Download THE RAMSEY GROWTH MODEL

THE RAMSEY GROWTH MODEL
Sarantis Kalyvitis
Contents:
- Introduction
- The decentralized problem
- The social planner problem
Some general comments about the Solow model (advantages and deficiencies):
• This is the first general equilibrium model with production side.
• It has very strong predictions that are empirically testable.
• Lacks microfoundations (saving is not determined exogenously)
• Exogenous technological progress explains all.
Introducing consumer optimization in the neoclassical model:
The Ramsey model
Advantages:
• Consumption (and saving) is determined endogenously
• The decentralized equilibrium (problem of households and firms) can be
compared to the socially desirable outcome (Pareto optimality)
But: stronger analytical tools are needed, because the problem of consumers is
intertemporal!
General setup of the decentralized problem:
Demand side: Households maximize intertemporal utility subject to their budget
constraint
Supply side: Firms maximize profits subject to the factor accumulation
constraint
Equilibrium: Supply equals demand (all in real terms)
General setup of the social planner problem:
The social planner maximizes utility of households subject to aggregate resource
allocation constraint of the economy.
⇒ If the solutions coincide, then the outcome is Pareto optimal.
1
The Household in the Ramsey model
Assumptions:
• One infinitely-lived household that maximizes intertemporal utility
• The household receives wage income in exchange for its labor services
and interest income for its accumulated assets
• The income acquired is directed in purchasing goods for consumption and
in savings through the accumulation of additional assets
• Population growth rate is constant (equal to n) and at time t=0 there is
only one individual in the economy (i.e. L0 = 1), so that the total
population at any time t is given by Lt = e nt
Budget constraint (the change in assets equals the sum of labor and interest
income less consumption):
•
B = wt Lt + rt Bt − Ct
Bt: assets
wt: wage rate and
rt: interest rate.
•
In per capita terms: b = wt + rt bt − nbt − ct
where bt =
Bt
Ct
, ct = .
Lt
Lt
One important constraint: households cannot borrow unlimited amounts to
finance arbitrarily high levels of consumption.
⇒ the present value of current and future assets must be asymptotically nonnegative
⇔ households cannot borrow infinitely until the end of their economic life cycle
⇔ the households’ debt cannot increase at a rate asymptotically higher than the
interest rate
t


− ∫ ( rs − n ) ds

lim  bt e 0
t →∞ 











≥0
2
Intertemporal utility:
∞
∫
U = u (ct ) Lt e
− ρt
∞
∫
dt = u (ct ) e nt e − ρt dt
0
0
ρ: rate of time preference. A higher ρ implies a smaller desirability of future
consumption in terms of utility compared to utility obtained by current
consumption.
Properties of instantaneous utility function u(ct):
(1) increasing (u´>0) and concave (u´´<0)
u´(c) = ∞ and lim u´(c) = 0 .
(2) satisfies the Inada conditions: lim
c →0
c →∞
Problem of the household: Maximize intertemporal utility subject to the
budget constraint (the Hamiltonian approach)
J = u (ct ) e − ( ρ − n )t + vt [wt + (rt − n) bt − ct ]
vt: present value of the shadow price of income in utility units.
First order conditions for the maximization of the Hamiltonian are:
∂J
= 0 ⇒ v = u ′(c ) e − ( ρ − n ) t
∂c
•
∂J
= −ν
∂b
•
⇒ ν = − ( r − n )v
Transversality condition:
lim(ν t bt ) = 0
t →∞
The value of the representative household’s assets must approach zero as time
approaches infinity ⇔ individuals do not prefer to hold assets perpetually or –if
they do- these assets should have no value in terms of the objective function.
3
From the first-order conditions we get:
•
 u′′(c) c  c
Euler equation: r = ρ − 

 u′(c)  c
.
Households choose consumption so to equate the rate of return to savings, r, to
the ‘rate of return’ to consumption (the r.h.s).
(or: return on future consumption = return on current consumption)
The higher r, the more willing households are to save and shift
consumption in the future.
The higher the rate of return to consumption is, the more willing
households are to sacrifice future consumption for more current
consumption and thereby less current saving.
⇒ Households are indifferent between consumption and saving if the associated
rates of return are equal.
Question: why do households prefer to bring some future consumption into the
present?
they value more current utility compared to the future one, as this is
reflected by the rate of time preference from the right-hand term of the Euler
equation.
•
the future consumption growth
c
c
increases the marginal utility of
consumption.
4
Parametric form for utility function:
constant relative risk aversion (CRRA) utility function: u (c) =
c 1−θ
1−θ
 u ′′(c) 
Elasticity of marginal utility with respect to consumption is − c ′  and
u (c ) 

reflects the coefficient of relative risk aversion.
For the CRRA utility function, this elasticity equals θ (constant for all c)
The CRRA cover interesting cases:
for θ→0, the utility function becomes linear in consumption,
for θ→1, the utility function approximates the logarithmic function log(c).
For the CRRA utility function, the Euler equation takes the simple form:
•
c 1
= (r − ρ )
c θ
Here
Euler equation for CRRA utility
1
gives ‘the elasticity of substitution between consumption at different
θ
time periods’ and reflects the individuals’ willingness to accept deviations from
a uniform pattern of consumption over time. The lower θ (i.e. the higher the
elasticity of substitution is), the slower the decline of the marginal utility in
response to consumption growth, and therefore the more willing are individuals
to accept variations in their consumption.
⇒ The optimizing behavior of individuals leads to a fluctuating growth rate of
per capita consumption, which equals the product between the elasticity of
substitution and the difference between the rate of return (‘return’ to future
consumption) and the rate of time preference (‘impatience’). Households save
more when the rate of return is high, that is, when they are less impatient to
consume today. Hence, more consumption will be brought forward in the future.
This tendency is encouraged when the elasticity of substitution is high (meaning
that future consumption is considered a good ‘substitute’ for actual
consumption.
5
Firms’ behavior in the decentralized economy
Assumptions:
• the economy consists of an large number of identical firms producing a
homogeneous consumption good Υ
• Each firm uses a production technology Υ = F(K,L) that satisfies the
neoclassical properties, and pays wages and rent in exchange for labor and
capital services respectively
• R is the rental price for a unit of capital ⇒ The net rate of return to capital
equals then R-δ=r, with δ denoting the capital depreciation rate, because r
is the interest rate on loans to other households, which are perfect
substitutes with capital.
Problem of the representative firm i: choose the amount of capital Κi and
labor input Li to maximize its profit πi (static problem)
π i = F ( K i , Li ) − (r + δ ) K i − wLi
First-order conditions:
∂π i
=0 ⇒
∂K i
f ′(k ) = r + δ
∂π i
=0 ⇒
∂Li
f (k ) − kf ′(k ) = w
where k =
(marginal product of labor = wage rate)
Ki K
=
since firms are identical.
Li
L
⇒ the marginal product of each input equals its associated cost.
6
Equilibrium in the Ramsey model
In equilibrium the stock of per capita assets equals the stock of capital per
worker, i.e. b=k:
•
⇒ k = f (k ) − (n + δ )k − c
Combining the problem of the firm and the Euler equation:
•
c 1
= ( f ′(k ) − δ − ρ )
c θ
This is a two-equation system that yields the market equilibrium.
In the steady-state growth rates of per capita consumption and the capital stock
•
•
per worker equal zero (i.e. k = c = 0 ) and we have:
c = f (k ) − (n + δ )k
f ′(k ) = δ + ρ
⇒ the growth rate of per capita output is also zero
Result. In the absence of technological progress the Ramsey model predicts
zero growth as in the simple Solow-Swan model.
7
Dynamics in the Ramsey model
•
∂k
= −1
Dynamics of capital accumulation:
∂c
c
•
k =0
k
⇒ an increase in consumption causes the capital accumulation rate to decline:
starting from a positive value, it reaches zero and ends up negative
•
∂c 1
= f ′′(k ) < 0
Dynamics of consumption growth rate:
∂k θ
⇒ when the capital stock increases, the consumption growth rate is diminishing.
c
•
c=0
k
k
8
Graphical solution of the system (phase diagram):
c
•
c=0
c*
c
•
k =0
k
k
•
•
Equilibrium: crossing point of the c = 0 and k = 0 schedules.
For any other point, the time paths of c and k are determined by the system of
equations.
Types of variables: consumption is a ‘jump’ variable (i.e. it is allowed to
take any value), whereas capital is a ‘state’ or ‘predetermined’ variable (i.e.
its current state is determined by the past level).
Type of equilibrium: for any capital stock level, there is a unique
consumption level that drives the economy to equilibrium (saddle path)
9
Variations in the rate of time preference
Example: a decrease in ρ denotes that households have become less impatient
and value future consumption higher.
⇒ households are willing to sacrifice present consumption (similar to assuming
that households increase their saving rate s in the Solow-Swan model).
A change in ρ affects consumption, but has no implications for capital
accumulation ⇒ ρ can only determine the pattern of consumption and saving,
whereas capital accumulation is driven by the production function and by
exogenous parameters, like the depreciation rate and the population growth rate.
Effects of a decline in the rate of time preference
c
•
c=0
•
c′ = 0
c′
•
k =0
c
k
k′
k
•
For a given level of k, a decrease in ρ lowers the consumption growth rate
c,
•
and hence, shifts the
c = 0 schedule to the right.
•
⇒ to maintain consumption constant ( c = 0 ), the capital stock must increase so
that, in the new long-run equilibrium, consumption, capital and output per capita
attain a higher level ( k ′ > k , c ′ > c ).
Driving force of equilibrium: consumption jumps along the saddle path (here
consumption declines), whereas capital per worker remains at the same level k .
The reduction in consumption boosts saving and thereby leads to a increase in
capital stock, until it reaches k ′ . ⇒ income per capita level rises, with positive
consumption growth rate until reaching the higher levels of consumption and
capital ( c ′ ,
k ′ ), and income.
10
The modified ‘golden rule’ of capital accumulation
‘Golden rule’ of capital accumulation in the Solow-Swan model: f ′(k * ) = n + δ .
‘Modified Golden Rule’ of capital accumulation in the Ramsey model:
f ′(k Ramsey ) = δ + ρ
For the transversality condition to hold we need ρ > n ,
Result. In the Ramsey model, the long-run equilibrium level of capital per
worker is lower than the steady-state level predicted by the Solow-Swan
model.
In the Ramsey model households save less than the level prescribed by the
‘golden rule’ since in the Solow-Swan model there in no discounting of future
utility.
c
•
c=0
c*
c
•
k =0
k
k*
k
11
The ‘social planner’ problem
Question: Is the decentralized equilibrium of the economy the first-best outcome
(Pareto optimal)?
The social planner aims at maximizing households’ intertemporal utility subject
to an aggregate budget constraint, which shows how the economy’s output is
allocated to different uses.
In a closed economy with no government, the aggregate outcome Y can then be
used either for consumption C or for investment in physical capital Ι:
•
•
Y = C + I ⇒ F ( K , L) = C + K + δK ⇒ k = f (k ) − (n + δ ) k − c
The Hamiltonian associated to the social planner’s problem is:
c 1−θ ( n − ρ ) t
J=
e
+ ν [ f (k ) − (n + δ ) k − c ]
1−θ
First-order conditions with respect to consumption and capital:
∂J
= 0 ⇒ ν = c −θ e ( n − ρ )t
∂c
•
∂J
= −ν
∂k
•
⇒ ν = ν [n + δ − f ′(k )]
After some manipulation:
•
c 1
= [ f ′(k ) − ( ρ + δ )]
c θ
•
•
Since c = k = 0 in the steady state, equations (6.23) and (6.28) become:
c = f (k ) − (n + δ )k
same as in decentralized equilibrium
f ′(k ) = ρ + δ
same as in decentralized equilibrium
Result. The competitive equilibrium in the Ramsey model is Pareto optimal.
12
Τhe ΑΚ model with consumption optimization
Under a linear production function, the system of differential equations is:
•
k = ( A − δ − n) k − c
•
c 1
= (A − δ − ρ)
c θ
⇒ the growth rate is constant and independent from the level of the capital stock
per worker.
⇒ capital and income per capita increase at the growth rate of consumption.
Result. In the equilibrium of the linear growth model the following relation
holds:
•
•
•
k c y 1
= = = (A − δ − ρ)
k c y θ
Proof: see appendix.
the higher the constant Α (reflecting the state of technology), the faster the
long-run growth is.
the growth rate is negatively affected by the depreciation rate δ (the more
capital is used up, the slower the pace of development)
the growth rate is negatively affected by the discount rate ρ,
the growth rate is negatively affected by the elasticity of the marginal utility
with respect to consumption θ.
Remark: the equilibrium of the AK model is Pareto optimal (replacing the
neoclassical production function with the linear form y=Ak does not distort the
competitive character of the market)