Question

Consider an individual whose preferences are described by the utility function U = X⁰·⁵Y⁰·⁵. If she consumes 2 units of X and 6 units of Y, then some level of utility will be experienced. If the individual instead consumes 4 units of X, how much of good Y must she consume in order to attain the level of utility associated with 2 units of X and 6 units of Y?

Answer 1

Using the utility function U = X^0.5Y^0.5, the individual must consume approximately 6 units of good Y to achieve the same utility level with 4 units of X as previously achieved with 2 units of X and 6 units of Y.

Step-by-step explanation:

To determine the amount of good Y that the individual must consume to attain the same level of utility with 4 units of X, we first calculate the original utility with 2 units of X and 6 units of Y. The utility function is U = X0.5Y0.5. Plugging in the original consumption levels, we get U = 20.5 × 60.5, which simplifies to U = 1 × 2.45, giving a utility level of approximately 4.9 utils.

Knowing the original utility, we can set up an equation with the new amount of X (4 units) and solve for the required amount of Y. The equation becomes 4.9 = 40.5Y0.5. Solving for Y gives us Y = (4.9 / 40.5)2, which is approximately 2.452, resulting in Y being approximately 6 units.

Therefore, to maintain the same utility level as originally attained with 2 units of X and 6 units of Y, the individual needs to consume the same amount of Y, which is 6 units, when consumption of X is increased to 4 units.