Cobb-Douglas Production Function Properties

The question asks to identify the statement that does *not* represent a property of the Cobb-Douglas Production Function, typically represented as $Q = AK^\alpha L^\beta$, where $Q$ is output, $K$ is capital, $L$ is labour, $A$ is total factor productivity, and $\alpha$ and $\beta$ are output elasticities.

Analysis of Options

• Option 1: Log-linear form.

  The multiplicative form $Q = AK^\alpha L^\beta$ can be transformed into a log-linear form by taking the natural logarithm: $\ln(Q) = \ln(A) + \alpha \ln(K) + \beta \ln(L)$. This is a standard transformation and thus a property.

• Option 2: Homogeneity.

  The statement claims power functions are *not* homogeneous because the sum of the exponents ($\alpha + \beta$) is not equal to 1. This is incorrect. A function is homogeneous of degree $k$ if $f(tK, tL) = t^k f(K, L)$. For the Cobb-Douglas function, $f(tK, tL) = A(tK)^\alpha (tL)^\beta = t^{\alpha+\beta} AK^\alpha L^\beta = t^{\alpha+\beta} Q$. The function is always homogeneous of degree $(\alpha + \beta)$. The degree of homogeneity does not need to be 1 for the function to be homogeneous. Therefore, this statement is *not* a property.

• Option 3: Elasticity Coefficient.

  The exponents $\alpha$ and $\beta$ directly measure the output elasticity with respect to capital and labour, respectively. For example, $\frac{\partial Q}{\partial K} \frac{K}{Q} = \alpha$. This is a key feature and thus a property.

• Option 4: Share of Inputs.

  When the function exhibits constant returns to scale (i.e., $\alpha + \beta = 1$), the parameters $\alpha$ and $\beta$ represent the proportion of total output attributed to capital and labour, respectively (often interpreted as income shares). This interpretation holds under the CRS condition.

Based on the analysis, the statement that misinterprets homogeneity is not a property of the Cobb-Douglas Production Function.