房租封顶之后，租到房的人会变多吗？Cap the rent. Do more people get an apartment?
一座城市的公寓市场租金是每月 $2,000，大约 1 万套租出去。市议会提议把租金上限定在 $1,500。左边的图把市场画成两条线：需求（每户愿意付多少）和供给（房东要多少才肯出租）。下面一排小方块是一户户家庭和一套套公寓。
A city’s apartments rent for $2,000 a month, and about 10,000 are rented. The council proposes capping rent at $1,500. The chart on the left draws the market as two lines: demand (what each household would pay) and supply (what each landlord needs before renting out). The squares below it are the individual households and apartments.
01预测Predict
02设定上限Set the cap
拖动图上的橙色线，或用下面的滑块，调整租金上限。看三个数：房东愿意出租多少（Qs），家庭想租多少（Qd），以及实际租出了多少。然后切到“长期”：几年之后，房东可以把公寓改成自住、出售，或者干脆不再新建。
Drag the orange line on the chart, or use the slider, to move the cap. Watch three numbers: how many apartments landlords offer (Qs), how many households want (Qd), and how many are actually rented. Then switch to the long run: after a few years landlords can convert units to condos, sell them, or stop building.
03解释：短边决定成交量Explain: the short side sets the quantity
一次出租需要两方都同意：一户想租，一个房东肯租。在 $1,500，想租的有 1.5 万户，但房东只肯出 （短期）或 （长期）。成交量 = min(Qd, Qs)，由数量更少的那一边，也就是“短边”决定。上限压低了价格，同时把成交量推到了供给一侧。
成交量 = min(Qd, Qs) = Qs （上限有约束时）
上限高于市场价，就什么都没变
把上限拖到 $2,000 以上试试：它不再“约束”任何人，租金照样停在 $2,000。只有低于市场价的上限才会起作用。
为什么短期看起来几乎没事
楼已经盖好了，短期里房东除了出租没有更好的用途，所以短期供给很陡：$1,500 时仍有约 9,500 套在租，几乎和原来一样。代价要过几年才出现：改作自住或出售、减少维护、不再新建，长期供给变平，同一个上限下只剩约 5,000 套。好处马上看得见，代价慢慢才来，这是限租政策在政治上很受欢迎的一个原因。
谁赢、谁输
租到房的家庭每月少付 $500，图上虚线框的“转移”从房东流向这些租客。没租到的家庭什么都没得到，还得继续找房。房东失去了转移的那部分，再加上不再出租的那些公寓原本能赚的钱。左下的长条始终把可能的总剩余分成四份：租客剩余 + 房东剩余 + 无谓损失 + 分配损失，四份加起来不变。
A rental takes two people who agree: a household that wants to rent and a landlord willing to rent. At $1,500, 15,000 households want an apartment, but landlords offer only (short run) or (long run). Quantity traded = min(Qd, Qs): whichever side is smaller, the “short side,” decides. The cap pushed the price down and, with it, pushed the quantity onto the supply side.
traded = min(Qd, Qs) = Qs (when the cap binds)
A cap above the market rent changes nothing
Drag the cap above $2,000: it no longer binds anyone, and rent stays at $2,000. Only a cap below the market price does anything.
Why the short run looks almost harmless
The buildings already exist, and in the short run a landlord has no better use for them than renting, so short-run supply is steep: at $1,500 about 9,500 units stay on the market, nearly the same as before. The cost arrives over years: conversions to condos or owner occupancy, less maintenance, no new construction. Long-run supply is flatter, and the same cap leaves only about 5,000 units. The benefit is visible at once; the cost comes slowly. That is one reason rent caps are politically popular.
Who wins and who loses
Households who get an apartment pay $500 less each month; the dashed “transfer” box moves that money from landlords to them. Households who miss out get nothing and keep searching. Landlords lose the transfer plus what the withdrawn apartments would have earned. The bar under the chart always splits the total possible surplus four ways: tenants + landlords + deadweight loss + rationing loss. The four always add up to the same total.
- 先看没有上限时的市场：两条曲线在 $2,000 相交。Start with the market without a cap: the curves cross at $2,000.
- 再看上限：$1,500 这条线压在交点下面，所以它有约束。Then the cap: the $1,500 line sits below the crossing, so it binds.
- 看上限线上的两个点：想租的 Qd 远多于房东肯出的 Qs，成交量只能是较小的 Qs。Now the two points on the cap line: Qd wanted far exceeds Qs offered, so only the smaller Qs gets rented.
- 看转移：租到房的家庭少付的钱，是房东少收的钱。Look at the transfer: what the lucky households save is what landlords lose.
- 最后看三角形：不再出租的那些公寓，本来能让双方都得益，这部分谁也没得到。Finally the triangle: the apartments no longer rented would have benefited both sides, and nobody gets that.
这个模型不衡量什么What this model does not weigh
剩余把每一美元算得一样重，不管它落到谁手里；它也不计算“能留在原来的社区”的价值。限租的支持者看重的正是这两点：分配，以及租客的稳定。实证研究两方面都看到了：Diamond、McQuade 和 Qian（2019，American Economic Review）研究旧金山 1994 年扩大的限租范围，发现受保护的租客明显更可能留在原住址；同时受管制楼宇的房东把出租供给减少了约 15%，作者估计这可能推高了全市其他房子的租金。
Surplus counts every dollar the same, whoever receives it, and it puts no value on staying in one’s neighborhood. Those are exactly what supporters of rent control care about: distribution and tenant stability. The evidence sees both sides. Diamond, McQuade and Qian (2019, American Economic Review) studied San Francisco’s 1994 expansion of rent control. Covered tenants were substantially more likely to stay at their address, while landlords of covered buildings reduced rental supply by about 15%, which the authors estimate likely raised rents elsewhere in the city.
04谁拿到那些公寓？Who gets the apartments?
短缺时，价格不再决定谁租到房，总得用别的办法分配。教科书里的无谓损失三角形默认最想要的人拿到房。换一种分配规则，看左下的长条和小方块。
With a shortage, price no longer decides who gets a unit; something else has to. The textbook deadweight triangle quietly assumes the households who value apartments most get them. Change the rule and watch the bar and the squares.
分配规则Rationing rule抽签：多出一块“错配”
想租的 Qd 户里随机抽 Qs 户。中签者的平均估值只是所有想租者的平均，有人只愿付 $1,550 却中了签，有人愿付 $2,900 却落空。损失多出一块：
错配损失 = b · Qs · (Qd − Qs) / 2
重新抽签只会换掉中签的人，不会改变这个期望损失。Glaeser 和 Luttmer（2003）在纽约的受管制公寓里就发现了大量这类“住错房”的错配。
排队：把转移烧掉
如果靠排队、托关系、花时间找房来分配，大家会一直“付出”到最后一个租到房的人刚好觉得不值，也就是到需求曲线在 Qs 处的高度。多出来的部分不是给了房东，而是花在了等待上。这里假设每个人的时间同样值钱、找房成本全部浪费，是一个极端情形。
Lottery: an extra misallocation loss
Qs winners are drawn at random from the Qd households who want a unit. The average winner values it only as much as the average applicant. Someone who would pay $1,550 wins while someone who would pay $2,900 misses out. That adds a loss:
misallocation = b · Qs · (Qd − Qs) / 2
Redrawing changes who wins, not this expected loss. Glaeser and Luttmer (2003) found substantial misallocation of this kind among New York City’s rent-controlled apartments.
Queue: the transfer goes up in smoke
If units go to whoever waits longest, knows someone, or searches hardest, people keep “paying” in time until the last household to get a unit is just indifferent: up to the demand curve’s height at Qs. The extra doesn’t reach landlords; it is spent waiting. This assumes everyone’s time is worth the same and all search cost is wasted, an extreme case.
05独立练习：这次没有图Independent practice: no chart this time
一座城市给面粉定了价格上限。开始后，左边的图、标注和读数都会隐藏，控件锁定。只用下面的方程算。
A city caps the price of flour. Once you start, the chart, its labels and the readouts on the left are hidden and the controls lock. Work from the equations alone.
06迁移：小麦的最低价Transfer: a minimum price for wheat
这个模型的适用范围Where this model applies
需求和供给都是直线，市场只有一种公寓。真实的住房按地段、面积、质量各不相同；实际的限租通常只覆盖一部分房子，还常有空置后重新定价等例外。这里的数字是示意，没有按任何城市校准；短期和长期两条供给线只用来说明“供给会随时间变得更有弹性”。
抽签和排队是两个极端情形。排队假设每个人的时间同样值钱、找房成本全部浪费。价格下限模式假设卖得出去的是成本最低的卖家，也没有计入卖不出去的产品本身的浪费。
参考：标准的供求与剩余分析见任一本经济学原理教材；Diamond, McQuade & Qian (2019), The Effects of Rent Control Expansion on Tenants, Landlords, and Inequality: Evidence from San Francisco, AER；Glaeser & Luttmer (2003), The Misallocation of Housing Under Rent Control, AER。
Demand and supply are straight lines, and the market has one kind of apartment. Real housing differs by location, size and quality; real rent control usually covers only part of the stock, often with exceptions such as resetting rent between tenants. The numbers are illustrative and not calibrated to any city; the short- and long-run supply lines only show that supply becomes more elastic over time.
Lottery and queue are two extremes. The queue assumes everyone’s time is worth the same and all search cost is wasted. Floor mode assumes the lowest-cost sellers are the ones who sell, and does not count the waste of goods produced but unsold.
References: supply, demand and surplus as in any principles textbook; Diamond, McQuade & Qian (2019), The Effects of Rent Control Expansion on Tenants, Landlords, and Inequality: Evidence from San Francisco, AER; Glaeser & Luttmer (2003), The Misallocation of Housing Under Rent Control, AER.
离线可用。用 Tab 切换控件，方向键调节滑块；图上的橙色线也可以用鼠标或手指拖动。Works offline. Use Tab to reach controls and arrow keys on the slider; the orange line on the chart can also be dragged with a mouse or finger.
可能的总剩余，分成四份Total possible surplus, split four ways
- 无谓损失Deadweight loss
- 分配损失Rationing loss