Cobb-Douglas Functions

A Cobb-Douglas function takes the form:

\[y=Ax_1^\alpha x_2 ^\beta\]

It is often used for utility functions \((y\) is utility, \(x_1\) and \(x_2\) are goods) and production functions \((y\) is output, \(x_1\) and \(x_2\) are factors of production, such as \(L\) and \(K).\)

It is often converted into logarithmic form by taking the natural logs:

\[ln \, y = ln \, A + \alpha \, ln \, x_1 + \beta \, ln \, x_2\]

This function is (surprisingly) easy to work with for its many useful properties:

1. Returns to scale
   • If \(\alpha+\beta > 1\): increasing returns to scale
   • If \(\alpha+\beta = 1\): constant returns to scale
   • If \(\alpha+\beta < 1\): decreasing returns to scale
2. Elasticities/Shares
   • In the constant returns to scale case (the exponents sum to 1), often rewritten as \(y=Ax_1^{\alpha}x_2^{1-\alpha}\):
     • \(\alpha\) represents the (fraction of income spent - in utility case; output elasticity - in production case) of each \(x_i\)
       • e.g. if \(u=x^{0.4}y^{0.6}\), consumer spends 40% of income on \(x\) and 60% on \(y\)
       • e.g. if \(y=K^{0.4}L^{0.6}\), a 1% increase in \(K\) generates a \(0.4\)% increase in \(y\)
       • Aggregate Production Function case: \(\alpha\) is the “share” of output or income going to each factor
3. Desirable shape of function
   • Convex
   • Positive but diminishing returns to each \(x_i\), \(\left(\frac{\partial y}{\partial x_i} > 0, \, \frac{\partial^2 y}{\partial x_i^2} < 0 \right)\)

See more on Cobb-Douglas functions from my ECON 306 - Microeconomic Analysis course notes:

• Cobb Douglas Utility Functions
• Constrained Optimization with Cobb-Douglas
• Deriving Demand from Cobb-Douglas Utility

For production, simply replace \(u\) with \(y\), and \(x\) and \(y\) with \(l\) and \(k\).