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 $\alpha$ is a parameter which lies on the interval (0,1).

Consumer demand function

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

$M = P_{x} x + P_y y$

where $M$ is the consumers income, $P_x$ is the price of good $x$ and $P_y$ is the price of good $y$.

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

$\mathcal{L} = x^{\alpha} y^{1-\alpha} + \lambda(M - P_x x + P_y y)$

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:

$<br /> U = X^\alpha Y^{1-\alpha}<br />$(1)

where $\inline U$ is the consumers utility, $\inline X$ is the quantity of commodity 1 and $\inline Y$ is the quantity of commodity 2.

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

$<br /> M = P<em>{X} X + P</em>{Y} Y<br />$(2)

where $\inline M$ is the consumers income, $\inline P<em>{X}$ is the price of commodity $\inline X$ and $\inline P</em>{Y}$ is the price of commodity $\inline Y$.

The demand for commodity $\inline X$ and commodity $\inline Y$ are derived from the cost minimising solution to the Lagrangian

$\mathcal{L} = X^\alpha Y^{(1-\alpha)} + \lambda[M - P_X X + P_Y Y]<br />$ (3)

The first-order conditions for the Lagrangian are:

$\frac{\partial \mathcal{L}}{\partial X}=\alpha \frac{U}{X}-\lambda P_{x}=0$(3)

$\frac{\partial \mathcal{L}}{\partial Y}=(1-\alpha) \frac{U}{Y}-\lambda P_{y}=0<br />$(4)

The two equations can be equated to find a relationship between $\inline X$ and $\inline Y$ such that

$\alpha \frac{U}{P<em>{X} X}= \frac{(1-\alpha) U }{P</em>{Y} Y}$(5)

which can be rearranged such that $\inline X$ is the subject

$X = (\frac{\alpha}{(1-\alpha)})(\frac{P_Y}{P_X})Y$(6)

Substituting this equation back into the budget constraint yields:

$M=P<em>{X} \frac{\alpha}{(1-\alpha)} \frac{P</em>{Y}}{P<em>{X}} Y +P</em>{y} Y$(7)

Solving for $\inline Y$ is completed via the following steps:

$M =\frac{\alpha}{1-\alpha}P<em>{Y} Y +P</em>{Y} Y$(8)

$(1-\alpha) M =\alpha P<em>{Y} Y + (1-\alpha) P</em>{Y} Y$(9)

$(1-\alpha) M = P_{Y} Y$(10)

The demand for $\inline Y$ is expressed:

$Y =\frac{(1-\alpha) M}{P_{Y}}$(11)

where:

$P_{Y} Y = (1-\alpha) M$(12)

Budget share

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

$(1-\alpha)=\frac{P_{Y} Y}{M}$(13)

The $\inline \alpha$ 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:

$\varepsilon=\frac{d Y}{d P_{Y}} \frac{P_Y}{Y}$(14)

Where $\frac{d y}{d p}$ is the derivative of demand for $x$ with respect to the price of $\inline X$.

$\frac{d y}{d P<em>{y}}=-(1-\alpha) \frac{M}{P</em>{y}} \cdot \frac{1}{P_{y}}$(15)

$\frac{d y}{d P<em>{y}}=\frac{-y}{P</em>{y}}$(16)

$\frac{d y}{d p<em>{y}} \cdot \frac{p</em>{y}}{y}=-1$(17)

Therefore, $\inline \varepislon = -1$ is the own-price elasticity.