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

Cuong Le-Van (CNRS and Paris School of Economics, Paris, France)
Binh Tran-Nam (School of Accounting, Auditing and Taxation, UNSW, Sydney, Australia)

Fulbright Review of Economics and Policy

ISSN: 2635-0173

Article publication date: 10 October 2023

Issue publication date: 6 December 2023

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Abstract

Purpose

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.

Design/methodology/approach

The paper utilizes a positivist research framework that emphasizes the causal relationships between the variables in each of the three models. Mathematical methods are employed to formulate and examine the three models under study. Since the paper is theoretical, it does not use any empirical data although numerical illustrations are provided whenever they are appropriate.

Findings

The Harrod-Domar model explains why countries with high rates of saving may also enjoy high rate of economic growth. Both the Solow and Ramsey models can be used to explain the medium-income trap. The paper examines the impact of Covid shocks on the macroeconomy. While the growth rate can be recovered, it may not always possible to recover the output level.

Research limitations/implications

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. For the Ramsey model with AK production function, both the private consumption and the outputs will be lowered. 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. If the country borrows to recover the Total Factor Productivity 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.

Practical implications

The economy can recover the growth rate after a Covid shock, but the recovery process will generally take many periods.

Social implications

This paper focuses on economic implications and does not aim to examine social implications of policy changes or Covid-type shock.

Originality/value

The paper provides a comparison of three basic growth models with respect to public spending, public debts and repayments and Covid-type shocks.

Keywords

Citation

Le-Van, C. and Tran-Nam, B. (2023), "Comparing the Harrod-Domar, Solow and Ramsey growth models and their implications for economic policies", Fulbright Review of Economics and Policy, Vol. 3 No. 2, pp. 167-183. https://doi.org/10.1108/FREP-06-2023-0022

Publisher

:

Emerald Publishing Limited

Copyright © 2023, Cuong Le-Van and Binh Tran-Nam

License

Published in Fulbright Review of Economics and Policy. Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) licence. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this licence may be seen at http://creativecommons.org/licences/by/4.0/ legalcode

1. Introduction

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.

2. The economy

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. ct: total consumption per capita

  2. cp,t: household consumption per capita

  3. St: saving per capita

  4. It: investment per capita

  5. kt: capital stock per capita

  6. yt: output per capita

  7. Gt: government expenditure per capita.

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

ct+St=yt
ct=cp,t+Gt

The models will compute the total consumption ct. 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

yt=F(kt)
i.e., the per capita output yt is obtained by using the per capita capital input kt with a production technology F. Dynamic capital accumulation is dictated by
kt+1=(1-δ)kt+It+1
with δ ∈ [0, 1] denoting the capital depreciation rate and we assume that It+1 equals St, i.e., all savings in the current period become investment in the next period.

3. The Harrod-Domar model

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

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

  2. yt = Akt 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

(1)ct+It+1=yt⇔ct+kt+1-(1-δ)kt=Akt
(2)yt=Akt
(3)ct=(1-s)yt
(4)It+1=syt
(5)kt+1=sAkt+(1-δ)kt

Solving the Harrod-Domar model

(6) Observe that equation (5) gives, for all t≥0,kt+1=(sA+1-δ)t+1k0
(7)hence yt=A(sA+1-δ)tk0
(8)It+1=sA(sA+1-δ)tk0
(9)ct=(1-s)(sA+1-δ)tk0

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

(10)g=sA-δ
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 (ct, kt+1, yt), the planner shares the total consumption ct between cp,t and Gt.

4. The Solow model

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

(11)ct+It+1=yt⇔ct+kt+1-(1-δ)kt=Aktα
(12)It+1=syt
(13)yt=Aktα
(14)kt+1=sAktα+(1-δ)kt
(15)ct=(1-s)yt

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).

kt=sAktα+(1-δ)kt

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

kt+1=ϕ(kt)

Starting with k0 > this equation gives an infinite sequence {k0, k1, …, kt, …} where kt+1 = ϕ(kt) 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

(16)kt+1=(sA)1-αt1-α(k0)αt

5. The Ramsey model

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

(17)max{ct,kt+1}∑t=0∞βtu(ct),0≤β<1
(18) for any t,ct+(kt+1-(1-δ)kt)≤yt=F(kt)
(19)⇔ct+kt+1≤F(kt)

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,

(20)u′(ct*)=βu′(ct+1*)(F′(kt+1*)+1-δ)
(21)⇔u′(ct*)=βu′(ct+1*)F′(kt+1*)

  1. and if it satisfies the transversality condition:

(22)limT↑+∞{βTu′(cT*)kT+1*}=0,

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

β(F′(k*s)+1-δ)=1⇔F′(k*s)=1+r
where β=11+r.

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

rt*→r

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

∑t=0∞βtu(ct)=∑t=0∞1(1+r)tu(ct)

If we measure the utility function u(ct) 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:

pt*=pt+1*(1+rt*)

  • 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∗.

5.1 The Ramsey model with AK production function

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

(23)max{ct,kt+1}∑t=0∞βt⁡ln(ct),0≤β<1
(24) for any t,ct+(kt+1-(1-δ)kt)≤yt=Akt

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

(25)kt+1*=[β(A+1-δ)]t+1k0
(26)ct*=βt(A+1-δ)t+1(1-β)k0

is optimal.

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

(27) The optimal investment is It+1*=kt+1*-(1-δ)kt*
(28)It+1*=[β(A+1-δ)]t[β(A+1-δ)-(1-δ)]k0
(29) the optimal investment rate is st*=It+1*Akt*
(30)st*=[β(A+1-δ)-(1-δ)]A

A special case: at period 0, the constraint is

(31)c0+k1≤B,B>0

The optimal solution is

(32)kt+1*=βt+1(A+1-δ)tB
(33)ct*=[(A+1-δ)β]t(1-β)B

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

5.2 The Ramsey model with the production function F(k) = Akα, 0 < α < 1

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

(34)kt*=(Aαβ)1-αt1-αk0αt
(35)ct*=A(1-αβ){kt*}α
(36)kt+1*=Aβα{kt*}α
(37)investment rate st*=kt+1*Ak*α=αβ

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).

6. First comparisons of these models

6.1 The Harrod-Domar model and the Ramsey model with AK production function

  1. The Harrod-Domar model: The investment rate is exogenous. Investment is always positive. The quantities output yt, capital stock kt, consumption ct, the investment It 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

It+1*=[β(A+1-δ)]t[β(A+1-δ)-(1-δ)]k0

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∗.

6.2 The Solow model and the Ramsey model with F(k) = Akα and δ = 1

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

kt=(sA)1-αt1-αk0αt

  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

kt*=(s*A)1-αt1-αk0αt

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.

6.3 Prices

  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

∑t=0∞pt*ct≤F′(k0)k0+π
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

pt*=pt+1*(1+rt*)
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.

7. Second comparisons of these models: economic policies

7.1 With the Harrod-Domar model

  • 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

c0+γ+I1=Ak0I1=sy0=sAk0for t≥1,ct+It+1=AktIt+1=syt=sAkt

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

It is easy to check that

kt=kt*,yt=yt*, for alltbut c0=c0*-γ,ct=ct*, for all t≥1

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

kt=kt*,yt=yt*,for all t but c0=c0*,ct=ct* for t≥2,c1=c1*-γ

  • 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

c0+I0=Ak0+DI0=sy0=sAk0c1+D(1+ρ)+I1=Ak1I1=sy1=sAk1c2+I2=Ak2I2=sy2=sAk2…

2.1. The government uses the debt only for consumption

In this case:

kt=kt*,yt=yt*, for all tc0=c0*+D,ct=ct*, for t≥2c1=c1*-D(1+ρ)

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:

c0+I1=Ak0′I1=sAk0′c1+Ak̃0(1+ρ)+I2=Ak1I2=sAk1c2+I3=Ak2I3=sAk3…

We obtain that

c0=(1-s)Ak0′>c0*c1=(1-s)A(sA+1-δ)k0′-Ak̃0(1+ρ)c1=(1-s)A(sA+1-δ)k0+ADζwith ζ=(1-s)(sA+1-δ)-(1+ρ)

We have to make sure that c1 > 0.

If ζ > 0 then. c1>c1∗>0

If ζ < 0 then we must have the condition

D<(1-s)A(sA+1-δ)k0-ζ

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

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

for all t,yt>yt*ct>ct*,for all t≠1,c1 may be larger or smaller than c1*

  • 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.

c0+I1=A′k0<c0*,k1=(sA′+1-δ)k0<k1*

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

c1+D(1+ρ)+I2=Ak1I2=sAk1c2+I3=Ak2I3=sAk2…

We have the following results

for all t≥1,kt+1=(sA+1-δ)t(sA′+1-δ)k0yt+1=A(sA+1-δ)t(sA′+1-δ)k0t≥1y1y0=As+1-δA′>y1*y0*yt+1yt=yt+1*yt*,t≥1

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

yt<yt*,t≥1

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

(1-s)A(sA+1-δ)sA′+1-δsA+1-δk0-D(1+ρ)>0

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

7.2 With the Ramsey model and AK production function

  • 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

c0+(k1-(1-δ)k0)=Ak0c1+(k2-(1-δ)k1)=Ak1-γc2+k(K3-(1-δ)k2)=Ak2…

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

c1+k2=B1

We solve first

max∑t≥1βt⁡ln(ct)c1+k2=B1ct+(kt+1-(1-δ)kt)=Akt,t≥2

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

k̃t=(A+1-δ)t-2βt-1B1c̃t=(A+1-δ)t-1βt-1B1

Define S1=∑t≥1βt⁡lnc̃t. We find

S1=β1-βlnB1=β1-βln((A+1-δ)k1-γ)

To find the optimal k̃1, we solve

maxk1ln((A+1-δ)k0-k1)+β1-βln((A+1-δ)k1-γ)

We find

k̃1=1-βγ+β(A+1-δ)2k0A+1-δ>k1*k̃t=A+1-δβt-2((A+1-δ)k̃1-γ)<kt*,t≥2

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

max{ct}∑t≥0βt⁡ln(ct)ct+kt+1-(1-δ)kt=Akt,t≥0

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

max{ct}∑t≥1βt⁡ln(ct)c1+k2=(A+1-δ)k1*-γct+kt+1-(1-δ)kt=Akt,t≥2

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

k^2=β(1+1-δ)2βk0-γ<k̃2
and
k^t<k̃t<kt*,t≥3
and k^1=k1∗<k̃1. In particular
∑t≥0βt⁡ln(c^t)<∑t≥0βt⁡ln(c̃t)<∑t≥0βt⁡ln(ct*)

  • 2. Public debt

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

c0+k1=(A+1-δ)k0+Dc1+D(1+ρ)+k2=(A+1-δ)k1c2+k3=(A+1-δ)k2…

Define k̃0 by D=(A+1-δ)k̃0.

The constraints become

c0+k1=(A+1-δ)k0′,k0′≡k0+k̃0c1+k2=B2≡(A+1-δ)(k1-(1+ρ)k̃0)c2+k3=(A+1-δ)k2…

Use the same technics as in the previous section to obtain

k̃t=[(A+1-δ)β]t-2βt-1B2c̃t=[(A+1-δ)β]t-1βt-1(1-β)B2

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

(A+1-δ)+k̃0(A-δ-ρ)>0

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:

k̃1=β(A+1-δ)k0+k̃0β(A+1-δ)+(1-β)(1+ρ)=k1*+k̃0β(A+1-δ)+(1-β)(1+ρ)>0c̃0=(1-β)(A+1-δ)k0+k̃0(A-δ-ρ)=c0*+(1-β)k̃0(A-δ-ρ)>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).

c0+k1=(A′+1-δ)k0c1+D(1+ρ)+k2=(A+1-δ)k1c2+k3=(A+1-δ)k2…

Define B3 = (A + 1 − δ)k1 − D(1 + ρ). Then

c1+k2=B3c2+k3=(A+1-δ)k2…

We obtain the solutions

k^t=[(A+1-δ)β]t-2βt-1B3,t≥2c^t=(A+1-δ)t-1βt-1(1-β)B3,t≥1

Let S1 = ∑t≥1βt ln ct. We find

S1=β1-βlnB3+H
where H is a constant.

Since B3 = (A + 1 − δ)k1 − D(1 + ρ) to obtain the optimal value k^1 we solve

maxk1{ln(c1)+S1}
⇔maxk1ln((A′+1-δ)k0-k1)+β1-βln((A+1-δ)k1-D(1+ρ))

Define k^0 by D=(A+1-δ)k^0. We obtain

k^1=(1-β)(1+ρ)k^0+β(A′+1-δ)k0

We have to check that B3 > 0. We find that B3>0⇔k^0<A′+1-δ1+ρk0. If this condition is satisfied then c^0>0,c^1>0.

If the cost to recover the productivity D, hence k^0, is too high, then the recovering process will last for more than one period.

Let us compute the growth rates of the output y^t. We find

1+g^1=y^1y^0=Ak^1A′k0=A(1-β)(1+ρ)k^0+β(A′+1-δ)k0A′k0>β(A′+1-δ)A′>β(A+1-δ)A=y1*y0*=1+g1*

But for t ≥ 2, g^t=gt∗.

8. Conclusion

The main results we obtain from the comparisons of the Harrod-Domar model, the Solow model and the Ramsey model can be summarized 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 which 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.

Note

1.

We suppose due to Covid, the TFP decreases for the first period, because Covid deteriorates the health of the workers and hence the TFP. The country will borrow to buy vaccin and medicine to help the workers recover their health and hence the TFP as well.

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Acknowledgements

This article is a revised version of a paper presented at the international conference From Economic Theory to Economic Policy: Contributions of Ngo Van Long, organized by the Fulbright Review of Economics and Policy., Fulbright University, Ho Chi Minh City, Vietnam, 22-23 February 2023. Valuable comments from two reviewers are gratefully acknowledged.

Corresponding author

Cuong Le-Van can be contacted at: Cuong.Le-Van@univ-paris1.fr

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