Question

Consider a two-person exchange economy where two goods, \( x \) and \( y \), are available in limited quantities of 50 and 100, respectively. The preferences of the two persons, Anil and Binod, are given by the utility functions: \[ U_{{Anil}}(x_{{Anil}}, y_{{Anil}}) = x_{{Anil}}^{0.4} y_{{Anil}}^{0.6} \] \[ U_{{Binod}}(x_{{Binod}}, y_{{Binod}}) = x_{{Binod}}^{0.6} y_{{Binod}}^{0.4} \] If they decide to share good \( y \) equally among themselves, the amount of good \( x \) Anil receives is ______________ (in integer).

Hint:

In an exchange economy, when goods are allocated between individuals with different preferences, you can use utility maximization techniques to determine the optimal allocation of goods. The optimal allocation depends on the utility functions and the available quantities of goods.

Answer 1

Step 1: Sharing the Good \( y \). Since Anil and Binod decide to share good \( y \) equally, each person receives half of the total available quantity of good \( y \). The total quantity of good \( y \) is 100, so each person will receive: \[ y_{{Anil}} = y_{{Binod}} = \frac{100}{2} = 50 \] Step 2: Maximizing Utility Using the Budget Constraint. The total quantity of good \( x \) is 50, so Anil and Binod together must share the total available amount of good \( x \). Let \( x_{{Anil}} \) be the amount of good \( x \) that Anil receives. The remaining amount of good \( x \), \( x_{{Binod}} \), is then: \[ x_{{Binod}} = 50 - x_{{Anil}} \] Next, we need to maximize their utility functions subject to the budget constraint. The general utility maximization problem involves setting up a Lagrangian function. We want to allocate goods \( x \) and \( y \) between Anil and Binod in such a way that their marginal utilities are proportional to the price ratio. 
Step 3: Utility Maximization and Solution. The solution to this problem can be derived by maximizing the utility functions subject to the budget constraint. After solving, we find that the optimal allocation of good \( x \) for Anil is: \[ x_{{Anil}} = 21 \]