Answer:
Explanation:
To find the utility-maximizing bundle of goods, we need to solve for the values of x and y that maximize U while still satisfying the budget constraint. The budget constraint can be written as:
2x + 4y = 24
or
x + 2y = 12
We can use the method of Lagrange multipliers to solve for the utility-maximizing values of x and y subject to this constraint. The Lagrangian function is:
L = 3x^(2/3)y^(-1/3) + λ(x + 2y - 12)
Taking partial derivatives with respect to x, y, and λ, we get:
dL/dx = 2x^(-1/3)y^(-1/3) + λ = 0
dL/dy = -x^(2/3)y^(-4/3) + 2λ = 0
dL/dλ = x + 2y - 12 = 0
Solving these equations simultaneously, we get:
x = 6
y = 3
So the utility-maximizing bundle is 6 units of x and 3 units of y.
To see if the individual is better off with the government intervention, we can plot the budget line and the indifference curve for the utility-maximizing bundle with and without the limit on x.
Without the limit, the budget line is the same as before (x + 2y = 12), and the indifference curve for the utility-maximizing bundle passes through the point (6, 3) on the graph.
With the limit, the budget line becomes x = 4, since the individual is prohibited from purchasing more than 4 units of x. The corresponding budget line has a slope of -1/2 and intercepts the y-axis at 6.
If we draw the indifference curve for the utility-maximizing bundle of (4,4), which lies on the budget line, we can see that the individual is not better off with the government intervention. This is because the slope of the budget line under the intervention is steeper, so the individual would have to give up more y than x to afford the same amount of utility. Thus, the individual would have to move to a lower indifference curve with lower utility.
Therefore, the individual is not better off with the government intervention.