Question

A country produces goods X and Y and has the following equation for its production possibilities

frontier: Y = 1000 – 90X - X2, 0 ≤ X ≤ 10

When the economy is producing 6 units of X and 424 units of Y, what is the opportunity cost of X?
A) 0
B) 12
C) 78
D) 82
E) 84
F) 90
G) 94
H) 96
I) 102
J) none of the above

Answer 1

When the economy is producing 6 units of X and 424 units of Y, the opportunity cost of one more unit of X, based on the given production possibilities frontier equation, is 102 units of Y forgone.

Step-by-step explanation:

The student has provided an equation for a country's production possibilities frontier (PPF): Y = 1000 – 90X – X2, where 0 ≤ X ≤ 10. At 6 units of X and 424 units of Y, to find the opportunity cost of producing one more unit of X, we look at the change in Y when X is increased by 1 unit from X to X+1. The slope of the PPF at any given point shows the opportunity cost of producing one more unit of X in terms of how much Y is given up.

The derivative of the PPF equation with respect to X gives us the slope of the PPF, which is the marginal rate of transformation (MRT) and thus the opportunity cost of X. Differentiating Y = 1000 – 90X – X2 with respect to X yields a derivative of – 90 – 2X. Plugging in X=6, we get – 90 – 2(6) = – 90 – 12 = – 102. The minus sign indicates that Y decreases as X increases, and since opportunity cost is a non-negative value, we take the absolute value which is 102. Therefore, the opportunity cost of producing one more unit of X is 102 units of Y forgone.