Here’s what “money is isomorphic to
transferable utility” means, and when it’s a
good approximation.
以下说明“货币与可转移效用（Transferable
Utility, TU）同构”的含义，以及在何种情形下
这是一种好的近似。
What transferable utility (TU) means
什么是可转移效用（TU）
- In TU models, you can convert one person’s
payoff into another’s without loss. Utility is
measured on a common, cardinal scale,
so “total utility” is meaningful and can be
reallocated via side payments.
在 TU 模型中，一个人的收益（效用）
可以无损地转移给另一个人。效用在一个共同的、
基数尺度上度量，因此“总效用”有意义，
并可通过侧支付（货币转移）在个体间重新分配。
- Formally (quasilinear preferences):
u_i(x, t_i) = v_i(x) + t_i, where x is the allocation
and t_i is a money transfer. A dollar changes
utility by a constant amount, independent
of x or wealth.
形式上（拟线性偏好）：u_i(x, t_i) = v_i(x) + t_i，
其中 x 是资源配置，t_i 是货币转移。每一美元带
来的效用增量是常数，不依赖于 x 或财富水平。
Why money can be treated as TU
为什么可以把货币当作 TU
- Money acts as a common numeraire: a
perfectly divisible, transferable good with
linear utility.
货币充当共同计价基准（numeraire）：
一种可无限可分、可转移、效用线性的“商品”。
- If transfers are allowed and enforceable,
we can choose allocations that maximize
total surplus and use money to make
everyone whole.
若转移可行且可强制执行，我们可以选择使
总剩余最大化的配置，并用货币转移让每个人
都不吃亏。
The “isomorphism” idea
(intuition + formalism)
“同构”的直觉与形式
- Intuition: A dollar to me is the same
“unit of welfare” as a dollar from you;
moving dollars just moves welfare between
us without changing the total.
直觉：对你和我来说，一美元代表同样的“
福利单位”；转移钱只是把福利在我们之间搬运，
总量不变。
- Formal: There’s a structure-preserving map
φ(m) = α m from dollars to utility units.
With a common scale (normalize α = 1),
dollars and utils align linearly, so side
payments transfer utility one-for-one.
形式：存在保持结构的映射 φ(m) = αm，
把美元数映射为效用单位。常将 α 归一化为 1，
使美元与效用线性对齐，从而侧支付可一对一
转移效用。
When it works (key assumptions)
何时成立（关键假设）
- Quasilinear/wealth-neutral preferences:
marginal utility of money is constant
(no big wealth effects).
拟线性/财富中性偏好：货币的边际效用近似
恒定（无显著财富效应）。
- Risk neutrality in money over the relevant
range.
在相关范围内对金钱近似风险中性。
- Side payments are feasible, legal, and
enforceable; low transaction costs.
侧支付可行、合法且可执行；交易成本低。
- No liquidity constraints; money is actually
available to transfer.
无流动性约束，确实可以完成转移。
- No externalities that money can’t price in;
the value of x is captured in v_i(x).
不存在货币无法充分定价的外部性；x 的价值
可由 v_i(x) 捕捉。
What it buys you (applications)
能带来什么（应用）
- Efficient allocation via transfers: Choose x
to maximize Σ_i v_i(x) and settle with
transfers t_i (auctions, VCG mechanisms).
通过转移实现效率：选择使 Σ_i v_i(x)
最大化的 x，并用 t_i 清算（如拍卖、VCG 机制）。
- Bargaining: With TU, the Nash bargaining
solution collapses to splitting surplus; who
gets what is just a transfer.
谈判：在 TU 下，纳什谈判解可简化为分割剩余；
谁得多少只需靠货币转移实现。
- Cooperative games: Coalition values can be
redistributed as cash; concepts like the
core/Shapley value apply cleanly.
合作博弈：联盟价值可用现金再分配；核心与
沙普利值等概念适用且更简洁。
Simple examples
简单例子
- Auction: A values a license at $300, B at $180.
Efficient winner is A. TU lets A pay, say, $200,
leaving A net +$100, B 0—total surplus $300 is
realized and split by money.
拍卖：A 对某许可证估值 300 美元，B 为 180。
有效配置是给 A。TU 允许 A 支付例如 200 美元，
A 净得 +100，B 为 0——总剩余 300 得以实现，
并由货币来分配。
- Rent split: Room A worth $1,200 to Alex,
$800 to Bailey. Give Room A to Alex and set a
side payment so both are at least as well off
as the next-best arrangement.
房租分摊：房间 A 对 Alex 的价值 1200 美元、
对 Bailey 为 800。把房间 A 给 Alex，并设定侧
支付，使两人至少不劣于下一佳方案。
Where it breaks down
何时失效
- Diminishing marginal utility of income or
strong wealth effects: A dollar doesn’t move
utility equally for everyone.
收入的边际效用递减或强财富效应：同一美元
对不同人的效用影响不等。
- Risk aversion or income uncertainty around
the transfer.
风险厌恶或围绕转移的收入不确定性。
- Credit/liquidity constraints, taxes,
illegal/blocked side payments,
transaction costs.
信用/流动性约束、税收、侧支付违法或受限、
交易成本显著。
- Externalities, public goods, or non-priced
dimensions (status, norms) that money can’t
fully compensate.
外部性、公共品或非价格维度（地位、规范）
使货币无法完全补偿。
- Behavioral factors (loss aversion, fairness)
that make equal-dollar transfers unequal in
perceived utility.
行为因素（损失厌恶、公平偏好）导致等额转移
在感知效用上并不等同。
Takeaway
要点
- Treating money as isomorphic to transferable
utility is a powerful modeling move:
it lets us turn resource allocation into
“maximize surplus, then use money to
make it fair.”
把货币视为与可转移效用同构，是一种强有力的
建模技巧：把资源配置问题变为“先最大化剩余，
再用货币实现公平”。
- It’s accurate when preferences are
quasilinear and transfers are feasible;
it’s misleading when wealth effects, risk,
constraints, or externalities bite.
当偏好近似拟线性且转移可行时，它相当准确；
当财富效应、风险、约束或外部性显著时，
则可能产生误导。