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Step-by-step mathematical derivation with typeset equations and expandable detail.
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The AK production function
The AK model replaces the neoclassical production function with a linear technology , where is a constant productivity parameter and represents a broad concept of capital that includes physical capital, human capital, knowledge, and organizational capital. The critical feature is constant returns to this broad capital aggregate: doubling exactly doubles , with no diminishing marginal product. This contrasts with the Solow model's where ensures diminishing returns to capital alone.
The justification for constant returns rests on treating as encompassing all reproducible factors. Physical capital accumulation generates learning-by-doing externalities that raise human capital; investment in research creates knowledge spillovers. When these complementarities are strong enough, the aggregate return to investment does not decline as the economy grows. In per-capita terms (normalizing labor to 1), the production function is simply , and the marginal product of capital is the constant .
AK production function: output is proportional to broad capital with no diminishing returns. The marginal product of capital is the constant .
Constant marginal product of capital: unlike the neoclassical model, additional capital always yields the same return , regardless of the capital stock.
Endogenous growth without convergence
Capital accumulates according to , where is the saving rate and is the depreciation rate. Substituting gives . Dividing both sides by yields the growth rate of capital, which is also the growth rate of output: . This growth rate is constant and independent of the level of .
The absence of diminishing returns eliminates the convergence mechanism that is central to the Solow model. In Solow, a poor country (low ) has a high marginal product of capital and therefore grows faster than a rich country, eventually catching up. In the AK model, the marginal product is always regardless of , so rich and poor countries grow at the same rate forever. There is no steady state to converge toward. Initial differences in capital persist indefinitely, and there is no tendency for cross-country income levels to equalize.
Constant endogenous growth rate: determined by the saving rate, productivity, and depreciation. Positive long-run growth requires .
No convergence: the growth rate does not depend on the capital stock, so poor and rich economies grow at the same rate permanently.
Policy implications: savings and long-run growth
The AK model's most striking implication is that the saving rate affects the long-run growth rate, not merely the level of income as in Solow. A permanent increase in raises permanently. This gives governments a direct lever on growth: policies that raise investment rates (tax incentives for R&D, education subsidies, infrastructure spending) translate into permanently faster growth rather than a temporary transition to a higher level.
This result also means that temporary policy shocks can have permanent effects on the level of output. If the saving rate rises for a finite period and then returns to its original value, the economy is permanently richer than it would have been. There is no mean-reversion as there would be in a neoclassical model. The flip side is that policies or shocks that reduce investment (conflict, institutional breakdown, capital flight) cause permanent growth losses that are never recovered.
Output path: exponential growth at rate . A higher raises both the exponent and therefore the entire trajectory permanently.
Growth impact of policy: a one-percentage-point increase in the saving rate raises the growth rate by percentage points, with no diminishing effect over time.