Solution | Q. No. - 2 | General Economics - I | 2009

2. Show how a demand function may be derived from the Cobb-Douglas utility function.

The Cobb-Douglas utility function can be written as follows: \[U = {x^\alpha }{y^\beta }\] Where x and y are quantity demanded of the two goods, say, Good X and Good Y. A consumer's objective is to maximize utility with the given budget constraints which can be written as: \[I = {p_x}x + {p_y}y\] Equlibrium Condition in this two commodity is given by: \[MR{S_{x,y}} = \frac{{{p_x}}}{{{p_y}}}\] Where, \[MR{S_{x,y}} = - \frac{{dy}}{{dx}} = \frac{{M{U_x}}}{{M{U_y}}}\] We can derive the MU of x and y from the utility function as follows: \[M{U_x} = \frac{{\partial U}}{{\partial x}}\] \[M{U_x} = \alpha {x^{\alpha - 1}}{y^\beta }\] \[M{U_x} = \frac{{\alpha {x^\alpha }{y^\beta }}}{x}\] \[M{U_x} = \frac{{\alpha U}}{x}\] Similarly, \[M{U_y} = \frac{{\beta U}}{y}\] Now, we can derive MRS from the above result: \[MR{S_{x,y}} = \frac{{M{U_x}}}{{M{U_y}}} = \frac{{\alpha U}}{x} \times \frac{y}{{\beta U}}\] \[MR{S_{x,y}} = \frac{{\alpha y}}{{\beta x}}\] From the results derived above, we can write equlibrium condition for Cobb-Douglas as: \[\frac{{\alpha y}}{{\beta x}} = \frac{{{p_x}}}{{{p_y}}}\] Rearranging the equation above, we can write: \[y = \frac{\beta }{\alpha }\frac{{{p_x}x}}{{{p_y}}}\] Substituting this value of y in the budget constraint, we get: \[I = {p_x}x + {p_y}\frac{\beta }{\alpha }\frac{{{p_x}x}}{{{p_y}}}\] We can remove py from the equation above: \[I = {p_x}x + \frac{\beta }{\alpha }{p_x}x\] Rearranging the equation above, we get: \[I = {p_x}x\left( {\frac{{\alpha + \beta }}{\alpha }} \right)\] Let \({\alpha + \beta = 1}\), \[x = \frac{{\alpha I}}{{{p_x}}}\] this is the demand function for Good X, which shows a negtaive relationship between quantity demanded and the price of the commodity. Similarly, by substituting the value of this demand function in the budget constraint, we can derive the demand function for Good Y as follows: \[y = \frac{{\beta I}}{{{p_y}}}\]

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