Comparing the Harrod-Domar, Solow and Ramsey growth models and their implications for economic policies Open Access

Economic growth can be considered to be the single most important long-term policy objective of any modern government, both at national and subnational levels. Not surprisingly, this topic has attracted the attention of many of the brightest economic theorists (see, for example, Ramsey, 1928; Harrod, 1939, 1948; Domar, 1946, 1947; Solow, 1956; Uzawa, 1961, 1963, 1965; Lucas, 1988; Romer, 1990). Economic growth was also a subject of interest to Ngo Van Long (see, for example, Long, 1982; Long & Wong, 1997; Long, Nishimura, & Shimomura, 1997; Long & Shimomura, 2004), to whom this paper is dedicated.

The principal aim of this paper is to review three basic theoretical growth models, namely the Harrod-Domar model, the Solow model and the Ramsey model and examine their implications for economic policies. We first introduce a closed economy which operates in a discrete infinite time horizon. In this context, the Harrod-Domar and Solow models are presented successively. We then consider a discrete-time version of Ramsey model (for continuous-time versions of the Ramsey model, refer to Cass, 1965; Koopmans, 1965). For simplicity and tractability, we assume the one-period utility function of the representative consumer in the Ramsey model has a very simple form u(c) = ln(c), c > 0.

For economic policies and economic shocks, we focus on:

1. Public spending. In particular, with the Ramsey model, we show that the impacts of public spending change with the date of the announcement of the policy

2. Public debt and its repayments

3. Covid-type shock

The main results we obtain from these comparisons are as follows:

1. In the Harrod-Domar model, the saving rate is given and positive. In the Ramsey model with AK production function, the optimal saving rate is constant over time but it can be negative.

2. In the Harrod-Domar model, the change of the rate of growth with respect to the TFP A equals the saving rate, while in the Ramsey Model with AK production function, the change is higher than the saving rate.

3. If both the Solow and Ramsey Models with Cobb-Douglas production function and with full depreciation of the capital, exhibit medium-income traps, a change of the TFP induces a larger change of these traps in the Ramsey Model than in the Solow Model.

4. Another important difference is we can calculate the prices of the consumption goods and capital goods for the Ramsey Model, while only the prices of capital goods can be obtained with the Solow model.

5. Impacts of the public spendings: For the Harrod-Domar Model, the public spending decreases the private consumption at the period 1, but there is no change in the capital stock and hence the production in subsequent periods. It is due to the saving rate that remains unchanged. In the Ramsey model with AK production function, since the saving rate is endogenous, both the private consumptions and the outputs will be lowered.

6. In both the Harrod-Domar and Ramsey models with Cobb-Douglas production function, if the debt is not high and the interest rate is sufficiently low, it is better to use public debt for production rather than for consumption. We get more outputs.

7. If the country borrows to recover the TFP after the Covid pandemic, both the Harrod-Domar and Ramsey models with Cobb-Douglas production function show that the rate of growth is higher for the year just after the pandemic but is the same as before the pandemic.

The economy we consider is a closed economy and lasts for an infinite number of time periods, denoted by t, with t = 0, 1, 2, …, + ∞. It starts with a population, a production technology and an initial per capita capital stock k₀ > 0. It is assumed that the population is stationary. There is a produced aggregate good, which can be consumed and/or used as capital input.

At date t, we denote by

1. c_t: total consumption per capita

2. c_p,t: household consumption per capita

3. S_t: saving per capita

4. I_t: investment per capita

5. k_t: capital stock per capita

6. y_t: output per capita

7. G_t: government expenditure per capita.

At any date t, the following two national accounting identities hold

The models will compute the total consumption c_t. The share between the households consumption and the public expenditure is exogenous. It can be considered as a tool of public policy.

The output market is described by the aggregate production function

i.e., the per capita output y_t is obtained by using the per capita capital input k_t with a production technology F. Dynamic capital accumulation is dictated by

with δ ∈ [0, 1] denoting the capital depreciation rate and we assume that I_t+1 equals S_t, i.e., all savings in the current period become investment in the next period.

The main additional assumptions in this model specify the saving and production function as follows:

1. S_t = sY_t for any period t where s(∈ (0, 1)) is exogenously given.

2. y_t = Ak_t where A(>0) is exogenously given.The parameter A measures the efficiency of the production technology.

v = 1/A is the usual capital coefficient. We present below the Harrod-Domar Model. For any period t ≥ 0

Solving the Harrod-Domar model

We see in this model all the quantities grow with a constant growth rate g given by

g increases when s or/and A increase. For instance, if δ = 0.05, A = 0.5 then g = 0.10 if s = 0.3. We can now understand why some countries may have 0.1 as growth rate of the GDP.

Once we have determined the path (c_t, k_t+1, y_t), the planner shares the total consumption c_t between c_p,t and G_t.

In the Solow model the production function is increasing and strictly concave. Here, we assume F(k) = Ak^α, 0 < α < 1. We list below the equations of the Solow model. For any period t ≥ 0, with the initial capital stock k₀ > 0

As in the Harrod-Domar model, the saving rate s is constant over time and exogenous, and we have saving = investment.

Solving the Solow model

As in the Harrod-Domar model, we first use the accumulation capital equation (14).

Define ϕ(x) = (1 − δ)x + Ax^α. We can rewrite as follows

Starting with k₀ > this equation gives an infinite sequence {k₀, k₁, …, k_t, …} where k_t+1 = ϕ(k_t) for every t ≥ 0. It is easy to prove that there exists $k¯=(sAδ)11-α$ such that:

1. If $0<k0<k¯$ then $kt<kt+1<k¯$ for any t > 1 and ${kt}↑k¯$⁠.

2. If $k0>k¯$ then $k¯<kt+1<kt$ for any t > 1 and ${kt}↓k¯$⁠.

3. If $k0=k¯$ then $kt=k¯$ for all t.

4. $k¯$ can be viewed as the medium-income trap.

When δ = 1 tedious computations give

There exists a benevolent social planner who maximizes the intertemporal utility of the representative consumer under the constraints that at any period t the consumer’s consumption plus her investment is less than or equal to the per capita output. Define $F(k)≡F(k)+(1-δ)k$⁠. Formally, with k₀ > 0

The production function is concave, strictly increasing and differentiable. The utility function u is strictly concave and satisfies the “Inada condition” u′(0) = +∞.

Existence of the solution is well-known (a simple proof: first show a solution exists at finite horizon T. This problem is easy: maximization under convex constraints of a concave function in finite dimension. And then, let T go to infinity. The limits of the quantities constitute the solution for the infinite-horizon model.

Some basic results: let ${ct∗,kt+1∗}t=0,…,+∞$ be the solution. Then,

1. if it satisfies the Euler Equation: for any t,

1. and if it satisfies the transversality condition:

then it is optimal.

1. The optimal sequence ${kt+1∗}$is monotonic (increasing or decreasing)

2. If the production function F is strictly concave and satisfies F^′(+∞) = 0 then the sequence ${kt+1∗}$ converges to a value k*^s (steady state) which satisfies

where $β=11+r$⁠.

     In this case, if we define $1+rt∗=F′(kt+1∗)$⁠, then

That means the returns of the capital in the long run equals the real interest rate.

1. If 0 < k₀ < k*^s then the optimal sequence ${kt+1∗}$ is increasing and converges k*^s. And if k₀ > k*^s then the optimal sequence ${kt+1∗}$ is decreasing and converges k*^s.

Comments

• 1. Let r denote the real interest rate. We suppose it constant over time. Define $β=11+r$⁠. Then

If we measure the utility function u(c_t) at period t in money of period 0, then the sum $∑t=0∞1(1+r)tu(ct)$ is the total value of the utilities measured in money of period 0.

• 2. About the Euler equation: We can define discounted prices $pt∗$ by $pt∗=βtu′(ct∗)$⁠. Define $1+rt∗=F′(kt∗)$⁠. Then, the returns obtained by investing a unit of capital is $rt∗$ $(F(kt∗)-F(0)=F(kt∗)-0≃F′(kt∗)kt∗)$⁠.

Since the capital good and the consumption good is assumed to be the same aggregate good, $pt∗$ is also the price of the capital at period t. Hence, the Euler equation is a no-arbitrage condition:

• 3. About the transversality condition, $limt↑+∞βtu′(ct∗)kt+1∗=0$⁠. We can write this condition $limt↑+∞pt∗kt+1∗=0$⁠. The value of the capital vanishes at infinity. Observe since the capital $kt+1∗$ is bought at period t we value it with the price at period t, $pt∗$⁠.

We will assume u(c) = ln(c). The model can be expressed as

We claim that the sequence ${kt+1∗,ct∗}t≥0$ which satisfies for any t ≥ 0

is optimal.

The claim is true if Euler Equation (20) and Transversality condition (22) are satisfied.

A special case: at period 0, the constraint is

The optimal solution is

To verify that, define k₀ by (A + 1 − δ)k₀ = B.

To make tractable the calculations, we assume full capital depreciation, δ = 1.

We assume again u(c) = ln(c). The optimal solutions are: for any t ≥ 0

To prove these claims, one uses Euler Equation (20) and Transversality condition (22).

Comments

1. From (25), we see that the sequence ${kt∗}$ converges when t goes to infinity, to a value $k∗s=(Aβα)11-α$⁠.

2. We can prove that if 0 < k₀ < k*^s then the optimal sequence ${kt∗}$ is increasing and converges to k*^s, and if if k₀ > k*^s then the optimal sequence ${kt∗}$ is decreasing and converges to k*^s. This value k*^s can be viewed as the medium-income trap, as in Solow Model.

3. The long term marginal productivity r* verifies: r* = 1 + r,(recall r is the real interest rate).

1. The Harrod-Domar model: The investment rate is exogenous. Investment is always positive. The quantities output y_t, capital stock k_t, consumption c_t, the investment I_t grow at the same rate g = sA − δ. This rate g is positive if sA > δ, negative if sA < δ.

2. The Ramsey model with AK production function: The quantities output, capital, consumption, investment grow also at constant rate $g∗=β(A+1-δ)-1$⁠. The investment equals

If βA > (1 − β)(1 − δ) then the investment is positive. Otherwise, it is negative or zero.

     The investment rate s* equals $[β(A+1-δ)-1-δ]A$⁠. It may be positive or negative. It increases with A and decreases when the real interest rate r decreases. We obtain that g* = s*A − δ as in the Harrod-Domar Model. However the impact of A on the growth rate differs. Indeed.

1. In the Harrod-Domar model, $∂g∂A=s$

2. In the Ramsey model, $∂g∗∂A=s∗+A∂s∗∂A>s∗$⁠.

In the Solow model there exists a medium-income trap $k¯=(sA)11-α$⁠. If we increase A then we enlarge the trap.

The sequence of capital stocks (k) is

1. In the Ramsey model, the investment rate s* equals αβ. It increases with the elasticity α of the production function. It decreases if the real interest rate r increases. The medium-income trap is $k∗s=(Aαβ)11-α=(As∗)11-α$⁠. Its expression is quite similar to the one in Solow model. It increases when A increases and increases with s*. The sequence of optimal capital stocks is

Consider the impact of the TFP, A.

We have

1. $∂⁡ln(k¯)∂A=11-α1A$

2. $∂⁡ln(k¯s∗)∂A=11-α1A+11-α1s∗∂s∗∂A>11-α1A$

Another important difference.

1. In the Solow model, we can obtain only the marginal production cost $rt∗=F′(kt∗)$

2. In the Ramsey model (see Le Van & Dana, 2003): If we define the discounted prices $pt∗$ by $pt∗=βtu′(ct∗)$ then

   • the sequence $(ct∗)$ maximizes the intertemporal utility of the representative consumer $∑t=0∞βt⁡ln(ct)$ under the budget constraint

where π denotes the maximal profit of firm,

• $π=max{k1,k2,…,kt,…}∑t=0∞pt*(Aktα-kt+1)-F′(k0)k0$⁠.

We have in particular the no-arbitrage condition

where $1+rt∗=F′(kt+1∗)=F′(kt+1∗)$⁠, since we assume δ = 1.

In the Ramsey model, one can define consumption prices because the social preferences are specified while in the Solow model, the demand side is missing.

• 1. Public spending

a. We suppose the government announces in period −1 it will increase its spending by γ > 0 in period 0.

The constraints will be

Let $(kt+1∗,ct∗)t$ be the sequences of initial capitals and consumptions (without public spending).

It is easy to check that

If the government announces at the beginning of period 0, there will be an increase of public expenditures in period 1 then

• 2. Public debt

The government borrows an amount D in period 0 and reimburses in period 1 the amount D(1 + ρ), ρ is the real interest rate. The constraints are

2.1. The government uses the debt only for consumption

In this case:

D and ρ must not be too large in order to ensure that c₁ > 0.

2.2. The government uses debt for production. It buys $k̃0$⁠, s.t. $D=Ak̃0$⁠. Define $k0′=k0+k̃0$⁠. The constraints will be:

We obtain that

We have to make sure that c₁ > 0.

If ζ > 0 then. $c1>c1∗>0$

If ζ < 0 then we must have the condition

Under this condition c₁ > 0 but $c1<c1∗$⁠.

Summing up, If D is not too large and ρ small enough then

• 3. Covid shock ^[1]

Question: what is the cost to recover from Covid-19 pandemic?

At t = 0, because of Covid pandemic, the TFP A passes to A′ < A.

The country borrows in period 0 an amount D to recover A in period 1. But it has to reimburse D(1 + ρ).

We have the following results

At date 1, the output growth rate is higher than the one before Covid. It becomes the same after period 1. However

We can recover the growth rate but not the level of the output.

But we have to ensure that c₁ > 0. It will be true if

If the cost D is very big, the TFP cannot be recovered in period 1. The recovering process will last for many periods.

• 1. Public spending

We will show the importance of the date of the announcement of the government.

• The government announces at period −1 that it will spend γ > 0 in period 1.

     The constraints of the economy will be

Define B₁ = (A + 1 − δ)k₁ − γ. We suppose (A + 1 − δ)k₁ − γ > 0. We will show actually the new optimal sequence of capitals will satisfy this condition.

     How do we process to solve the model? Observe the constraint at period 1 becomes

We solve first

Go back to (31). The solution is given by

Define $S1=∑t≥1βt⁡lnc̃t$⁠. We find

To find the optimal $k̃1$⁠, we solve

We find

We show that $(A+1-δ)k̃1-γ>0$ if, and only if (A + 1 − δ)²k₀ > γ. We have to assume that γ must be lower to (A + 1 − δ)²k₀.

• Now suppose the government announces at the end of period 0 that it will spend γ > 0 in the next period, period 1. It is a surprise for the consumer. This one has already planned the optimal investments, hence the optimal capital $k1*$ by solving

The consumer, at the beginning of period 1 will solve a new program

We have to assume now (A + 1 − δ)²βk₀ > γ to have $(A+1-δ)k1∗-γ>0$⁠.

     We have the results for the new optimal sequences ${k^t}$⁠, for t ≥ 2

and

and $k^1=k1∗<k̃1$⁠. In particular

• 2. Public debt

We suppose the government borrows in order to increase the initial capital. The constraints are

Define $k̃0$ by $D=(A+1-δ)k̃0$⁠.

The constraints become

Use the same technics as in the previous section to obtain

We have to make sure B₂ > 0. The necessary and sufficient condition is

This condition is satisfied if.

1. either A > δ + ρ (ρ is not large)

2. or, if A < δ + ρ, then $k̃0<(A+1-δ)k0δ+ρ-A$ (⁠$k̃0$⁠, hence D, not too large)

We then have $c̃0>0$⁠, $c̃1>0$⁠:

$c̃0$ may be higher or smaller than $c0∗$⁠.

• 3. Covid shock

At t = 0, because of Covid-pandemic, the TFP A passes to A′ < A.

The country borrows in period 0 an amount D to recover A in period 1. But it has to reimburse D(1 + ρ).