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ECO 2543: Macroeconomic Theory II
Task 2
Professor: Fabrice Dabiré
University of Ottawa, Winter 2024
1. Explain, in simple terms, what is meant by labor productivity.
The quantity of production or labor that a worker can do in a specific length of time is referred
to as labor productivity. It is a gauge of the productivity of laborer’s in producing products or services. To
put it another way, it's similar to gauging how much work or items a person can complete in a day or an
hour. A worker's labor productivity for an hour is 10 widgets, for instance, if they can produce 10 widgets
in the same hour. Workers are more productive in creating products or services when their labor
productivity is higher.
1
, 2. The Black Death: The Black Death was a pandemic that decimated between 30 and
60 percent of Europe's population in the mid-14th century. This problem asks you
to analyze what Solow's model would predict after a major population reduction.
Suppose the economy is characterized by the Solow model, with labor-increasing
pro- ductivity. Suppose there is no population growth. The value at period t of
population/total labor is Nt and is given exogenously. Future population/work
values are expected to be equal to the present value. The key model equations for
this problem are as follows:
Kt+1 = sZ Kt t
α (A N )tt1−α + (1 -
ZtKα 1-α
δ)Kt Yt = t (AtNt)
At = (1 + a)t
Nt+h = Nt
Nt given exogenously
(a) Define
ke t = Kt
as capital per efficient unit of worker (and
AtNt
similarly, for production, ye, and all other vart iables in the
model). Reformulate the capital accumulation equation in terms of t+1
ke
and
ket and similarly for the production function to derive a
expression for ye in terms of ke . Find an expression for ie and ce.
t t t t
kt= kt / AtNt is the capital per efficient unit of worker
yt= Yt / AtNt is the output per efficient unit of worker
capital accumulation equation: Kt+1=sZKtt^α(AtNt)^1−α+(1−δ) Kt
reformulated in terms of k = A t+1 Nt+1 kt+1=s Zt^αkt + (1−δ) kt At Nt
a
sZ t 1−δ
Kt+1 = + kt
At + 1 Nt +1 At +1
Production function
Yt= ZtKt^α(AtNt)^(1-a)
Reformulated in terms of y
a
Yt ZtKt Zt
Yt= = = ktα
AtNt AtNt AtNt
(b) Find the algebraic expressions for capital per efficient unit of labor in the steady
state and output per efficient unit of labor assuming that At is constant at a
certain value A∗.
2
, In the steady state, kt+1=kt=k∗ and yt=y∗, where k∗ and y∗ denote the steady state levels.
Given that at is constant at A*, we have:
K*= ¿^(1/1-α) T
Y*= ¿ ^(1/1-α)
(c) The real wage is equal to the marginal product of labor, ∂Y t. Derive an
∂Nt
expression for the real wage and write it in terms of the capital stock per
efficient unit of labor. Argue (using mathematics) that when ke → (k )et∗ ,
real wages increase at rate a.
The real wage is equal to the marginal product of labor t+1
∂ Yt
Wt =
∂ Nt
Differentiating the production function with respect to Nt
∂ Yt Yt
=(1−α)
∂ Nt Nt
Using the expression for Yt in terms of kt:
Zt
Wt = (1-α) Kt ∝
AtNt
When kt→k∗, wt increases at rate a because At = (1+a) ^ t.
(d) Draw the main Solow diagram for this model (dynamics of ke
as a function of kte ) and assume that the economy is initially in a state
of
stationary. Suppose the Black Death occurs and reduces the population by1 (so
that2 Nt decreases and is expected to remain at this lower level). Show
graphically what should happen to the capital stock per efficient unit of labor
both during the Black Death period and dynamically.
Equation for the Balanced Growth Path (k∗):
y=x
Equation for the Capital Accumulation Curve:
F(x)= szx^{a}/ At+1Nt+1 + (1-a) x)/ At+1
3
ECO 2543: Macroeconomic Theory II
Task 2
Professor: Fabrice Dabiré
University of Ottawa, Winter 2024
1. Explain, in simple terms, what is meant by labor productivity.
The quantity of production or labor that a worker can do in a specific length of time is referred
to as labor productivity. It is a gauge of the productivity of laborer’s in producing products or services. To
put it another way, it's similar to gauging how much work or items a person can complete in a day or an
hour. A worker's labor productivity for an hour is 10 widgets, for instance, if they can produce 10 widgets
in the same hour. Workers are more productive in creating products or services when their labor
productivity is higher.
1
, 2. The Black Death: The Black Death was a pandemic that decimated between 30 and
60 percent of Europe's population in the mid-14th century. This problem asks you
to analyze what Solow's model would predict after a major population reduction.
Suppose the economy is characterized by the Solow model, with labor-increasing
pro- ductivity. Suppose there is no population growth. The value at period t of
population/total labor is Nt and is given exogenously. Future population/work
values are expected to be equal to the present value. The key model equations for
this problem are as follows:
Kt+1 = sZ Kt t
α (A N )tt1−α + (1 -
ZtKα 1-α
δ)Kt Yt = t (AtNt)
At = (1 + a)t
Nt+h = Nt
Nt given exogenously
(a) Define
ke t = Kt
as capital per efficient unit of worker (and
AtNt
similarly, for production, ye, and all other vart iables in the
model). Reformulate the capital accumulation equation in terms of t+1
ke
and
ket and similarly for the production function to derive a
expression for ye in terms of ke . Find an expression for ie and ce.
t t t t
kt= kt / AtNt is the capital per efficient unit of worker
yt= Yt / AtNt is the output per efficient unit of worker
capital accumulation equation: Kt+1=sZKtt^α(AtNt)^1−α+(1−δ) Kt
reformulated in terms of k = A t+1 Nt+1 kt+1=s Zt^αkt + (1−δ) kt At Nt
a
sZ t 1−δ
Kt+1 = + kt
At + 1 Nt +1 At +1
Production function
Yt= ZtKt^α(AtNt)^(1-a)
Reformulated in terms of y
a
Yt ZtKt Zt
Yt= = = ktα
AtNt AtNt AtNt
(b) Find the algebraic expressions for capital per efficient unit of labor in the steady
state and output per efficient unit of labor assuming that At is constant at a
certain value A∗.
2
, In the steady state, kt+1=kt=k∗ and yt=y∗, where k∗ and y∗ denote the steady state levels.
Given that at is constant at A*, we have:
K*= ¿^(1/1-α) T
Y*= ¿ ^(1/1-α)
(c) The real wage is equal to the marginal product of labor, ∂Y t. Derive an
∂Nt
expression for the real wage and write it in terms of the capital stock per
efficient unit of labor. Argue (using mathematics) that when ke → (k )et∗ ,
real wages increase at rate a.
The real wage is equal to the marginal product of labor t+1
∂ Yt
Wt =
∂ Nt
Differentiating the production function with respect to Nt
∂ Yt Yt
=(1−α)
∂ Nt Nt
Using the expression for Yt in terms of kt:
Zt
Wt = (1-α) Kt ∝
AtNt
When kt→k∗, wt increases at rate a because At = (1+a) ^ t.
(d) Draw the main Solow diagram for this model (dynamics of ke
as a function of kte ) and assume that the economy is initially in a state
of
stationary. Suppose the Black Death occurs and reduces the population by1 (so
that2 Nt decreases and is expected to remain at this lower level). Show
graphically what should happen to the capital stock per efficient unit of labor
both during the Black Death period and dynamically.
Equation for the Balanced Growth Path (k∗):
y=x
Equation for the Capital Accumulation Curve:
F(x)= szx^{a}/ At+1Nt+1 + (1-a) x)/ At+1
3
