Question

Consider a small country populated by individuals with identical preferences given by U= X04y0.6. The PPF of this country is described by X2+ Y² = 5000.

1. In autarky, find the relative price px=Px/ Py. Draw the corresponding diagram and label it fully. What is the utility at autarky?

Answer 1

To find the relative price px in autarky, take the derivative of Y with respect to X. Draw the PPF curve by calculating Y values for different X values. Find the utility at autarky by plugging in the X and Y values into the utility function.

Step-by-step explanation:

To find the relative price px in autarky, we need to look at the slope of the PPF curve. The equation of the PPF is X^2 + Y^2 = 5000.1. Solving for Y, we get Y = sqrt(5000.1 - X^2). The slope of the curve at any point can be found by taking the derivative of Y with respect to X. In this case, it would be dY/dX = (-X) / sqrt(5000.1 - X^2). This derivative represents the relative price px.

To draw the diagram, we can plot the PPF curve by assigning different values to X and calculating the corresponding Y. Choose a range of values for X, such as 0 to 100. Plug each value of X into the equation Y = sqrt(5000.1 - X^2) to determine Y. Plot each pair of X and Y points on the graph. Label the axes as X and Y, and include units if necessary.

To find the utility at autarky, we can plug in the values of X and Y from the autarky point into the utility function U = X^0.4 * Y^0.6. For example, if the autarky point is (X*, Y*), the utility would be U = (X*)^0.4 * (Y*)^0.6.