2. 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
3. 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.
4. 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!
5. 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.
6. 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]
7. 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
8. 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.
9. 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
10. 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.
11. 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
12. 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.
13. 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.
14. 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.
15. 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
16. • 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.
17. • 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
18. 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)).
19. • 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:
20. 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.
21. 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.
22. • 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:
23. • 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.
24. 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
25. • 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.
26. 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 ( ).
27. 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
28. • 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
30. 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
31. 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
32. • 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
34. • 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
35. • 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.
36. • 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∗.
