On a PPC (The Production Possibilities Curve) more of one good means less of the other. How much less is the opportunity cost, and you can read it off the graph as a slope, the idea from Slope as Rate of Change.
Key idea
Moving along a PPC, the opportunity cost of the extra units of $x$ is the amount of $y$ given up. Per unit: $$\text{opportunity cost of one } x = \frac{\text{amount of } y \text{ given up}}{\text{amount of } x \text{ gained}},$$ the size of the slope. A straight PPC has a constant opportunity cost. A PPC bowed out from the origin has an increasing opportunity cost.
Why it works
Why does the curve bow out? Resources are not equally good at everything. When a farm country starts making robots, it first moves the people and land least useful for farming, so little wheat is lost. As it makes more robots, it has to move its best farmland and most skilled farmers, so each extra batch of robots costs more wheat. If resources were equally suited to both goods, every step would cost the same and the PPC would be a straight line.
Worked example: A straight-line PPC
A country can make either 40 robots or 120 tons of wheat, or any combination on the straight line between. What is the opportunity cost of one robot? Of one ton of wheat?
- Moving from all wheat to all robots gains 40 robots and gives up 120 tons.
- One robot costs $\dfrac{120}{40} = 3$ tons of wheat.
- One ton of wheat costs $\dfrac{40}{120} = \dfrac{1}{3}$ robot, the reciprocal.
Answer: 3 tons of wheat per robot; $\tfrac{1}{3}$ robot per ton of wheat.
Worked example: Rising cost along a table
Using the table in the picture, find the opportunity cost per robot of moving from 10 to 20 robots, and from 20 to 30 robots.
- From 10 to 20 robots, wheat falls from 90 to 75: 15 tons for 10 robots, or $\dfrac{15}{10} = 1.5$ tons per robot.
- From 20 to 30 robots, wheat falls from 75 to 50: 25 tons for 10 robots, or $2.5$ tons per robot.
- The cost per robot rises as more robots are made: increasing opportunity cost.
Answer: 1.5 tons per robot, then 2.5 tons per robot.
Common mistake
Two errors are common: reporting what was gained instead of what was given up, and flipping the ratio, as in "a robot costs $\tfrac{1}{3}$ ton" when it costs 3. Check with units: the cost of a robot is measured in wheat, so wheat goes on top: tons of wheat per robot. And the two costs must be reciprocals of each other.