The Encyclopedia of Economic Principals

Chapter 39

Innovation Incentives, Creative Destruction, and Patent Races

Count the entrants a prize attracts and see why incumbents and entrants value the same invention differently.

Four of the chapter's worked examples, made interactive: rent dissipation in a patent race, preemption by an incumbent, Arrow's replacement effect and effort in a prize contest. Change one value at a time and watch the figure, the numbers and the hand calculation respond.

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

How many labs enter a patent race?

With free entry and one prize, how much of the prize is burned in entry costs?

Each entrant compares its share of the prize with its own cost and ignores that it lowers the others' chances. Entry continues until the expected share barely covers the cost, so total spending approaches the prize.

Equation, written in LaTeX: 90-20=70.

Equation, written in LaTeX: \frac{90}{4}-20=2.5.

Scroll sideways for the whole equation

V is the patent prize and F the cost of joining the race. With n equally likely labs each expects V/n - F. A lab enters while that is not negative; a lab that exactly breaks even is counted.

Predict first. If entry gets cheaper (cost 15 instead of 20, prize 90), does total spending on the race go up or down?

Your prediction

Choose an example

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Figure: How many labs enter a patent race? Bars of expected net payoff per lab for 1 to 8 labs with prize 90 and entry cost 20. Payoffs stay at or above zero up to 4 labs, which spend 80 in total.
Patent prize: 90, Entry cost: 20
Constructed example: the chapter's hypothetical race (prize 90 and 60, entry cost 20); a prize of 120 and an entry cost of 15 are added for comparison.

Calculated values

Labs that enter
4
Payoff per entrant
2.5
Payoff of lab 5
-2
Total entry spending
80
Rent left after entry costs
10

With 4 labs, each expects 90 / 4 - 20 = 2.5, while lab 5 would expect 90 / 5 - 20 = -2 and stays out. Together the entrants spend 4 x 20 = 80 chasing a prize of 90, leaving 90 - 80 = 10.

Worked steps

  1. n = 4: 90 / 4 - 20 = 2.5
  2. n = 5: 90 / 5 - 20 = -2, below zero
  3. Spending = 4 x 20 = 80
  4. Rent left = 90 - 80 = 10

Use the idea

When many teams chase one reward, compare their combined spending with the reward before praising the race as efficient.

Where the conclusion applies

Equal win chances, no reusable knowledge from losing programs and no speed gain from extra entrants; entry is counted in whole labs.

Check your understanding: With prize 120 and entry cost 20, how many labs enter and how much rent survives?
120 / 6 - 20 = 0 and 120 / 7 - 20 is negative, so 6 labs enter, spend 6 x 20 = 120 and leave no rent.

Chapter 39 source: section "Patent-race rent dissipation".

Demonstration 2 of 4

Why incumbents race harder

Why does an incumbent pay for research an entrant would skip?

The incumbent races to avoid losing its current business as well as to win the new one. The more it would lose if the entrant won, the more a gain in winning probability is worth to it.

Equation, written in LaTeX: P_I=100-25=75.

Equation, written in LaTeX: 0.20(75)=15>12.

Equation, written in LaTeX: 0.20(40)=8<12.

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The incumbent's business is worth 100 if it wins and keeps the stated value if the entrant wins, so its prize P_I is the difference. The entrant's prize is 55. A program adds 0.20 to a firm's chance of winning at the stated cost.

Predict first. If licensing lets the incumbent keep more value when the entrant wins, does it race harder or less?

Your prediction

Choose an example

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Figure: Why incumbents race harder. Two bars: the incumbent's expected benefit 15 and the entrant's 11, against a program cost line at 12.
Incumbent value kept if the entrant wins: 25, Program cost: 12
Constructed example: the chapter's hypothetical sensor race (worth 100, kept 25 or 60, entrant prize 55, gain 0.20, cost 12); a kept value of 40 and a cost of 10 are added for comparison.

Calculated values

Incumbent prize P_I
75
Incumbent benefit
15
Entrant benefit
11
Program cost
12
Incumbent
Buys
Entrant
Declines

The incumbent's prize is 100 - 25 = 75, so the program is worth 0.20 x 75 = 15 > 12; the entrant's is worth 0.20 x 55 = 11 < 12. So the incumbent buys and the entrant declines.

Worked steps

  1. P_I = 100 - 25 = 75
  2. Incumbent: 0.20 x 75 = 15 > 12
  3. Entrant: 0.20 x 55 = 11 < 12

Use the idea

To predict who invests in a race, measure what each firm loses if the other wins, not only what each gains by winning.

Where the conclusion applies

A fixed probability gain from the program, risk neutrality and a single discrete choice. The kept value of 40 gives an exact tie with cost 12, shown as indifference.

Check your understanding: With a kept value of 40 and a cost of 10, who buys the program?
Incumbent 0.20(60) = 12 > 10 and entrant 0.20(55) = 11 > 10: both buy.

Chapter 39 source: section "Patent-race preemption".

Demonstration 3 of 4

Arrow's replacement effect

Why does an entrant gain more from the same invention than the incumbent?

An incumbent's invention replaces its own profit, so its gain is the increment over what it already earns. An entrant has nothing to replace and gains the whole new rent.

Equation, written in LaTeX: Q_M(c)=\frac{100-c}{2}.

Equation, written in LaTeX: (80-60)20=\$400.

Equation, written in LaTeX: 1{,}600-400=\$1{,}200.

Equation, written in LaTeX: (60-20)40=\$1{,}600.

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Inverse demand is P = 100 - Q. At unit cost c a monopolist produces Q_M(c). Maya earns her old profit with cost 60 and 1,600 with the new cost 20; entrant Eli earns 1,600 from the new process and nothing otherwise. The outlay is the development cost.

Predict first. If competition erodes Maya's old profit, does she become more or less willing to innovate?

Your prediction

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Figure: Arrow's replacement effect. Two bars: Maya's gain 1,200 and Eli's rent 1,600, against the development outlay 1,400.
Maya's old profit: 400, Development outlay: 1,400
Constructed example: the chapter's hypothetical process innovation (P = 100 - Q, costs 60 and 20, old profit 400 or 100, outlay 1,400); an old profit of 250 and outlays of 1,000 and 1,700 are added for comparison.

Calculated values

Maya's old profit
400
Maya's gain
1,200
Eli's rent
1,600
Outlay
1,400
Who invests
Eli invests

With cost 20 Maya produces (100 - 20) / 2 = 40 at price 60 and earns (60 - 20) x 40 = 1,600, so her gain is 1,600 - 400 = 1,200, below the outlay of 1,400. Eli has no old profit to replace: his rent is (60 - 20) x 40 = 1,600, above the outlay. Eli invests.

Worked steps

  1. Q_M(20) = (100 - 20) / 2 = 40, price 60
  2. Maya's new profit = (60 - 20) x 40 = 1,600
  3. Maya's gain = 1,600 - 400 = 1,200 vs 1,400
  4. Eli's rent = (60 - 20) x 40 = 1,600 vs 1,400

Use the idea

To judge who will adopt a new process, subtract the profit each firm would give up from the profit it would earn.

Where the conclusion applies

Linear demand, constant costs, a certain innovation and a stipulated entrant rent at the old competitive price of 60.

Check your understanding: With an old profit of 250 and an outlay of 1,000, who invests?
Maya: 1,600 - 250 = 1,350 > 1,000; Eli: 1,600 > 1,000. Both clear the outlay.

Chapter 39 source: section "Arrow replacement effect".

Demonstration 4 of 4

Prize size and number of contestants

How do the prize and the number of teams change effort in a contest?

More rivals lower each team's chance of winning, so each spends less, but the sum across teams rises toward the prize. A bigger prize scales all effort up in proportion.

Equation, written in LaTeX: e^*=\frac{rV(n-1)}{cn^2}.

Equation, written in LaTeX: e^*=\frac{100(4-1)}{4^2}=18.75.

Equation, written in LaTeX: e^*=\frac{100(2-1)}{2^2}=25.

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V is the prize, n the number of identical teams, r = 1 how strongly effort shifts the winning chance and c = 1 the cost per unit of effort. e* is each team's equilibrium effort.

Predict first. Does adding contestants raise each team's effort?

Your prediction

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Figure: Prize size and number of contestants. Effort per team and aggregate effort against the number of teams for a prize of 100. With 4 teams each spends 18.75 and together 75.
Prize: 100, Number of teams: 4
Constructed example: the chapter's hypothetical purification contest (V 100 and 200, n 4 and 2); fields of 3 and 5 teams are added for comparison.

Calculated values

Effort per team e*
18.75
Expected prize per team
25
Expected payoff per team
6.25
Aggregate effort
75

e* = 100 x (4 - 1) / 4^2 = 300 / 16 = 18.75. Each team expects 100 / 4 = 25 of the prize, so its payoff is 25 - 18.75 = 6.25. All 4 teams together spend 4 x 18.75 = 75 of effort for one prize of 100.

Worked steps

  1. e* = 100 x 3 / 16 = 18.75
  2. Expected prize = 100 / 4 = 25
  3. Payoff = 25 - 18.75 = 6.25
  4. Aggregate = 4 x 18.75 = 75

Use the idea

Choose the field size for a prize by what you value: more teams buy more total search, fewer teams leave more rent with each contestant.

Where the conclusion applies

Identical risk-neutral teams, a Tullock contest with r = 1, linear effort cost and a symmetric interior equilibrium. Losing effort is counted as cost, not as useful knowledge.

Check your understanding: With V = 100 and n = 5, what is aggregate effort?
e* = 100 x 4 / 25 = 16, so aggregate effort is 5 x 16 = 80.

Chapter 39 source: section "Prize Incentives and Contest Design".