Download A two-period closed economy with sticky prices

Politicas macroeconomicas, handout, Miguel Lebre de Freitas ([email protected])
10. The two-period economy with sticky prices
Index:
10. The two-period economy with sticky prices............................................................1
9.1
Introduction........................................................................................2
9.2
Basic model........................................................................................2
9.2.1 Main assumptions ......................................................................................2
9.2.2 Equilibrium ................................................................................................4
9.3
The well functioning versus the Keynesian cases..............................6
9.3.1 The classical case.......................................................................................6
9.3.2 The Keynesian case....................................................................................7
9.3.3 The Keynesian cross ..................................................................................8
9.3.4 The saving investment locus......................................................................9
9.3.5 What do we mean by an exogenous interest rate? ...................................10
9.4
Fiscal policy .....................................................................................11
9.4.1 Increase in government expenditures today.............................................11
9.4.2 Anticipated increase in government expenditures ...................................12
9.4.3 Permanent increase in government expenditures.....................................12
9.5
Productivity slowdown ....................................................................13
9.5.1 Well functioning case ..............................................................................13
9.5.2 Keynesian case.........................................................................................14
9.5.3 Analysis using the IS curve......................................................................14
9.5.4 Animal spirits...........................................................................................15
9.6
Policy implications...........................................................................16
9.7
Conclusions......................................................................................17
9.7.1 What next? ...............................................................................................18
Further reading.............................................................................................18
Review questions and exercises...................................................................19
Review questions .................................................................................................19
Exercises ..............................................................................................................19
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Politicas macroeconomicas, handout, Miguel Lebre de Freitas ([email protected])
9.1
Introduction
A key insight of the Keynesian doctrine is that the price mechanism is too slow to
ensure the equilibrium in the money market each moment in time. When prices are sticky, as
it happens in the short term, the variable that clears the money market is the nominal interest
rate, not the price level. Then, because the interest rate is determined in the money market, an
alternative variable is required to ensure the equality between planned savings and planned
investment: this variable is real income.
In this note, we review this argument, in the context of a two-period closed economy
where the interest rate is exogenously given. Analytically, the approach is similar to that of a
small open economy, the difference being that the exogenous interest rate is determined in
the (omitted) money market, rather than in the international economy. Since the economy is
closed to financial trade, there can be no mismatches between savings and investment: in the
context of the closed economy, if the price (real interest rate) fails to clear the market for
goods and services, then quantity (income) has to do the job.
This note proceeds as follows. In Section 10.2, we present the basic model. In Section
10.3, we show how the model solves when prices are flexible and when prices are sticky. In
Sections 10.4 and 10.5, we discuss how the equilibrium changes in response to changes in the
exogenous fiscal and productivity parameters.
9.2
Basic model
9.2.1
Main assumptions
Consider a two-period closed economy, where output in the current period is pre-
determined, and output in the second period depends on current investment only (the capital
stock depreciates fully each year). In this economy, there is a government that purchases
goods from firms, financing these purchases with lump taxes.
The preferences of the representative household’s are as follows
U  lnC1  
lnC2 
1 
(10.1)
In this economy, the current (year 1) “natural” output is pre-determined,
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Q1  Q1n
(10.2)
Future output depends on current investment, according to:
Q2  z 2 F K 2 
(10.3).
Capital is assumed to depreciate fully each year. Also for simplicity, we rule out
financial market frictions: all domestic agents can borrow or lend any amount of output at the
prevailing interest rate. We retain however the assumption that the economy is closed to
capital flows1.
Firms maximize the net present value of investment:
V1   K 2 
z 2 F K 2 
1  r1
(10.4)
The optimal investment I1  K 2 is implicitly defined by the first order condition of
that problem:
z 2 FK K 2   1  r1
(10.5)
Implying:
  
I1  K 2  z 2 , r1 


(10.5a)
The government budget constraint is given by:
T1 
T2
G
 G1  2
1  r1
1  r1
(10.6)
The household maximizes utility (10.1) subject to the budget constraint:
C1 
C2
 1
1  r1
(10.7)
1
Although the assumption of a closed economy might look restrictive, one should remember that
economies are composed by tradable and non-tradable good sectors. Hence, the analysis below may be thought
as applying for at least some fraction of any open economy.
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Where life-time wealth is given by:
1  Q1  T1  V1 
T2
1  r1
(10.8)
The first order conditions of the household problem deliver the Euler equation
C2 1  r1

C1 1  
(10.9)
This condition, together with (10.7) and (10.8) determine the household’ optimal
consumption plan for the current period:
C1 
1 
2 

T2 
Q1  T1  V1 

1  r1 

(10.10)
Equation (10.10) implies that current consumption depends positively on current
output and on future productivity, and negatively on the interest rate and taxes. From (10.10),
taking the partially derivative in respect to Q1 , we obtain the marginal propensity to
consume:
C1 1  
1

Q1 2  
(10.11)
Combining (10.10) with (10.6) we see that in this framework, the Ricardian
equivalence holds: what matters for consumption is current and future government spending,
not the timing of tax collection. This is obviously a limitation of the model, arising from the
fact that we are postulating lump sum taxation.
Using (10.5.a) and (10.6), total absorption (or aggregate demand) in this economy
becomes:
A1  C1  I1  G1

1 
2 
z 2 F K 2   G2 
1


I1 z2 , r1   G1 
Q1 

1 r
 2 
(10.12)
This expression stresses the impact of future events on current spending decisions.
9.2.2
Equilibrium
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Equilibrium in the output market requires the equality between income Q1 and
absorption, that is,
     
Q1  A1  G1 , G2 , r1 ,  , z2 


(10.13)
Using (10.12), the definition of equilibrium (10.13) implies:
 z F K 2   G2 
Q1  1    2
  G1  I1 z 2 , r1 
1  r1


(10.14)
This equation summarizes the negative relationship between income and the interest
rate, holding fixed all other variables, such as government expenditures, investment and
productivity. This relationship is known as the IS curve, and is depicted in Figure 1. It is
negatively sloped, because when the interest rate increases, both consumer and investment
spending decline, determining a lower level of income.
Figure 1: The IS curve
This curve is labelled “IS” because it implies the equality between an economy’ total
savings and investment. Indeed, another way of writing (10.13) is:

   
 
S  Q1 , r ,  , T1 ,..   S1G  I1  z 2 , r , 






P
1
(10.15)
Where
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S1P  Q1  T1  C1 
Q1  T1
2 
S1G  T1  G1
 1   
T 
 V1  2 
 
1  r1 
 2   
(10.16)
(10.17)
A key feature in (10.15) is that private savings depend positively on current output.
This opens a channel through which aggregate savings may adjust endogenously to equal
investment, whenever the interest rate fails to do the job.
9.3
The well functioning versus the Keynesian cases
The model above can be solved in two different manners. In the classical case, current
output is at its natural level, and the endogenous variable is the interest rate. In the Keynesian
case, the interest rate is exogenous, and current output is exogenous.
9.3.1
The classical case
In the well functioning case, the real interest rate adjusts so as to assure that output is
always at its natural level. Defining the natural interest rate as r1n , the equilibrium is such
that:
     
Q1n  A1  G1 , G2 , r1n ,  , z2 


(10.13a)
In alternative, one can look at the same equilibrium, using the savings and investment
relationship:
   
  
S1P  Q1n , r1n ,..   S1G  I1  z 2 , r1n , 




(10.15a)
In any case, the solution for the endogenous interest rate is:
 z F K   G2 
C
1  r1n  1    2 n 2
  1    2
C1
 Q1  I1  G1 
(10.14a)
This is no more than the Euler equation: remember that we are in a closed economy,
so the real interest rate shall adjust to the level determined by the slope of the consumer’
indifference curve.
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The equilibrium in the well functioning case is described in Figure 2 by point 1. In
this point, given the level of output (described by the vertical aggregate supply LR), the real
interest rate is determined so as to equal output and spending. The interest rate in point 1 is
labelled the “natural real interest rate”, because it is the one that equals savings and
investment when output is at the “natural” level.
Figure 2: Equilibrium: short run and long run
9.3.2
The Keynesian case
Instead of assuming that the interest rate adjusts to clear the market of goods and
services, assume instead that prices are sticky. When prices are sticky, imbalances in the
money market cannot be cleared by the price level. In that case, the variable that adjusts to
clear the money market is the interest rate: for instance, when money supply exceeds money
demand, this gives rise to an excess demand for bonds, driving the price of bonds upwards
and by then the interest rates down.
Since with sticky prices, the interest is determined in the money market, the free
variable to clears the market for goods and services is current output. This is illustrated in
Figure 2 by point 1’. This equilibrium corresponds to the intersection of the exogenous
interest rate (defined by the horizontal schedule MP, that stands for “monetary policy”) with
the IS curve. Because in this case the the actual interest rate is higher than the natural interest
rate ( r1  r1n ),equilibrium output falls short the level of natural output, Q1  Q1n , giving rise to
a “negative output gap”.
Analytically, setting r1  r1 in (10.14), the equilibrium level of output, Q1 , is such
that:
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 z F K 2   G2 
Q1  1    2
  G1  I1
1  r1


(10.14b)
In the savings-investment framework, the equivalent condition is:
   
  
S1P  Q1 , r1 ,..   S1G  I1  z2 , r1 , 




9.3.3
(10.15b)
The Keynesian cross
Figure 3 offers an alternative representation of the equilibrium in the goods market,
based on equations (10.12) and (10.13). Equation (10.12) implies a slope equal to 1 in the
Q1 , A1 
space. Equation (10.12) also implies a positive relationship between aggregate
demand and current income, but the slope (the marginal propensity to consume, 7.11), is less
than 1.
The well functioning case is described by point 1. At point 1, output level is at the
natural level, Q1  Q1n , determined by the supply side of the economy. The key feature of the
model with flexible prices is that the interest rate (price of current output) adjusts
instantaneously to ensure that aggregate demand equals the existing aggregate supply (note
that changes in the real interest rate make the aggregate demand schedule in Figure 3 shift up
and down). Because the real interest rate moves so that aggregate demand equals full
employment, in this model the Say’s Law holds: supply determines its own demand.
The Keynesian equilibrium is described by point 1’. In that case, the real interest rate
is too high, and it fails to adjust to clear the goods market. Hence, aggregate demand will be
too low. Since firms will not produce unless there is demand, output will be only Q1  Q1n .
Thus, output reaches an equilibrium that falls short the full employment level.
In the Keynesian version of the model, current production responds passively to the
level of aggregate demand. Now, the Say’s law holds in reverse: it is the demand that
determines output, not the other way around. Since the price (interest rates) fails to adjust, the
equilibrium is met through changes in quantity (income).
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Figure 3: The Keynesian cross
9.3.4
The saving investment locus
The distinction between the well functioning case and the case with sticky prices can
also be described with the help of Figure 4. Suppose that initially the economy is at point 0.
In point 0, given the interest rate, planned savings (at the natural output level) exceed planned
investment by the distance 01’. In a classical world, the excess savings over investment
would cause the real interest rate to adjust downwards, until the equilibrium between savings
and investment was met, in point 1.
Now suppose that the real interest rate remains unchanged, at r1  r1n . In that case, the
only way of achieving the balance between savings and investment is through a fall in current
output. Graphically, the fall in output implies a shift of the savings curve to the left. The
Keynesian equilibrium is described by point 1’. In this point, the interest rate is too high,
giving rise to an output gap equal to Q1  Q1n  0 . Also note that, in point 1’ total investment
is less than in point 1: the price stickiness generating a recession today has implications for
future output as well.
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Figure 4: Savings and investment in the classical and in the Keynesian models
9.3.5
What do we mean by an exogenous interest rate?
In the analysis above, we abstract from changes in monetary policy, assuming that the
real interest rate remains constant. This assumption must be clarified.
In fact, an “unchanged monetary policy” can have different meanings. This includes,
an unchanged money supply, an unchanged interest rate, or an unchanged inflation target, for
instance.
In our discussion, we are not assuming that the money supply remains constant 2 .
Instead, we assume that the central bank is following some interest rate rule, and it will do
“whatever it takes” to keep the real interest rate on track. This includes changing the interbank lending rates, using open market operations or whatever policy the central bank may
find appropriate to achieve its target for the real interest rate.
2
The assumption that the central bank targets the money supply underlies the traditional IS-LM model.
In the IS-LM model, the LM curve describes the equilibrium in the money market and is positively sloped in the
output-interest rate locus, because with a constant money supply, an increase in output translates into a higher
demand for money for transaction purposes, creating an excess demand for money and by then an increase in the
interest rate.
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The assumption that the central bank targets the interest rate is more in accordance to
the real world facts: in today’s world, most central banks act to make the real interest rate
behaving in a certain way as a function of key variables, such as inflation and output3.
In Figure 2, the assumption that the central bank is targeting the interest rate
materializes with a horizontal curve, labelled MP (standing for “Monetary Policy) 4 . This
curve may shift upwards or downwards, depending on the central bank intentions5. By setting
the MP curve unchanged, we abstract from monetary policy considerations.
9.4
Fiscal policy
9.4.1
Increase in government expenditures today
An increase in government expenditures today shifts the IS curve rightwards, on a
one-to-one basis. As long as the interest rate remains unchanged, the impact on output will
also be one-to-one6:
Q1
1
G1
Thus, with sticky prices, a fiscal expansion fully financed with (current or future)
lump-sum taxes will be able to drive the economy to full employment. Most important, in
3
Note that the central bank does not determine the real interest rate. But as long as prices are sticky, the
central bank can adjust the inter-bank lending rate so as to induce the desired change in the real interest rate.
This is the tactics most central banks follow in our days.
4
Romer, D., 2000. Keynesian macroeconomics without the LM curve. Journal of Economic
Perspectives, 14(2), 149-169.
5
In general, most central banks are concerned with inflation, so they set the real interest rate to increase
whenever inflation is foreseen to increase above target. In practice, central banks may also react to changes in
output, setting the real interest rate to decline when output is below potential, and vice-versa.
6
The particular case in which the additional government spending is financed with current taxes
( G1  T1 ) is known as the Haavelmo Theorem [T. Haavelmo: Multiplier Effects of a Balanced Budget, in:
Econometrica 13, 311-318, 1945].
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contrast to the well functioning case, private consumption is not crowded out ( C1  0 ): one
on hand, the increase in government spending, by inducing an expansion in current output,
Q1 , increases the household life-time wealth (10.8); on the other hand, the extra expenditure
will come along with higher current or future taxes, by the same current value; on balance
individual wealth does not change at al.
9.4.2
Anticipated increase in government expenditures
Equation (10.14b) also shows that an increase in future government expenditures has
a deflationary impact on current output:
Q1
1 

G2
1  r1
The reason is that government expenditures impact negatively on private wealth,
reducing consumption today. Whether consumption will rise more or less than proportionally,
it depends on how the interest rate relates to the rate of time preference.
9.4.3
Permanent increase in government expenditures
When government expenditures increase permanently, the government budget
constraint changes according to:
T1 

T2
1 

 G1 1 
1  r1
 1  r1 
The multiplier becomes:
Q1
G1

G1  G2
r1  
1  r1
In this case, the IS curve may shift up or down, depending on whether private
consumption falls more or less than the increase in government spending. This, in turn,
depends on the relative size of the interest rate versus the rate of time preference.
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9.5
Productivity slowdown
In this section, we examine the implications of a fall in the productivity parameter,
z 2 . In terms of our model, the first impact will be on investment: the fact that productivity
declines implies that firms will find it profitable to invest less, for any given level of the
interest rate. A lower investment, in turn, will reduce the household life-time wealth, so
current consumption will decline too. Given the current output, the households’ savings
schedule expands to the right.
9.5.1
Well functioning case
The well functioning case is described in Figure 5, with a move from point 0 to point
1. The fall in productivity causes investment to decline and savings to increase, impacting
negatively on the (natural) interest rate.
The fact that the natural interest rate declines implies that current consumption
becomes less expensive and creates incentives for investment, despite the productivity fall.
This effect is just the enough to assure that aggregate demand remains unchanged, at the level
determined by the current natural output (the Say’s law holds).
Figure 5: productivity slowdown: classical case
Whether total investment and savings increase or decrease, it depends on the utility
function. In the particular case we are dealing with, the move in the interest rate exactly
offsets the shifts in the curves, so the final levels of savings and investment are the same as
before the shock. Note however that this is not a general case.
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9.5.2
Keynesian case
Figure 6 describes the alternative case in which the real interest rate fails from
declining.
Point 1 cannot occur, because that would imply an excess savings over investment or,
what is the same, an excess supply in the goods market. In order for savings to match
investment at the prevailing interest rate, output has to decline, moving the economy to point
2: the saving schedule adjusts leftwards to meet the new investment schedule at the
exogenous interest rate.
Figure 6: Productivity slowdown: Keynesian case
Figure 6 also illustrates an important proposition of the Keynesian thinking, labelled
as “the Paradox of thrift”: from point 0 to point 1, the fact that the households’ lifetime
wealth declined with the productivity shock lead agents to increase their planned savings.
However, because this caused an excess of savings over investment - or insufficient demand current output declined, reducing the total savings. In the end, individual attempts to save
more failed to produce higher savings at the aggregate level.
9.5.3
Analysis using the IS curve
Figure 7 explains what happens in terms of the IS curve. Initially, the economy lies in
point 0, with full employment and the interest rate at its natural level. Point 1 depicts the
classical case, in which the interest rate adjusts downward to its new natural level, and full
employment is maintained. Point 1’ describes the Keynesian case in which the interest rate
fails from declining.
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Figure 7: productivity slowdown and the IS curve
9.5.4
Animal spirits
In the section above, we discussed the case in which future productivity is expected to
slow down. Proponents of the neo-classical school believe that agents’ expectations regarding
the future are in general well grounded on fundamentals and economic signs that people
accurately interpret. Forecasting errors are possible, but people learn with past mistakes, so
systematic expectation errors cannot occur.
Proponents of the Keynesian school content instead that forecasting errors may be
large, because expectations are not necessarily rational. Expectations are driven by
psychological factors and changes in collective mood ( “animal spirits”). Hence, expectations
may deviate significantly from reality. Changes in collective mood give rise to upward and
downward swings in aggregate demand, which, together with price stickiness, will cause
output to fluctuate around its natural level, even when no fundamental reason underlies the
changes in expectations.
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Referring to Figure 7, the economy may move from full employment to depression
only because people became pessimistic regarding the future7.
9.6
Policy implications
The Keynesian school advocates that, if the market fails to do drive the economy to its
natural rate in a reasonable time, the government should help.
A first avenue to achieve this would be to use monetary policy. Referring to Figure 7,
if the central bank, by expanding the money supply, successfully achieved a reduction in the
interest rate to its new natural level, then the economy would approach the full employment.
In real life, however, this may not be that simple. First, because the central bank may
be constrained by a mandate that cares only with price stability. Second, even if the central
bank wanted the real interest rate to fall, the central bank influences the short term nominal
interest rates, while what matters for spending decisions are long term real interest rates.
In some contexts, the central bank may find it hard to drive the real interest rates
down below a given level. This is the case of the zero lower bound, in which the nominal
interest rate reaches the value of zero and inflation expectations are to small to allow further
declines in the real interest rate.
In that case, the alternative instrument is a fiscal expansion. In terms of our model, an
increase in government spending today or a decrease in government spending tomorrow will
move the IS curve to the right, causing output to expand, for each level of the interest rate.
This is illustrated in Figure 8, by a move from 1’ to 28.
7
Keynes argued that investment spending collapsed during the Great Depression because of an
exogenous decline in the entrepreneurs’ expectations regarding future profitability.
8
Note that in the neoclassical model, a fiscal expansion departing from point 1 would result in an
upward increase in the equilibrium interest rate, from 1 to 2, without any impact on output.
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Figure 8: Fiscal expansion
Of course, the claim that governments should use fiscal and monetary policies to
stabilize the economy is not free of criticism. Sceptics of macroeconomic stabilization
policies contend that private shocks to the economy are hard to identify. Identification lags
and intervention lags reduce the effectiveness of stabilization. On the other hand, activist
policies are subject to pressures and electoral interests, so policymakers with discretionary
power may be tempted to move away from the public interest.
9.7
Conclusions
In this chapter, we departed from the well functioning two-period economy, by
assuming that the real interest rate does not evolve so as to clear the market for loanable
funds. The rationale is that prices are sticky and the nominal interest rate is determined in the
money market. If, in plus, the central bank is committed to some interest rate rule, then the
real interest rate can be seen as exogenous to the goods market.
A real interest rate lying above the level that clears savings and investments implies
an excess savings over investment. In an open economy, this gap can be met with a surplus in
the trade balance. In a closed economy, the excess supply of goods cannot be absorbed, so the
only solution for firms is to produce what is actually demanded. Thus, the economy will be
operating below its potential.
In contrast to the classical model, according to which more savings brings more
investment and thereby more income, in the Keynesian model an increase in desired savings
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Politicas macroeconomicas, handout, Miguel Lebre de Freitas ([email protected])
has a deflationary impact in the economy. For instance, the anticipation of a productivity fall
will depress current consumption and demand, causing a recession today. Because people
have sentiments, these swings may happen just because of expectations, even if the
underlying fundamentals are sound. The government can fix a low demand, either by
expanding the monetary policy so as to decrease the real interest rate, or by increasing
expenditures on a temporary basis.
9.7.1
What next?
When prices are sticky, the nominal interest is determined in the money market. This
gives the central bank the power to influence economic activity. If, for instance, the real
interest rate increases above its natural level, the central bank may adjust its policy tools to
drive down the short term interest rates and by then, to achieve a reduction in the long term
rates closer to the natural level.
Driving the long term real interest rate down may not be however that easy, especially
during financial crisis. First, the nominal interest rate may reach the zero lower bound,
making it hard to drive the real interest rate further down without inflation. Second, because,
in a context of high uncertainty, real interest rates in private contracts may depart
significantly from the interest rates set by the central bank in the money market. Third, in a
context of high uncertainty, lenders may refrain from lending, regardless the level of interest
rates. In the next handouts, we review these arguments.
Further reading
Andolfatto, D., Macroeconomic Theory and Policy, Chapter 7.
Sachs, J., Larrain, F., Macroeconomics in the global economy. Prentice Hall, 1993, Chapter
12.
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Politicas macroeconomicas, handout, Miguel Lebre de Freitas ([email protected])
Review questions and exercises
Review questions
Exercises
9.1.
(Endowment economy) Consider an economy where full employment
current and future output are initially given Q1n  Q2n  1000 . Further assume that in
this economy, the preferences of the representative consumer are given by
U  ln C1   ln C 2  . Initially, there are no government taxes and expenditures.
a) (Natural interest rate). From the household optimization problem, find out the
natural interest rate in this economy. Represent the equilibrium in the
Keynesian cross and find the expression of the IS curve.
b) (Dark future). Suppose that people expect instead future output to be
Q2n  800 . How much will be the natural interest rate in this case? Assuming
that the interest rate is free to move, explain what happens using the savings
schedule, the Keynesian cross and the IS curve.
c) (Keynesian case) Now consider the case in which the real interest rate cannot
fall below zero. Explain explain the adjustment of the economy referring to
the savings schedule, the Keynesian cross and the IS curve.
d) (Fiscal expansion) Departing from (c), examine the implications of a
government intervention amounting to G1  200 . Explain how the savings
schedule, aggregate expenditure and the IS curve will shift. Will it make a
difference if these expenditures are financed with current taxation of with
future taxation?
9.2.
(Investment economy) Consider an economy where natural output in the
current period is Q1n  1200 . Output in the second period depends on current
investment, according to Q2  z 2 K 20.5 . The preferences of the representative consumer
are given by U  ln C1   ln C 2  . Initially, there are no government taxes and
expenditures.
a) (Initial state). Suppose that z 2  40 . Find out the optimal investment,
consumption, and the natural interest rate in this economy. Depict the
equilibrium in the savings-investment schedule.
b) (Productivity slowdown). Examine the implications of a fall in expected
productivity to z 2  20 , considering two cases: (b1) the interest rate is free to
adjust. (b2) the interest rate cannot fall below zero. In particular, show what
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Politicas macroeconomicas, handout, Miguel Lebre de Freitas ([email protected])
happens in the saving-investment diagram, in the Keynesian cross and to the
IS curve.
c) (Paradox of thrift): Comparing (a) with (b2), explain the Paradox of thrift.
d) Returning to the initial case, with z 2  40 , examine the implications of a
n
n
positive transitory shock on current output, from Q1  1200 to Q1  1875 .
Again, compare the well functioning case with the case in which the real
interest rate cannot fall below zero. In particular, show what happens in the
saving-investment diagram, in the Keynesian cross and to the IS curve.
e) Departing from (d), explain how fiscal policy could be used to drive the
economy to full employment.
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