Question

Maria has a utilty function u(x,y) = min{ 2x, 0.5y} and a labor endowment of 120 units. The production functions are x = 4L_x and y = 12L_y. In this case, what is the optimal labor that should be allocated to producing good x?

1. 51.4
2. 54.5
3. 56.5
4. 62.3

Answer 1

Answer

The correct answer is (a) 51.4

Maria has a utility function u(x,y) = min{2x, 0.5y}

Whenever we have utility function of the form U = min(ax,by), this means that she considers x and y as perfect complements. For such function in order to maximize Utility(u) a consumer consumes at a point where kink of the indifference curve(IC) will occur. Kink of this function will occur at a point where ax = by.

So kink of the IC of Maria will occur at a point where 2x = 0.5y => y = 4x

It is given that x = 4L_x and y = 12L_y

So, y = 4x => 12L_y = 4(4L_x) => L_y = (4/3)L_x -------------------------(1)

It is given that total labor endowment is 120 => L_x + L_y = 120 => L_y = 120 - L_x ---------------------(2)

From (1) and (2) we have :

(4/3)L_x = 120 - L_x

=> (7/3)L_x = 120

=> L_x = 51.4

Thus,  the optimal labor that should be allocated to producing good x is 51.4 units.

Hence, the correct answer is (a) 51.4