Cobb-Douglas function

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 />

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 />

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 />

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 />

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)


 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.

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