Download Problem 1. The Mundell

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Problem 1. The Mundell-Tobin effect
Consider the following full-employment IS-LM model:

M

Y  C  Y  T , P   I  r   G

M  L r , Y


P


Y  Y
(1) Notice that investment depends on the real interest rate, r, but money demand
depends on nominal interest rate, r   . In addition, consumption depends on
real money balances. Explain why each of these assumptions is reasonable.
(2) In this model, changes in rate of inflation affect the real interest rate. Hence,
even though prices are fully flexible, money is not superneutral; a change in the
growth rate of money alters some real variables. Solve for ddr in this model.
What is its sign? Explain.
(3) Is this effect, which is called the Mundell effect important quantitatively? That is,
do you think ddr is large or small? In answering this question, you might need
some of the following:
interest elasticity of investment = 0.8;
interest elasticity of money demand = 0.1;
income elasticity of money demand = 1;
investment/GNP ratio = 0.15;
marginal propensity to consume out of income = 0.5;
money/GNP ratio = 0.1;
marginal propensity to consume out of wealth = 0.05.
(1)
Basic model assumptions
Investment should be carried on when it promises to bring some extra profit. This
happens whenever marginal profits (MPK, marginal product of capital) exceed
marginal costs (r, real interest rate). This reasoning suggests that investment depends
(negatively) on real interest rate r.
The demand for money, on the contrary, depends on the nominal interest rate, i,
because it is nominal interest rate which measures the profits of money deposits
(and, thus, the alternative costs of holding money).
The amount of consumption depends on the level of disposable income of consumers
– the higher the income the more they consume – and the costs of transfers which
accompany the process of consumption – the costs are less the more real money
balances the consumers have.
(2)
Change in the rate of growth of money
Differentiating the system of equations by  , yields:
page: 1/10
 Y C Y C W dI r
   Y   W   dr   0

r  L Y
 W L 


1 

i    Y 
 
 Y
0

 
The assumption is that government policy (G and T) is exogenous.
Letter W denotes real money balances, or wealth of the consumer: W 
This system may be easily solved for
r

M
P
.
:
r
1

0
I r

1
C L

W i
Why the sign? In the first item of the problem it was reasoned that I r should be
negative, C W – positive, and L i – negative. This means that the
denominator of the fraction is positive, and hence the derivative
(3)
r

is negative.
Quantitative estimation of the magnitude of effect
1
1
1


I 
 I r  I
I



0.8








r

r
I
r



r
r
   1 
  1 
   1 
 
C L 
L
C  L i  L 





0.05

0.1




 

W i 
i 


 W  i L  i 
1
1
1
0.8Ii 
I GNP 
0.15 
1



  1 
   1  160
   1  160 0.1    241  0.004
L
GNP
 0.005 Lr 


The derivative is taken at point   0 , where i  r . If the rate of inflation is positive
(   0 ) then i  r and Mundell effect is even smaller in magnitude.
So, in ordinary situations, Mundell effect is negligible.
But if the society experiences deflation (for example, if Fisher was successful in
implementation of his rule i  0 ), then r  1 and effect is quite significant. In the
case of even higher deflation, it may even become positive, but it is very unlikely to
happen.
page: 2/10
Problem 2. Expectations, monetary policy,
and the business cycle.
Assume an economy, which is described by the two equations:
e

 yt    pt  pt 


 yt  mt  pt
where the first equation is aggregate supply equation and the second equation is
aggregation demand equation. Here yt , mt , and pt are (logarithms of) output, nominal
money, and prices; pte is time t  1 expectation of time t prices. Also, there is a monetary
authority, who can use open-market operations to conduct monetary policy. Assume that
nominal money follows
mt   mt 1   t ,
where  is the policy choice of the authority. In particular,   0 means that the
authority acts to keep nominal money as stable as possible;   1 means that the
authority does not intervene and lets for unrestricted drifts in money. The shocks  t are
identically and independently distributed with mean zero and variance  2 . One can
interpret these shocks as being the result of fluctuations in inside (i.e. generated by the
banking system) money, which is beyond the grasp of policy makers. The policy objective
is to minimize unconditional variance of output in every period t , i.e. to dampen the
business cycle.
(1) Assume that individuals who live in that economy are smart and form rational
expectations,
pte  Et 1 pt .
What is the St. Louis equation of the monetary authority? Solve for the
unconditional variance of output as a function of model parameters. What is the
optimal (i.e. variance minimizing) value of policy  ?
(2) Assume that there is a sudden shift in expectation formation, so that individuals
become stupid and form static expectations,
pte  pt 1 .
What is the St. Louis equation now? Is it different from that in part (1)? Give an
intuition behind your answer.
(3) Solve for output as a function of current and past shocks to money. What is the
unconditional variance of output? What value of  is optimal now? Does the
optimal policy under static expectations allow to achieve a greater stability of
output compared to that under rational expectations? Why or why not? Explain.
(4) Assume now that a bit of common sense gets back into the heads of these people,
so that they start forming adaptive expectations,
pte  pte1    pt 1  pte1  ,
page: 3/10
where  , 0    1 , is the fraction of surprise that passes onto
expectations (assume that  cannot be influenced by monetary authority).
What is the St. Louis equation in that case? How is it different from above?
Explain your answer.
(5) Redo part (3) for the case of adaptive expectations.
(6) Suppose now that the authority can manipulate public opinion about how much
of innovation to prices is permanent (i.e. can influence  ). What is the message
the authority will try to convince the public? What will be the corresponding
optimal monetary policy? Give an intuitive explanation as a part of your answer.
Would that policy yield more output stability that could be achieved under
rational expectations?
The evolution of money stock:
mt   mt 1   t   t    t 1   mt 2   ...   t   t 1   2 t 2  ...   k  t k  ...
The evolution of prices:
m

mt  pt   pt   pte
pt  t 
pte

1  1 
The level of output:
m


yt  mt  pt  mt  t 
pte 
 mt  pte 
1  1 
1 
(1)
Rational expectations
The rational expectations are: pte  Et 1 pt . Then
mt
m



pte  t 
Et 1 pt
1  1 
1  1 
E m

1
Et 1 pt  t 1 t 
Et 1 Et 1 pt 
 Et 1mt   Et 1 pt 
1  1 
1 
Et 1 pt  Et 1mt  Et 1   mt 1   t    mt 1  mt   t
pt 

 mt  pet     mt  Et 1 pt     t
1 
1 
1 

The St. Louis equation: yt 
 mt   mt 1 
1 
The unconditional variance of the level of output is:
2
2
 
   
  
Var yt  Var 
t   
Var





t
 1    1  
 1  
This variance does not depend on the value of policy  , so the authority is free to
choose it whatever they like.
yt 
(2)
Static expectations
Assume that the individuals not only become stupid in a moment, but also forget
about the time of their being intelligent – so that when they are forming their
page: 4/10
expectations in a backward-looking manner, they think that they were always
stupid and their past decisions where due to static expectations, not due to rational
expectations model.
Static expectations: pte  pt 1 .
The evolution of output:


  mt 1




yt 
mt 

pt  2  
mt 
mt 1 
 mt  pt 1  

1 
1 
1  1  1 
 1 
1   2


    mt  2


pt 3   ...
 
1


1


1



 

The St. Louis equation:
2
k

1 

  
  
yt 
mt 1  
m

...

  mt 
 t 2

 mt  k  ... 
1  
1 
 1  
 1  

The equation differs from that in part (1). In the case of rational expectations the
individuals have the complete information about the situation in the economy, and
do their best at predicting the level of prices, output and money growth. Thus, the
only difference in the level of yt comes out of “surprise” – the unexpected shocks in
money demand. So, the level of output depends only on the magnitude on that
“surprise” and neglects all the other parameters of the economy.
In the case of static expectations, they are formed, on the contrary, by the prices in
the previous period. But the previous prices were depended on previous
expectations, and so on. As a result, the current level of output depends on all of the
previous parameters of the economy; in particular, all the shocks in the previous
history do contribute to the variance of the current output.
2
(3)
Output as a function of shocks
Recall that mt   t   t 1   2 t  2  ...   k  t  k  ...
Then
2
k

1 

  
  
yt 

m

m

m

...

 t

 t 2

 mt  k  ...  
t 1
1  
1 
1  
1  

1
1





 t 1 
 t 1  ...  
 t   t 1   2 t  2  ... 

1 
1  1 
1 

2

1   


  t  2  ...  
1  1  

2

 2
1 
 


   
 
 t    
  t 1    
   t  2  ...
1  
1  
1 

1   


The k-th member of this series is:


   k 2
   
k
 k 1  
   ...  
  1    
1 
 1  
 1  
2
k   k 
k
page: 5/10
2
k







 
  1 

  ...    1      

 
  1      1    
k
k 1
k 1




 
 
k  
1 







 
 1     
 1    
  k 1    
  k 1    1 



 1      

1




 1 


  k    1 1       k 
1

  







Finally,
k
k

       1 1      1 

yt 
 t  k 


1   k 0 
    

The variance is (since all  t are independent and identically distributed):
  
Var yt  

1  
  k    1 1       k
1







k 0 
2 
2
2
 


  
 
  1           

1   2
1
1 
2
2


     1 1   

2


1
1





1     1   2   2
1  2


 

2    1

1  


  1 







 
        1   1     1  2        
2
2
2


2       
2 2 2
 
 
 1    1    1       1    1   1     
The minimum of the variance is at the point
  12
1 ,

   2
  12

1,
 2 3 2
1


  ,
 1  2 
2
Var yt  
2
1
 
2

1  2  ,
2
This variance is always bigger than that in point (1). This is not a surprise: the cases
of static and rational expectations have identical first members in the equation of
yt  t ,  t 1 ,  t  2 ,... , but the latter has no other members, while the former does,
which increase its variance.
page: 6/10
(4) Adaptive expectations
The additional assumption is the same as in point (2).
Adaptive expectations: pte  pte1    pt 1  pte1 
pte   pt 1  1    pte1 

1 
 

mt 1  
 1    pte1
1 


  e

mt 1  1 
 pt 1
1 
 1  
The evolution of output will be:



  
  e 

yt 
mt 
pte 
mt 
mt 1  1 
 pt 1  

1 
1 
1 
1  1 
 1  

pte 



 
  e


mt 
mt 1 
mt 
mt 1 
1 
 pt 1 
2
1 
1   1  
1 
1   
1   2
 
  
  e 

mt  2  1 
1 

 pt  2   ...
1   1  1 
1




The St. Louis equation:
2

 

 
 
 
 
yt 
m

m

1

m

1

 t

 t 2

 mt  2  ...
t 1
1  
1 
1   1  
1   1  


(5)
Output as a function of shocks (adaptive expectations)
Substitute series mt   t   t 1   2 t  2  ...   k  t  k  ... :

 

 
 
 
 
mt 1 
 mt 
1 
 mt  2 
1 
 mt  2  ... 
1  
1 
1   1  
1   1  



 t   t 1   2 t  2   3 t 3  ...     t 1   t  2   2 t 3  ... 
1 
1  1 
2

 
 

 
 

1 
   t  2   t  3  ... 
1 
   t  3  ...  ... 
1  1   1  
1  1   1  
2
yt 


 
 2 
 
 


  t 1    
1 
   t  2  ...
 t    
1  
1  1   1   




The k-th member of this series is:


1 
k   k 

 
  k 2
 
 
 k 1 
1 
 
1 

1 
1   1  
1   1  
k 1
 k 
 1     1    

 1    
 k 1 1 

 ...  


1 
  1      1    
  1    
 k 
  1     k   1     

 k 1 1  
  1 

1 
   1        1    
2

k 1



page: 7/10
 
 1    

 k    1 1      1 


1


1





k
  

   1 1     
   1 1     
So,
k
 

  

k
yt 
     1 1      1 
   t k

1        1 1       k  0 
 1   
k
k
k 
The variance is:
2
k
2
 k
   



2       1 1       1 




  
 1     
1 
Var yt  

 
 




 1    k  0 
   1 1    
     1 1     
1
1
2
2


    1 1    1   2  2    1 1    1     1       


 2

1   2
  

2
2
1     1     



2    1



  1 












    1 1       1   1   1      2 1     
2




  1   1    1       2 1   






1    1     
2 1      
    1 1        1  
2
 1    1   2    2   1   





 1     1    1      


2 1     

2
2
2
2 1    1     2 1     2     2    1    1      2  2     2 







    1 1     
(
2
1     1    1        2 1      
2
 1    1   2    2  1   




)

1     1    1        2 1           1 1      
2
2
2
2


 2 1    1      1     4  4    2




    1 1       1     1    1        2 1      
2
2  1    1      






    1 1       1     1    1        2 1      
2
2 2 2
 2  2    1    1     1      
Finally,
2 2 2
Var yt 
 2  2    1       2 1      

page: 8/10

Authority maximizes this expression by  , and chooses

1
 2 1     ,
 
1,

 8 2 1      2
 ,

3
  2  2   
Var yt  
2

2
   2  2     ,
(6)
3
2
3
   1
2
   1
3
2
   1
3
2
   1
Optimal choice of λ
The authority can try to convince the public about the level of  they should have in
order to minimize the variation of output.
This target level of  can be found from a minimization problem:
Var yt  min

This function has many potential solutions:
1)   2  2     0

  1   1
  32  32  1  0  1    always
Var yt
 1

2
2
1   2
3
2
 1        2  2     3  2  2    1     
2) 


3
 2  2   6
  2  2    
3  3  3  2  2  
1    2


0

4
 2  2   
 2  2   4
1 


2
  34  34   1    1  always
8 2 12 1    2 32  2
Var yt  1 
 
2
3
2
27 
27



2
1


1


8
2
3)   1   
3
Var yt
4)   1
  23 1 

2
9 2
2


2
2
2
4


1


1


8


1


3
3
page: 9/10
 2 3
1


 ,
 1  2 
2
Var yt  1  
2

1


1  2 ,
2
The first of these solutions looks wonderfully, but it does not satisfy the condition of
0    1 (the last does not either). Out of two solutions left, number two is better.
So, the authority should convince the public to use
1 
 
2
Then the government will choose
1
 
2
32  2 2

Var yt 
27 1   2
This solution is the best what consumers can achieve under the adaptive expectations
model. Still, it is worse than in the case of rational expectations (in order to have the
same efficiency as with the rational expectations, the consumers should use adaptive
expectations with   1   , which is beyond their mental capabilities).
Why the rational expectation solution is better? Rationality means that consumers
guess the future level of prices and money stock using every bit of information they
have. In particular, they know past levels of prices and money stocks, and thus can
use this information in case they need it. So, the adaptive guess is only one of the
possibilities which have a rational consumer. She can use any other method of
guessing if she thinks it would result in a better accuracy of predictions. Therefore,
rational expectations cannot give worse result.
This passage answers the questions in 3, 5, and 6 items.
page: 10/10