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Step-by-step mathematical derivation with typeset equations and expandable detail.
Sections
Capital accumulation and the savings-investment identity
The Harrod-Domar model is a foundational post-Keynesian growth framework that links the economy's growth rate to its saving rate and the productivity of capital. The model assumes a closed economy with no government sector, so the national income identity reduces to and the goods-market equilibrium condition is . Saving is a constant fraction of income: .
Output is produced with a fixed capital-output ratio (also called the incremental capital-output ratio, ICOR). This means that units of capital are required to produce one unit of output, with no substitutability between capital and labor. Capital accumulates through investment: , where is the depreciation rate. In the simplest version, depreciation is set to zero so that net investment equals gross investment: .
Aggregate saving: a constant fraction of national income.
Fixed capital-output ratio: units of capital produce one unit of output.
Capital accumulation: net investment equals saving minus depreciation.
The warranted rate of growth
The warranted growth rate is the rate at which output must grow for planned saving to equal planned investment, keeping the capital-output ratio at its desired level. Since , differentiating with respect to time gives . In equilibrium, , so . Dividing both sides by yields the warranted growth rate .
This is the central result of the Harrod-Domar model. It says that the equilibrium growth rate of output is determined by the ratio of the saving rate to the capital-output ratio. An economy that saves 20% of its income and requires 4 units of capital per unit of output grows at 5% per period. With depreciation, the formula adjusts to . The result carries a powerful policy implication for developing economies: to grow faster, either raise the saving rate (through domestic saving or foreign aid) or lower the capital-output ratio (through more efficient investment).
The warranted growth rate: saving rate divided by the capital-output ratio.
Warranted growth with depreciation: net of capital decay.
The knife-edge instability
The most striking feature of the Harrod-Domar model is its instability: if the economy departs even slightly from the warranted growth path, it does not self-correct but instead diverges further. Suppose output grows faster than . Firms find their capital insufficient to meet demand (capital utilization exceeds the desired ratio), so they invest more aggressively. But higher investment raises income (through the multiplier), which generates even more saving and investment, pushing growth further above the warranted rate. The boom feeds on itself.
Conversely, if growth falls below , firms discover they have excess capacity (capital exceeds what is needed for current output). They cut investment, which reduces income, lowers saving, and further depresses growth. The slump deepens. This knife-edge property arises because the model has no built-in stabilizer: the fixed capital-output ratio means there is no price adjustment (no flexible interest rate clearing the loanable-funds market) and no factor substitution to absorb the imbalance. The economy balances on the warranted path like a ball on a razor's edge. This instability motivated Solow's neoclassical growth model, which introduces a variable capital-output ratio through a smooth production function, eliminating the knife-edge.
Above-warranted growth generates excess demand for capital, which stimulates further investment and accelerates growth away from equilibrium.
Below-warranted growth creates excess capacity, discouraging investment and deepening the contraction.