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:

$$
U = X^\alpha Y^{1-\alpha}
$$(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:

$$
M = P{X} X + P{Y} Y
$$(2)

where $$\inline M$$ is the consumers income, $$\inline P{X}$$ is the price of commodity $$\inline X$$ and $$\inline P{Y}$$ 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

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

The first-order conditions for the Lagrangian are:

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

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

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

$$ \alpha \frac{U}{P{X} X}= \frac{(1-\alpha) U }{P{Y} Y}$$(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:

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

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

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

$$(1-\alpha) M =\alpha P{Y} Y + (1-\alpha) P{Y} Y $$(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 $$\inline X$$.

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

$$\frac{d y}{d P{y}}=\frac{-y}{P{y}} $$(16)

$$\frac{d y}{d p{y}} \cdot \frac{p{y}}{y}=-1$$(17)

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

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