ILE-HSG
Dr. Stefan Legge
2,150,1.00
Macroeconomics I
Spring Semester 2025
Problem Set 4
The Labor Market & Inflation
Introduction. To combat the Corona crisis, the government took on enormous debts.
These debts must now be paid off. It is therefore planned to increase of the profit tax. What
medium-term effects will such a structural measure have on the labor markets?
Problem 1 shows you how to determine the equilibrium wage and the equilibrium unemployment rate. In Problem 2, you will deal with the dynamics of labor markets. The Phillips
curve and its variants are the subject of Problem 3. In Problem 4, you will again deal
with the well-known optimization problem of a monopolist. In particular, the connection
between the price elasticity of demand in the profit maximum and the markup (µ) has to
be determined.
Problem 1
Wages and market prices
The determination of the wage level is an important issue in the economy, since most people
are employees, i.e. wage earners. The level of the wage implicitly determines the level of
consumption and thus also the prosperity of an individual. Of course, there is not one wage
level, but many different ones. Nevertheless, it is an important step to establish a theory for
determining the wage level. Consider the following functions. Production function is given
by
Y = F (N ) = N,
price-setting relation by
P = (1 + µ) ⋅ W,
2,150,1.00
Macroeconomics I
Dr. Stefan Legge
Problem Set 4
and wage-setting relation by
W = P ⋅ F (u, z).
Let µ = 0.2
1. Explain each component of the wage-setting equation and its effect on wages.
2. Determine the real wage ( W
P ) in the economy.
Let F (u, z) = 1 − 2 ⋅ u and thus
W
= 1 − 2 ⋅ u.
P
3. Determine the natural rate of unemployment (un ). Illustrate the equilibrium in a graph
with the unemployment rate on the horizontal and the real wage on the vertical axis.
4. Now assume that companies have to pay a payroll tax of 25%. The payroll tax is used
for government spending. The market structure and the markup remain the same.
Determine the new natural rate of unemployment and the new real wage. Illustrate
the result in an appropriate graph.
5. Does it matter for the level of the real wage whether employers or employees pay taxes
or social insurance contributions (also called payroll taxes)?
Solution to Problem 1
1. The price level P has a positive effect on the wage. This is because for employees and
employers the real wage is the decisive factor, not the nominal wage. If P increases, the
required nominal wage increases as well, so that the real wage remains constant. The
unemployment rate u has a negative effect on the wage. As unemployment increases,
the bargaining position of workers is weakened because the employer can more easily
hire new workers, this leads to a lower wage. The variable z includes all other variables
that have an impact on the wage (e.g., unemployment insurance, minimum wage or
employment protection). All these things strengthen the bargaining position of workers
and they can demand a higher wage. Thus, z has a positive effect on the wage.
1
2. Rewriting the price-setting relation gives us W
P = 1+µ . For µ = 0.2 we then get a real
wage of
W
1
1
5
=
=
= = 0.83̄.
P
1 + µ 1 + 0.2 6
3. We now use the wage-setting relation and the calculated real wage. In equilibrium we
then have
(
W
=)
P
1−2⋅u =
1
1+µ
(=
W
)
P
2
⇔
un =
µ
1
=
= 0.083̄.
2 ⋅ (1 + µ) 12
2,150,1.00
Macroeconomics I
Dr. Stefan Legge
Problem Set 4
W
P
●
1
(1+µ)
A
P S (price-setting relation)
W S (wage-setting relation)
u
un
Figure 1: Problem 1.3
4. The pricing-setting relation is now
P = (1 + µ) ⋅ (1 + τ ) ⋅ W.
It follows for the natural rate of unemployment ūn :
(
W
=)
P
(1 − 2 ⋅ u) =
1
(1 + µ) ⋅ (1 + τ )
(=
W
)
P
The payroll tax thus (i) raises the natural rate of unemployment to un = 61 = 0.16̄, and
2
(ii) lowers the real wage to W
P = 3 = 0.6̄.
W
P
1
(1+µ)
●
A
PS
●
1
1+µ̄
B
P S′
WS
un
u¯n
u
Figure 2: Problem 1.4
5. Even with non-competitive labor markets (as long as markups remain constant),
the employee in the medium term bears the full burden of social security contributions/payroll taxes, regardless of who pays them. In the short run, the employer bears
the “employer’s share”, but in the medium run, higher non-wage labor costs lead to
lower wage growth.
3
2,150,1.00
Macroeconomics I
Dr. Stefan Legge
Problem 2
Problem Set 4
Inflow, outflow and the unemployment rate
Let L be the labor force in an economy, N the number of employed and U the number of
unemployed. Assume that every month p = 5% of the employed lose their job, while s = 15%
of the unemployed find a new job.
1. Determine the equilibrium unemployment rate (u∗ )
2. Let L = 1, 000. Determine the monthly outflow and inflow.
3. Through improved job placements, s increases to = 0.2. Determine the new equilibrium
unemployment rate (ũ)
4. Let s = 0.1. Determine the probability of still being unemployed after one month.
What is the probability after six months?
Solution to Problem 2
1. In equilibrium, entries into unemployment equals exits from unemployment:
p⋅N = s⋅U
⇔
p ⋅ (L − U ) = s ⋅ U → since L ≡ N + U
U
L U
⇔p⋅( − ) = s⋅
L L
L
⇔ p ⋅ (1 − u) = s ⋅ u
p
0.05
1
⇔ u∗ =
=
= = 25%.
p + s 0.05 + 0.15 4
2. Since L = 1, 000 and u∗ = 14 we find
U ∗ = u∗ ⋅ L = 250
and thus N = 750. The monthly outflow is thus equal to
s ⋅ U ∗ = 0.15 ⋅ 250 = 37.5
and the inflow is equal to
p ⋅ N ∗ = 0.05 ⋅ 750 = 37.5.
3. In equilibrium we find:
p⋅N = s⋅U
0.05
1
p
=
= = 20%.
⇔ ũ =
p + s 0.05 + 0.2 5
Thus, the equilibrium unemployment rate is decreasing due to the improved job placement.
4
2,150,1.00
Macroeconomics I
Dr. Stefan Legge
Problem Set 4
4. The probability of having found a new job after one month (t = 1) is s = 10%. Thus, the
probability of still being unemployed after one month is 1 − s = 90%. After 6 months
(t = 6) the probability of still being unemployed is (1 − s)t = 0.96 ≈ 53.14%.
Problem 3
Phillips curve
From the price-setting relation and the wage-setting relation, a function can be derived that
describes the relationship between the price level, the expected price level and production
for P ≠ P e (see Blanchard 2021, Chapter 8.1):
P = P e (1 + µ) ⋅ F (u, z),
(1)
where F (u, z) = 1 − α ⋅ u + z.
1. Derive the Phillips curve from equation (1) as a function of inflation, expected inflation,
α, µ, z and ut . Interpret your result.
2. Determine the natural rate of unemployment (un ).
3. How does the natural rate of unemployment (un ) react to changes of
(i) µ,
(ii) α,
(iii) z.
Interpret your result.
4. Derive a version of the Phillips curve that relates inflation, expected inflation, natural
unemployment rate and current unemployment rate.
5. Let α = 2, µ = 0.12 and z = 0.04.
(a) Determine the natural rate of unemployment (un ).
(b) Let π0 = 0 and the target unemployment rate ū = 5%. Determine the inflation in
the years t = 1 and t = 2 for the case of static expectations (here: πte = 0) and for
the case of adaptive expectations (here: πte = πt−1 ).
(c) Determine the Phillips curve for the case of wage indexation. Interpret your
result.
Solution to Problem 3
1. Starting from equation (1) and utilizing time subscript for the price level, the expected
price level and the unemployment, we have
Pt = Pte ⋅ (1 + µ) ⋅ (1 − α ⋅ ut + z)
→
⇔
Dividing both sides by the price level in t − 1 yields
Pte
Pt
=
⋅(1 + µ) ⋅ (1 − α ⋅ ut + z)
Pt−1
Pt−1
±
±
I
II
5
2,150,1.00
Macroeconomics I
Dr. Stefan Legge
Problem Set 4
Rewrite I:
Pt − Pt−1 + Pt−1
Pt − Pt−1
=1+
= 1 + πt .
Pt−1
Pt−1
Rewrite II:
P e − Pt−1
Pte − Pt−1 + Pt−1
=1+ t
= 1 + πte .
Pt−1
Pt−1
Hence, it follows from equation (2) that
1 + πt = (1 + πte ) ⋅ (1 + µ) ⋅ (1 − α ⋅ ut + z)
→
⇔
divide by (1 + πte ) ⋅ (1 + µ) yields
(1 + πt )
= 1 − α ⋅ ut + z
(1 + πte ) ⋅ (1 + µ)
→
For small values of πt , πte , ut this can be rearranged to
⇔ πt = πte + (µ + z) − α ⋅ ut .
(2)
Thus, if inflation expectations remain constant, an increase in the unemployment rate
by one percentage point will lead to an increase in inflation by 0.01α. An increase in
the markup (µ), the composite variable (z), or inflation expectations (πte ) leads to an
increase in inflation (by the same amount) since the Phillips curve shifts upwards.
2. The natural rate of unemployment is given by equation (2) if expected inflation is
equal to current inflation:
πt = πte
⇔
un =
µ+z
.
α
(3)
3. (a) An increase in markup leads to a decrease in the real wage (via the price-setting
relation). For workers to accept this real wage decrease, unemployment must increase, since higher unemployment lowers workers’ wage demands (see the wagesetting relation). Thus, an increase in markup leads to higher natural unemployment:
∂un 1
= > 0.
(4)
∂µ α
(b) The parameter α indicates how strongly workers respond to changes in unemployment (i.e., the slope of the wage-setting relation). Thus, an increase in α
causes unemployment to have a stronger effect on wage demands. Therefore,
lower natural unemployment is sufficient for workers to accept the existing real
wage:
∂un
µ+z
= − 2 < 0.
(5)
∂α
α
6
2,150,1.00
Macroeconomics I
Dr. Stefan Legge
Problem Set 4
(c) An increase in the composite variable z leads workers to demand a higher real
wage (shifting the wage-setting relation upward) because they are in a better
bargaining position. However, the real wage is determined by the price-setting
relation, so the better bargaining position does not lead to a higher real wage,
but to more unemployment:
∂un 1
= > 0.
(6)
∂z
α
4. From equation (3) it follows that
µ + z = α ⋅ un .
So one can manipulate (2) as follows:
πt = πte + (µ + z) − α ⋅ ut
⇔
πt = πte + α ⋅ un − α ⋅ ut
⇔
πt = πte − α ⋅ (ut − un ).
5a) For α = 2, µ = 0.12 and z = 0.04 we get a natural rate of unemployment (cf. eq. (2)) of
un =
µ + z 0.12 + 0.04
=
= 0.08.
α
2
5b) Let π0 = 0 and ū = 0.05.
Static expectations. Utilizing equation (2) and πte = 0 yields
t=1∶
π1 = 0 + 0.16 − 2 ⋅ ū = 0.16 − 2 ⋅ 0.05 = 0.06,
t=2∶
π2 = 0 + 0.16 − 2 ⋅ ū = 0.16 − 2 ⋅ 0.05 = 0.06.
In each period the inflation is 6%, because the expected inflation in each period is zero.
Adaptive expectations. Utilizing equation (2) and πte = πt−1 yields
t=1∶
π1 = 0 + 0.16 − 2 ⋅ ū = 0.16 − 2 ⋅ 0.05 = 0.06,
t=2∶
π2 = 0.06 + 0.16 − 2 ⋅ ū = 0.22 − 2 ⋅ 0.05 = 0.12.
In each period, inflation must increase by 6 percentage points. Compared to static
expectations, inflation has to increase by the same amount in each period because the
expected inflation is the same as last year’s.
According to this analysis, not only does unemployment have an effect on inflation,
but inflation can also have an effect on unemployment, depending on how you look
at it. It seems as if the government can achieve any unemployment rate if only it
sets inflation correctly. However, this trade-off between inflation and unemployment
should be taken with caution. The empirical relationship is not (anymore) stable
and fine-tuning inflation and unemployment is an impossibility (see Blanchard (2017),
pp. 165-172.).
7
2,150,1.00
Macroeconomics I
Dr. Stefan Legge
Problem Set 4
5c) With wage indexation, a share λ (with 0 ≤ λ < 1) of the labour contracts is indexed.
Nominal wages in those contracts move one-for-one with variations in the actual price
level. A share 1 − λ of labour contracts is not indexed. Nominal wages are set on the
basis of expected inflation. We additionally assume, as the textbook suggests, that
πte = πt−1 . Hence, equation becomes:
⇔
πt = λ ⋅ πt + (1 − λ) ⋅ πt−1 − α ⋅ (ut − un )
ut − un
πt = πt−1 − α ⋅
1−λ
The larger the share λ, the larger the inflationary reaction to changes in the unemployment rate (ut ). Furthermore, the larger the share of λ, the larger (smaller) the
inflation if ut < un (ut > un ).
Problem 4
Labour demand
The inverse demand function on a market is given by P (Y ) = a − b ⋅ Y and the production
function by Y = F (N ) = A ⋅ N . We additionally assume that a, b, w, A > 0 and (a ⋅ A) > w,
where w represents the wage rate.
1. Determine the price elasticity of demand (εYp ) at the monopolist’s profit maximum.
2. Determine the markup (µ) on the marginal costs in the profit maximum. How does
the markup change when w → 0 or w → a ⋅ A? Interpret your result.
3. Determine the functional relationship between the price elasticity of demand (εYp ) and
the markup (µ) in the profit maximum.
Solution to Problem 4
1. Cost function. The costs of the firm are w ⋅ N . However, to get the cost function we
need the employment (N ) that is contingent on output (Y ). We get this relation by
taking the inverse of the production function:
N (Y ) = F −1 (Y ) =
Y
A
Thus, the cost function is equal to
C(Y ) = w ⋅ N (Y ) =
w
⋅ Y.
A
Output and price at the profit maximum. The profit of the monopolist is equal
to
w
Π(Y ) = P (Y ) ⋅ Y − C(Y ) = (a − b ⋅ Y ) ⋅ Y − ⋅ Y.
A
8
2,150,1.00
Macroeconomics I
Dr. Stefan Legge
Problem Set 4
In the monopolist’s profit maximum the following holds
Π′ (Y ) = 0
⇔
a−2⋅b⋅Y −
w
=0
A
⇔
YM =
a⋅A−w
.
2⋅A⋅b
Plugging YM into the inverse demand function P (Y ) gives us the monopoly price:
pM = P (YM ) =
a⋅A+w
.
2⋅A
Price elasticity of demand. The price elasticity of demand measures the percentage
change in demand associated with a one percent change in the market price. It is
defined as
dY (p)
p
εYp ≡
⋅
.
dp
Y (p)
The demand function Y (p) is obtained by utilizing the inverse of the inverse demand
function (P (Y )):
a−p
P (Y ) = a − b ⋅ Y ⇔ Y (p) =
b
The price elasticity of demand is thus equal to
εYp =
dY (p)
p
1 p⋅b
p
⋅
=− ⋅
=−
.
dp
Y (p)
b a−p
a−p
In the profit maximum of the monopolist we have:
εYP = −
−1
a ⋅ A + w 2 ⋅ a ⋅ A − (a ⋅ A + w)
pM
=−
⋅(
)
a − pM
2⋅A
2⋅A
=−
a⋅A+w
.
a⋅A−w
2. At the monopolist’s profit maximum, the market price (pM ) is equal to marginal costs
times some parameter. The larger the percentage markup on the marginal costs, the
larger the monopolist’s market power. If we use µ as the markup on marginal costs,
we get
pM = (1 + µ) ⋅
w
A
⇔
a⋅A+w
w
= (1 + µ) ⋅
2⋅A
A
⇔
µ=
a⋅A−w a⋅A 1
=
− .
2⋅w
2⋅w 2
For w → 0 we get
lim µ → ∞.
w→0
For w = 0 the marginal costs are zero and the monopoly price pM = a2 . Thus, the
markup goes to infinity.
For w → a ⋅ A we get
lim µ = 0.
w→a⋅A
For w → a ⋅ A the monopoly price goes to the prohibitive price (P (0) = a) and the
marginal costs also go to a. Thus the markup µ = 0, i.e. the market price is equal to
the marginal costs.
9
2,150,1.00
Macroeconomics I
Dr. Stefan Legge
Problem Set 4
3. We know that at the profit maximum, both
pM ⋅ (1 +
w
1
w
)=
as well as pM = (1 + µ) ⋅ applies.
Y
εp
A
A
Therefore, the following also holds
1+
1
εYp
=
1
1+µ
⇔
1+µ=
εYp
1 + εYp
⇔
µ=−
1
.
1 + εYp
There is a clear functional relationship between the price elasticity of demand (εYp )
and the markup (µ): the larger the absolute value of the price elasticity of demand,
the smaller the markup, et vice versa. For example, if µ = 0 then εYp → −∞.
Problem 5
Conclusion
After lectures 5 (The Labour Market) and 6 (Inflation) and after reading chapters 7, 8, 23.2
and 23.3 of Blanchard 2021, you should be able to answer the following questions, among
other
1. What are efficiency wages?
2. What factors other than price levels and unemployment rates influence real wages?
3. What is NAIRU?
4. What are the benefits and costs of inflation?
References
Blanchard, Olivier (2021). Macroeconomics. 8 (Global Edition). Harlow, UK: Harlow: Pearson
Education Limited.
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