Exercise 2: Why the Frontier's Slope Represents Opportunity Cost — Possible Solution
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A production possibilities frontier simply showing what an economy is
CAPABLE of would only need to mark out the boundary of achievable
combinations - a flat listing of "yes, this combination is reachable"
versus "no, that one isn't," with no particular meaning attached to
moving along that boundary in one direction versus another.

But the frontier's real DOWNWARD slope carries more specific
information than that. Moving along the frontier from one point to
another - say, producing more butter - necessarily means producing
less of the other good (guns), precisely because the economy is
already using every one of its resources at full capacity along that
boundary. There's no slack left to draw from; any resource shifted
toward producing more butter has to be a resource taken directly away
from producing guns.

That specific trade-off - gaining some amount of one good only by
giving up some real amount of the other - is exactly what opportunity
cost means at the scale of a whole economy. The STEEPNESS of the
slope at any given point even tells you HOW MUCH of one good has to be
sacrificed for a given gain in the other, which is a genuinely
quantitative measure of opportunity cost, not just a boundary marking
what's reachable.

ANSWER: The frontier's downward slope represents opportunity cost
because moving along it - producing more of one good - necessarily
requires giving up some real amount of the other good, since every
point on the frontier already uses all available resources at full
capacity; the specific amount given up (the slope's own steepness) is
a direct, quantitative measure of that trade-off, not merely a
boundary showing which combinations are achievable.

WHY THIS WORKS AS AN ANSWER
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This explains the real mechanical reason (full resource use at every
point on the frontier) that movement along the boundary necessarily
trades one good for another, rather than simply asserting the
connection to opportunity cost without justification.