Harrod-Domar Growth Model

Lord Harrod and Mr. Domar Keynesian based models Saving rates Capital/Output ratio or

Capital Productivity Capital stock GDP

Personal Consumption Gross Savings Gross Investment Net Investment, or Capital

Accumulation Depreciation Dynamic models

In growth models, we will encounter the following terms:

What is a Keynesian Growth Model?

Keynes’ model and Keynesian models were developed to explains business cycles

» A short run phenomena As such they attribute a major role to aggregate

expenditures (demand side) Regarding the supply side, they assume that there

is unemployment: production responds fast to increases in aggregate demand because capital and labor is unemployed.

Aggregate Demand, AD– AD = C + I + G + X-M– C, Consumption expenditures– I, Investment expenditures– G, Government expenditures– X-M, Foreigners’ Expenditures

Aggregate Supply– AS < ASfe – Aggregate Supply, at full employment

Macroeconomic Equilibrium– AS = AD– Or – S = I

A Keynesian Model

A Keynesian growth model takes a long run perspective.– Aggregate demand (or savings=investment)

still is important, but– It also includes the aggregate supply

» Investment has two impacts: On expenditures (in the short run) On capital stock (in the long run)

Trends and business cyclesR

eal G

DP

Years

One business cycle

Trend

Main Propositions Economic growth can be accelerated by

– changing the saving rate – improving technology.

Saving rates and technology can be changed – government interventions without

consideration to prices

v= K/Y ora=Y/K

Capital

Capital/Output Ratio or Productivity

GDP

s

Saving Rate

C

Sd

Depreciation Rate

D

In

Ig

Production function

Harrod-Domar Growth Model A Flow chart model

Factors Explaining the growth rateAccording to Harrod-Domar model

g

s

a

d

Saving rate

Capital productivity

Capital depreciation

Rate ofEconomicGrowth

Explained variable Explanatory Variables

+

+

_

Arithmetic specificationWithout Depreciation

a=dY/dK

Y=K.a

S=Y.s s=dS/dY

I

K

dK

If we know the initial capital stock K; and we

know a, (how much output increases when

capital increases 1 unit) then we know what will

total output Y be.

If we know output Y, and we know s, which is the saving rate, then we know total savings S.

If we know total savings S, we know how much we can invest (I) in new

capital (dK)

If we know dK and a, we know growth

of output dY

dY

Numerical specificationWithout Depreciation

a=.20

Y=1

S=.10 s=.10

.10

K=5

dK=.10

If we know the initial capital stock K=5; and

we know its productivity a=.20 then we know

total output Y=1

If we know output Y=1 and we know the saving rate s=.10, then we know

total savings S=.10

If we know total savings S=.10 we know that we can

invest I=.10 in new capital (dK=.10)

If we know dK=.10 and a=.20, we know

growth of output dY=0.02=2%

dY=.02=2%

Or …dY = s.a

By approximation: dY/Y=s.a/Y

g=s.aSince Y=1

Economic growth formulaAccording to Harrod-Domar the rate of economic growthis defined by the formula:

g = s.a – d

that is, if s=10% and a=0.20 and d =1%, then

g=0.10*0.20 - 0.01 = 0.02 -0.01 = 0.01 = 1%

What happens if the rate of saving (s) increases to 20% ?What happens if the productivity of capital (a) increases to 0.40?What happens if the depreciation (d) rate is 2% ?

.Conventional Keynes’ Model

Specification

Saving function (demand side) S = s.Y where s is the average propensity to save or average saving rate.

In the conventional short run Keynesian model investment (I) is given. I = Ia

In equilibriumS = I

Solving the models.Y = Ia

Y = 1/s.Ia = m.Ia where m is the investment multiplier

Mathematical derivation of Harrod-Domar model

In this model national GDP increases because the autonomous demand (I)

increases. It is assumed that aggregate supply responds as to producewhat is demanded. But, what will happen if the economy was at full employment? The only way for production to increase will be an increasein the capital stock. With more capital (and labor) the economy will produce more GDP.

Mathematical derivation of Harrod-Domar model (2)Keynes’ Model Expanded to Consider Growth

Harrod and Domar explained how the aggregate supply expands. For them, investment has two effects, one on the aggregate demand side (businesses expend more) and another in the aggregate supply side (more investment increases capital stock and thereby businesses produce more the next period).

We, therefore, need to add a production function:

Y = a.K production function

Where a is the productivity of capital: Y/ K, which is constant

Now we can determine how a change in capital changes income.

Y = a.K

Mathematical derivation of Harrod-Domar model (3)What we need to know is how capital changes. It changes by businesses, and government investment:

K = Ia

We are assuming that capital doesn’t ware out, i.e. there is not depreciation.

Returning to the equilibrium condition (S=I) we solve the model again for the long run case

s.Y = Ia = K, but we know that K = Y/a, then

s.Y = Y/a

s.a = Y/Y

Calling Y/Y = g : rate of GNP growth

Mathematical derivation of Harrod-Domar model IV

g = s.a

If we recognize that capital depreciates:

g = s.a – d

Where d is the depreciation rate per year.

Notice that in this model the rate of growth (g) is constant. Why?

Harrod’s way:K = v.Y where v = 1/ag = s/vAnd with depreciationg = s/v - d

Production function

K

NGDP1

GDP2> GDP1

Production function

GDP

K

GDP

Productivity rate

To growth model

Assumptions– Labor/capital proportions are fixed

– Saving rate is given

K

NGDP1

GDP2> GDP1

S

S

Income = GDP

Production function

Saving function

GDP

K

GDP

Saving rate

Productivity rate

To growth model

Non-existence of equilibrium

K

Y,S,D,I Y

S

DIn

C

D

Review

Technology or capital/output ratio

Saving rate

Depreciation rate Capital accumulation Growth rate

You should now be familiar with the following terms: