The Harrod-Domar Growth Model

The Harrod-Domar growth model gives some insights into the dynamics of growth. We want a method of determining an equilibrium growth rate g for the economy. Let Y be GDP and S be savings.

The level of savings S is a proportion of the level of output Y, say S = sY where s is the propensity to save. The level of capital K needed to produce an output Y is given by the equation K=σY where σ designates the capital/output ratio K/Y.

Investment is a very important variable for the economy because Investment has a double role. Investment I represents an important component of the demand for the output of the economy as well as an increase in the capital stock. Thus Investment I is DK = σDY.

For equilibrium there must be a balance between the supply of savings and the demand for increasing the capital stock. This balance reduces to I = S.

Thus, I = ΔK = σΔY

and I = S = sY

so
σΔY = sY.

Therefore the equilibrium rate of growth ΔY/Y which we call g, is given by:

ΔY/Y = s/σ = g

In other words, the equilibrium growth rate of output is equal to ratio of the propensity of savings s = S/Y and of the capital/output ratio σ = K/Y. This is a very significant result. It tells us that growth in the capacity of the economy to produce is determined by matching the supply of savings and the demand for increasing the capital stock of the economy.

Consider the following numerical example. Suppose the economy is currently operating at an output level of Y/L = 30 000 per year and has a capital-output ratio σ = K/Y = 3. This means the capital stock is 90 000. Assume the savings rate s=25% and the propensity to consume output Y is 75%. Savings rate s=25% includes business and public savings as well as household savings.

The Harrod-Domar growth model tells that the equilibrium growth rate is g = 25%/3 = 8.33%; i.e., the economy can grow at 8.33% per year. We can now check this result with actual aritmetic calculations. We shall calculate from two ends:

Matching of savings and demand for increasing the capital stock of the economy

At the current output level of Y/L = 30000, the level of saving is 25%*30000=7500. The growth of output is g = 8.33%*30000 = 2500 and with a capital-output ratio of 3 the additional capital required to produce the additional output of 2500 is 3*2500=7500. This is the investment required in order to increase capacity by the right amount and, this is equal to the amount of saving available in the economy ie. 7500. So the balanced growth model requires that Investment equals Saving and the new output Y/L is 30000+2500=32500.

From the consumption or demand end

We must check that there is adequate aggregate demand next year to absorb the new production level of 30000+2500=32500. At that level of income, consumer demand is 75% of 32500, so 24375. The level of investment the next year under the assumed equilibrium growth conditions is derived as above. The 8.33% growth of production Y of 32500 requires 2707 increase of output which, with a capital-output ratio of 3, demands an increase in the capital stock of 8125. Thus next year's investment will be 8125. This, added to the consumer demand of 24375 gives an aggregate demand of 32500. So the balanced growth model verifies that aggregate demand equals output.

Sensitivity

For contrast, let us consider what would happen if the demand for increased capital stock were to be higher, say 10000 instead of 8125. This, combined with consumption demand of 24375 generates a aggregate demand of 34375, which is greater by 1875 (5.8%) than 32500, the capacity of the economy to produce. The demand of investment of 10000 corresponds to 10000/3 = 3333 of increase of output, instead of 2500 that the economy can achieve. In other words the rate of growth would have to be 3333/32500 = 10.26% instead of 8.33%. There is an irresolvable excess of demand in the economy. From this there should result an increase in the general level of prices in order to return to the equilibrium.

On the other hand, suppose that the demand for investment were to be lower than 8125, say 7000 corresponding to an increase of ouput of 7000/3=2333. Now the aggregate demand is only 24375+7000=31375, less by 1125 than the 32500 capacity of the economy. If output Y/L falls to 31375, capital stock required is 31375/3=10458 whereas at the level of production of 32500, capital stock is already 32500/3=10833; thus there is an excess of capital of 10833-10458=375 and so there is no need for any investment. Thus aggregate demand falls as investment drops to zero and consumer demand drops along with investment. There results an irresolvable deficiency of demand. The growth rate should be 2333/32500 = 7.18% instead of 8.33%. This must result in decreasing household consumption; but in the real world, some of the households will resist this decreas of consuption, in particular those that are in a position of strength vis à vis their employers; therefore the decrease of consumption will affect those in a weak bargaining position eg. young unqualified people, senior workers near retirement, workers in the private sector hit by the decrease of demand and whose jobs will be shed. The net result will be an increase of unemployement, precarious jobs, and a general stagnation or decrease of purchasing power.

The equilibrium of the Harrod-Domar model is a razor-edge equilibrium. If the economy deviates from it in either direction there will be an economy calamity.