Summary Macroeconomics (Master)
Chapter 5 – Real-Business Cycle Theory
5.1 Introduction: Some Facts about Economic Fluctuations
Understanding the causes of aggregate fluctuations is a central goal of macroeconomics. Important
facts:
They do not exhibit any simple regular or cyclical pattern;
Fluctuations are distributed very unevenly over the components of output;
No large asymmetries between rises and falls in output.
So main message:
- Business cycles vary in length;
- Business cycles occur infrequently;
- Business cycles occur vary in severity.
5.2 An Overview of Business-Cycle Research
We begin the study of aggregate fluctuations by asking whether they can be understood using a
Walrasian model – that is, a competitive model without any externalities, asymmetric information,
missing markets, or other imperfections. The Ramsey model (Chapter 2) is the natural Walrasian
baseline model of the aggregate economy: the model excludes not only market imperfections, but also
all issues raised by heterogeneity among households. The incorporation of aggregate fluctuations
requires modifying the model in two ways:
1. There must be a source of disturbances: without shocks, a Ramsey economy converges to a
balanced growth pat hand then grows smoothly;
2. Allow for variations in employment.
Recap Ramsey Model
Components
The economy consists of a large number of identical, price taking firms and a large number of
identical price taking infinitely lived households. Profit maximising firms choose production factors
K and L. Utility maximising consumers choose C, always work full-time. For simplicity, no
uncertainty (we will add this later), no international trade (major shortcoming, but easy to solve), no
population growth (not relevant on a business cycle frequency).
Firms
Production determined by a constant returns to scale production technology with diminishing returns
to each production factor. The law of diminishing returns states that in all productive processes,
adding more of one factor of production, while holding all others constant, will at some point yield
lower incremental per-unit returns. Adding more workers causes problems at some point such as
workers getting in each other's way or frequently finding themselves waiting for access to a part. In all
of these processes, producing one more unit of output per unit of time will eventually cost
increasingly more, due to inputs being used less and less effectively.
Y t = At F ( K t , Lt ) , F K ( ., . ) >0, F KK <0, F L >0, F¿ <0Their capital stock depreciates at rate δ and
grows with investments.
K t +1=( 1−δ ) K t + I t 0< δ<1
1
,Present value of future and current profits:
( )
∞ t
1
φ=∑ ∏ 1+r [ A t F ( K t , Lt ) −w t Lt−I t ]
t=0 τ =0 τ
Maximize the present value of future profits given the current capital stock (K), technology (A),
wages (w) and interest rate (r).
(∏ )[
∞ t
1
max φ=¿ ∑ A t F ( K t , Lt ) −w t Lt−I t ] ¿ subject to K t +1=( 1−δ ) K t + I t
t=0 τ =0 1+r τ
Solution: Substitute investment out of the equation.
(∏ )[
∞ t
1
max φ=∑ A t F ( K t , Lt )−w t Lt−K t +1+ ( 1−δ ) K t ]
t=0 τ=0 1+r τ
Firms: First Order Conditions
Capital demand: Differentiation with respect to K t +1gives:
(∏ )[
t t+1
∂φ 1 1
=0: ∏ = A t+ 1 F K ( . , . )+ ( 1−δ ) ] → r t + δ =A t F K ( K t , Lt ) for each
∂ K t+ 1 τ=0 1+ rτ τ =0 1+r τ
t=1,2,3 , …( NOT t=0)
Firms face a trade-off: using more capital costs interest rate plus discount rate (marginal cost of
capital). The benefits are the marginal producitivity of capital. Profit maximisation implies that
marginal costs equal marginal benefits. If the marginal product of capital exceeds the costs, firms want
to invest more, increasing the demand for loanable funds, increasing the interest rate and lowering the
marginal productivity of capital.
Labour demand: Differentiation with respect to Lt gives
t
∂φ 1
=0 : ∏ [ At F L ( K t , Lt )−wt ]=0 → wt = At F L ( K t , Lt ) for each t=0,1,2 , …
∂ Lt τ =0 1+r τ
Marginal cost of labour (the wage) equals the marginal productivity of labour. If the marginal
product of labour exceeds the costs, firms hire more labour, which lowers marginal productivity and
(potentially) increases wages.
Consumers
Optimisation: Consumers maximise present value of utility given current assets, current and future
income and interest rates. Labour supply is inelastic and standardised to one ( Lt =1). The only savings
possibility is capital, which generates a return of r t .
∞
For log-utility we have max ¿ ∑ β ln C t subject to K t +1=( 1+r t ) K t + w t −C t
t
t=0
−ρ
Note: the book uses e instead of β .
The Langragian: One objective, with a constraint for every (future) period. Denote the Lagrangian
∞
multiplier of period t’s constraint by λ t. L=∑ β lnC t −¿ λt [ K t+1 −( 1+r t ) K t −w t +C t ] ¿.
t
t =0
2
, ∂L βt
FOC: =0: =λ t
∂C t Ct
∂L ∂L
=0: λt =λt +1 ( 1+ r t +1 ) =0: K t+1 =( 1+r t ) K t +w t−C t
∂ K t+ 1 ∂ λt
Consumers: Solution
The Euler equation: Combining the first and second first order conditions gives:
1
Ct
=β
1
( )
C t +1
(1+r t +1)LHS: the marginal utility of one unit current consumption.
RHS: the expected marginal utility of saving one unit current consumption and using the savings next
period.
This difference equation describes the optimal path of consumption. It is usually called the Euler
equation. Euler says that someone must be indifferent between consuming one unit more today and
saving that unit and consuming it in the future.
Equilibrium
Labour market: Everybody supplies one unit of labour. Standardizes mass of people to one. Lt =1
Goods market: Production is used for consumption, investment and government consumption:
Y t =C t + I t +G t Combined with capital accumulation identity:
K t +1=( 1−δ ) K t −Ct −G t Capital market equilibrium already imposed (and it clears by Walras’ law).
The Full Model
Labour demand: w t = A t F L ( K t , Lt )
Capital demand: r t + δ =A t F K ( K t , Lt )
1 1
Euler equation:
Ct
=β
Ct +1
( 1+r t +1 )
Production function: Y t = A t F ( K t , Lt )
Goods market: K t +1=( 1−δ ) K t +Y t−C t −G t
Labour market: Lt =1
with K 0 , A t and G t (t=0,1,2 , …) exogenous.
A minor problem is that given K 0 , A t and G t these six equations determine Y t , w t , r t , K t +1 , Lt and
C t. Unfortunately, an explicit solution is very hard to find.
Solutions:
- Graphical analysis (phase diagram, only qualitative answers);
- Special cases;
- Approximate (log-linearize) general solution;
- Approximate using computer.
Graphical Analysis (Phase Diagram)
Core model: two dynamic equations. Drop Lt ( ¿1), simplify and the dynamics of this system are
given by the Euler equation and the capital accumulation equation.
3
, Ct +1
Ct [ ]
=β ( At +1 F K ( K t +1 ) +1−δ ) Phase Diagram
Dynamics of consumption: The Euler equation:
Ct +1
Ct [ ]
=β ( At +1 F K ( K t +1 ) +1−δ ) There is one level of the capital stock K* for which consumption is
¿
stable according to the Euler equation, so the C t+ 1=C t line is vertical. To the left, K < K , so F ' ( K ¿ ).
In order for the Euler equation to hold it must be that C t+ 1>C t , so consumption goes up. To the right
the opposite must hold and consumption goes down.
Dynamics of the capital stock: K t +1−K t =−δ K t + A t F ( K t ) −C t−Gt
Start with the K t +1=K t line: C t= A t F ( K t )−δ K t−G t . This line is humped shaped because of the
properties of the production function. Above this line: C> F ( K )−δK +G ; society consumes too much
and cannot fully replace all depreciations, so the capital stock diminishes ( K t +1 < K t ). Below this line
the opposite happens and the capital stock increases.
Construction:
Saddle Path:
The line SS is a stable manifold, a saddle path. Any
point no on this line is not attainable, a waste of
resources or unsustainable.
4
Chapter 5 – Real-Business Cycle Theory
5.1 Introduction: Some Facts about Economic Fluctuations
Understanding the causes of aggregate fluctuations is a central goal of macroeconomics. Important
facts:
They do not exhibit any simple regular or cyclical pattern;
Fluctuations are distributed very unevenly over the components of output;
No large asymmetries between rises and falls in output.
So main message:
- Business cycles vary in length;
- Business cycles occur infrequently;
- Business cycles occur vary in severity.
5.2 An Overview of Business-Cycle Research
We begin the study of aggregate fluctuations by asking whether they can be understood using a
Walrasian model – that is, a competitive model without any externalities, asymmetric information,
missing markets, or other imperfections. The Ramsey model (Chapter 2) is the natural Walrasian
baseline model of the aggregate economy: the model excludes not only market imperfections, but also
all issues raised by heterogeneity among households. The incorporation of aggregate fluctuations
requires modifying the model in two ways:
1. There must be a source of disturbances: without shocks, a Ramsey economy converges to a
balanced growth pat hand then grows smoothly;
2. Allow for variations in employment.
Recap Ramsey Model
Components
The economy consists of a large number of identical, price taking firms and a large number of
identical price taking infinitely lived households. Profit maximising firms choose production factors
K and L. Utility maximising consumers choose C, always work full-time. For simplicity, no
uncertainty (we will add this later), no international trade (major shortcoming, but easy to solve), no
population growth (not relevant on a business cycle frequency).
Firms
Production determined by a constant returns to scale production technology with diminishing returns
to each production factor. The law of diminishing returns states that in all productive processes,
adding more of one factor of production, while holding all others constant, will at some point yield
lower incremental per-unit returns. Adding more workers causes problems at some point such as
workers getting in each other's way or frequently finding themselves waiting for access to a part. In all
of these processes, producing one more unit of output per unit of time will eventually cost
increasingly more, due to inputs being used less and less effectively.
Y t = At F ( K t , Lt ) , F K ( ., . ) >0, F KK <0, F L >0, F¿ <0Their capital stock depreciates at rate δ and
grows with investments.
K t +1=( 1−δ ) K t + I t 0< δ<1
1
,Present value of future and current profits:
( )
∞ t
1
φ=∑ ∏ 1+r [ A t F ( K t , Lt ) −w t Lt−I t ]
t=0 τ =0 τ
Maximize the present value of future profits given the current capital stock (K), technology (A),
wages (w) and interest rate (r).
(∏ )[
∞ t
1
max φ=¿ ∑ A t F ( K t , Lt ) −w t Lt−I t ] ¿ subject to K t +1=( 1−δ ) K t + I t
t=0 τ =0 1+r τ
Solution: Substitute investment out of the equation.
(∏ )[
∞ t
1
max φ=∑ A t F ( K t , Lt )−w t Lt−K t +1+ ( 1−δ ) K t ]
t=0 τ=0 1+r τ
Firms: First Order Conditions
Capital demand: Differentiation with respect to K t +1gives:
(∏ )[
t t+1
∂φ 1 1
=0: ∏ = A t+ 1 F K ( . , . )+ ( 1−δ ) ] → r t + δ =A t F K ( K t , Lt ) for each
∂ K t+ 1 τ=0 1+ rτ τ =0 1+r τ
t=1,2,3 , …( NOT t=0)
Firms face a trade-off: using more capital costs interest rate plus discount rate (marginal cost of
capital). The benefits are the marginal producitivity of capital. Profit maximisation implies that
marginal costs equal marginal benefits. If the marginal product of capital exceeds the costs, firms want
to invest more, increasing the demand for loanable funds, increasing the interest rate and lowering the
marginal productivity of capital.
Labour demand: Differentiation with respect to Lt gives
t
∂φ 1
=0 : ∏ [ At F L ( K t , Lt )−wt ]=0 → wt = At F L ( K t , Lt ) for each t=0,1,2 , …
∂ Lt τ =0 1+r τ
Marginal cost of labour (the wage) equals the marginal productivity of labour. If the marginal
product of labour exceeds the costs, firms hire more labour, which lowers marginal productivity and
(potentially) increases wages.
Consumers
Optimisation: Consumers maximise present value of utility given current assets, current and future
income and interest rates. Labour supply is inelastic and standardised to one ( Lt =1). The only savings
possibility is capital, which generates a return of r t .
∞
For log-utility we have max ¿ ∑ β ln C t subject to K t +1=( 1+r t ) K t + w t −C t
t
t=0
−ρ
Note: the book uses e instead of β .
The Langragian: One objective, with a constraint for every (future) period. Denote the Lagrangian
∞
multiplier of period t’s constraint by λ t. L=∑ β lnC t −¿ λt [ K t+1 −( 1+r t ) K t −w t +C t ] ¿.
t
t =0
2
, ∂L βt
FOC: =0: =λ t
∂C t Ct
∂L ∂L
=0: λt =λt +1 ( 1+ r t +1 ) =0: K t+1 =( 1+r t ) K t +w t−C t
∂ K t+ 1 ∂ λt
Consumers: Solution
The Euler equation: Combining the first and second first order conditions gives:
1
Ct
=β
1
( )
C t +1
(1+r t +1)LHS: the marginal utility of one unit current consumption.
RHS: the expected marginal utility of saving one unit current consumption and using the savings next
period.
This difference equation describes the optimal path of consumption. It is usually called the Euler
equation. Euler says that someone must be indifferent between consuming one unit more today and
saving that unit and consuming it in the future.
Equilibrium
Labour market: Everybody supplies one unit of labour. Standardizes mass of people to one. Lt =1
Goods market: Production is used for consumption, investment and government consumption:
Y t =C t + I t +G t Combined with capital accumulation identity:
K t +1=( 1−δ ) K t −Ct −G t Capital market equilibrium already imposed (and it clears by Walras’ law).
The Full Model
Labour demand: w t = A t F L ( K t , Lt )
Capital demand: r t + δ =A t F K ( K t , Lt )
1 1
Euler equation:
Ct
=β
Ct +1
( 1+r t +1 )
Production function: Y t = A t F ( K t , Lt )
Goods market: K t +1=( 1−δ ) K t +Y t−C t −G t
Labour market: Lt =1
with K 0 , A t and G t (t=0,1,2 , …) exogenous.
A minor problem is that given K 0 , A t and G t these six equations determine Y t , w t , r t , K t +1 , Lt and
C t. Unfortunately, an explicit solution is very hard to find.
Solutions:
- Graphical analysis (phase diagram, only qualitative answers);
- Special cases;
- Approximate (log-linearize) general solution;
- Approximate using computer.
Graphical Analysis (Phase Diagram)
Core model: two dynamic equations. Drop Lt ( ¿1), simplify and the dynamics of this system are
given by the Euler equation and the capital accumulation equation.
3
, Ct +1
Ct [ ]
=β ( At +1 F K ( K t +1 ) +1−δ ) Phase Diagram
Dynamics of consumption: The Euler equation:
Ct +1
Ct [ ]
=β ( At +1 F K ( K t +1 ) +1−δ ) There is one level of the capital stock K* for which consumption is
¿
stable according to the Euler equation, so the C t+ 1=C t line is vertical. To the left, K < K , so F ' ( K ¿ ).
In order for the Euler equation to hold it must be that C t+ 1>C t , so consumption goes up. To the right
the opposite must hold and consumption goes down.
Dynamics of the capital stock: K t +1−K t =−δ K t + A t F ( K t ) −C t−Gt
Start with the K t +1=K t line: C t= A t F ( K t )−δ K t−G t . This line is humped shaped because of the
properties of the production function. Above this line: C> F ( K )−δK +G ; society consumes too much
and cannot fully replace all depreciations, so the capital stock diminishes ( K t +1 < K t ). Below this line
the opposite happens and the capital stock increases.
Construction:
Saddle Path:
The line SS is a stable manifold, a saddle path. Any
point no on this line is not attainable, a waste of
resources or unsustainable.
4
