Download The two period production economy

Politicas macroeconomicas, handout Miguel Lebre de Freitas ([email protected])
The two period production economy
Index:
The two period production economy .............................................................................1
10.1
Introduction........................................................................................2
10.2
The Representative Agent Two Period Production Economy (RATPPE) 2
10.2.1 The representative firm’ problem ..............................................................3
10.2.2 The household problem..............................................................................3
10.2.3 The government budget constraint.............................................................5
10.3 Aggregate demand .............................................................................6
10.3.1 The IS curve...............................................................................................6
10.3.2 Shifts in aggregate demand........................................................................7
10.4 Aggregate supply ...............................................................................9
10.4.1 Labour market equilibrium ........................................................................9
10.4.2 Partial equilibrium: Shifts in labour supply and demand.........................10
10.4.3 The aggregate supply ...............................................................................12
10.4.4 Shifts in aggregate supply........................................................................12
10.5 The natural interest rate ...................................................................13
10.6
Adjustment to economic shocks ......................................................14
10.6.1 What happens when future productivity increases?.................................14
10.6.2 What happens when current productivity increases?...............................15
10.6.3 What happens when government consumption increases temporarily? ..17
10.7 The Heterogeneous Agent Two Period Production Economy (HATPPE) 18
10.8
The case with sticky prices ..............................................................18
Further reading.............................................................................................20
Review questions and exercises...................................................................20
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10.1
Introduction
In previous lectures, we addressed the cases of (a) a static economy with endogenous
labour, and (b) a two period economy with endogenous savings. In this note, we put the two
pieces together, to discuss a more general case in which agents face simultaneously the
choice between consumption and leisure and the choice between consumption and savings.
This is setup is, of course, more realistic, but also more complex. Because of this, and to keep
the model tractable, we make a set of simplifying assumptions. In particular, we consider a
model without capital and where the utility function is additively separable. The model cane
be labelled as the Representative Agent Two Period Production Economy (RATPPE). With
this model, we are able to sketch out some key proposition of the business cycle model
without worrying with the algebra too much.
This note is organized as follows. In Section 2, we present the main model and the
implied optimality conditions. In Section 3 we describe the aggregate demand, as a
relationship between spending and the real interest rate. In Section 4 we describe the
aggregate supply, as a positive relationship between output and the real interest rate. In
Section 5 we solve for the closed economy equilibrium, to find out the endogenous interest
rate. In Section 6 we use the model to analyse the impact of shifts in productivity and in
government spending. In Section 7, we introduce heterogeneity in the model, assuming that
some fraction of the households is constrained on borrowing. Finally, in Section 8 we analyse
the case in which the real interest rate fails to adjust to balance the goods market. In that case,
government intervention may help the economy to meet the full employment.
10.2
The Representative Agent Two Period Production Economy (RATPPE)
Consider an economy with a large number of equal firms and households. In this
economy, the government buys from firms the amount of output it desires, and finances these
purchases with a lump-sum tax T on households. Each consumer is endowed with a given
amount of time, h. Firms hire labour from households, paying a wage rate w, and produce a
final good that can be used for private consumption or for government spending. The
production function is linear in labour and there is no capital.
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10.2.1 The representative firm’ problem
Production takes place using a Ricardian production function:
Qt  zt N t
t=1,2
(1)
Where z is a productivity term, N is labour input, and t is a time suffix. Firms hire
working time from households at the wage rate w and sell output (numeraire). The
representative firm’ profits in period t are:
 t  Qt  wt N t .
(2)
Firms take the wage rate as given and choose N so as maximize profits, (2). This leads
to the following demands for labour in period 1 and in period 2:
w1  z1
(3)
w2  z 2
(4)
These conditions, together with (2) imply that profits are zero:
1   2  0
(5)
10.2.2 The household problem
Households are endowed with h units of time. Time can be split into working hours,
N, and leisure, l:
h  N t  lt
(6)
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The preferences of the representative consumer are assumed additively separable1:
U  ln C1  ln C2  ln l1  ln l2
(7)
Where ct denotes for private consumption in moment t, and lt for leisure in moment
t. Households maximize their life-time utility function subject to an inter-temporal budget
constraint of the form:
C1  w1l1 
C2  w2l2
hw   2  T2
 hw1  1  T1  2
1 r
1 r
(8)
The right hand side of (8) is the representative agent’ life-time wealth:
1  hw1  1  T1 
hw1   2  T2
1 r
(9)
The first order conditions of the household maximization problem imply:
C1
 w1
l1
(10)
C2
 w2
l2
(11)
C2
 1 r
C1
l1
w
 1  r  2 .
w1
12
(12)
(13)
The first two equations state that the marginal rate of substitution between
consumption and leisure each period shall be equal to the real wage in that period. Condition
1
This is the simplest possible formulation of the utility function to make the point. It is assumed that all
goods are weighted equally in the utility function. In Exercise 3, you will workout a slightly more general case.
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(12) states that the marginal rate of substitution between future consumption and present
consumption shall be equal to the real interest rate.
Condition (13) shows that the consumer can substitute inter-temporally not only
current consumption for future consumption but also current leisure for future leisure: an
increase in the interest rate increases the price of current leisure relative to the price of future
leisure, and all else equal the consumer optimally decides to postpone leisure. This effect is
labelled inter-temporal substitution of leisure. Inter-temporal substitution effects also arise
from variations of real wage over time. For instance, when the current wage rate rises relative
to the future wage rate, this means that current leisure becomes relatively more expensive,
inducing the household to postpone leisure and work more today. Note that this intertemporal substitution effect refers to the price incentive only: as long as changes in wages
come along with changes in life-time wealth, there will be income effects that may reinforce
or offset the inter-temporal substitution effects.
The demands for consumption and leisure in both periods are all implied by (10)-(12)
together with (8):
C1  w1l1 
C2
wl

 22  1
1 r 1 r
4
(14)
It is important to note that, whenever the price of one good declines (say w1 ), the
quantity demanded of that good ( l1 ) increases, but the share of that good in total spending
remains unchanged. This is a direct consequence of postulating a unit elasticity of
substitution between each two goods in the utility function. If the elasticity of substitution
between any two goods was different from one, a change in the price of any good would
deliver uncertain results in the demands for the other goods.
10.2.3 The government budget constraint
Assuming that lump-sum taxes are available, the government inter-temporal budget
constraint is given by:
G1 
G2
T
 T1  2
1 r
1 r
(15)
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10.3
Aggregate demand
10.3.1 The IS curve
Since this economy is closed to capital flows and there is no investment, aggregate
demand in this economy shall be equal to:
Q1d  C1  G1
(15)
Using (14), (9), (5) and (15), the demand for goods and services becomes:
Q1d 
1
 G1
4
or
Q1d 
1
z h  G2 
z1h  G1  2
 G1

4
1  r 
(16)
This is the equation for aggregate demand - or the IS curve. It establishes a negative
relationship between the demand for goods and services and the real interest rate: when the
real interest rate increases, the current value of the household’ life-time wealth declines.
Consequently, the consumer optimally decides to reduce its consumption expenditures2.
The negative relationship between the real interest rate and the level of aggregate
demand in (16) is displayed in figure 1.
2
In a more general case accounting for investment decisions, there is an additional reason why the
aggregate demand slopes down: when the real interest rate increases, the optimal capital stock decreases and
hence the firm reduced its current investment spending.
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Figure 1: The aggregate demand
10.3.2 Shifts in aggregate demand
From equation (16), it is easy to see how the aggregate demand shifts in response to
changes in the exogenous parameters.
Consider first the effect of a temporary increase in productivity. In this case, the
partial derivative is:
Q1d h
 0
4
z1
The reason is that the increase in productivity makes the household richer, and
accordingly she decides to spend more in current consumption. Thus, everything else
constant, the aggregate demand shifts to the right, as described in Figure 1. Note that the
impact is only 4 to 1: since the consumer values four different goods in the same manner
(current consumption, future consumption, current leisure and future leisure), the additional
wealth brought about with the productivity shock is equally divided by these four goods.
Also note that the increase in productivity comes along with an increase in income,
and only half of this increase is spend this year: the other half is spend in future consumption
and future leisure. Hence, the household’ savings increases.
Having analysed this case, one shall remember a limitation of the exercise: we are
assuming two periods only. Thus, the wealth effect implied by the productivity shift is split
into two periods only. Obviously, if we allowed the household to live more periods, the
impact of a temporary productivity shock on current consumption and leisure would be much
smaller and close to negligible.
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Consider now the effect of a future increase in productivity When agents anticipate a
higher productivity in the future, the effect on aggregate demand will be:
Q1d
h

0
z2 41  r 
In this case, the household starts spending before the productivity change actually
takes place. Thus, there will be a fall in private savings today.
Now, we consider the two shocks together: as we already know, a permanent increase
in productivity has a much stronger impact on aggregate demand than when shocks are
temporary:
Q1d
z1

dz1  dz 2
h2r 

0
4  1 r 
In this case, because the shock is of permanent nature, there is no need to smooth it.
Hence, the effect on savings will be negligible. Note that the multiplier is close to ½, because
the increase in wealth also gives rise to more leisure, today and in the future.
An increase in government spending is like a negative productivity shock: when the
household anticipates an increase in government spending in period 2, she will reduce
consumption because its life-time wealth declines:
Q1d
1

0
G2 41  r 
Finally, when current government expenditures increase, there are two effects: on one
hand, consumer spending declines by one fourth. On the other hand, the government
spending adds to aggregate demand on a one-to-one basis. The net effect is:
Q1d 3
 0
G1 4
In sum, aggregate demand expands whenever productivity increases –more if the
effect is permanent – when current government expenditure increases and when future
government expenditure decline.
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10.4
Aggregate supply
To find out the aggregate supply function, remember that the production function is
linear in labour (eq. 1). Employment, in turn, is determined by the demand for labour and the
supply of labour.
10.4.1 Labour market equilibrium
The demand for labour, given by equation (3), is represented by the horizontal line in
Figure 2. It is a horizontal curve, because we are assuming constant returns on labour.
The supply of labour is determined by the resource constraint, (6), and the demand for
leisure implied by (9) and (14). This gives:
N1 
S
T 
h  w2  1 
3h
  

 T1  2 
4 41  r   w1  4 w1 
1 r 
(17)
Given the government budget constraint, and the fact that period-2 wages are equal to
productivity, the optimal supply of labour as a function of the wage rate is:
N1 
S
G 
h  z2  1 
3h
  

 G1  2 
4 41  r   w1  4 w1 
1 r 
(17a)
The equilibrium in the labour market is depicted in Figure 2. In Figure 2, the supply
of labour is upward sloped. The reason is that an increase in current wages, by increasing the
opportunity cost of leisure today, leads households to postpone leisure3. Because the model
assumes constant returns on labour, the wage rate is only determined by productivity. The
3
You may verify that the wage rate enters twice in equation (17). For an increase in the wage rate to
have a positive impact on labour supply, the following (reasonable) condition must holed: hw2  T1 1  r   T2 .
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labour supply then determines the equilibrium level of employment, given the values of the
other parameters.
The equilibrium level of employment in period 1 is:
N1 
G 
h  z2  1 
3h
  

 G1  2 
4 41  r   z1  4 z1 
1 r 
(18)
Figure 2: The labour market equilibrium
10.4.2 Partial equilibrium: Shifts in labour supply and demand
To see how the labour market adjusts to different types of shocks, consider first an
increase in productivity today. This case is described in Figure 3. On the firms’ side, the
productivity increase implies an upward shift in the demand for labour, determining an
increase in the real wage rate. On the households’ side, there is a move along the labour
supply function: a higher wage rate today leads households to consume less leisure through
the inter-temporal substitution effect, and hence to work more. All in all, the employment
level increases and the real wage rate increases.
Figure 3: Temporary productivity shock
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Now consider a rise in future productivity, z 2 . This case is analysed in Figure 4.
When future productivity increases, the demand for labour remains unchanged, but the labour
supply shifts to the left due to a wealth effect: the household understands he got richer and
increases spending in all goods, including in current leisure. As a result, the level of
employment declines. Note that if the demand for labour was negatively slopped, this shock
should come along with an increase in the wage rate today as well.
Figure 4: Anticipated productivity shock
Finally, consider what happens when government spending decreases, today or in the
future. From (17) it is easy to see that the present value of government spending impacts
negatively on the labour supply. The reason is that lower government expenditures come
along with an increase in household’ wealth, inducing more consumption and leisure. Thus, a
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decrease in the government life-time spending has an impact similar to that of anticipated
productivity shock, as described in Figure 4.
Finally, consider a decrease in the interest rate. When the interest rate decreases, it
pays for the household to spend more today. This is valid for all goods, including
consumption and leisure. Thus, the labour supply curve shifts to the left, determining a lower
level of employment. Since a lower level of employment comes along with a lower level of
output, there is a positive relationship between the real interest rate and output that underlies
the aggregate supply curve.
10.4.3 The aggregate supply
Given the production function and the equilibrium level of employment, the output
supply in this economy becomes:
Q1 
S
1
G 
hz

3hz1  2   G1  2 

4
1 r 
1  r 
(19)
The aggregate supply is described in Figure 5. It slopes positively because when the
interest rate increases, households optimally decide to reduce current leisure, through intertemporal substitution. Thus, the labour supply expands, and so will do the employment level
and output.
Figure 5: Aggregate supply
10.4.4 Shifts in aggregate supply
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From the discussion above, it should now be clear how the aggregate supply shifts in
response to different types of shocks.
In brief, consider first the case of a productivity increase today. In that case, we know
from Figure 3 that the labour supply expands and the employment level increases. On the
other hand, because current productivity increases, the two terms in (1) are on the rise. The
joint effect is a shift in the aggregate supply to the right, as depicted in Figure 5.
The remaining cases operate through the effect on labour supply only: because the
labour supply expands when future productivity declines and when the present value of
government spending increases, the same will happen to the output supply curve.
10.5
The natural interest rate
We now close our model, by setting the aggregate supply equal to aggregate demand.
Because we are in a closed economy, the variable that adjusts to make these two equal is the
real interest rate. The equilibrium in the model is described in Figure 6.
Figure 6: The natural interest rate
Solving together the aggregate demand (16) and the aggregate supply (19), we obtain
the natural interest rate in this economy:
1 r 
hz 2  G2
hz1  G1
(20)
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Substituting this in (16) or in (19), is easy to see that the equilibrium (natural) level of
output in this economy will be
Q1 
hz1  G1
2
(21)
Using (21) and the similar condition to period 2 in (20), we see that:
1 r 
Q2  G2
Q1  G1
(20a)
Note that this is no more than the Euler equation.
10.6
Adjustment to economic shocks
Putting the pieces together, we now analyse how the economy adjusts to different
types of shocks.
10.6.1 What happens when future productivity increases?
The case of an anticipated productivity shift is analysed in Figure 7. In this case, we
know that the aggregate demand expands (individuals will feel richer and will spend more)
and aggregate supply contracts (individuals will devote more time to leisure).
Clearly, the effect on the real interest rate will be positive: because output in the
future rises relative to output today, the real interest rate must rise.
Figure 7: The effect of an anticipated productivity surge
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In the labour market, there are two opposing effects: on one hand, the fact that future
productivity increases makes individuals richer and hence they will consume more leisure
today: in figure 7 this is represented by a shift in the labour supply curve to the left, that
underlies the contraction of the aggregate supply curve. Then, there is a general equilibrium
effect through the change in the real interest rate: as the real interest rate increases to the new
natural level, current leisure more expensive, inducing households to work more (the labour
supply curve moves to the right again). In the particular model we are using, these two effects
exactly cancel each other.
10.6.2 What happens when current productivity increases?
The effect of a temporary productivity shift is examined in Figure 8. In this case, there
is a strong effect on aggregate supply, because workers are more productive. On the other
hand, there are wealth effects that give rise to more consumption and leisure today. Note
however that these effects are of lower magnitude: the increase in the household life-time
wealth impacts of current consumption on a 1:4 basis, only (and the impact would be even
lower, in a model with many periods). Since the effect on aggregate supply dominates the
expansion in the aggregate demand, the real interest rate declines unambiguously.
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In the labour market, there are three different effects: on one hand, the higher demand
for labour pressures the real wage up, inducing a higher work effort (movement along the
initial labour supply). On the other hand, two other effects imply a shift of the labour supply
to the left: first, the increase in the household lifetime wealth induces a higher demand for
leisure; second, the fall in the real interest rate induces households to consume more leisure
today through the inter-temporal substitution effect. On balance, the increase in real wage and
the shift in the labour supply curve exactly cancel out, so employment does not change at all4.
Figure 8: The effect of a temporary productivity shift
4
Note that this prediction of the model is inconsistent with the real world fact that employment is pro-
cyclical.
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10.6.3 What happens when government consumption increases temporarily?
Now, consider what happens when there is a temporary increase in government
spending.
First, let’s consider the impact on households’ choices. Because a higher government
spending implies an increase in life-time taxation, the household will get poorer. Thus,
through a wealth effect, the household will reduce leisure, expanding the labour supply. By
the same token, private consumption will increase, expanding the aggregate demand. These
effects are however of second order. In a more realistic setup with many periods, the implied
shifts in the AD and in the AS curve should be close to negligible.
In contrast, the impact of government expenditures on aggregate demand is of first
order and dominates the wealth effects. Hence, the overall effect shall be a slight expansion
of aggregate supply and a large expansion of aggregate demand, causing the interest rate to
increase. The increase in the interest rate amounts to the required to keep the household in
shape with the Euler equation after the partial crowding out of private consumption by
government consumption.
Note that the prediction of the model that private consumption declines when
government expenditures and output increase does not square well with the real world facts:
the empirical evidence reveals that private consumption is pro-cyclical.
Figure 9: The effect of a temporary increase in government spending
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In the labour market, there are two reinforcing effects: First, that fact that the increase
in government expenditures comes along with a lower life-time wealth, the household
reduces its current leisure and works more. Thus, the labour supply shifts to the right through
a wealth effect. Second, the increase in the real interest rate turns current leisure more
expensive relative to future leisure. Thus, through an inter-temporal substitution effect, the
household will be induced to work more today, resulting in a further expansion of the labour
supply to the right. On balance, the labour supply will increase unambiguously. As for the
wage rate, it remains constant in our model because we are assuming constant returns: in a
more general model with diminishing returns on labour, the real wage rate should decline.
10.7
The Heterogeneous Agent Two Period Production Economy (HATPPE)
10.8
The case with sticky prices
In the sections above, it is assumed that the real interest rate adjusts to clear the goods
market. This will be the case when prices are flexible and adjust to clear the money market. If
however prices are sticky, the interest rate is determined in the money market (or influenced
by the central bank). As long as the real interest rate is determined somewhere else, it will
hardly be such as to balance the goods market each moment in time. In other words, the real
interest rate may differ from the natural interest rate and the economy will found itself in an
equilibrium that is different from full employment.
The case with sticky prices is illustrated in Figure 9. In the figure, let r0 refer to a real
interest rate that is exogenous is respect to the goods market equilibrium. In that case,
because the real interest rate is to high relative to the natural interest rate, the aggregate
demand will fall short the level of aggregate supply. Since there is no point for firms to
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produce when there is no demand, output will be determined by aggregate demand, at Q0 . In
other words, output will fall short the full employment level.
Of course, if nothing was done, in the long run the real interest rate should fall: the
excess supply in the goods market would imply a fall in prices, and therefore an increase in
real money, driving the interest rate downwards. The problem, however, is that this process
may be too slow, dooming the economy to a long period of unemployment. Thus, there might
be a scope for government intervention.
In examining the policy alternatives, consider first the monetary policy. If the interest
rate is too high, a natural avenue would be to expand the money supply. The real money
supply would increase, driving down the nominal interest rate. Given the inflation
expectations, such move will in general allow the real interest rate to decrease, and the
economy to move from point 0 to point 1 in Figure 10.
General cases have exceptions, and a quite dramatic one occurs when the natural real
interest rate falls below zero, rn  0 . This case is known as Secular Stagnation: a negative
natural real interest rate constitutes a policy challenge, because the nominal interest rate can
hardly be driven below zero.
Of course, a negative natural real interest rate would not be a problem if inflation
expectations were high enough. But, having an eye on the situation of Europe today, suppose
inflation expectations are equal to zero and the nominal interest rate is also zero. That is,
r0  0 , while rn  0 . In this case, unless the central bank could credibly commit with very
high inflation in the future (to drive the real interest rate below zero), little can be achieved by
a monetary expansion.
In this case, the most obvious policy avenue to achieve full employment is to expand
government expenditures. From what we learned in Figure 8, an increase in government
spending not only increases the level of output, it also has the desired implication of moving
the natural interest rate up, eventually eliminating the secular stagnation problem.
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Figure 10: Keynesian unemployment
Further reading
Williamson, Macroeconomics, Chs 10, 13.
Review questions and exercises
1. Explain why aggregate supply is a positive function of the real interest rate.
2. Explain how an anticipated productivity slowdown shifts the aggregate demand and the
aggregate supply. What will happen to the interest rate? What would happen if prices
were sticky and the real interest rate could not adjust?
3. Consider the following utility function for the representative agent:
U   ln c1  1   ln l1   ln c2  1   ln l2  1    . Find out the optimal demands
for current consumption and leisure as a function of life-time wealth, 1 .
4. (Aggregate demand) Consider a two-period economy with a large number of firms
and households. Each household is endowed with h=24 units of time, that can be sold
to firms at the wage rate w. The preferences of the representative consumer are given
by U  ln c1  ln c2  ln l1  ln l2 . In this economy, the production function each period
is given by qt  zt N t , with t=1,2. There is also a government, which spends Gt each
period and levies a lump sum tax Tt on consumers.
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Politicas macroeconomicas, handout Miguel Lebre de Freitas ([email protected])
a) (Firms’ problem) From the firm optimization problem, find out the
expressions of the labour demand.
b) (Household’ problem) From the household optimization problem, find out the
expression for optimal consumption in period 1.
c) (Aggregate demand) Find out the expression for aggregate demand in this
economy, as a function of the exogenous parameters. Then, consider the
following parameterization of the model: Z1  Z 2  10 , G1  T1  40 , G2  0 .
Find out the expression of aggregate demand and represent it in a graph. In
particular, find out what the level of aggregate demand when: (c1) 1+r=1.2;
(c2) 1+r=1; (c3) 1+r=1.5. Explain the intuition.
d) (temporary fiscal expansion) Departing from c), suppose the government
decides to increase the current expenditure and taxes to G1  T1  80 .
Describe, quantifying, what will happen to aggregate demand. At a constant
interest rate (say, 1+r=1.2) what will happen to private consumption?
e) (announced fiscal expansion) Departing from c), suppose instead that the
government decided to increase the future expenditure and taxes to
G2  T2  48 . Describe, quantifying, what will happen to aggregate demand.
At a constant interest rate (say, 1+r=1.2) what will happen to private
consumption?
f) (Fiscal cut) Departing from c), suppose instead that the government decided to
cut the current taxes to zero: T1  0 , keeping its spending unchanged..
Describe, quantifying, what will happen to aggregate demand. At a constant
interest rate (say, 1+r=1.2) what will happen to private consumption?
g) (Temporary productivity change) Departing from c), suppose that current
productivity decreased to Z1  8 . Describe, quantifying, what will happen to
aggregate demand. At a constant interest rate (say, 1+r=1.2) what will happen
to private consumption?
h) (Anticipated productivity shift) Departing from c), suppose that future
productivity increased to Z 2  13 . Describe, quantifying, what will happen to
aggregate demand. At a constant interest rate (say, 1+r=1.2) what will happen
to private consumption?
i) (Permanent productivity shift) Departing from c), suppose that both private
and future productivity increased to Z1  Z 2  12 . Describe, quantifying, what
will happen to aggregate demand. At a constant interest rate (say, 1+r=1.2)
what will happen to private consumption?
j) Discussion): suppose that, instead of living two periods, households leaved 50
periods. Explain how this would alter the impact of temporary (one-period)
changes of exogenous variables.
5. (Aggregate Supply) Consider a two-period economy with a large number of firms
and households. Each household is endowed with h=24 units of time, that can be sold
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Politicas macroeconomicas, handout Miguel Lebre de Freitas ([email protected])
to firms at the wage rate w. The preferences of the representative consumer are given
by U  ln c1  ln c2  ln l1  ln l2 . In this economy, the production function each period
is given by qt  zt N t , with t=1,2. There is also a government, which spends Gt each
period and levies a lump sum tax Tt on consumers.
a) (labour supply) From the households’ optimization problems, find out the
expression of the optimal supply of labour as function of the exogenous variables.
b) (aggregate supply) Using the labour market equilibrium condition and the
production function, find out the expression of aggregate supply as a function of
the interest rate.
c) Consider the following parameterization of the model: Z1  Z 2  10 , G1  T1  40 ,
G2  0 . Find out the expression of aggregate demand and represent it in a graph.
In particular, find out what the level of aggregate supply when: (c1) 1+r=1.2; (c2)
1+r=1; (c3) 1+r=1.5. Explain the intuition.
d) (temporary fiscal expansion) Departing from c), suppose the government decides
to increase the current expenditure and taxes to G1  T1  80 . Describe,
quantifying, what will happen to aggregate supply. At a constant interest rate (say,
1+r=1.2) what will happen to labour supply and demand?
e) (announced fiscal expansion) Departing from c), suppose instead that the
government decided to increase the future expenditure and taxes to G2  T2  48 .
Describe, quantifying, what will happen to aggregate supply. At a constant interest
rate (say, 1+r=1.2) what will happen to labour supply and demand?
f) (Fiscal cut) Departing from c), suppose instead that the government decided to cut
the current taxes to zero: T1  0 , keeping its spending unchanged.. Describe,
quantifying, what will happen to aggregate supply. At a constant interest rate (say,
1+r=1.2) what will happen to labour supply and demand?
g) (Temporary productivity change) Departing from c), suppose that current
productivity decreased to Z1  8 . Describe, quantifying, what will happen to
aggregate supply. At a constant interest rate (say, 1+r=1.2) what will happen to
labour supply and demand?
h) (Anticipated productivity shift) Departing from c), suppose that future
productivity increased to Z 2  13 . Describe, quantifying, what will happen to
aggregate supply. At a constant interest rate (say, 1+r=1.2) what will happen to
labour supply and demand?
i) (Permanent productivity shift) Departing from c), suppose that both private and
future productivity increased to Z1  Z 2  12 . Describe, quantifying, what will
happen to aggregate supply. At a constant interest rate (say, 1+r=1.2) what will
happen to labour supply and demand?
j) (Discussion): suppose that, instead of living two periods, households leaved 50
periods. Explain how this would alter the impact of temporary (one-period)
changes of exogenous variables.
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Politicas macroeconomicas, handout Miguel Lebre de Freitas ([email protected])
6. (Real Business Cycle Model) Consider a two-period economy with a large number of
firms and households. Each household is endowed with h=24 units of time, that can
be sold to firms at the wage rate w. The preferences of the representative consumer
are given by U  ln c1  ln c2  ln l1  ln l2 . In this economy, the production function
each period is given by qt  zt N t , with t=1,2. There is also a government, which
spends Gt each period and levies a lump sum tax Tt on consumers.
a)
(Aggregate demand) Find out the expression of aggregate demand as a function
of the exogenous parameters.
b)
(Aggregate supply) Find out the expression of aggregate supply as a function of
the exogenous parameters.
c)
Consider the following parameterization of the model: Z1  Z 2  10 ,
G1  G2  0 . Find out the equilibrium real interest rate. Then, find out how much
is the employment level and private consumption. Represent the equilibrium
graphically.
d)
(Anticipated productivity shift) Departing from c), examine the implications of a
future productivity shift to Z 2  12.5 . Find out the new equilibrium real interest
rate and quantify the effects on output, private consumption and employment.
e)
(Temporary productivity change) Departing from c), examine the implications
of a productivity shift to Z1  12.5 . Find out the new equilibrium real interest rate
and quantify the effects on output, private consumption and employment.
f)
(permanent productivity change) Departing from c), examine the implications of
a permanent productivity change to Z1  Z 2  12.5 . Find out the new equilibrium
real interest rate and quantify the effects on output, private consumption and
employment.
g)
(temporary fiscal expansion) Departing from c), examine the implications of an
increase in current expenditure and taxes to G1  T1  40 . Find out the new
equilibrium real interest rate and quantify the effects on output, private
consumption and employment.
h)
(announced fiscal expansion) Departing from c), examine the implications of an
increase in future expenditure and taxes to G2  T2  40 . Find out the new
equilibrium real interest rate and quantify the effects on output, private
consumption and employment.
i)
(Tax cut) Departing from c), suppose the government decided to cut the current
taxes to zero: T1  0 , keeping spending unchanged.. Find out the new equilibrium
real interest rate and quantify the effects on output, private consumption and
employment.
7. (Heterogeneous agents) Consider a two-period economy with a large number of
firms and households. Each household is endowed with h=24 units of time, that can
be sold to firms at the wage rate w. In this economy, 50% of the households are
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Politicas macroeconomicas, handout Miguel Lebre de Freitas ([email protected])
Ricardian (have access to financial markets), while the other 50% are “Hand-tomouth” (they cannot borrow or lend). Ricardian households maximize
U R  ln c1  ln c2  ln l1  ln l2 subject to a conventional inter-temporal budget
constrain. “Hand-to-mouth” households solve the one-period problem, maximizing
U HTM  ln c1  ln l1 subject to a static budget constraint. Further assume that: the
production function is Q  10 N in both periods; government expenditures are zero in
both periods; taxes, T1R , T1HTM , T2R , T1HTM (transfers, if negative) may apply.
a)
(Ricardian household) Find out the expressions for optimal consumption and
optimal demand for labour of the Ricardian household.
b)
(Hand-to-mouth) Find out the expressions for optimal consumption and
optimal demand for labour of the Hand-to-Mouth household.
c)
(Government budget constraint): write down the expression of the
government inter-temporal budget constrain.
d)
(Aggregate demand): Find out the expression of aggregate demand as a
function of the fiscal parameters and of the interest rate.
e)
(Aggregate supply): Find out the expression of aggregate supply as a function
of the fiscal parameters and of the interest rate.
f)
R
R
HTM
 T2HTM  0 . Find out the equilibrium
(Baseline): assume that T1  T2  T1
levels of output, interest rate and employment in each group [A: r=0; Q=120;
N=12].
g)
(Tax cut): assume now that the government sets a uniform tax in the first period
T1R  T1HTM  30 , which proceeds are to be returned uniformly to households next
R
HTM
 0 ). Find out the impact of such policy on: aggregate supply;
period ( T2  T2
aggregate demand; interest rate; output; employment and consumption of each
group. Explain what will happen in the debt markets. [A: r=0.8; Q=120,
N R  10.5 , N HTM  13.5 ].
h)
(Income transfer today) Departing from g), examine the implications of a
redistributive policy, from HTM to Ricardian consumers, fully financed within the
HTM
 60  T1R . Find out the impact of this policy on: aggregate
period, that is T1
supply; aggregate demand; interest rate; output; employment and consumption by
R
HTM
 15 ].
each group. [A: r=0.8; Q=120, N  9 , N
i)
(Income transfer tomorrow) Departing from g), examine the implications of a
redistributive policy in the future, from HTM to Ricardian consumers, fully
HTM
 60  T2R . Find out the impact of this
financed within that period, that is: T2
policy on: aggregate supply; aggregate demand; interest rate; output; employment
and consumption by each group.
8. (Sticky prices) Consider a two-period economy with a large number of firms and
households. Each household is endowed with h=24 units of time, that can be sold to
firms at the wage rate w. The preferences of the representative consumer are given by
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Politicas macroeconomicas, handout Miguel Lebre de Freitas ([email protected])
U  ln c1  ln c2  ln l1  ln l2 . In this economy, the production function each period is
given by qt  zt N t , with t=1,2. There is also a government, which spends Gt each
period and levies a lump sum tax Tt on consumers.
a) Consider the following parameterization of the model: Z1  Z 2  10 , G1  G2  0 .
Find out the equilibrium real interest rate. Then, find out how much is the
employment level and private consumption. Represent the equilibrium
graphically.
b) (Anticipated productivity shift) Departing from a), examine the implications of a
future productivity shift to Z 2  8 . Find out the new equilibrium real interest rate
and quantify the effects on output, private consumption and employment.
c) (Sticky prices) Suppose the real interest rate in this economy could not fall below
zero. Given the change in (b), how would the economy adjust? Quantify and
describe in a graph.
d) (Fiscal expansion) Could the government use fiscal policy to achieve full
employment? How? Quantify and illustrate graphically.
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