Classical monetary model - ComUniTedibartolomeo.comunite.it/courses/polmon/slides/classics.pdf · Classical monetary model Giovanni Di Bartolomeo [email protected]

Classical monetary model

Giovanni Di Bartolomeo

[email protected]

Sapienza University of Rome

Department of economics and law

Advanced Monetary Theory and Policy EPOS 2013/14

mailto:[email protected]

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


• Asset market clearing


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


• 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


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)


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


• 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


• 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


• 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



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)


• 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


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


• 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


• See forward solution slides

• Moreover, the price level is also uniquely determined

(given some initial value).


• 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



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


• 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 …


• … we obtain

𝑝𝑡 = 𝑚𝑡 +

1 +

𝑘


• 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


• Both the price level and the nominal interest rate are

uniquely determined


• 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


• 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)!!!

Documents

Classical monetary model - ComUniTedibartolomeo.comunite.it/courses/polmon/slides/classics.pdf · Classical monetary model Giovanni Di Bartolomeo [email protected]