AK Model and Linear Production Functions

Core Idea

The AK model assumes a linear production function Y = A·K with constant returns to capital and no diminishing returns. This generates perpetual growth: if agents save a constant fraction of output, capital and consumption grow at a constant rate indefinitely without requiring exogenous technological progress.

Explainer

The Solow model, which you encountered earlier in your study of endogenous growth theory's motivations, has a famous limitation: long-run growth in output per worker eventually stops unless technology improves exogenously. The reason is diminishing returns to capital — each additional unit of capital produces less additional output than the last. As an economy accumulates more machines, factories, and infrastructure, the marginal product of capital falls, investment just barely covers depreciation, and growth grinds to a halt. The AK model asks: what if diminishing returns never set in?

The AK production function is strikingly simple: Y = A·K, where A is a positive constant representing productivity and K is the broad capital stock. Output is directly proportional to capital with no diminishing returns — double the capital and you exactly double output. This linearity is the model's defining feature. The "A" captures not just physical productivity but also human capital, knowledge, and organizational capacity embedded in the capital stock. Under this interpretation, K is not just machines but the entire stock of productive assets including education, R&D, and institutional capacity. Because these forms of capital generate positive externalities (a more educated workforce raises everyone's productivity), the aggregate production function can exhibit constant returns to capital even if individual firms face diminishing returns.

The growth implications are dramatic. With a constant savings rate *s* and depreciation rate *δ*, the growth rate of capital (and therefore output) is simply *sA − δ*. As long as *sA > δ* — as long as the return to saving exceeds what depreciation destroys — the economy grows at a constant, positive rate forever. There is no convergence to a steady state, no need for exogenous technological progress, and no prediction that poor countries will catch up to rich ones. The growth rate depends on the savings rate and the productivity parameter, both of which can differ permanently across countries. This is a sharp contrast with the Solow model, where the savings rate affects the *level* of income but not the long-run growth rate.

The AK model is powerful because it demonstrates the minimum theoretical ingredient needed for endogenous growth: eliminate diminishing returns to the accumulable factor. But this simplicity is also its weakness. The model predicts that countries with higher savings rates grow permanently faster — an extreme prediction that fits some cross-country data but not all. It also lacks a mechanism for explaining *why* A differs across countries or how policy might change it. More sophisticated endogenous growth models (Romer, Lucas) build on the AK insight by modeling the micro-foundations of knowledge creation and human capital accumulation explicitly. The AK model remains valuable as the cleanest illustration of the core logic: sustained growth requires that the engine of accumulation never runs into diminishing returns.