Suppose an economy is characterized by a linear production possibility function for its two goods \((X \text { and } Y)\) of the form \\[ X+2 Y=180 \\] There are two individuals in this economy, each with an identical utility function for \(X\) and \(Y\) of the form \\[ \boldsymbol{U}(\boldsymbol{X}, \boldsymbol{Y})=\sqrt{\boldsymbol{X} \boldsymbol{Y}} \\] a. Suppose \(Y\) production is set at \(10 .\) What would the utility possibility frontier for this economy be? b. Suppose \(Y\) production is set at \(30 .\) What would the utility possibility frontier be? c. How should \(Y\) production be chosen so as to ensure the "best" utility possibility frontier? d. Under what conditions (contrary to those of this problem) might your answer to part (c) depend on the point on the utility possibility frontier being considered?

We are given that the production of Y is fixed at 10. We plug this value into the production possibility function: \[X + 2(10) = 180\]

Solving for X: \[X = 180 - 20 = 160\]

Now we have the production levels for X and Y. We can use these values to find the utility for each individual with the given utility function: \[U(X, Y) = \sqrt{XY}\] \[U(160, 10) = \sqrt{160*10} = \sqrt{1600} = 40 \] So, the utility possibility frontier with Y production set at 10 is (40, 40). #b. Utility possibility frontier with Y production at 30#

We are now given that the production of Y is fixed at 30. We plug this value into the production possibility function: \[X + 2(30) = 180\]

Solving for X: \[X = 180 - 60 = 120\]

Now we have the production levels for X and Y. We can use these values to find the utility for each individual with the given utility function: \[U(120, 30) = \sqrt{120*30} = \sqrt{3600} = 60 \] So, the utility possibility frontier with Y production set at 30 is (60, 60). #c. Optimal Y production level for the best utility possibility frontier#

We need to maximize the utility function with respect to Y production, subject to the linear production possibility constraint: Maximize \(U(X, Y) = \sqrt{XY}\) Subject to \(X + 2Y = 180\)

Solving the production function for X, we get: \(X = 180 - 2Y\) Now, substitute this expression into the utility function: \[U(180-2Y, Y) =\sqrt{(180 - 2Y)Y}\]

Differentiate the utility function with respect to Y: \[ \frac{dU}{dY} = \frac{1}{2} \frac{180Y - 2Y^2 - 180Y + 2Y^2}{\sqrt{(180 - 2Y)Y}} = 0 \] Setting the first order condition equal to zero and simplifying the equation, we get: \[Y = 45\]

Now that we have the optimal Y production, we can find the optimal X production by plugging Y value back into the production possibility function: \[X + 2(45) = 180\] \[X = 180 - 90 = 90\]

With the optimal production levels of X and Y, we can calculate the utility possibility frontier: \[U(90, 45) = \sqrt{90*45} = \sqrt{4050} ≈ 63.6\] The optimal Y production is 45, resulting in the best utility possibility frontier of (63.6, 63.6). #d. Conditions for different optimal production choices# The answer to part (c) may depend on the point on the utility possibility frontier being considered in cases where: 1. The utility functions of the individuals are not identical. 2. The production possibility frontier is not linear. 3. There are external factors such as differing preferences or the presence of public goods that influence the optimal production decision and utility levels. In these situations, the optimal production choice may vary depending on the individual's preferences or external factors, and it may not be possible to find a single optimal production choice to maximize the utility possibility frontier.