$©$ September 21, 2020, Christopher D. Carroll RamseyCassKoopmans

The Ramsey/Cass-Koopmans (RCK) Model

Ramsey (1928), followed much later by Cass (1965) and Koopmans (1965), formulated the canonical model of optimal growth for an economy with exogenous ‘labor-augmenting’ technological progress.

1 The Budget Constraint

The economy has a perfectly competitive production sector that uses a Cobb-Douglas aggregate production function

α 1− α Y = F(K, L ) = K (ZL )

(1)

to produce output using capital and labor.¹ Labor hours (the same as population) increases exogenously at a constant rate²

˙L ∕L = ξ

(2)

and $Z$ is an index of labor productivity that grows at rate

Z˙∕Z = ϕ.

(3)

Thus, technological progress allows each worker to produce perpetually more as time goes by with the same amount of physical capital.³ The quantity $ZL$ is known as the number of ‘eﬃciency units’ of labor in the economy.

Aggregate capital accumulates according to

K˙ = Y − C − δK.

(4)

Lower case variables are the upper case version divided by eﬃciency units, i.e.

y = Y ∕(ZL ) = K α(ZL )1− α∕ (ZL ) α = (K ∕ZL ) α = k .

(5)

Note that

( ) ( ) ˙ ˙ ˙ ˙ dk- KZL---−--K-(ZL---+-Z-L-)- k ≡ dt = (ZL )2 = K˙ ∕ZL − k(Z˙∕Z + L˙∕L ) = K˙ ∕ZL − (ϕ + ξ)k

(6)

which means that (4) can be divided by $ZL$ and becomes

K˙ ∕ZL = y − c − δk ˙k + (ϕ + ξ)k = f(k) − c − δk ˙k = f(k) − c − (ϕ + ξ + δ)k.

(7)

A steady-state will be a point where $˙k = 0$ .

Equation (7) yields a ﬁrst candidate for an optimal steady-state of the growth model: It seems reasonable to argue that the best possible steady-state is the one that maximizes $c$ . This is the “golden rule” optimality condition of Phelps (1961), an article well worth reading; this is one of the chief contributions for which Phelps won the Nobel prize.

2 The Social Planner’s Problem

Now suppose that there is a social planner whose goal is to maximize the discounted sum of CRRA utility from per-capita consumption:

∫ ∞ ( (C ∕L )1− ρ ) max ----------- e− 𝜗t. 0 1 − ρ

(8)

But $C ∕L = Z(C ∕ZL ) = Zc$ . Recall that for a variable growing at rate $ϕ$ ,

Zt = Z0e ϕt

(9)

so if the economy started oﬀ in period 0 with productivity $Z0$ , by date $t$ we can rewrite

C ∕L = cZ = cZ eϕt. 0

(10)

Using (10) and the other results above, we can rewrite the social planner’s objective function as

∫ ∞ ( ) ∫ ∞ ( ) (Zc-)1−-ρ − 𝜗t 1− ρ -c1− ρ- − 𝜗t (1− ρ)ϕt e = Z0 e e 0 1 − ρ ∫ 0 1 − ρ 1− ρ ∞ ((1− ρ)ϕ− 𝜗)t = Z0 u(c)e . 0

(11)

Thus, deﬁning $ν = 𝜗 − (1 − ρ)ϕ$ and normalizing the initial level of productivity to $Z0 = 1$ , the complete optimization problem can be formulated as

∫ ∞ max u (c)e− νt 0

(12)

subject to

k˙= k α − c − (ξ + ϕ + δ)k,

(13)

which has a discounted Hamiltonian representation

ℋ (k, c,λ ) = u (c) + (k α − c − (ξ + ϕ + δ)k)λ.

(14)

The ﬁrst discounted Hamiltonian optimization condition requires $∂ ℋ ∕ ∂c = 0$ :

− ρ c = λ − ρ− 1 − ρc ˙c = ˙λ.

(15)

The second discounted Hamiltonian optimization condition requires:

λ˙= ν λ − ∂ℋ ∕∂k ′ = ν λ − λ(f (k ) − (ξ + ϕ + δ )) ˙ ′ λ ∕λ = (ν + (ξ + ϕ + δ) − f (k )) ( − ρ− 1 ) −-ρc-----c˙ = (ν + (ξ + ϕ + δ) − f′(k )) c− ρ ′ ρ˙c∕c = (f (k ) − ν − (ξ + ϕ + δ )) − 1 ′ ˙c∕c = ρ (f◟ (k)-−--(ξ◝◜+--ϕ-+-δ-)◞− ν ) ≡ ´r

(16)

where the deﬁnition of $´r$ is motivated by thinking of $′ f (k) − (ξ + ϕ + δ )$ as the interest rate net of depreciation and dilution.

This is called the “modiﬁed golden rule” (or sometimes the “Keynes-Ramsey rule” because it was originally derived by Ramsey with an explanation attributed to Keynes).

Thus, we end up with an Euler equation for consumption growth that is just like the Euler equation in the perfect foresight partial equilibrium consumption model, except that now the relevant interest rate can vary over time as $f′(k)$ varies.

Substituting in the modiﬁed time preference rate gives

c˙∕c = ρ− 1(f ′(k ) − (ξ + δ + ϕ ) − 𝜗 + (1 − ρ)ϕ ) = ρ− 1(f ′(k ) − (ξ + δ) − 𝜗 − ρ ϕ),

(17)

and ﬁnally note that deﬁning per capita consumption $χ = C ∕L$ so that $c = χZ − 1$ ,

˙− 1 − 1 − 1 ˙ ˙c = (χZ ) = χ˙Z − (χZ )Z ∕Z ˙− 1 − 1 c˙∕c = (χZ )∕(χZ ) = χ˙∕χ − ϕ

(18)

and since (17) can be written

− 1 ′ c˙∕c = ρ (f(k ) − (ξ + δ) − 𝜗) − ϕ,

(19)

we have

− 1 ′ ˙χ∕ χ = ρ (f (k) − (ξ + δ ) − 𝜗 )

(20)

so the formula for per capita consumption growth (as a function of $k$ ) is identical to the model with no growth (equation (17) with $ϕ = 0$ ). Any important diﬀerences between the no-growth model and the model with growth therefore must come through the channel of diﬀerences in $k$ .

3 The Steady State

The assumption of labor augmenting technological progress was made because it implies that in steady-state, per-capita consumption, income, and capital all grow at rate $ϕ$ .⁴

$c˙∕c = 0$ implies that at the steady-state value of $ˇk$ ,

′ˇ f(k ) = 𝜗 + ξ + δ + ρϕ α ˇkα− 1 = 𝜗 + ξ + δ + ρϕ ( ) α1−1 ˇk = 𝜗-+--ξ +-δ-+--ρϕ-- α ( ) -1- α 1−α ˇk = ------------------ 𝜗 + ξ + δ + ρϕ

(21)

Thus, the steady-state $k$ will be higher if capital is more productive ( $α$ is higher), and will be lower if consumers are more impatient, population growth is faster, depreciation is greater, or technological progress occurs more rapidly.

4 A Phase Diagram

While the RCK model has an analytical solution for its steady-state, it does not have an analytical solution for the transition to the steady-state. The usual method for analyzing models of this kind is a phase diagram in $c$ and $k$ . The ﬁrst step in constructing the phase diagram is to take the diﬀerential equations that describe the system and ﬁnd the points where they are zero. Thus, from (7) we have that $˙k = 0$ implies

c = f(k) − (ϕ + ξ + δ )k

(22)

and we have already solved for the (constant) $ˇk$ that characterizes the $c˙∕c = 0$ locus. These can be combined to generate the borders between the phases in the phase diagram, as illustrated in ﬁgure 1.

5 Transition

Actually, as stated so far, the solution to the problem is very simple: The consumer should spend an inﬁnite amount in every period. This solution is not ruled out by anything we have yet assumed (except possibly the fact that once $k$ becomes negative the production function is undeﬁned).

Obviously, this is not the solution we are looking for. What is missing is that we have not imposed anything corresponding to the intertemporal budget constraint. In this context, the IBC takes the form of a “transversality condition,”

∫t lim λte(ϕ+ ξ)t− 0 rτdτkt = 0. t→ ∞

(23)

The intuitive purpose of this unintuitive equation is basically to prevent the capital stock from becoming negative or inﬁnity as time goes by. Obviously a capital stock that was negative for the entire future could not satisfy the equation. And a capital stock that is too large will have an arbitrarily small interest rate, which will result in the LHS of the TVC being a positive number, again failing to satisfy the TVC.

Figure 2 shows three paths for $c$ and $k$ that satisfy (17) and (7). The topmost path, however, is clearly on a trajectory toward zero then negative $k$ , while the bottommost path is heading toward an inﬁnite $k$ . Only the middle path, labelled the “saddle path,” satisﬁes both (17) and (7) as well as the TVC (23).

6 Interactive Notebooks

An explicit numerical solution to the Ramsey problem, with a description of a solution method and its mathematical/computational underpinnings, is available here.

References

Cass, David (1965): “Optimum growth in an aggregative model of capital accumulation,” Review of Economic Studies, 32, 233–240.

Grossman, Gene M., Elhanan Helpman, Ezra Oberfield, and Thomas Sampson (2016): “Balanced Growth Despite Uzawa,” Working Paper 21861, National Bureau of Economic Research.

Koopmans, Tjalling C. (1965): “On the concept of optimal economic growth,” in (Study Week on the) Econometric Approach to Development Planning, chap. 4, pp. 225–87. North-Holland Publishing Co., Amsterdam.

Phelps, Edmund S. (1961): “The Golden Rule of Accumulation,” American Economic Review, pp. 638–642, Available at http://teaching.ust.hk/~econ343/PAPERS/EdmundPhelps-TheGoldenRuleofAccumulation-AfableforGrowthMen.pdf.

Ramsey, Frank (1928): “A Mathematical Theory of Saving,” Economic Journal, 38(152), 543–559.

Figure 1: $c˙∕c = 0$ and $k˙= 0$ Loci

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Figure 2:Transition to the Steady State

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