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.