Question

In: Economics

Consider the following utility function: u(x, y) = x2/3y1/3. Suppose that Px = 4, Py =...

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.

Expert Solution

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 .

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.

Rahul Sunny answered 2 years ago

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