PROBLEM SET 3, MACROECONOMICS: POLICY, 31E23000
1. Take the medium-term model (determining the price competitiveness and
output together with trade balance). One medium term issue the model as such
cannot dela with is the public sector budget balance. A curve for it can be added
to the diagram by looking at the budget balance condition
(1)
ty − g = d¯
Here g = public expenditure, t = tax rate and d¯ is the target level for primary
balance, based e.g. on medium/long term public debt sustainability estimations.
Add the line implied by this requirement to the ERU − AD − BT − diagram!
In this enhanced model describe what happens to the budget balance in addition
to the other variables if exports increase due to increased demand in the export
markets! Assume that in the initial situation primary deficit is larger than the one
required by the budget balance equation!
Notice that the price competitiveness variable q does not directly enter the budget balance equation given above. Can you think of any reasons why it should enter
the equation? If so, what would be the sign of the effect?
A: With the basic case given above there is a unique level of income at which
the target surplus is reached
d¯ + g
(2)
yb =
t
This is a vertical line, denoted by BB in the next figure, on the left hand side of
it the surplus is below the target on the RHS it is above the target. The next figure
gives the situation for the case where at the initial equilibrium 0 the the surplus is
smaller/deficit larger than the one required for the budget balance:
An increase in exports shifts both the AD and BT curve downwards, more
the BT-curve (the direct impact of the increase in exports on both curves AD
and BT is the same while the increase in output, at any given level of competitiveness, is smaller for AD (as any given increase in output increases demand by
(1 − c (1 − t) + m) dy and for BT an increase in output reduces trade balance surplus by mdy. Since the ERU-curve does not move, the new equilibrium is at point 1,
where the new AD-curve and ERU-curve intersect. As this point is now above the
BT-curve (the competitiveness, for the output at point 1, and output, for the level
of competitiveness at point 1, are higher than needed for trade balance equilibrium.
Notice that export expansion reduces competitiveness.
The budget balance curve BB is not affected directly by the increase in exports.
What happens now is that since output has increased, budget deficit declines (or
surplus increases) moving the economy closer to the target (or even to the other
side, with budget surplus becoming excessive, this depends on the exact location
of th BB-curve).
The second part is harder, as we are looking for possibilities where relative prices
matter for the budget balance equation. One such possibility comes directly into
mind: public sector can, like the private sector, buy both domestic and foreign
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PROBLEM SET 3, MACROECONOMICS: POLICY, 31E23000
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BB
q
AD
ERU
BT
0
1
y
goods. What exactly happens depends on budgeting practices. One example is the
the case where the government takes decisions on the volume of public expenditure
measured in terms of the price index relevant for public sector purchases. One can
think of the public sector producing public goods with a “production function”
G1−κ (G∗ )κ . This gives the public sector price index GP I ≡ P 1−κ (EP ∗ )κ .
In this case the nominal value of government expenditure is GP I × Ĝ ≡ G1−κ (G∗ )κ
I
κ
and the expenditure in terms of domestic output is then GP
P Ĝ = Q Ĝ. Linearizing
this gives g = κĝ. The budget balance equation then becomes
(3)
ty − κĝ = d¯
This gives and upward sloping BB-curve: higher output increases tax income
and expenditure must increase. Improved competitiveness increases expenditure in
dq
= κt . If the share of
terms of domestic output. The slope of the BB curve is dy
foreign goods in government consumption, κ is very small relative to the marginal
tax rate, we are close to the case studied above. You can see easily, by just draving
an upward sloping curve in the previous figure that the conclusion drawn above
still holds.
One could think that also the tax revenue can be affected by the competitiveness as the real income consumers are concerned with is the income in terms of
the consumer price index where also foreign are included. But given our assumptions this does not seem likely. Consumer income in terms of the consumer priPy
ce index is CP
I and gives total expenditure in terms of the consumption basket
PROBLEM SET 3, MACROECONOMICS: POLICY, 31E23000
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SugarSync/Luennot/Macro2016/PS4F2.pdf
growth rate
(dk/dt)/k
sk-α
n+δ
k
k'
Ĉ ≡ c
Py
CP I
Py
− t CP
I
k*
Py
= c (1 − t) CP
I . But we need this in terms of domestic output
I
and this is given CP
P Ĉ = c (1 − t) y. Note in particular that tax income in terms
of domestic production is ty.
2. Use the Solow growth model to study what happens in an economy in which
the labor force increases suddenly, there is a discrete increase in L! Assume that the
economy is initially at the steady state! What happens to the income per capita in
the short run? What happens to the growth rate? What happens to the long run
income per capita?
A: Note that the question is about the discrete, one time increase in the population/labor force size, not about a change in the population growth rate. Thus we
assume that these new people have the same birth rate as the original people.
On impact, since the current capital stock is given by past decisions, the capitallabor ratio declines. In Figure 2 I assume that initially the economy is in the steady
state (note also that the analysis applies to all incarnations of the Solow-model)
capital-labor ratio declines from k ∗ to k 0 . Two things happen on the impact: the
growth rate increases and the income per capita declines. The economy begins to
converge back to its initial equilibrium, the one-time increase in population does
not have any effect on the long run equilibrium. But notice that during the whole
adjustment process income per capita is lower than it would have been without the
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sudden inflow of the people. Note that this is an example of a case where increased
growth and decline in income can be observed simultaneously.
3. During the last years there has been a lot of discussion about increased income
inequality, possibly partly due to technological change. Some of the changes can then
be crudely captured in the simple Solow growth model by assuming that labor share
in aggregate income has declined, i.e. α has declined. To study the implications for
growth of this decline, assume that the economy is initially in the steady state with
higher α and study both the short and long term impacts of a decline in α! What
would happen, if realistically also simultaneously the rate of technological progress
x increases!
A: Let us first look at what happens if the labor share alone declines. Figure 3
tells the story. The decline in the labor share is equivalent to α becoming smaller,
and thereby 1 − α becoming larger. This implies that at any capita-labor-ratio the
saving curve shifts up: Increase in the capital share is equivalent to capital becoming
more productive (the marginal product of capital increases at given capital-labor
ratio), thus increasing output per capita. If the economy is initially in the steady
state there will be a new period of growth to reach the higher capital-output ratio,
in other initial situations there is also a temporary boost in the growth rate.
Let us now move to the case where also the rate of productivity growth, x,
increases simultaneously with the technological change reducing labor share. The
line (δ + n + x) k̂ rotates upwards, Figure 4 shows what happens. I have drawn
PROBLEM SET 3, MACROECONOMICS: POLICY, 31E23000
5
the case where the increase in the rate of productivity growth reduces the capitalefficient labor ratio. There is therefore a period of “degrowth” in the sense that
capital stock is being effectively decumulated, it seems as if the role of physical
capital as a source of growth diminishes or disappears. There is naturally growth
because the labor productivity grows.
The most interesting item is what happens to (real) wages. As we assume the
economy to be competitive, we know that the wage equals the marginal product of
labor. As the production function relevant for this case is Yt = Kt1−α (At Lt )α the
wage rate is
(4)
wt = αAt Kt1−α (At Lt )α−1 = αAt k̂t1−α , k̂ ≡
K
AL
Taking a derivative of this with respect to the labor share α gives (in doing this
you may want to use the fact that k̂t1−α = e(1−α)lnk̂t )
(5)
∂w
= [1 − α (1 − α)] At k̂t1−α
∂α
This is positive (as α (1 − α) < 1 for α < 1) implying that the impact effect
of a reduction in the wage share is a decline in real wage. Real wage also declines
further if the rate of productivity growth reduces the steady state capital intensity
of production. With a change in the labor share alone the long run capital intensity
increases counteracting the immediate effect. If you want, you can try whether the
effect is strong enough to lead to an increase in the real wage by using the formula
for the steady state capital intensity. And we must remember that productivity
growth definitely goes on and increases the real wage in the end.
4. Germany implemented during the previous decade the so called Hartz reforms to increase equilibrium employment and reduce equilibrium unemployment.
A summary of the reforms can be found e.g. in this VoxEU column http://www.
voxeu.org/article/german-labour-reforms-unpopular-success. Use the medium term model to analyze the impacts of these reforms!
A: As stated in the article given as reference to the reforms, they improved the
matching between the unemployed persons and firms (removing frictions), reduced
unemployment benefits (for the long-term unemployed), and introduced wage subsidies paid to firms (and organizations) to reduce hiring costs. The first two set of
reforms can be seen as shifting the WS-curve down, they can be seen as moderating
wage claims. The textbook contains the discussion on the impacts of improving the
matching, the impact of unemployment benefits has been treated in the lectures.
With better matching there will be less open vacancies taming the pressure on wages. better matching also gives an incentive to stay in the workforce, increasing the
labor supply and making the WS-curve also shift down
The last set of measures, wage subsidies to firms to increase hiring clearly has an
effect on firms and on the PS-curve. Formally its effects are from the firms’ point
of view the same as an increase in labor productivity, allowing firms’ to pay higher
wages and satisfy the profit-margin requirements. Thus, the reforms altogether shift
the WS-curve down and the PS-curve up, Figure 5 shows this.
The problem now is that if the assumption of horizontal PS-curve is kept then
the implication will be that the real wages increase, against the evidence. To see
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PROBLEM SET 3, MACROECONOMICS: POLICY, 31E23000
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this, recall that the PS-curve determines the (real) wage firms can offer to workers,
it is derived from the equation P = (1 + µ) W
λ where µ = mark-up, λ = labor
productivity, and W = wage the firm pays to the workers. Now, in the case of wage
subsidy the wage the firm faces is (1 − s) W where s = the wage subsidy. Thus we
(1−s)W
λ
have P = (1 + µ)
from which we can solve W
λ
P = (1+µ)(1−s) . An increase in
the subsidy allows the firm pay higher wage, with horizontal PS-curve the worker
gets the whole benefit from the subsidy.
With a downward sloping curve it is possible both for employment to increase
and wages to decline, which conforms with the evidence. So assume the PS-curve to
be downward sloping. Even without this assumption, since WS-curve shifts down
we get the result that the ERU-curve shifts to the right. Figure 5 summarizes the
story:
The impact on employment and on the unemployment rate are clear: Employment increases and unemployment rate falls. The impacts on the wage rate are
ambiguous, in principle at least, since the PS-curve shifts up. Assuming a downward sloping PS-curve the wage also declines.
If this holds, then the ERU curve in the aggregate model, our AD-ERU-BTlanguage shifts to the right: the output at any given level of competitiveness increases, or, at any given level of output competitiveness improves, Figure 6. In the
picture I have assumed that in the pre-reform situation, point 0, Germany’s trade
balance was in deficit, as it was.
PROBLEM SET 3, MACROECONOMICS: POLICY, 31E23000
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SugarSync/Luennot/Macro2016/PS4F5.pdf
w
WS
PS
0
1
PS*
y
As a consequence of the reforms (point 1 in the figure), output and employment
increase, unemployment is reduced, competitiveness improves and trade balance
deficit is increased, in the figure it turns into surplus. Given the increase in output
it is possible that part of the initial decline in (real) wages is reversed, but full
recovery need not happen (and has not, yet at least happened).
Due date: April 14, 2:15 PM.
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SugarSync/Luennot/Macro2016/PS4F6.pdf
AD
q
1
BT
0
ERU
y