The AK model of economic growth is an endogenous growth model used in the theory of economic growth, a subfield of modern macroeconomics. In lieu of the diminishing returns to capital implied by the usual parameterizations of a Cobb–Douglas production function, the AK model uses a linear model where output is a linear function of capital. Its appearance in most textbooks is to introduce endogenous growth theory.

Origin of the name

The name AK model originates in the mathematic representation of the production function in this model. This production function is a special case of a Cobb–Douglas function with constant returns to scale.

Basic Thoery

This simple AK growth model highlights the link between investment rates and growth rates that this class of models predicts. As we shall see, the simplest versions of AK models imply a tight positive relationship between investment as a share of output and the growth rate of output. This model is a representative household that chooses per capita consumption c and per capita investment x in each period to maximize utility U; that is, (1) $max{ct,xt}t=0btU(ct)\,$

for 0 < b < 1, where t is an index for time. The optimization problem (1) is subject to a resource constraint, a capital accumulation constraint, and inequality constraints: (2) $ct+xt=Akt\,$

(3)Failed to parse (syntax error): {\displaystyle kt+1 = (1−d)kt + xt\,}

(4)Failed to parse (syntax error): {\displaystyle ct ³ 0 and xt ³ 0\,}

given k0, where kt is the stock of capital at time t, A is the level of technology, and d is the rate of depreciation of the capital stock. Per capita output in this model is simply (5) $yt=Akt\,$

The production technology in equation (2) has constant returns to scale; clearly, doubling the stock of capital doubles output.Without diminishing returns to scale, a country with a high stock of capital will continue to invest and continue to grow. To justify the constant returns assumption, we typically interpret the capital stock as a broad measure that includes not only physical capital, but also human capital and intangible capital. If the level of technology does not change over time, then in this simple version of the model, the growth rate of output equals the growth rate of the capital stock. If we divide both sides of equation (3) by the current capital stock kt, then we have (6)Failed to parse (syntax error): {\displaystyle gt = 1 − d + xt/kt\,}

(7)Failed to parse (syntax error): {\displaystyle gt = 1 − d + Axt/yt\,}

where gt is the growth rate of capital and of output at time t. Equation (7) illustrates the tight link predicted between the investment rate and output growth. This theory predicts that sustained increases in the investment/output ratio should be accompanied by sustained increases in the growth rate of output.^[1]

Y=AK^{a}L^{1-a}\,

This equation shows a Cobb–Douglas function where Y represents the total production in an economy. A represents total factor productivity, K is capital, L is labor, and the parameter $a$ measures the output elasticity of capital. For the special case in which $a=1$ , the production function becomes linear in capital and does not have the property of decreasing returns to scale in the capital stock, which would prevail for any other value of the capital intensity between 0 and 1.