THE RAMSEY OR CASS-KOOPMANS MODEL (RCK MODEL)

THE NEOCLASSICAL INFINITE-HORIZON AND OVERLAPPING-GENERATIONS MODELS

THE RAMSEY OR CASS-KOOPMANS MODEL (RCK MODEL)

History:-
This model was developed by Ramsey (1928), Koopmans (1965) and Cass(1965).in this model Competitive firms rent capital and hire labour to produce and sell output, and a fixed number of in- finitely lived households supply labour, hold capital, consume, and save. It avoids all market imperfections and all issues raised by heterogeneous households and links among generations.

Assumptions
Firms
There are a large number of identical firms. Each has access to the pro- duction function Y = F (K,AL) . The firms hire workers and rent capital in competitive factor markets, and sell their output in a competitive output market. Firms take A as given; as in the Solow model, A grows exogenously at rate g. The firms maximize profits. They are owned by the households, so any profits they earn accrue to the households.
Households
There are also a large number of identical households. The size of each household grows at rate n. Each member of the household supplies 1 unit of labour at every point in time. The households have initial capital holding equal to K(0)/H. where K(0) is the initial amount of capital in the economy and H is the number of households. There is no depreciation of capital. Also the households divides its income at each point of time between consumption and saving so as to maximise its lifetime utility.
The households utility function takes the form:-

C(t):- consumption of each member of the household at time t.
U(.):- Each members utility at given date. (Instantaneous utility function).
L(t):- Total population of the economy.
ρ :- the discount rate. The greater is ρ, the less the household values future consumption relative to current consumption. ( ρ > n)
L(t)/H :- The number of members of the household.
The instantaneous utility function takes the form


This function form is required for the economy to converge at balanced growth path. It is known as Constant-relative-risk-aversion(CRRA) utility.
The reason for the name is that the coefficient of relative risk aversion (which is defined as −Cu′′(C)/u′(C)) for this utility function is θ, and thus is independent of C.
The Behaviour of Households and Firms
Firms
At each point in time they employ the stocks of labour and capital, pay them their marginal products, and sell the resulting output. As production function has constant returns and the economy is competitive, firms earn zero profit.
As there is no depreciation, the real rate of return on capital equals its earnings per unit time. Thus real interest rate at time t is
R(t) = f’(k(t)) ………………….(3)
Labour’s marginal product is ∂F(K,AL)/∂L which equals A∂F(K,AL)/∂AL.
Thus real wage at t equals,
W(t) = A(t)[f(k(t)) – k(t)f’(k(t))] ………………(4)
The wage per unit of effective labour is therefore
W(t) = f(k(t)) - k(t)f’(k(t)) ………………….(5)
Let us denote the asset holding of the representative household at time t by A(t).
Then the law of motion for the total assets of the household is
Ȧ(t) = r(t)A(t) + w(t)L(t) - c(t)L(t) ……………..(6) (The LHS is A dot (t))
Where c(t)= consumption per capital of the household.
r(t) = risk free market rate of return on assets
w(t)L(t) = flow of labour income earning of the household.
Per capital asset = a(t) = A(t)/L(t)
Dividing eqn 6 by L(t) and using the fact that population grows at constant rate of n we get:
ȧ(t) = (r(t)-n)a(t) + w(t) -c(t) …………………(7)
As in this model there is no role of govt. bond so B(t) =0, and thus mkt clearing implies that assets per capital must be equal to the capital stock per capital.
a(t) = k(t) ………………………(8)
As the capital depreciates at rate
the mkt rate of return on assets is
r(t) = R(t) -δ ………………(9)
The natural debt limit and the No-Ponzi game condition
The natural debt limit requires that a(t) should never become so negative that the household cannot repay the debt even if it choose zero consumption.
So, the natural debt limit is given as,

And the No-Ponzi condition states that the representative household does not asymptotically tend to a negative wealth. The equation is given as:-

Definition of Equilibrium:-
A competitive equilibrium of the neoclassical growth model consists of path of consumption (C), capital stock (K), wage rates (W) and rental rates of capital (r), such that the representative household maximises its utility given initial assets holding K(0)>0 and taking the time path of price of given. firms maximises profits taking the time path of factor prices as given and factor prices are such that all markets clear.
Household Maximization :-
Here we will set the equation in Hamiltonian form. Hamiltonian is a sum of per period discounted utility from consumption and value of asset accumulation.
To know about the HAMILTONIAN form in detail refer to Daron Acemoglu’s chapter-7, page-329-340 of Introduction to Modern Economic Growth.
By using the equation 1 and the equation of law of motion of capital (7) and the condition for No-Ponzi game (11) we set the current-value Hamiltonian as:-

Here a is state variable. (A state variable is one of the set of variables that are used to describe the mathematical "state" of a dynamical system).
And C is control variable. (A variable that is held constant in order to assess or clarify the relationship between two other variables.)
Now by taking out the FOC with respect to C and a in equation 12 we get.

Using the transversality condition which is written in current value cost we get:

This condition (eqn- 20) states that consumption per worker is rising if the real return exceeds the rate at which the house- hold discounts future consumption, and is falling if the reverse holds.
Where

Equation 21 is known as the elasticity of the marginal utility u’(c(t)).
Since a(t) = k(t), the transversality condition of the representative household can alternatively be written as

It emphasises that the transversality condition requires the discounted market value of the capital stock in the very far future to be equal to zero.
Also, using equation 3 and 9 we get,
r(t) = f’(k(t)) - δ …………….(23)
substituting eqn 22 into household maximisation problem we obtain

The Dynamics of the Economy:
The Dynamics of C:-

Figure 1. The Dynamics of C
Since all households are the same, equation 24 describes the evolution of c not just for a single household but for the economy as a whole. Thus c ̇ is zero when f ′(k) equals
Let k∗ denote this level of k. When k exceeds k∗, f ′(k) is less than
and so c ̇ is negative; when k is less than k∗, c ̇ is positive.
The Dynamics of K:

Figure 2. The Dynamics of K
k ̇(t ) = f (k (t )) − c (t ) − (n + δ)k (t ).
For a given k ,the level of c that implies k ̇=0 is given by f(k)−(n+δ)k; k ̇ is zero when consumption equals the difference between the actual output and break-even investment lines. This value of c is increasing in k until f ′(k) = n +δ (the golden-rule level of k) and is then decreasing. When c exceeds the level that yields k ̇ = 0, k is falling; when c is less than this level, k is rising. For k sufficiently large, break-even investment exceeds total output, and so k ̇ is negative for all positive values of c.
The Phase Diagram :-

Figure 3. The Dynamics of c and K.
Figure 3 combines the information in Figures 1 and 2. The arrows now show the directions of motion of both c and k. To the left of the c ̇ = 0 locus and above the k ̇ = 0 locus, for example, c ̇ is positive and k ̇ negative. Thus c is rising and k falling, and so the arrows point up and to the left. The arrows in the other sections of the diagram are based on similar reasoning. On the c ̇= 0 and k ̇= 0 curves, only one of c and k is changing. On the c ̇= 0 line above the k ̇ = 0 locus, for example, c is constant and k is falling; thus the arrow points to the left. Finally, at Point E both c ̇ and k ̇ are zero; thus there is no movement from this point.
The Saddle Path:-
For any positive initial level of k, there is a unique initial level of c that is consistent with households’ intertemporal optimization, the dynam- ics of the capital stock, households’ budget constraint, and the requirement that k not be negative. The function giving this initial c as a function of k is known as the saddle path; it is shown in Figure 4 For any starting value for k, the initial c must be the value on the saddle path. The economy then moves along the saddle path to Point E.

Figure 4. the saddle path

Limitations of RCK model

While this model is an important milestone in our study of the mechanics of economic growth, as with the Solow growth model, the focus is on the proximate causes of these differences–we are still looking at differences in saving rates, investments in human capital and technology, perhaps as determined by preferences and other dimensions of technology (such as the rate of labor-augmenting technological change). It is therefore important to understand that this model, by itself, does not enable us to answer questions about the fundamental causes of economic growth. What it does, however, is to clarify the nature of the economic decisions so that we are in a better position to ask such questions. 

It offers no economic explanation for persistent cross-country differences in growth, other than to say that some countries must have a faster rate of technological progress than others. 

References:-

Acemoglu, Daron (2009). "The Solow Growth Model". Introduction to Modern Economic Growth. Princeton: Princeton University Press.

Romer, David (2006). Advanced Macroeconomics. McGraw-Hill.