Show that Cobb-Douglas Production Function exhibits both Hicks and Harrod neutral technical progress.

The Cobb-Douglas Production Function

The general form of the Cobb-Douglas production function is:

Y = A * K^α * L^β

Where:

• Y = Output
• A = Total Factor Productivity (TFP) – represents the level of technology
• K = Capital
• L = Labor
• α and β = Output elasticities of capital and labor, respectively (0 < α < 1 and 0 < β < 1)

Hicks-Neutral Technical Progress

Hicks-neutral technical progress implies that the marginal rate of technical substitution (MRTS) between capital and labor remains constant. This means that a given change in technology affects the efficiency of both capital and labor equally. Mathematically, this is represented by an increase in ‘A’ in the Cobb-Douglas function.

If ‘A’ increases to A’, the new production function becomes:

Y’ = A’ * K^α * L^β

The MRTS is given by: MRTS = (MP_K) / (MP_L) = (αK/L) / (βK/L) = α/β. Notice that the MRTS remains unchanged when A changes, demonstrating Hicks neutrality. The increase in A simply scales up the output for any given combination of K and L.

Harrod-Neutral Technical Progress

Harrod-neutral technical progress implies that the amount of labor required to produce a given level of output decreases, while the capital-labor ratio remains constant. This is equivalent to an increase in the effective labor supply. We can represent this by considering the production function in terms of effective labor.

Let’s define effective labor (L*) as: L* = E * L, where E represents labor-augmenting technological progress.

Substituting L* into the Cobb-Douglas function, we get:

Y = A * K^α * (E * L)^β = A * E^β * K^α * L^β

Now, if E increases to E’, the new production function becomes:

Y’ = A * E’^β * K^α * L^β

Here, the increase in E directly increases output for any given levels of K and L. The capital-labor ratio (K/L) remains unchanged, demonstrating Harrod neutrality. The increase in E effectively increases the productivity of labor without affecting the productivity of capital directly.

Demonstrating Both Neutralities

The Cobb-Douglas function can represent both types of technical progress because of its specific functional form. An increase in ‘A’ represents Hicks-neutral progress, while an increase in ‘E’ (labor-augmenting technology) represents Harrod-neutral progress. The function’s inherent structure allows for these distinct forms of technological advancement to be modeled separately or even simultaneously.

Economic Intuition

Hicks-neutral progress is often associated with inventions that improve the efficiency of both capital and labor equally, such as general-purpose technologies like electricity. Harrod-neutral progress, on the other hand, is often linked to innovations that specifically enhance labor productivity, such as computer software or improved worker training.