + 1 − 𝑛𝑡

• where 𝑎𝑡 is a stochastic process (e.g. 𝑎𝑡 = 𝑎𝑎𝑡−1 + 𝑡,

𝑡𝑁0,𝑎2)

• 4 equation for 5 unknowns (𝑦𝑡, 𝑐𝑡, 𝑛𝑡, 𝑖𝑡, 𝑡)!!!

• But…

Real variables (nt and yt)

• By using good market clearing (𝑦𝑡 = 𝑐𝑡) and the

aggregate production relationship 𝑦𝑡 = 𝑎𝑡 + 1 − 𝑛𝑡 in

the labor market clearing

𝑐𝑡 + 𝑛𝑡 = 𝑎𝑡 − 𝑛𝑡 + 𝑙𝑜𝑔 1 −

• We obtain

𝑎𝑡 + 1 − 𝑛𝑡 + 𝑛𝑡 = 𝑎𝑡 − 𝑛𝑡 + 𝑙𝑜𝑔 1 −

• Solving the above expression, we get

𝑛𝑡 = 𝑛𝑎𝑎𝑡 + 𝑛

𝑦𝑡 = 𝑦𝑎𝑎𝑡 + 𝑦

• Where 𝑛𝑎 =1−

; 𝑛 =log 1−

; 𝑦𝑎 =

1+

; 𝑦 =

1 − 𝑛; = + + 1 −

Real variables (rt and wt)

• Asset market clearing is

𝑦𝑡 = 𝐸𝑡 𝑦𝑡+1 −1

𝑖𝑡 − 𝐸𝑡 𝑡+1 −

𝑟𝑡 = 𝑖𝑡 − 𝐸𝑡 𝑡+1 = 𝐸𝑡 𝑦𝑡+1 − 𝑦𝑡 +

𝑟𝑡 = 𝐸𝑡 𝑦𝑡+1 + = 𝑦𝑎𝐸𝑡 𝑎𝑡+1 +

• Recall 𝑦𝑡 = 𝑦𝑎𝑎𝑡 + 𝑦

• Real wage is

w𝑡 = 𝑤𝑡 −𝑝𝑡 = 𝑎𝑡 − 𝑛𝑡 + 𝑙𝑜𝑔 1 −

w𝑡 = w𝑎𝑎𝑡 + 𝑙𝑜𝑔 1 −

• Where 𝑛 =log 1−

; 𝑦𝑎 =

1+

; 𝑦 = 1 − 𝑛;

w𝑎 =+

; = + + 1 −

Real variables dynamics

• Solving the above expression, we get (IRFs)

𝑛𝑡 = 𝑛𝑎𝑎𝑡 + 𝑛

𝑦𝑡 = 𝑦𝑎𝑎𝑡 + 𝑦

𝑟𝑡 = 𝑦𝑎𝐸𝑡 𝑎𝑡+1 +

w𝑡 = w𝑎𝑎𝑡 + 𝑙𝑜𝑔 1 −

• Steady states (for 𝑎 = 0, i.e. 𝐴 = 1)

𝑛 = 𝑛

𝑦 = 𝑦

𝑟 =

w = 𝑙𝑜𝑔 1 −

• What are the effects of a shock? What are the effects if 𝑎

growth a constant rate (i.e., 𝑎𝑡+1 = 𝑔𝑎)?

• And nominal variables?

Real business cycle (RBC)

• Both labor and output (and consumption) are driven by

log technology

• For example, if log technology is a random walk then

labor and output will be random walk as well.

• Notice that employment will go down with technology if

> 1, go up if < 1 and stay the same if = 1. It shows

the substitution and income effects for labor supply

• Recall

𝑛𝑡 = 𝑛𝑎𝑎𝑡 + 𝑛

• with 𝑛𝑎 =1−

++ 1− (note the denominator is always

positive)

Dynare (RBC code)

• var n, c, a; varexo e;

• parameters sigma, delta, alpha, rhoa, nss;

• sigma = 0.9; delta = 1; alpha = 0.7; rhoa = 0.7; nss = log(1-

alpha)/(sigma+delta+(1-sigma)*alpha);

• model;

• sigma*c + delta*n = a - alpha*n + log(1-alpha);

• c = a + (1-alpha)*n;

• a = rhoa * a(-1) + e;

• end;

• initval; n = nss; c = (1-alpha)*nss; a = 0; end;

• steady;

• shocks;

• var e; stderr 0.01;

• end;

• stoch_simul(irf=20, order=1);

Outcomes

• Try to modify the Dynare code to replicate the figure

Summing up

• Policy neutrality: real variables determined

independently of monetary policy

• Neoclassical dichotomy between real and monetary

sector: real values only depends on relative prices,

money instead determines the aggregate level of price

• Optimal policy: undetermined.

• A specification of monetary policy is needed to

determine nominal variables

Monetary policy specification

• Three solutions

– An exogenous path for the nominal interest rate

– A simple inflation-based interest rate rule (cashless

economy)

– An exogenous path for the money supply

• Consider an ad hoc simple money demand (transaction/

opportunity cost of holding money)

𝑚𝑡 − 𝑝𝑡 = 𝑦𝑡 − 𝑖𝑡

• Remember the Fisher equation

𝑟𝑡 = 𝑖𝑡 − 𝐸𝑡 𝑡+1

Nominal interest rate exogenous path

• We assume an exogenous path for the nominal interest

rate, i.e. an exogenous stationary process for 𝑖𝑡 with

mean , in this case 𝑟𝑡 is determined independently of

𝑖𝑡 and we have (Fisher equation):

𝐸𝑡 𝑡+1 = 𝑟𝑡 − 𝑖𝑡

• Any path for the price level which satisfies

𝑝𝑡+1 = 𝑝𝑡 +𝑟𝑡 −𝑖𝑡 + 𝑡+1

• is consistent with the equilibrium (with 𝐸𝑡 𝑡+1 = 0 for all

t ). Actual inflation can be anything and price can be

anything as well

• We call 𝑡+1 a sunspot shock, i.e. it has nothing to do

with the model but can really blow things up. We have an

indeterminate equilibrium and price level indeterminacy

Nominal interest rate exogenous path

• We assume an exogenous path for the nominal interest

rate, i.e. an exogenous stationary process for 𝑖𝑡 with

mean , in this case 𝑟𝑡 is determined independently of

𝑖𝑡 and we have (Fisher equation):

𝐸𝑡 𝑡+1 = 𝑟𝑡 − 𝑖𝑡

• Any path for the price level which satisfies

𝑝𝑡+1 = 𝑝𝑡 +𝑟𝑡 −𝑖𝑡 + 𝑡+1

• is consistent with the equilibrium (with 𝐸𝑡 𝑡+1 = 0 for all

t ). Actual inflation can be anything and price can be

anything as well

• We call 𝑡+1 a sunspot shock, i.e. it has nothing to do

with the model but can really blow things up. We have an

indeterminate equilibrium and price level indeterminacy

Nominal interest rate exogenous path

• The implied path for the money supply is:

𝑚𝑡 = 𝑝𝑡 + 𝑦𝑡 − 𝑖𝑡

• and hence it inherits the indeterminacy of prices ( 𝑝𝑡 )

• In other words, the central bank fixes the interest rate

and let money be determined endogenously. But since

we have undetermined price, money is undetermined as

well

A simple inflation-based interest rate rule

• Four equation and five unknowns (𝑦𝑡, 𝑐𝑡, 𝑛𝑡, 𝑖𝑡, 𝑡)

𝑦𝑡 = 𝑐𝑡

𝑐𝑡 + 𝑛𝑡 = 𝑎𝑡 − 𝑛𝑡 + 𝑙𝑜𝑔 1 −

𝑦𝑡 = 𝐸𝑡 𝑦𝑡+1 −1

𝑖𝑡 − 𝐸𝑡 𝑡+1 −

𝑦𝑡 = 𝑎𝑡 + 1 − 𝑛𝑡

• where 𝑎𝑡 is a stochastic process

• Adding a rule for the interest rate 𝑖𝑡 = + 𝑡

• Five equation and five unknowns!!!

• Remember the model dichotomy: the first four equations

independently determines the real values, the last the

nominal ones (now we focus on this last)

A simple inflation-based interest rate rule

• Rule for the interest rate 𝑖𝑡 = + 𝑡

• Consider > 0 this policy matches our common

sense: when inflation is high the central bank raises

interest rate to “cool the economy down,” and vice versa

• We refer to > 0 as the Taylor principle, the bank

should react “aggressively” to inflation

• Plugging it into the Fisher equation, we get

𝑡 = 𝐸𝑡 𝑡+1 + 𝑟𝑡

• It is a stochastic difference equation. Two cases:

– > 0 we can get a stationary solution for inflation

by repeated forward substitution

– < 0 in this case it has a backward solution

Forward solution

• Given 𝑡 =1

𝐸𝑡 𝑡+1 +

1

𝑟𝑡

• Then 𝑡+1 =1

𝐸𝑡+1 𝑡+2 +

1

𝑟𝑡+1

• And thus

𝑡 =1

𝐸𝑡

𝐸𝑡+1 𝑡+2

+

𝑟𝑡+1

+

𝑟𝑡

• Moreover as 𝑡+2 =𝐸𝑡+2 𝑡+3

+

𝑟𝑡+2

, it follows

𝑡 =1

𝐸𝑡

1

𝐸𝑡+1

𝐸𝑡+2 𝑡+3

+

𝑟𝑡+2

+

𝑟𝑡+1

+

𝑟𝑡

• Note: 𝐸𝑡 𝐸𝑡+1 𝑥 = 𝐸𝑡 𝑥 (law of iterated expectations)

Forward solution

• Note that 𝑡 =1

𝐸𝑡

1

𝐸𝑡+1

𝐸𝑡+2 𝑡+3

+

𝑟𝑡+2

+

𝑟𝑡+1

+

𝑟𝑡

can be written as

𝑡 =1

3𝐸𝑡 𝑡+3 +

𝐸𝑡 𝑟𝑡+2

3

+𝐸𝑡 𝑟𝑡+1

2

+𝑟𝑡

• Continuing the forward substitutions …

𝑡 =𝐸𝑡 𝑡+𝑇+1

𝑇+1

+ 1

𝑘+1𝑇

𝑘=0𝐸𝑡 𝑟𝑡+𝑘

• … and continuing:

𝑡 = 1

𝑘+1

𝑘=0𝐸𝑡 𝑟𝑡+𝑘

• If the model is stable 𝐸𝑡 = (in our case = 0)

A simple inflation-based interest rate rule

• Rule for the interest rate

𝑖𝑡 = + 𝑡

• Combined with the definition of the real rate (𝑟𝑡 = 𝑖𝑡 −𝐸𝑡 𝑡+1 ) gives 𝑡 = 𝐸𝑡 𝑡+1 + 𝑟𝑡 i.e.

𝑡 =1

𝐸𝑡 𝑡+1 +

1

𝑟𝑡

• If > 1, unique stationary solution:

𝑡 = 1

𝑘+1

𝑘=0𝐸𝑡 𝑟𝑡+𝑘

• See forward solution slides

• Moreover, the price level is also uniquely determined

(given some initial value).

A simple inflation-based interest rate rule

• Rule for the interest rate

𝑖𝑡 = + 𝑡

• Combined with the definition of the real rate (𝑟𝑡 = 𝑖𝑡 −𝐸𝑡 𝑡+1 ) gives:

𝑡 = 𝐸𝑡 𝑡+1 + 𝑟𝑡

• If < 1, any process t satisfying

𝑡+1 = 𝑡 − 𝑟𝑡 + 𝑡+1

• is consistent with a stationary equilibrium (where

𝐸𝑡 𝑡+1 = 0 for all t )

– Price level indeterminacy

– Taylor principle, stability requires > 1, a central

bank should respond to an increase in with an even

greater increase in 𝑖 (so that the 𝑟 rate rises).

The classic model with an interest rate rule

• Five equation and five unknowns (𝑦𝑡, 𝑐𝑡, 𝑛𝑡, 𝑖𝑡, 𝑡)

𝑦𝑡 = 𝑐𝑡

𝑐𝑡 + 𝑛𝑡 = 𝑎𝑡 − 𝑛𝑡 + 𝑙𝑜𝑔 1 −

𝑦𝑡 = 𝐸𝑡 𝑦𝑡+1 −1

𝑖𝑡 − 𝐸𝑡 𝑡+1 −

𝑦𝑡 = 𝑎𝑡 + 1 − 𝑛𝑡

𝑖𝑡 = + 𝑡 + 𝑒𝑡

• where 𝑎𝑡 and 𝑒𝑡 are a stochastic processes

𝑎𝑡 = 𝑎𝑎𝑡−1 + 𝑡, 𝑡𝑁0,𝑎2

𝑒𝑡 = 𝑒𝑒𝑡−1 + 𝑡, 𝑡𝑁0,𝑒2

• Productivity (real) shock 𝑎𝑡

• Monetary shock 𝑒𝑡

Dynare (classic model code)

• var n, c, i, pi, a, e; varexo e1, e2;

• parameters sigma, delta, alpha, rho, rhoa, rhoe, xipi, nss;

• sigma=0.9; delta=1; alpha=0.7; rhoa=0.7; rhoe=0.7; rho=0.99;

xipi=1.5; nss=log(1-alpha)/(sigma+delta+(1-sigma)*alpha);

• model;

• sigma*c + delta*n = a - alpha*n + log(1-alpha);

• c = a + (1-alpha)*n;

• c = c(+1) - (i - pi(+1) - rho)/sigma;

• i = rho + xipi*pi + e;

• a = rhoa * a(-1) + e1;

• e = rhoe * e(-1) + e2;

• end;

• initval;

• n=nss; c=(1-alpha)*nss; pi=0; i=rho; e=0; a=0;

• end;

• shocks; var e1; stderr 0.01; var e2; stderr 0.01; end;

• stoch_simul(irf=20, order=1);

Productivity shock

• Check the outcomes of a monetary shock (note that Dynare does not plot

the IRF of variables that do not change)

An exogenous path for the money supply

• Now we assume that the money supply follows an

exogenous path 𝑚𝑡

• Consider an ad hoc money demand:

𝑚𝑡 − 𝑝𝑡 = 𝑦𝑡 − 𝑖𝑡

• Combining money demand and Fisherian equations:

𝑝𝑡 =

1 + 𝐸𝑡 𝑝𝑡+1 +

1

1 + 𝑚𝑡 + 𝑢𝑡

• where 𝑢𝑡 =𝑟𝑡−𝑦𝑡

1+ evolves independently of monetary

policy

An exogenous path for the money supply

• Given 𝑝𝑡 =

1+𝐸𝑡 𝑝𝑡+1 +

1

1+𝑚𝑡 + 𝑢𝑡, assuming > 0

and solving forward, we obtain:

𝑝𝑡 =

1 +

1 +

𝑘

𝑘=0𝐸𝑡 𝑚𝑡+𝑘 + 𝑢′𝑡

• where 𝑢′𝑡 =

1+

𝑘𝑘=0 𝐸𝑡 𝑢𝑡+𝑘 again evolves

independently of monetary policy

• As 1

1

𝑘𝑘=0 𝐸𝑡 𝑥𝑡+𝑘 = 𝑥𝑡 +

1

𝑘𝑘=1 𝐸𝑡 𝑥𝑡+𝑘 , we

obtain …

An exogenous path for the money supply

• … we obtain

𝑝𝑡 = 𝑚𝑡 +

1 +

𝑘

𝑘=1𝐸𝑡 𝑚𝑡+𝑘 + 𝑢′𝑡

• Where 𝑣𝑡= 𝑦𝑡 + 𝑢′𝑡 /

• Moreover, by using the money demand

𝑖𝑡 =𝑦𝑡 − 𝑚𝑡 − 𝑝𝑡

• The implied nominal interest rate is

𝑖𝑡 =1

1 +

𝑘

𝑘=1𝐸𝑡 𝑚𝑡+𝑘 + 𝑢′𝑡

• Both the price level and the nominal interest rate are

uniquely determined

An exogenous path for the money supply

• Consider as example

𝑚𝑡 = 𝑚𝑚𝑡−1 + 𝑚,𝑡

• Assume for simplicity 𝑦𝑡 = 𝑟𝑡 = 0.

• Price response:

𝑝𝑡 = 𝑚𝑡 +𝑚

1 + 1 − 𝑚𝑚𝑡

• Result: large price response

• The above result gives a rather strange implication.

Empirically, we have 𝑚 > 0, that money growth is

positively correlated over time. Now for each unit

increase in 𝑚𝑡, we have a more than one unit increase in

𝑝𝑡, which contradicts the data remarkably: price

responds very, very slowly on the data

An exogenous path for the money supply

• Consider as example

𝑚𝑡 = 𝑚𝑚𝑡−1 + 𝑚,𝑡

• Assume no real shocks (𝑦𝑡 = 0).

• Price response:

𝑝𝑡 = 𝑚𝑡 +𝑚

1 + 1 − 𝑚𝑚𝑡

• Large price response

• Nominal interest rate response:

𝑖𝑡 =𝑚

1 + 1 + 𝑚𝑚𝑡

• Result: no liquidity effect (𝑚𝑡 and 𝑖𝑡)!!!