Suppose a production possibilities frontier can be expressed as $9 X ^ { 2 } + Y ^ { 2 } = 81$ what is the opportunity cost of going from 1 unit of X to 2 units of X (in terms of units of Y)?
A) 45
B) $\sqrt { 45 }$
C) $\sqrt { 72 } - \sqrt { 45 }$
D) 1

Question

Quizplus · Accepted Answer

First, we need to find the initial production point with 1 unit of X. By plugging in X = 1 into the production possibilities frontier equation, we get:

9(1)^2 + Y^2 = 81
Y^2 = 72
Y = sqrt(72)

So the initial production point is (1, sqrt(72)).

Next, we need to find the production point with 2 units of X. By plugging in X = 2 into the production possibilities frontier equation and solving for Y, we get:

9(2)^2 + Y^2 = 81
Y^2 = 9
Y = 3

So the production point with 2 units of X is (2, 3).

The opportunity cost of going from 1 unit of X to 2 units of X is the change in Y divided by the change in X.

Change in Y = 3 - sqrt(72)
Change in X = 2 - 1 = 1

Opportunity cost = (3 - sqrt(72))/1 = 3 - sqrt(72)

Simplifying, we get:

Opportunity cost = sqrt(9) - sqrt(72) = 3 - 2sqrt(2)

Therefore, the best choice is C, $\sqrt { 72 } - \sqrt { 45 }$, because it correctly expresses the opportunity cost of going from 1 unit of X to 2 units of X in terms of units of Y.

Suppose a Production Possibilities Frontier Can Be Expressed As 9X2+Y2=819 X ^ { 2 } + Y ^ { 2 } = 819X2+Y2=81

Suppose a Production Possibilities Frontier Can Be Expressed As $9 X ^ { 2 } + Y ^ { 2 } = 81$