Final answer:
The optimal choice of x for Consumer A is found by equating the marginal utility to price ratio of both goods using the given utility function and budget constraint. Partial derivatives of the utility function for X and Y are used to solve for the quantities that maximize utility within the budget constraint.
Step-by-step explanation:
To find the optimal choice of x for Consumer A, we can use the concept of utility maximization under a budget constraint. We know the utility function for Consumer A is given as Uᴬ=X⁰.⁵ Y⁰.⁵, and the budget constraint is I=Px(X)+Py(Y) with prices Px=3 and Py=6 and income I=100. By using the marginal utility to price ratio method, we set the marginal utility of X (MUX) over the price of X (Px) equal to the marginal utility of Y (MUY) over the price of Y (Py) as follows:
MUX / Px = MUY / Py
Calculating the partial derivatives of the utility function will give us MUX = 0.5*X-0.5Y⁰.⁵ and MUY = 0.5*X⁰.⁵Y-0.5. Substituting these into the equation above and using the given prices, we can solve for the quantities of X and Y that maximize utility subject to the budget constraint. This will determine the optimal choice of X for Consumer A. To find the optimal choice of x for Consumer A, we need to compare the ratios of the marginal utility of good X to the price of good X (3) and the marginal utility of good Y to the price of good Y (6). At the optimal choice, these ratios should be equal.
The utility function for Consumer A is Uᴬ = X⁰.⁵Y⁰.⁵. To find the ratios, we can take the derivative of the utility function with respect to X and Y:
dUᴬ/dX = 0.5X^(-0.5)Y^(0.5)
dUᴬ/dY = 0.5X^(0.5)Y^(-0.5)
Setting the ratio dUᴬ/dX divided by Px equal to the ratio dUᴬ/dY divided by Py, we get:
(dUᴬ/dX) / Px = (dUᴬ/dY) / Py
Plugging in the values for Px, Py, and the derivatives, we have:
(0.5X^(-0.5)Y^(0.5)) / 3 = (0.5X^(0.5)Y^(-0.5)) / 6
Simplifying this equation, we can cross multiply and rearrange to solve for X:
0.5X^(-0.5)Y^(0.5) = (0.5X^(0.5)Y^(-0.5)) / 2
0.5X^(-0.5)Y^(0.5) = 0.25X^(0.5)Y^(-0.5)
At this point, we can cancel out the common terms and solve for X:
2X^(-0.5)Y^(0.5) = X^(0.5)Y^(-0.5)
2Y = X
Therefore, the optimal choice of X for Consumer A is equal to 2Y.