The Encyclopedia of Economic Principals

Chapter 4

Opportunity Cost, Specialization, Scale, Scope, and Technology

Count what a choice gives up, and let the size of the market decide how to produce.

Four of the chapter's worked examples, made interactive: short- and long-run labor demand, the market size that pays for specialization, gains from comparative advantage, and the value of time in cooking or buying meals.

Every example here is a constructed teaching example: it uses the hypothetical numbers of the chapter's worked examples, plus a few values added for comparison and labelled as such in each panel. Nothing here measures a real market, firm or household.

Demonstration 1 of 4

Short run vs long run labor response

Why does labor respond more to a wage rise once machines can adjust?

The Le Chatelier principle says a response is at least as large when more choices can adjust. With machines fixed, the output target pins labor; once machines are free, the firm moves along the isoquant toward the input that became relatively cheaper.

Equation, written in LaTeX: q=10\sqrt{LK}.

Equation, written in LaTeX: L_{SR}=\frac{291{,}600}{270}=1{,}080.

Equation, written in LaTeX: L_{LR}=540\sqrt{\frac{40}{15}}\approx882, K_{LR}=540\sqrt{\frac{15}{40}}\approx331.

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q = 5,400 meal kits, L labor hours and K machine hours. The machine rental price is r = 40 and the wage starts at 10. In the short run K stays at 270.

Predict first. Does the short-run labor response depend on the wage at all?

Your prediction

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Figure: Short run vs long run labor response. Isoquant for 5,400 kits with the starting point (1,080, 270) and the long-run choice at wage 15: 882 labor hours and 331 machine hours.
New wage: 15, Horizon: Long run
Constructed example: the chapter's hypothetical meal-kit plant (wage 10 rising to 15, r = 40); wages of 20 and 25 are added for comparison.

Calculated values

Labor hours
882
Machine hours
331
Labor change from 1,080
-198
Cost at the new wage
26,454
Horizon
long run

With both inputs free, 15L = 40K and LK = 291,600 give L = 540 x sqrt(40/15) = 882 and K = 540 x sqrt(15/40) = 331. Labor falls by 1,080 - 882 = 198 hours. Since 15L = 40K, cost is 2 x 40 x K = 80 x 330.681 = 26,454.

Worked steps

  1. Tangency: 15L = 40K
  2. Isoquant: LK = 291,600, so sqrt(LK) = 540
  3. L = 540 x sqrt(40/15) = 882
  4. K = 540 x sqrt(15/40) = 331
  5. Labor change = 882 - 1,080 = -198
  6. Cost = 15L + 40K = 80 x 330.681 = 26,454

Use the idea

Expect a larger labor response to a lasting wage change than to a temporary one, because capital has time to adjust.

Where the conclusion applies

A fixed output target, Cobb-Douglas technology q = 10 sqrt(LK), known prices and no adjustment costs. Irreversible capital or financing limits can block the long-run move.

Check your understanding: With the wage at 25, what is long-run labor?
L = 540 x sqrt(40 / 25) = 540 x 1.265 = 683 hours.

Chapter 4 source: section "Le Chatelier principle".

Demonstration 2 of 4

How big a market justifies specialization?

How many reachable sales does a dedicated process need before it beats general production?

The specialized line starts higher (the setup cost) but rises more slowly. It wins once sales pass the crossing point. Transport and border costs steepen it and push the crossing out.

Equation, written in LaTeX: q^*=\frac{\$24{,}000}{\$90-\$30}=400\text{ devices}.

Equation, written in LaTeX: \$24{,}000+(\$30+\$3)(750)=\$48{,}750.

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General production costs $90 per unit. The specialized process costs $30 per unit after $24,000 of setup, plus any delivery or border cost per unit. q is accessible annual sales.

Predict first. With the $35 delay added to delivery, at what sales level would specialization break even?

Your prediction

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Figure: How big a market justifies specialization? Total cost lines for the general and specialized methods with an extra cost of 3 per unit. At 750 devices they cost 67,500 and 48,750.
Accessible sales: 750, Extra cost per unit: $3 delivery
Constructed example: the chapter's hypothetical device maker (sales 250 and 750, delivery 3, delay 35); 1,000 sales and zero extra cost are added, and 400 is the book's break-even.

Calculated values

General cost
$67,500
Specialized cost
$48,750
Saving from specializing
$18,750
Break-even sales
421.1 devices
Choice
specialized

Specialized: 24,000 + (30 + 3) x 750 = 48,750. General: 90 x 750 = 67,500. Specialization saves $18,750. Break-even sales are 24,000 / (90 - 30 - 3) = 421.1 devices.

Worked steps

  1. Specialized = 24,000 + 33 x 750 = 48,750
  2. General = 90 x 750 = 67,500
  3. Difference = 67,500 - 48,750 = 18,750
  4. Break-even = 24,000 / 57 = 421.1 devices

Use the idea

Judge whether a market is big enough by the transactions you can actually reach after delivery costs, not by headcount.

Where the conclusion applies

Linear costs, a single specialized process and all accessible sales served. Congestion, variety or entry that splits sales would change the threshold.

Check your understanding: With 1,000 sales and only the $3 delivery cost, which method wins and by how much?
Specialized costs 24,000 + 33 x 1,000 = 57,000 against 90,000 general, so specialization saves 33,000.

Chapter 4 source: section "Smith extent-of-the-market theorem".

Demonstration 3 of 4

Trading cloth for wine

How can both regions gain when one is better at producing everything?

Each region specializes in the good with the lower opportunity cost. Trading at a price between the two opportunity costs lets each consume a bundle beyond its own frontier.

Equation, written in LaTeX: OC_A(C)=2/4=0.5W

Equation, written in LaTeX: OC_B(C)=6/8=0.75W

Equation, written in LaTeX: (C_A,W_A)=(16,5), (C_B,W_B)=(8,7).

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Region A needs 2 hours per cloth and 4 per wine and has 48 hours; region B needs 6 and 8 and has 96 hours. OC is the opportunity cost of one cloth in wine. The price is wine per cloth and A exports cloth.

Predict first. At the edge price of 0.5 wine per cloth, who captures all the gain?

Your prediction

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Figure: Trading cloth for wine. Two production possibility frontiers. Region A moves from (16, 4) to (16, 5.00) and region B from (8, 6) to (8, 7.00) at 0.625 wine per cloth.
Wine per cloth: 0.625, Cloth exported by A: 8
Constructed example: the chapter's hypothetical regions (price 0.625, 8 cloth exported, bounds 0.5 and 0.75); exports of 4 and 12 are added for comparison.

Calculated values

A consumes (cloth, wine)
(16, 5.00)
B consumes (cloth, wine)
(8, 7.00)
Hours A would need alone
52.0
Hours B would need alone
104.0
Price within 0.5 to 0.75
yes

A makes 48 / 2 = 24 cloth and B makes 96 / 8 = 12 wine. A sends 8 cloth for 0.625 x 8 = 5.00 wine, keeping 16 cloth; B keeps 12 - 5.00 = 7.00 wine. A's bundle would take 2 x 16 + 4 x 5.00 = 52.0 hours alone, beyond its frontier; B's would take 6 x 8 + 8 x 7.00 = 104.0 hours, beyond its frontier.

Worked steps

  1. A: cloth 24 - 8 = 16, wine 0.625 x 8 = 5.00
  2. B: cloth 8, wine 12 - 5.00 = 7.00
  3. A alone: 2 x 16 + 4 x 5.00 = 52.0 hours (has 48)
  4. B alone: 6 x 8 + 8 x 7.00 = 104.0 hours (has 96)

Use the idea

Assign work by opportunity cost, not by who is absolutely faster.

Where the conclusion applies

Labor is the only input, unit costs are constant, trade is voluntary and costless. Transport costs, adjustment and learning can shrink or redistribute the gain.

Check your understanding: At 0.75 wine per cloth and 8 cloth exported, what does each region consume?
A: 24 - 8 = 16 cloth and 0.75 x 8 = 6 wine. B: 8 cloth and 12 - 6 = 6 wine. B gains nothing; A gains 2 wine.

Chapter 4 source: section "Division of Labor and Comparative Advantage".

Demonstration 4 of 4

Cook or buy: what is an hour worth?

At what value of time does buying prepared meals beat cooking?

Time used at home has a price: what the hour would be worth elsewhere. Adding it to cash cost gives two full-cost lines; the steeper one, cooking, wins only below the crossing.

Equation, written in LaTeX: FC_H=\$18+1.5v, FC_P=\$42+0.25v.

Equation, written in LaTeX: v^*=\frac{\$42-\$18}{1.5-0.25}=\$19.20\text{ per hour}.

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v is the value of one hour in dollars. Cooking needs $18 of groceries and 1.5 hours; prepared meals cost $42 and 0.25 hour. FC is full cost: cash plus hours times v.

Predict first. Which option is chosen at exactly $19.20 per hour?

Your prediction

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Figure: Cook or buy: what is an hour worth? Full cost lines for cooking and buying against the value of an hour, crossing at 19.20. At v = 30.00 cooking costs 63.00 and buying 49.50.
Value of an hour ($): $30
Constructed example: the chapter's hypothetical household (values of $30 and $12 and the $19.20 break-even); $45 is added for comparison.

Calculated values

Full cost, cook
$63.00
Full cost, buy
$49.50
Break-even value of time
$19.20
Choice
buy
Difference
$13.50

Cook: 18 + 1.5 x 30.00 = 63.00. Buy: 42 + 0.25 x 30.00 = 49.50. Buying is cheaper in full terms. The break-even is (42 - 18) / (1.5 - 0.25) = 24 / 1.25 = 19.20 per hour.

Worked steps

  1. Cook = 18 + 1.5 x 30.00 = 63.00
  2. Buy = 42 + 0.25 x 30.00 = 49.50
  3. v* = 24 / 1.25 = 19.20
  4. Choice: buy

Use the idea

Value your unpaid time at its best alternative use, which may be well below your wage if extra paid hours are not available.

Where the conclusion applies

A constant value per hour, no enjoyment or dislike of cooking, and fixed prices and times. Fixed schedules or fatigue can make the hourly value hard to pin down.

Check your understanding: At v = 45, how much more does cooking cost than buying?
Cooking costs 18 + 67.50 = 85.50 and buying 42 + 11.25 = 53.25, so cooking costs 32.25 more.

Chapter 4 source: section "Opportunity Cost and Household Production".