1. GROWTH ECONOMICS and Fund-raising in international cooperation SECS-P01, CFU 9 Economics for Development academic year
6. THE HARROD-DOMAR MODEL OF GROWTH
Roberto Pasca di Magliano
Fondazione Roma Sapienza-Cooperazione Internazionale

2. Harrod-Domar Model introduction
Development and growth are natural phenomena
The modern theory of growth has been developed indipendently by two the economists:
Roy Harrod in his article “An Essay in Dynamic Theory” (1939),
Evsey Domar in his article “Capital expansion, rate of growth and employment” (1946)
Both inspired by the nascent Keynesian doctrine.
They developed what was then known as the Harrod-Domar model of growth as dynamic extension of the Keynesian analysis of static equilibrium.
The model explain that growth rate is influenced by the level of saving and the capital productivity.
Meanwhile, Robert Solow was developing the neoclassical model of growth, inspired to the dominant influence of Alfred Marshall’s Principles of Economy (1890).
Roberto Pasca di Magliano

3. Harrod-Domar Model Main questions
If the var. Y => Δ I, which is the growth rate of Y ensuring equality between planned I and S to garantee a balance in the long term?
Is there any certainty prevailing growth rate needs to ensure such equality? Otherwise, what happens?
In the static keynesian model, temporary gap between I and S are compensated through the automatic adjustment garanteed by multiplier. Instead, according Harrod, if overall productivity growth rate do not increase enough, what happens?
Answers
Actual growth rate: the real rate increase in a country's GDP per year
Warranted growth rate: the rate of growth at which the economy does not expand indefinitely or go into recession.
Natural growth rate: the growth of an economy required to maintain full employment.
Roberto Pasca di Magliano

4. Harrod-Domar Model simplified concepts
The H-D model is the easiest way to start learning about growth in the long run
Main concepts used in the model:
1. Income, saving and consumption:
Y = C + S. as S = I -> Y = C + I
All income is either saved or consumed:
S = sY -> C = (1-s)Y
2. Capital accumulation:
K t+1 = It + K(1-d) where d -> depreciation
The model use the concept of capital-output ratio (efficiency of the production system measured in term of capital):
cr = Kt / Yt
This to show that an economy can produce a lot of output with a little capital (and viceversa).
3. Rate of growth:
g = s / cr - d
Roberto Pasca di Magliano

5. Harrod-Domar Model Growth Rates
The three growth rates
Actual rate of growth (g) (i.e. the real income change):
g = s / c = (Y / Y) = var. Y / Y where: s -> propensity to save
I / Δ Y c -> quantity of capital need to produce one unity of production
g is then equal to the ratio between the propensity to save and the current capital- output ratio
Warranted rate of growth (gw) (i.e. the growth that leaves everyone satisfied with an increase in production (no more, no less) needed to pursue better resource’s allocation, by impling a necessary increase ininvestments)
gw = Δ Y / Y = s / cr where: s -> propensity to save
cr -> extra quantity of capital needed
gw is then equal to the ratio between planned and propensity to save and the extra capital required per unit of product
Natural growth rate (gn) (i.e. one that ensures growth that absorbs the available labor force in relation to its production capacity) as:
Y = L (Y / L)
Roberto Pasca di Magliano

6. Harrod-Domar Model Actual rate of growth (Harrod)
g = s / c = (Y / Y) / (I / Δ Y) = Δ Y / Y
s -> propensity to save
cr -> incremental capital-output ratio, i.e. :
K / Δ Y = I / Δ Y, provided that S = I
So, since S = I, the rate of increase of the product is:
g = (S / Y) / (I / Δ Y) = Δ Y / Y
Roberto Pasca di Magliano

7. Harrod-Domar Model Warranted rate of growth (Harrod)
gw = Δ Y / Y = S / cr
According to the static Keynesian model:
S = sY (propensity to save)
cr = Kr / Δ Y = I / Δ Y (capital-output ratio, ie, the amount of additional capital needed to produce additional product units at a given interest rate and given the technological conditions)
Then, the question is which var. Y ensure that the planned I is equal to S needed to increase Y:
sY cr = Δ Y
Therefore:
Δ Y / Y = s / cr = gw
In order to obtain dynamic equilibrium, the product should grow at this rate, i.e. consumer spending must equal the value of production.
But, in presence of an external shock -> deviation from equilibrium could happen, i.e. deficiencies in equipment etc.. In this case the current rate can growth beyond the guaranteed (c> cr), then surplus capital, and fall even greater growth rate.
Roberto Pasca di Magliano

8. Natural rate of growth Domar’s contribution
Domar introduces the natural rate of growth (gn)
Y = L (Y / L)
Two components, both exogenous:
growth of the labor force (L)
growth of labor productivity (Y / L)
A change in the level of I, Δ demand: Δ Yd = Δ I /S and I increases if the same offering: Δ Ys = Ip (p, capital productivity, Δ Y / I)
In order to have Δ Yd= Δ Ys, it is necessary that:
Δ I /s = Ip or Δ I / I = sp
(i.e. It has to grow at a rate such that it matches the propensity to save
and the productivity of capital)
The natural rate of growth is sp (equal 1/cr equilibrium Harrod)
But, even if the growth ensures full utilization of capital, it also to ensures full employment labor.
Roberto Pasca di Magliano

9. Natural rate of growth (Domar’s contribution)
Importance of the model:
Defining the rate of growth of production capacity that ensures the long-term equilibrium between S and I in order to reachifull employment
Fixing the upper limit of the current rate of growth that would lead to a useless accumulation.
If g > gw:
g can continue to diverge until it reaches gn when all the labour force is absorbed
but, g can never exceed gn as there are not enough labour force
In the long run, the relationship between gw and gn is crucial
Full employment of capital and labor requires:
g = gw = gn
and the ennsuring the famous "golden age” studied by Cambridge’s economist Joan Robinson
Roberto Pasca di Magliano

10. Natural rate of growth (Domar’s contribution)
Deviations between gw and gn
gw > gn, excess capital and savings, tendency to depression due to lack of work (g fails to stimulate growth in demand, i.e. the amount of savings that match with job)
Typical aspects of the crisis of '29 and maybe of recent financial crises
gw < gn -> overwork, inflation (g grows more that is necessary to match savings for labor), unemployment and lack of capital investment
Typical aspects of developing countries
example:
If Δ population (2%) and productivity Δ L (3%) -> Δ workforce in terms of efficiency (5%) while Δ propensity saving (9%), requires a Δ K / Y (3%):
gw = 6 (gn = 5)
Consequences: Δ work efficiency> Δ capital accumulation (rising unemployment) and Δ saving> Δ I (inflationary pressure)
Unemployment and inflation together is not a paradox, but it indicates that there are opportunities to increase investment in order to increase capital (Δ K / Y) up to 4, so that gw and gn can be equal in the long run
Roberto Pasca di Magliano

11. Natural rate of growth (Domar’s contribution)
Vertical axis: grow rate. Horizontal axis: savings and investment
- Growth and investment are related to K / Y (cr)
- Propensity to save is independent from the growth
To seek for the balance the policies have to:
reduce labor supply or productivity so as to reduce gn to gw
adopt expansionary monetary or fiscal policies to move S / Y to the right or even stimulate labor-intensive techniques, so as to raise gw gn
Roberto Pasca di Magliano

12. Policy contributions following the H-D model
Not only analysis but elements to design economic policy actions
eg. if country sets target growth rate of 5% and if the ratio K / Y is 3, the needed level of saving and investment has to be 15% of GDP
Roberto Pasca di Magliano

13. Theoretical debate following the H-D model
Concerning automatic adjustment related to the fact that labour productivity, savings and demand for K are determined independently while the model itself admits that in the long run propensity savings may vary, although it tends towards adjustment (in depression -> S may fall, in inflation -> can grow)
In depression (i.e. gw < gn), the profit share is reducing and this reduces the overall propensity to save and then reduces the level of gn to gw
In inflation (gn > gw), profit share will increase and this increases propensity to save and then increaasing the level of gw to gn
In both cases, there are limits: the fall in profits acceptable for businesses, the fall in wages acceptable for workers
Cambridge School (Robinson, Nicholas Kaldor, Richard Kahn, Luigi Pasinetti) -> emphasizes on the functional distribution
Roberto Pasca di Magliano