If an individual with the utility function U = X0.5Y0.5 consumes 2 units of X and 8 units of Y, how much Y is needed for 4 units of X to maintain the same utility?

To determine how much Y is needed for 4 units of X while maintaining the same utility level as when consuming 2 units of X and 8 units of Y, we first need to find the initial utility level with the given consumption.

The utility function is given as ( U = X^{0.5}Y^{0.5} ).

1. Calculate Initial Utility: Substitute ( X = 2 ) and ( Y = 8 ) into the utility function: [ U = (2)^{0.5} \times (8)^{0.5} ] This can be simplified to: [ U = \sqrt{2} \times \sqrt{8} = \sqrt{16} = 4 ]

2. Set Up for New Utility: Now, we want to find the value of Y when X is 4. We want to maintain the same utility level, which we have calculated to be 4. So we set up the equation: [ 4 = (4)^{0.5} \times (Y)^{0.5} ] Simplifying