Ramsey-Cass-Koopmans model.pptx

Ramsey-Cass-Koopmans model
Haile Girma (Assistant Professor)
Department of Economics
Salale University

Ramsey-Cass-Koopmans growth model
(consumption smoothening)
 In the Solow-Swan model, saving rate and,hence, the ratio of
consumption to income are exogenous and constant.
 The overall amount of investment in the economy was still
given by the saving of families, and that saving remained
exogenous.
 Not useful to study how the economy reacted to changes in
interest rates, tax rates, or other variables.
 We need a complete picture of the process of economic growth
– allow for the path of consumption and, hence, the saving
rate to be determined by optimizing households and firms
that interact on competitive markets.
 Households choose consumption and saving to maximize
utility subject to an inter-temporal budget constraint
• Names: Frank Ramsey, Tjalling Koopmans, David Cass

Key idea:
 Replace ad hoc savings [consumption] function by forward-
looking theory based utility maximization
 Specification of consumer behavior is a key element in the
Ramsey growth model, as constructed by Ramsey (1928) and
refined by Cass (1965) and Koopmans (1965).
– Hence the name Ramsey-Cass-Koopmans
 This model differs from the Solow-Swan growth model only in
one crucial respect:
– It explicitly models the consumer side and endogenizes
savings.
 In other words, it allows consumer optimization.

1. Representative consumer (RC)
Assumptions:
 Infinitely lived households;
 Identical households; each hh:
– has the same preference parameters,
– faces the same wage rate (because all workers are equally
productive),
– begins with the same assets per person, and has the same rate
of population growth.
 Use of representative-agent framework, in which the equilibrium
derives from the choices of a single household-heterogeneity
issues!

 A representative household with instantaneous utility function
 With properties:
– u(c(t)) is strictly increasing, concave, twice continuously
differentiable
– positive but diminishing marginal felicity of consumption.
 u(c) satisfies Inada conditions:
 Labour supply is exogenous and grows exponentially (with initial
labour equals 1):
and
 All members of the household supply their labor inelastically.
 Each adult supplies inelastically one unit of labor services per unit
of time.

 Households hold assets in the form of ownership claims on
capital or as loans.
– Negative loans represent debts.
 Households can lend to and borrow from other households, at
interest rate, r (t)
– but the representative household will end up holding zero net
loans in equilibrium.
 Households are competitive in that each takes as given the
interest rate, r(t), and the wage rate, w(t), paid per unit of labor
services.
 Sources of income:
– interest income plus wage income
 Use of income:
– consumption plus savings [asset accumulation]

 The household is fully altruistic towards all of its future
members, and always makes the allocations of consumption
(among household members) cooperatively.
 This implies that the objective function of each household at time
t = 0, U(0), can be written as:
Where, c(t) is consumption per capita at time t, i.e.
– Each household member will have an equal consumption
 ρ is the subjective discount rate and is assumed to be the same
across generations
– The effective discount rate is ρ−n

Notice that:
– the household will receive a utility of u(c(t)) per household
member at time t, or a total utility of
– Utility at time t is discounted back to time 0 with a discount rate
of .
 We also assume throughout that
- Ensures that in the model without growth, discounted utility is finite.
- Otherwise, the utility function would have infinite value, and standard
optimization techniques would not be useful in characterizing optimal
plans.
 A positive value of ρ (ρ>0) indicates parental “selfishness”
– Suppose that starting from a point at which the levels of
consumption per person in each generation are the same.
– Then parents prefer a unit of their own consumption to a unit of
their children’s consumption.

 No technological progress
 Factor and product markets are competitive.
 Production possibilities set of the economy:
 Standard constant returns to scale and Inada assumptions still
hold.
 Per capita production function f(.)
 Where

 Competitive factor markets imply:
 And
 Households use the income that they do not consume to
accumulate more assets
 Denote asset holdings of the representative household at time t by
A(t).
 Then,
 r(t) is the risk-free market flow rate of return on assets, and
 w(t)L(t) is the flow of labor income earnings of the household.

 Defining per capita assets as:
 To get:
 Household assets can consist of capital stock, K(t), which they
rent to firms and government bonds, B(t).
 With uncertainty, households would have a portfolio choice
between K(t) and riskless bonds.
 With incomplete markets, bonds allow households to smooth
idiosyncratic shocks. But for now no need. Why?
– There is no government!
 Market clearing condition:
 
 
 
 
 
 
 
 
 
A t a t A t L t
a t
L t a t A t L t
   

 No uncertainty and depreciation rate of δ, the market rate of
return on assets is:
The Budget Constraint
The differential equation:
Is a flow constraint.
• It is just an identity;
– hhs could accumulate debt indefinitely
• If the household can borrow unlimited amount at market interest
rate, it has an incentive to pursue a Ponzi-game.
– The household can borrow to finance current consumption and
then use future borrowings to roll over the principal and pay
all the interest.

 In this case, the household’s debt grows forever at the rate of
interest,r(t).
 To rule out chain-letter possibilities, we assume that the credit
market imposes a constraint on the amount of borrowing.
 The appropriate restriction turns out to be that the present value of
assets must be asymptotically nonnegative:
 Consider the case of borrowing by households
 Infinite-lived households tend to accumulate debt by borrowing
and never making payments for principal or interest.

 Naturally, the credit market rules out this chain-letter finance
schemes in which a household’s debt grows forever at the rate r
or higher.
– In order to borrow on this perpetual basis, households would
have to find willing lenders
– Other households that were willing to hold positive assets that
grew at the rate r or higher.
 Households will be unwilling to absorb assets asymptotically at
such a high rate.
– It would be suboptimal for households to accumulate positive
assets forever at the rate r or higher,
– because utility would increase if these assets were instead
consumed in finite time.

 Household maximization
• Set up the current value of Hamiltonian function
• with state variable a, control variable c and current-value costate
variable μ.
• It represents the value of an increment of income received at time t
in units of utils at time 0.
• FOCs:
(i)
(ii)
 
 
   
 
Ĥ
t t r t n
a t
 

    


• The transversality condition is:
• What is transversality condition?
– The transversality condition for an infinite horizon dynamic
optimization problem is the boundary condition determining a
solution to the problem's first-order conditions together with the
initial condition.
– The transversality condition requires the present value of the
state variables to converge to zero as the planning horizon
recedes towards infinity
• Intuition:
• The transversality condition ensures that the individual would
never want to ‘die’ with positive wealth.
– An optimizing agents do not want to have any valuable assets
left over at the end.

• From (ii), we obtain,
• The multiplier changes depending on whether the rate of return
on assets is currently greater than or less than the discount rate of
the household.
• The first necessary condition above implies that

The Euler Equation
• Differentiate equation (i) with respect to time and divide by), ,
we get the basic condition for choosing consumption over time:
• Upon substitution into (ii), we get the famous consumer Euler
equation:
• Where
• is the elasticity of the marginal utility u’(c(t)).

• Consumption will grow over time when the discount rate is less
than the rate of return on assets.
• It also specifies the speed at which consumption will grow in
response to a gap between this rate of return and the discount rate.
• Elasticity of marginal utility is the inverse of the intertemporal
elasticity of substitution.
• The elasticity between the dates t and s > t is defined as:
• As s approaches t, we get:

Equilibrium Prices
• The market rate of return for consumers, r (t), is given by:
• Substituting this into the consumer’s problem, we have:
• It is simply the equilibrium version of the consumption growth
equation.

Optimal Growth
• Capital and consumption path chosen by a benevolent social
planner trying to achieve a Pareto optimal outcome.
• The optimal growth problem simply involves the maximization of
the utility of the representative household subject to technology
and feasibility constraints.
• Subject to
• and k (0) > 0.

• Set up the current-value Hamiltonian:
• With state variable k, control variable c and current-value costate
variable μ.
• The necessary conditions for an optimal path are:

• It is straightforward to see that these optimality conditions imply:
• The transversality condition
• Both are identical with the previous results
– This establishes that the competitive equilibrium is a Pareto
optimum
– The equilibrium is Pareto optimal and coincides with the
optimal growth path maximizing the utility of the
representative household.

Steady-State Equilibrium
• Characterize the steady-state equilibrium and optimal allocations
• A steady state equilibrium is an equilibrium path in which capital-
labor ratio, consumption and output are constant.
• Since f(k∗) > 0, we must have a capital-labor ratio k∗ such that:
• The steady-state capital-labor ratio only as a function of the
production function, the discount rate and the depreciation rate.
• This corresponds to the modified golden rule:
• The interest rate equals:  
 
' KR
f k t r
   
    

• The modified golden rule involves a level of the capital stock that
does not maximize steady-state consumption
– This is due to discounting (i.e. earlier consumption is preferred
to later consumption).
– The objective is not to maximize steady state consumption
rather giving higher weight to earlier consumption
• Given k∗, the steady-state consumption level is:
• Which is similar to the consumption level in the basic Solow
model.

Transitional Dynamics
• Unlike the Solow-Swan model, equilibrium is determined by two
differential equations:
• Moreover, we have an initial condition k(0) > 0, also a boundary
condition at infinity:
• The intersection of and define the steady state (next
slide).
• The former is vertical since a unique level of k* can keep per
consumption constant ( ).

Dynamics of c
• since all households are the same, the evolution of C for the entire
economy is:
• There are two ways for to be zero:
(i) c(t)=0; corresponds to the horizontal axis
(ii) which is a vertical line at k*.
• Ignore the first case, and focus on the second.
• This provide the optimal level of k, denoted by k*
 
 
' 0
f k t  
  
 
 
'
f k t  
 

• When k exceeds k*,
• The opposite holds when k is less than k*.
• This is summarized in Figure 1 (next slide)
• c is rising if k<k* and declining if k>k*.
• The occurs at k=k* and c is constant for this value of k.
 
 
' 0
f k t c
 
   
0
c 

Dynamics of k
• The dynamics of the economy is given by:
• Notice that implies that
• Consumption equals the difference between actual output and
break-even investment.
• c is increasing in k until
• The last expression gives the golden rule of capital per worker (k*)
• When exceeds that yields , k is decreasing, and vice versa .
  0
k t 
   
     
c t f k t n k t

  
 
   
'
f k t n r t n
 
    
 
 
0
dc t
dk t
 
  0
k t 

Figure 2: Dynamics of k
When k is large and break-even investment exceeds total output,
for all positive values of c.
  0
k t 

• The dynamics of c and k: bringing the two together
• The arrows show direction of motion of c and k
• Consider the following: points to the left of and above
• The former is positive the latter is negative
• c is rising while k is falling
• On the curves, only one of c and k is changing.
• Example: on the line and above locus, c is constant
and k is falling.
• At point E when holds, there is no movement.
  0
c t    0
k t 
    0
c t k t
 
  0
c t    0
k t 
    0
c t k t
 

• The economy can converge to this steady state if it starts in two
of the four quadrants in which the two schedules divide the
space.
• Given this direction of movements, it is clear that there exists a
unique stable arm, the one-dimensional manifold tending to the
steady state.
• All points away from this stable arm diverge, and eventually
reach zero consumption or zero capital stock

• Consider the following:
• If initial consumption, c(0), started above this stable arm, say at
c’(0), the capital stock would reach 0 in finite time, while
consumption would remain positive.
• But this would violate feasibility.
– Therefore, initial values of consumption above this stable
arm cannot be part of the equilibrium
• If the initial level of consumption were below it, for example, at
c’’(0), consumption would reach zero.

• Thus capital would accumulate continuously until the maximum level
of capital (reached with zero consumption)
Continuous capital accumulation towards with no consumption
would violate the transversality condition.
• There exists a unique equilibrium path starting from any k(0)>0 and
converging to the unique steady-state (k∗, c∗) with k∗.

Ramsey-Cass-Koopmans model.pptx

Ramsey-Cass-Koopmans model.pptx

More Related Content

What's hot (20)

Similar to Ramsey-Cass-Koopmans model.pptx (20)

Recently uploaded (20)

Ramsey-Cass-Koopmans model.pptx