tThe Cobb-Douglas function is used to measure consumers preferences between different bundles of goods or how a firm can combine inputs to produce output.

# Cobb-Douglas utility function

Consider a world which only has two goods. The Cobb-Douglas utility function is usually defined as:

`U= X^\alpha Y^{(1-\alpha)}`

where U represents a consumers utility function, X is the quantity of good 1, Y is the quantity of good 2 and is a parameter which lies on the interval (0,1).

## Consumer demand function

Consider the Cobb-Couglas utility function subject to the budget constraint

where is the consumers income, is the price of good and is the price of good .

The consumer demand equation is found by maximising the consumers utility function subject to their budget constraint. The Langrangian function is defined as

The first-order conditions are:

```
\frac{\partial p}{\partial x}=\alpha \frac{u}{x}-\lambda P_{x}=0 \\
\frac{\partial \alpha}{\partial x}=(1-\alpha) \frac{u}{y}-\lambda P_{y}=0
```

# Cobb-Douglas function

The Cobb-Douglas function is used to represent consumers preferences or the production function of a firm.

## Consumer demand equations

In a world where there are two commodities, the Cobb-Douglas utility function for an individual is expressed:

(1)

where is the consumers utility, is the quantity of commodity 1 and is the quantity of commodity 2.

The budget constraint for an individual with fixed income is represented:

(2)

where is the consumers income, is the price of commodity and is the price of commodity .

The demand for commodity and commodity are derived from the cost minimising solution to the Lagrangian

(3)

The first-order conditions for the Lagrangian are:

(3)

(4)

The two equations can be equated to find a relationship between and such that

(5)

which can be rearranged such that is the subject

(6)

Substituting this equation back into the budget constraint yields:

(7)

Solving for is completed via the following steps:

(8)

(9)

(10)

The demand for is expressed:

(11)

where:

(12)

## Budget share

The budget share represents the fraction of income that is spent on each commodity. For the commodity , the budget share can be derived by re-arranging () such that:

(13)

The parameters of the Cobb-Douglas function turn out to be the budget shares for an individual.

## Own-price elasticity

The own-price elasticity is defined as:

(14)

Where is the derivative of demand for $x$ with respect to the price of .

(15)

(16)

(17)

Therefore, is the own-price elasticity.