In: Economics
Consider the following utility function: u(x, y) = x2/3y1/3. Suppose that Px = 4, Py = 2 and the income is I = 30. Derive the optimal choice for both goods.
The utility function is given as , prices are and , while income .
The the optimal choice for goods x and y is where given the budget constraint, utility is maximized. Budget constraint can be arranged as , assuming all income is exhausted on the commodities. Hence, we have as the budget constraint. We will optimize the problem with Lagrangian multiplier.
The problem is subject to .
The lagrangian function is hence, . We will do partial differentiation with respect to lambda, x and y.
or or . Equating , we have or , the constraint itself.
or or or . Equating , we have or or .
or or or . Equating , we have or or .
Equating the last two lambda's, we have or or or or .
Putting the value in the constraint, we have or or ; and or or .
Hence, the optimal choice of maximizing the given utility in the given budget constraint is , ie 5 units of x and 5 units of y.