An Economic Model of the L1-L2 Interaction

A deep dive into the "How?"

Workflow of a transaction

on a technical level, a blockchain protocol must solve the "double-spend" problem
this requires three components

transactions cannot be faked $\Rightarrow$ cryptography/digital signatures

the past cannot be altered
$\Rightarrow$ hash-linking of past transaction records

the past cannot be made to disappear
$\Rightarrow$ must not be able to create many blocks predictably

solved by economics: need to deploy an expensive resource to create blocks

Proof-of-Stake

Validators, Committees, Proposers

ETH owners lock up funds ($\ge32$ETH) in a special smart contract

validators selected at random based on proportional ownership of total stake

Slot 1

proposer

Slot 2

proposer

Slot 32

proposer

Blockchain organization

Slot 1

Slot 2

Slot 32

...

12 seconds

epoch$_t$

epoch$_{t+1}$

epoch$_{t-1}$

Block inclusion

Committee #1 of validators

Slot 1

Slot 2

Slot 32

...

proposer #1 assembles transactions in block and proposes to a committee

attests validity

Block inclusion

Committee #2 of validators

Slot 1

Slot 2

Slot 32

...

proposer #2 assembles transactions in block and proposes to a committee

attests validity

Block inclusion

Committee #32 of validators

Slot 1

Slot 2

Slot 32

...

proposer #32 assembles transactions in block and proposes to a committee

attests validity

Blockchain finality rules

Slot 1

Slot 2

Slot 32

...

12 seconds

epoch$_t$

epoch$_{t+1}$

epoch$_{t-1}$

checkpoint

Blockchain finality rules

...

epoch$_t$

epoch$_{t+1}$

epoch$_{t-1}$

checkpoint

all validators vote on validity

(for clarity: checkpoint voting happens over time, by all validators)

Blockchain finality rules

...

epoch$_t$

epoch$_{t+1}$

epoch$_{t-1}$

checkpoint

valid?
$\Rightarrow$ "justified"

last justified block becomes "final"

valid?
$\Rightarrow$ "justified"

last justified block becomes "final"

a checkpoint is "justified" once $\frac{2}{3}$ of all validators vote for validity
once a checkpoint is justified, the last justfied checkpoint becomes final

valid?
$\Rightarrow$ "justified"

Validators, Committees, Proposers

Slot 1

proposer

Slot 2

proposer

Slot 32

proposer

validation
committee

128 validators per committee ($\Rightarrow$ 32 x 128) are selected at random based on proportional ownership of total stake

Security?

To sabotage block production or to censor transactions, an attacker generally needs at least 33% of the total stake (so no 2/3 majority can be reached)
To create your own blocks only you need the 2/3 majority

on Sept 8, 2025: 34M ETH x $4,300=$146B

How much is that?

you need:

$50B to stop the network and
$100B to take it over (though likely not much would be left then)

Why does a cryptocurrency have value?

running of code costs cryptocurrency
demand for compute
$\Rightarrow$ demand for cryptocurrency
demand creates value
value makes 1/3 and 2/3 attacks expensive
$\to$ value creates security

Text

Basic idea of a Rollup

Ethereum is "full"

Source: Etherscan but with re-scaling (the 100% is actually 50%, because 50% is the target)

The process explained

users lock up funds in a special smart contract

a

b

e

f

g

perform operations or transact

sequencer creates a digest of bundled/rolled-up transactions

sequencer posts digest to mainnet

NB: blocks now have dedicated space for L2 data "blobs" (kept for limited time)

"account abstraction"
- = smart contract wallets
- no gas payments in crypto ("sponsored")
- great simplification

Some things you can do with a rollup

high throughput applications
- trading platforms
- payments systems
- reward systems

AML/KYC checks
- determine who gets to play on your rollup
- be regulatory compliant

create "local" tokens
non-finance applications
private transactions using zero-knowledge proofs
digital ID applications

a

b

e

f

g

user kindly asks the platform to return assets

How is a rollup different from a crypto-trading platform?

crypto platform: ownership transferred to platform

crypto platform: transactions enabled on a seperate system with no proftable relation to blockchain

rollup: ownership transferred to smart contract; retain control within rules of smart contract

rollups:

transactions follow blockchain rules (digitally signed)
blockchain mainnet checks vality
sequencer subject to punishment for violations
sequencer has no direct control over funds

What's happening?

This Paper

Research question: How do Layer-1 (L1) blockchains and their associated Layer-2 (L2) solutions interact economically?
Main finding:
- Even when L1 hosts value-generating dApps, L1 investment and cryptoasset value can vanish if L2 becomes sufficiently attractive.
- Sustained L1 development is essential for L1 survival. Without it, L1 eventually fails

Research Question & Main Finding

Key mechanism:

activity driven:
L2 attracts usage
→ lowers returns on L1
→ reduces incentives to invest in L1.

Discrete-time, infinite-horizon setting with risk-neutral investors.
Two layers:
- L1: dApp investment + staking (Proof-of-Stake).
- L2: dApp investment only.
dApp value depends on own-layer investment and cross-layer investment (externalities).
Productivity processes $A_{L1},A_{L2}$ $A_{1,t}, A_{2,t}$ drive long-run growth of each layer
based on John, Rivera, and Saleh, RFS 2025

Key Model Ingredients

Key Model Ingredients, Decision Variables, and Effects

L1 capital allocation

L2 Investment

dapps

L2 Investment

these compete for activity

dapps

staking

in principle pay fees but "not enough"

model based on John, Rivera, and Saleh, RFS 2025

Main Result: key conditions

strength of L2’s negative spillover
$\times$ L2 productivity growth

degree of concavity in L2 returns $\times$ L1 productivity growth

$>$

$?$

Proposition 2 (Failure):
L1 disappears if any of these hold:

L2 productivity growth $>>$ L1 productivity growth.
L2 returns decrease too slowly in scale (concavity $\to 0$).
L2 imposes a strong negative externality on L1 (spillover large)

Proposition 3 (Survival):

If L1 productivity growth is sufficiently large and above a threshold relative to L2 productivity growth, then both L1 and L2 can grow indefinitely.

My thoughts

extremely important to think about L1 and L2 interaction
very helpful model to formalize the need for L2s to internalize their
growth externality on L1

some limitations
- L2 value reaches L1 value only through fees
- fees are kept very low in the model
- what if they rise over time with L2 growth? How much do they have to rise?

policy choices (?)
- L1 tax on L2 (= Pigouvian revenue sharing)?
- cross staking requirement of L2 sequencers?

another thought
- in model: L1 and L2 are substitutes
- what if L1 and L2 are differentiated (e.g. L2 specialized payments)( $=\beta_{2,1}$ small)
- L1 users may have higher willingness to pay and demand L1 service

On my last point, for instance: Celestia Blockchain

L1: data availability and consensus
L2: execution enviroment (dapps)

The paper:
- L1 and L2 both host dApps and compete for the same activity.
- The negative externality channel (β₂,₁ < 0) drives the “L1 collapse” risk
In Celestia’s design, product differentiation is built-in:
- L1 = data availability & security layer.
- L2 = execution environments.
Celestia eliminates the competitive externality by construction
(I'm not saying that's optimal, there are other concerns that separation creates
$\to$ but blockchain folks come up with all kinds of clever designs...)

An Economic Model of the L1-L2 Interaction

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