Question

Suppose that there are two individuals in an economy, A and B, and that their utility possibility set is given by uB = 400 − 0.01u2A. Consider the point (uA,uB) = (100,300), is this point Pareto Efficient? Explain. Determine a linear social welfare function that has the point (100,300) as its maximum. Hint: the slope of the utility possibility frontier is given by −0.02uA. Start by determining the isowelfare line that goes through (100, 300).

Answer 1

Suppose that there are two individuals in an economy, A and B,

The utility possibility set is given as

.........(1)

Where, uA = utility of A and uB = utility of B

This is also known as Isowelfare Line.

Now, any combination of (uA, uB) will ne called pareto efficient if and only if the two points lies on the utility possibility frontier of UPF (As UPF is the locus of all pareto efficient points).

In other words, the point must satisfy equation (1).

Let us consider the point, (uA, uB) = (100, 300).

Putting, uA = 100 in equation (1) we get

or, uB = 400 - 100

or, uB = 300

Hence, the points are on the utility possibility frontier which is the locus of all pareto efficient points.

Hence, the point (100, 300) must be pareto efficient.

The point (uA, uB) = (100, 300) is pareto efficient.

Now, we have to determine a linear social welfare function that has the point (100,300) as its maximum.

Now, let us follow the hint and solve.

The slope of the utility possibility frontoer is given as

Slope = duB/duA = -0.02.uA

As, the point (100, 300) is the maximum point of the isowelfare line, hence it must be a tangent line to the previous utility possibility frontier (1) at point (100, 300).

Hence, slope of the isowelfare line at point (100, 300) is

Slope = -0.02.uA = -0.02×100

or, Slope = -2

Now, the equation of the isowelfare line passing through point (100, 300) with slope -2 is

(uB - 300)/(uA - 100) = -2

or, uB - 300 = -2.uA + 200

or, 2.uA + uB = 500

This is the required lineae social welfare function.

Hence, the linear social welfare function is

2.uA + uB = 500

Hope the solution is clear to you my friend.