The Cobb-Douglas utility function is a mathematical representation of consumer preferences used in microeconomics. It is widely used to model how consumers allocate their spending across different goods and services. The utility function is named after economists Paul Douglas and Charles Cobb, who first introduced it in 1928.

Utility Function:
The Cobb-Douglas utility function is expressed as:

U(x1, x2) = x1^α * x2^β

Where:
- U represents the level of utility or satisfaction derived from consuming goods x1 and x2.
- x1 and x2 represent the quantities consumed of goods 1 and 2, respectively.
- α and β are parameters that determine the shape of the utility function. They are positive constants that reflect the consumer's preferences for the two goods.

Properties of Cobb-Douglas Utility Function:
1. Homogeneity: The Cobb-Douglas utility function exhibits constant returns to scale. This means that if all inputs (x1 and x2) are multiplied by a positive constant, the resulting utility will also be multiplied by the same constant.

2. Diminishing Marginal Rate of Substitution (MRS): The MRS measures the rate at which a consumer is willing to substitute one good for another while maintaining the same level of satisfaction. In the Cobb-Douglas utility function, the MRS is given by the ratio of the marginal utilities of the two goods:

MRS = (MU1/MU2) = -αx2/(βx1)

The MRS is negative because the consumer is typically willing to trade off some quantity of one good for more of the other.

Graphical Representation:
To visualize the Cobb-Douglas utility function, we can plot indifference curves on a graph. Indifference curves represent combinations of goods that yield the same level of utility.

Examples:
1. Suppose a consumer has a Cobb-Douglas utility function with α = 0.5 and β = 0.5. This implies that the consumer values both goods equally. The indifference curves will be downward sloping straight lines with a constant slope of -1, indicating a constant rate of substitution between the two goods.

2. If we have α = 0.6 and β = 0.4, the consumer has a slight preference for good 1. The indifference curves will be convex to the origin, indicating a decreasing rate of substitution as more of good 1 is consumed.

Overall, the Cobb-Douglas utility function provides a useful tool for analyzing consumer behavior and understanding how individuals make choices based on their preferences and the relative prices of goods.