Inada Conditions

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Inada Conditions

Economists usually posit that production can be represented by a mathematical function that relates output to input factors. One assumes that aggregate production (Y ) depends on the two input factors capital (K ) and labor (L ). Formally, the process relating output to the inputs is described by the function F (K, L ) = Y, where more input raises output. The function F (·) is assumed to be linearly homogenous, so that production can be written as Y = L · F (K/L, 1), stating that raising capital and labor by a certain amount leads to a proportional rise of production. Given this property, production per labor input, y = Y/L, can be written as y = f (k), with k = K/L capital per labor. The function f (·) is increasing in k continuously, meaning that jumps are excluded, and it is continuously differentiable, implying that the derivative of the function exists and is itself a continuous function.

In addition, f (k ) satisfies the following conditions, called the Inada conditions, which are ideal assumptions about its shape:

The value of the function at 0 is zero, f (0) = 0
The function is strictly increasing in k, f'(k) > 0, with ' denoting the derivative
The derivative of the function is decreasing so that the function is strictly concave, f”(k) < 0
The limit of the derivative approaches plus infinity when k goes to zero, lim f'(k) = ∞, for k → 0
The limit of the derivative approaches zero when k goes to infinity, lim f'(k) = 0, for k → ∞

These conditions were named after the Japanese economist Ken-Ichi Inada (1925–2002), who wrote a number of important papers on welfare economics, economic growth, and international trade.

The Inada conditions guarantee that a unique steady state in the neoclassical growth model exists and is stable. The Inada conditions are purely technical assumptions and are hardly relevant for empirical economics. The neoclassical growth model is described by the differential equation dk (t )/dt = s · f (k(t )) –(n +δ ) · k (t ), where s is the constant savings rate, n is the growth rate of labor, δ is the depreciation rate of capital, and t denotes time. This

equation states that the change in capital per labor equals savings per labor in the economy, which are equal to investment, minus that part of capital per labor that is used up due to depreciation and due to the growth of the labor force. A steady state is defined as a situation with dk (t )/dt = 0 so that capital per labor is constant over time. Total investment, I (t ), in the economy then equals (n + δ ) · K (t ), so that the total capital stock grows at the same rate as labor. Given the Inada conditions, there exists a unique value k * > 0 solving dk (t )/dt = 0, because s · f (·) is larger than (n + δ ) · k for values of k < k *, and it is smaller for values of k > k *. Figure 1 illustrates the situation graphically. In addition, k < k * is stable because for k < k * capital per labor rises, dk (t )/dt > 0, and for k > k * capital declines, dk (t )/dt < 0.