![Page 1: Classical monetary model - ComUniTedibartolomeo.comunite.it/courses/polmon/slides/classics.pdf · Classical monetary model Giovanni Di Bartolomeo giovanni.dibartolomeo@uniroma1.it](https://reader031.vdocument.in/reader031/viewer/2022022514/5af310467f8b9aa916912da4/html5/thumbnails/1.jpg)
Classical monetary model
Giovanni Di Bartolomeo
Sapienza University of Rome
Department of economics and law
Advanced Monetary Theory and Policy EPOS 2013/14
![Page 2: Classical monetary model - ComUniTedibartolomeo.comunite.it/courses/polmon/slides/classics.pdf · Classical monetary model Giovanni Di Bartolomeo giovanni.dibartolomeo@uniroma1.it](https://reader031.vdocument.in/reader031/viewer/2022022514/5af310467f8b9aa916912da4/html5/thumbnails/2.jpg)
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.
![Page 3: Classical monetary model - ComUniTedibartolomeo.comunite.it/courses/polmon/slides/classics.pdf · Classical monetary model Giovanni Di Bartolomeo giovanni.dibartolomeo@uniroma1.it](https://reader031.vdocument.in/reader031/viewer/2022022514/5af310467f8b9aa916912da4/html5/thumbnails/3.jpg)
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
![Page 4: Classical monetary model - ComUniTedibartolomeo.comunite.it/courses/polmon/slides/classics.pdf · Classical monetary model Giovanni Di Bartolomeo giovanni.dibartolomeo@uniroma1.it](https://reader031.vdocument.in/reader031/viewer/2022022514/5af310467f8b9aa916912da4/html5/thumbnails/4.jpg)
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
![Page 5: Classical monetary model - ComUniTedibartolomeo.comunite.it/courses/polmon/slides/classics.pdf · Classical monetary model Giovanni Di Bartolomeo giovanni.dibartolomeo@uniroma1.it](https://reader031.vdocument.in/reader031/viewer/2022022514/5af310467f8b9aa916912da4/html5/thumbnails/5.jpg)
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
![Page 6: Classical monetary model - ComUniTedibartolomeo.comunite.it/courses/polmon/slides/classics.pdf · Classical monetary model Giovanni Di Bartolomeo giovanni.dibartolomeo@uniroma1.it](https://reader031.vdocument.in/reader031/viewer/2022022514/5af310467f8b9aa916912da4/html5/thumbnails/6.jpg)
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.
![Page 7: Classical monetary model - ComUniTedibartolomeo.comunite.it/courses/polmon/slides/classics.pdf · Classical monetary model Giovanni Di Bartolomeo giovanni.dibartolomeo@uniroma1.it](https://reader031.vdocument.in/reader031/viewer/2022022514/5af310467f8b9aa916912da4/html5/thumbnails/7.jpg)
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
![Page 8: Classical monetary model - ComUniTedibartolomeo.comunite.it/courses/polmon/slides/classics.pdf · Classical monetary model Giovanni Di Bartolomeo giovanni.dibartolomeo@uniroma1.it](https://reader031.vdocument.in/reader031/viewer/2022022514/5af310467f8b9aa916912da4/html5/thumbnails/8.jpg)
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
![Page 9: Classical monetary model - ComUniTedibartolomeo.comunite.it/courses/polmon/slides/classics.pdf · Classical monetary model Giovanni Di Bartolomeo giovanni.dibartolomeo@uniroma1.it](https://reader031.vdocument.in/reader031/viewer/2022022514/5af310467f8b9aa916912da4/html5/thumbnails/9.jpg)
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 −
![Page 10: Classical monetary model - ComUniTedibartolomeo.comunite.it/courses/polmon/slides/classics.pdf · Classical monetary model Giovanni Di Bartolomeo giovanni.dibartolomeo@uniroma1.it](https://reader031.vdocument.in/reader031/viewer/2022022514/5af310467f8b9aa916912da4/html5/thumbnails/10.jpg)
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!!!
![Page 11: Classical monetary model - ComUniTedibartolomeo.comunite.it/courses/polmon/slides/classics.pdf · Classical monetary model Giovanni Di Bartolomeo giovanni.dibartolomeo@uniroma1.it](https://reader031.vdocument.in/reader031/viewer/2022022514/5af310467f8b9aa916912da4/html5/thumbnails/11.jpg)
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…
![Page 12: Classical monetary model - ComUniTedibartolomeo.comunite.it/courses/polmon/slides/classics.pdf · Classical monetary model Giovanni Di Bartolomeo giovanni.dibartolomeo@uniroma1.it](https://reader031.vdocument.in/reader031/viewer/2022022514/5af310467f8b9aa916912da4/html5/thumbnails/12.jpg)
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 −
![Page 13: Classical monetary model - ComUniTedibartolomeo.comunite.it/courses/polmon/slides/classics.pdf · Classical monetary model Giovanni Di Bartolomeo giovanni.dibartolomeo@uniroma1.it](https://reader031.vdocument.in/reader031/viewer/2022022514/5af310467f8b9aa916912da4/html5/thumbnails/13.jpg)
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 −
![Page 14: Classical monetary model - ComUniTedibartolomeo.comunite.it/courses/polmon/slides/classics.pdf · Classical monetary model Giovanni Di Bartolomeo giovanni.dibartolomeo@uniroma1.it](https://reader031.vdocument.in/reader031/viewer/2022022514/5af310467f8b9aa916912da4/html5/thumbnails/14.jpg)
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?
![Page 15: Classical monetary model - ComUniTedibartolomeo.comunite.it/courses/polmon/slides/classics.pdf · Classical monetary model Giovanni Di Bartolomeo giovanni.dibartolomeo@uniroma1.it](https://reader031.vdocument.in/reader031/viewer/2022022514/5af310467f8b9aa916912da4/html5/thumbnails/15.jpg)
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)
![Page 16: Classical monetary model - ComUniTedibartolomeo.comunite.it/courses/polmon/slides/classics.pdf · Classical monetary model Giovanni Di Bartolomeo giovanni.dibartolomeo@uniroma1.it](https://reader031.vdocument.in/reader031/viewer/2022022514/5af310467f8b9aa916912da4/html5/thumbnails/16.jpg)
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);
![Page 17: Classical monetary model - ComUniTedibartolomeo.comunite.it/courses/polmon/slides/classics.pdf · Classical monetary model Giovanni Di Bartolomeo giovanni.dibartolomeo@uniroma1.it](https://reader031.vdocument.in/reader031/viewer/2022022514/5af310467f8b9aa916912da4/html5/thumbnails/17.jpg)
Outcomes
• Try to modify the Dynare code to replicate the figure
![Page 18: Classical monetary model - ComUniTedibartolomeo.comunite.it/courses/polmon/slides/classics.pdf · Classical monetary model Giovanni Di Bartolomeo giovanni.dibartolomeo@uniroma1.it](https://reader031.vdocument.in/reader031/viewer/2022022514/5af310467f8b9aa916912da4/html5/thumbnails/18.jpg)
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
![Page 19: Classical monetary model - ComUniTedibartolomeo.comunite.it/courses/polmon/slides/classics.pdf · Classical monetary model Giovanni Di Bartolomeo giovanni.dibartolomeo@uniroma1.it](https://reader031.vdocument.in/reader031/viewer/2022022514/5af310467f8b9aa916912da4/html5/thumbnails/19.jpg)
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
![Page 20: Classical monetary model - ComUniTedibartolomeo.comunite.it/courses/polmon/slides/classics.pdf · Classical monetary model Giovanni Di Bartolomeo giovanni.dibartolomeo@uniroma1.it](https://reader031.vdocument.in/reader031/viewer/2022022514/5af310467f8b9aa916912da4/html5/thumbnails/20.jpg)
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
![Page 21: Classical monetary model - ComUniTedibartolomeo.comunite.it/courses/polmon/slides/classics.pdf · Classical monetary model Giovanni Di Bartolomeo giovanni.dibartolomeo@uniroma1.it](https://reader031.vdocument.in/reader031/viewer/2022022514/5af310467f8b9aa916912da4/html5/thumbnails/21.jpg)
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
![Page 22: Classical monetary model - ComUniTedibartolomeo.comunite.it/courses/polmon/slides/classics.pdf · Classical monetary model Giovanni Di Bartolomeo giovanni.dibartolomeo@uniroma1.it](https://reader031.vdocument.in/reader031/viewer/2022022514/5af310467f8b9aa916912da4/html5/thumbnails/22.jpg)
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
![Page 23: Classical monetary model - ComUniTedibartolomeo.comunite.it/courses/polmon/slides/classics.pdf · Classical monetary model Giovanni Di Bartolomeo giovanni.dibartolomeo@uniroma1.it](https://reader031.vdocument.in/reader031/viewer/2022022514/5af310467f8b9aa916912da4/html5/thumbnails/23.jpg)
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)
![Page 24: Classical monetary model - ComUniTedibartolomeo.comunite.it/courses/polmon/slides/classics.pdf · Classical monetary model Giovanni Di Bartolomeo giovanni.dibartolomeo@uniroma1.it](https://reader031.vdocument.in/reader031/viewer/2022022514/5af310467f8b9aa916912da4/html5/thumbnails/24.jpg)
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
![Page 25: Classical monetary model - ComUniTedibartolomeo.comunite.it/courses/polmon/slides/classics.pdf · Classical monetary model Giovanni Di Bartolomeo giovanni.dibartolomeo@uniroma1.it](https://reader031.vdocument.in/reader031/viewer/2022022514/5af310467f8b9aa916912da4/html5/thumbnails/25.jpg)
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)
![Page 26: Classical monetary model - ComUniTedibartolomeo.comunite.it/courses/polmon/slides/classics.pdf · Classical monetary model Giovanni Di Bartolomeo giovanni.dibartolomeo@uniroma1.it](https://reader031.vdocument.in/reader031/viewer/2022022514/5af310467f8b9aa916912da4/html5/thumbnails/26.jpg)
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)
![Page 27: Classical monetary model - ComUniTedibartolomeo.comunite.it/courses/polmon/slides/classics.pdf · Classical monetary model Giovanni Di Bartolomeo giovanni.dibartolomeo@uniroma1.it](https://reader031.vdocument.in/reader031/viewer/2022022514/5af310467f8b9aa916912da4/html5/thumbnails/27.jpg)
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).
![Page 28: Classical monetary model - ComUniTedibartolomeo.comunite.it/courses/polmon/slides/classics.pdf · Classical monetary model Giovanni Di Bartolomeo giovanni.dibartolomeo@uniroma1.it](https://reader031.vdocument.in/reader031/viewer/2022022514/5af310467f8b9aa916912da4/html5/thumbnails/28.jpg)
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).
![Page 29: Classical monetary model - ComUniTedibartolomeo.comunite.it/courses/polmon/slides/classics.pdf · Classical monetary model Giovanni Di Bartolomeo giovanni.dibartolomeo@uniroma1.it](https://reader031.vdocument.in/reader031/viewer/2022022514/5af310467f8b9aa916912da4/html5/thumbnails/29.jpg)
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
![Page 30: Classical monetary model - ComUniTedibartolomeo.comunite.it/courses/polmon/slides/classics.pdf · Classical monetary model Giovanni Di Bartolomeo giovanni.dibartolomeo@uniroma1.it](https://reader031.vdocument.in/reader031/viewer/2022022514/5af310467f8b9aa916912da4/html5/thumbnails/30.jpg)
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);
![Page 31: Classical monetary model - ComUniTedibartolomeo.comunite.it/courses/polmon/slides/classics.pdf · Classical monetary model Giovanni Di Bartolomeo giovanni.dibartolomeo@uniroma1.it](https://reader031.vdocument.in/reader031/viewer/2022022514/5af310467f8b9aa916912da4/html5/thumbnails/31.jpg)
Productivity shock
• Check the outcomes of a monetary shock (note that Dynare does not plot
the IRF of variables that do not change)
![Page 32: Classical monetary model - ComUniTedibartolomeo.comunite.it/courses/polmon/slides/classics.pdf · Classical monetary model Giovanni Di Bartolomeo giovanni.dibartolomeo@uniroma1.it](https://reader031.vdocument.in/reader031/viewer/2022022514/5af310467f8b9aa916912da4/html5/thumbnails/32.jpg)
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
![Page 33: Classical monetary model - ComUniTedibartolomeo.comunite.it/courses/polmon/slides/classics.pdf · Classical monetary model Giovanni Di Bartolomeo giovanni.dibartolomeo@uniroma1.it](https://reader031.vdocument.in/reader031/viewer/2022022514/5af310467f8b9aa916912da4/html5/thumbnails/33.jpg)
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 …
![Page 34: Classical monetary model - ComUniTedibartolomeo.comunite.it/courses/polmon/slides/classics.pdf · Classical monetary model Giovanni Di Bartolomeo giovanni.dibartolomeo@uniroma1.it](https://reader031.vdocument.in/reader031/viewer/2022022514/5af310467f8b9aa916912da4/html5/thumbnails/34.jpg)
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
![Page 35: Classical monetary model - ComUniTedibartolomeo.comunite.it/courses/polmon/slides/classics.pdf · Classical monetary model Giovanni Di Bartolomeo giovanni.dibartolomeo@uniroma1.it](https://reader031.vdocument.in/reader031/viewer/2022022514/5af310467f8b9aa916912da4/html5/thumbnails/35.jpg)
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
![Page 36: Classical monetary model - ComUniTedibartolomeo.comunite.it/courses/polmon/slides/classics.pdf · Classical monetary model Giovanni Di Bartolomeo giovanni.dibartolomeo@uniroma1.it](https://reader031.vdocument.in/reader031/viewer/2022022514/5af310467f8b9aa916912da4/html5/thumbnails/36.jpg)
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)!!!