The frontier: every cake costs loaves
導入済み| English | 意味 |
|---|---|
| scarcity/ˈskeəsɪti/ | The condition in which available resources are insufficient to satisfy all competing wants. |
| opportunity cost/ɒpəˈtjuːnɪti kɒst/ | The value of the best alternative forgone when a choice is made. |
| production possibilities frontier | production possibilities frontier |
Twelve baker-hours, two products, one choice
- Nimbus has 12 baker-hours a day and can spend them on celebration cakes or sandwich loaves: 6 loaves an hour, or cakes that take ever more of the day's skill.
- Wanting both is not a problem; scarcity 稀缺 is. This lesson builds the diagram economists use to make a trade-off visible before pricing it.
Cost is the alternative you give up
- The opportunity cost 机会成本 of a choice is the value of the next-best alternative it displaces — the loaves the cake hours would have baked, not the flour bill.
- Flour bought last month is irrelevant either way: it is sunk. Only the forgone alternative counts, and it must be the next-best one, not all the others together.
The opportunity cost of the second cake is the flour and butter it uses.
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Opportunity cost is the next-best alternative given up — the 8 loaves those hours would have baked. Ingredients bought either way are not the cost.
Nimbus bakes one cake a day. What does the second cake cost?
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The frontier falls from 66 to 58 loaves. The 6-loaf segment belongs to the first cake; the third would cost 11.
The frontier through the bakery's day
- With cakes $c$ and loaves $l$, the day's possibilities are exactly: $(0,72), (1,66), (2,58), (3,47), (4,30), (5,0)$. Plot them and you have a production possibilities frontier 生产可能性边界.
- Every point on the curve uses all 12 hours. Points inside waste time; points beyond need more hours or better skill than the bakery has today.
On, inside or beyond the frontier?
Place each plan against the bakery's verified frontier (0,72), (1,66), (2,58), (3,47), (4,30), (5,0). Only points on the curve use all twelve hours.
The second cake costs more than the first
- Read the frontier's slope: giving up $72-66=6$ loaves for cake 1, then 8, then 11, then 17, then 30. Increasing opportunity cost is the bow in the curve made numerical.
- The reason is mismatched skill: the first cake is made by the baker best at cakes; the fifth would need the baker who is brilliant at loaves.
Why does each cake cost more loaves than the one before?
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The first cake is made by the baker best at cakes. By the fourth, the hours come from bakers who are excellent at loaves — increasing mismatch, increasing cost.
Match each frontier feature to its meaning.
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The curve is the efficient boundary; inside is waste, the slope is cost, and the bow is rising cost.
Price the trade-off
- At £4 a loaf, the second cake's opportunity cost is $8 \times 4 = £32$. Bake it only if a cake earns more than £32 plus its ingredients.
- The rule generalises: expand an activity until its marginal benefit falls to its opportunity cost — the exact cousin of the marginal rule met in lesson 0304.
Loaves sell at £4. What is the opportunity cost of the FOURTH cake, in £?
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The fourth cake gives up $47 - 30 = 17$ loaves, worth $17 \times 4 = £68$.
Growth shifts the menu, not the need to choose
- A training grant lifts everyone to 7 loaves an hour: the frontier becomes $(0,84), (1,77), (2,68), (3,56), (4,35), (5,0)$ — the dashed curve.
- Check what did NOT change: the curve still binds and cost still rises — cake 3 now costs 12 loaves (68 − 56) where it cost 11. A bigger menu is not the end of scarcity.
After the training grant (7 loaves an hour), what does the third cake cost in loaves?
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The new frontier runs $(0,84), (1,77), (2,68), (3,56)$: the third cake gives up $68 - 56 = 12$ loaves.
Three slips on the frontier
- Counting money instead of alternatives. The cake's cost is the 8 loaves, not the butter bill.
- Drawing a straight frontier while saying "increasing cost". A straight line has constant cost; the bow is the whole point.
- Calling one point "the efficient one". Every point on the curve is efficient; customers, not the curve, choose among them.
A straight-line frontier between cakes and loaves would mean what?
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A constant slope is a constant cost per cake. Increasing cost needs the bowed-out shape — that is why the model draws it bowed.
Recap
- Opportunity cost = the next-best alternative given up, valued properly.
- The frontier's slope is opportunity cost; the bow is increasing cost.
- Growth shifts the frontier outward; choosing never stops.