Political Economy 3 · Video 1 of rent-seeking

Learn the Tullock derivation as the lecturer builds it

This is one continuous worked lesson. We use the same three-player, €120 example from the first probability calculation through the Nash equilibrium and welfare conclusion.

Optional: highlights the concept currently being spoken. Student-paced derivation remains available below.Automatically pause when the lecturer compresses a key step

Follow the lectureEvery major transition has a recording timestamp.

Audit the algebraEach new line names the rule that makes it valid.

Keep the numbersn = 3 and V = €120 remain visible throughout.

1 · The question

Why does the lecture begin with monopoly?

At , the lecturer says that rent-seeking starts from monopoly theory. The contrast is not “profit is bad.” It is about how the profitable position is created.

Type P · productive monopoly

A firm researches a genuinely new product. Its temporary monopoly comes with new consumer possibilities and potential social value.

Type R · rent-created monopoly

A similar product already exists abroad. The firm spends resources obtaining import protection, reducing competition and redirecting surplus.

Annotated lecture slide contrasting profit-seeking and rent-seeking — Recording slide at 03:37 · matches page 62 of the local 2025 deck

The model’s question

If several firms compete for one policy-created rent worth V, how much real effort will they rationally spend trying to win it?

2 · Set up the contest

Translate the lecturer’s story into symbols

n

There are n players competing. In our example, n = 3.

V

There is one rent worth V. In our example, V = €120.

B_i

Player i chooses effort Bᵢ: lobbying money, time or other costly influence.

B_{-i}

B₋ᵢ is the total effort of everybody except player i.

Winner takes all

One player receives V. The other n−1 players receive no prize, but everybody still pays their own effort cost.

How the lecturer builds the lottery idea

3 · Build the winning probability

Each euro of effort acts like a lottery ticket

own effort

p_i=\\frac{B_i}{B_i+B_{-i}}

total effort by everyone

Running example: three firms choose €20, €30 and €10

Firm 1€20Firm 2€30Firm 3€10

1. Add every firm’s effort

20+30+10=60

2. Divide Firm 1’s effort by the total

p_1=\frac{20}{60}=0.333=33.3\%

Firm 1 owns 33.3% of the “tickets,” so it has a 33.3% chance of winning.

4 · Build expected profit

The prize is uncertain; the effort cost is certain

How the expected-profit equation is spoken

Expected prize

p_iV

−

Certain effort bill

B_i

Expected net profit

EP_i

EP_i(B_i)=V\\left(\\frac{B_i}{B_i+B_{-i}}\\right)-B_i

Continue the same Firm 1 example

Winning probability

p_1=0.333

Expected prize

0.333\times€120=€40.00

Effort paid for sure

-€20

Expected net profit

EP_1=€20.00

5 · Derive the equilibrium

Keep the whole chain visible

We now follow the lecturer from “differentiate expected profit” to the final equilibrium. Reveal one transformation at a time. Earlier lines stay on screen so the argument never breaks apart.

ADifferentiate probabilityExpected profit → FOC

BFind the best responseFOC → player choice

CImpose Nash symmetryPlayer choice → B*

DAdd every playerB* → total dissipation

Student pace is active: use Reveal next step below. Hear context plays only the lecturer segment you choose; reached equations stay visible.

Lecturer's annotated slide showing the expected profit derivative — Recorded working at 15:50 · local deck page 64

Derivation A · Expected profit → first-order condition

09:43

Start from the player’s objective

What the lecturer is doing: The lecturer first asks how much one player should spend. The prize is uncertain, but the lobbying bill is paid whether the player wins or loses.

Rule used · Definition of expected net profit

EP_i(B_i)=V\left(\frac{B_i}{B_i+B_{-i}}\right)-B_i

The fraction is player i’s probability of winning.
Multiplying by V converts that probability into an expected prize.
The final −Bᵢ is not probability-weighted because the effort is paid for certain.

Same €120 exampleIn our running example V = €120. At the symmetric solution, player i will spend €26.67 and face €53.33 of rival effort.

6 · Work any numerical example

Substitute first; calculate second

Only use the final formula after you understand where it came from. Change n and V below and follow every numerical line.

Number of players, n 3Value of rent, V €120

1 · Effort of one player

B^*=\\frac{n-1}{n^2}V

=\frac{3-1}{3^2}\times€120

=€26.67

2 · Total contest effort

nB^*=3\times€26.67

=€80.00

66.7% of the rent is dissipated.

3 · Expected prize per player

p_iV=\frac{1}{3}\times€120

=€40.00

4 · Expected net profit per player

EP_i^*=€40.00-€26.67

=€13.33=\frac{V}{n^2}

Arithmetic check

€80.00+3\times€13.33=€120.00

Total contest effort plus all players’ combined expected net profit equals the available rent.

effort €80

profit €40

How the lecturer interprets the derived result

Annotated lecture slide summarising rent dissipation — Rent-dissipation results at 23:46 · local deck page 66

7 · Return to the economic question

Why does the lecturer call the spending social waste?

Annotated lecture monopoly graph showing deadweight loss and rent-seeking waste — Completed welfare diagram at 29:17 · local deck page 67

Standard monopoly analysisThe Harberger triangle is lost because monopoly output is below the competitive level.

The monopoly rent VWithout rent-seeking, this rectangle is mainly a transfer from consumers to the producer.

Tullock’s additionFirms spend close to V trying to obtain the rectangle. Those real resources must be counted too.

Opportunity costLobbyists and lawyers are skilled and already employable elsewhere; using them to redirect rents displaces other valuable work.

Economic conclusion

\\text{social cost}=\\text{deadweight loss}+\\text{rent-seeking resources}

Rent-seeking does not create another rent or another market. It uses scarce inputs to change who receives an existing rent.

8 · Reconstruct it yourself

The exam-safe route through the derivation

Write $p_i=B_i/(B_i+B_{-i})$ .
Write $EP_i=p_iV-B_i$ and explain why Bᵢ is paid for certain.
Differentiate the probability with the quotient rule.
Set $\\partial EP_i/\\partial B_i=0$ .
Apply symmetry: $B_{-i}=(n-1)B$ and total effort $nB$ .
Solve $B^*=(n-1)V/n^2$ .
Multiply by n to get $nB^*=(n-1)V/n$ .
Interpret: more players dissipate a larger share of the rent.

Check one calculation

Four identical players compete for V = €160. What is B* for each player?

What to remember when the notation disappears

Chance = your effort share. Expected profit = chance × prize − certain effort. Optimise one player, impose symmetry, then add everybody.