Case Study 1 — Surplus Analysis of the Millbrook Housing Market

In Chapter 5 we used the Millbrook off-campus housing market as an extended example of supply and demand. In Chapter 7 we analyzed what would happen if Walden County imposed rent control. Now we have the tools — consumer surplus, producer surplus, and the deadweight loss framework — to ask a sharper question: how much value is the Millbrook housing market creating, and how much would be destroyed by various policies?

This case study walks through a stylized welfare analysis of three Millbrook scenarios: the unregulated market, the rent control proposal, and a housing voucher alternative. The numbers are illustrative — designed to be simple enough to follow — but the structure and the conclusions reflect what real welfare analyses look like.

Setting up the model

Suppose the off-campus housing market in Millbrook can be approximately described by:

Demand: $P = 1800 - 0.05 Q$ where P is the monthly rent in dollars and Q is the number of 1-bedroom-equivalent units.

Supply: $P = 600 + 0.10 Q$

At equilibrium: $1800 - 0.05Q = 600 + 0.10Q$, so $1200 = 0.15Q$, and $Q^* = 8000$ units. Equilibrium price $P^* = 1800 - 0.05(8000) = 1800 - 400 = \$1400$ per month.

This roughly matches what we said about Millbrook in Chapter 5: equilibrium rent around $1,400, equilibrium quantity around 8,000 units. Now we can use these curves to compute surpluses.

Scenario 1 — The unregulated market

At equilibrium ($P^* = \$1400$, $Q^* = 8000$):

Consumer surplus: $\frac{1}{2} \times 8000 \times (1800 - 1400) = \frac{1}{2} \times 8000 \times 400 = \$1,600,000$ per month

Producer surplus: $\frac{1}{2} \times 8000 \times (1400 - 600) = \frac{1}{2} \times 8000 \times 800 = \$3,200,000$ per month

Total surplus: $1,600,000 + 3,200,000 = \$4,800,000$ per month

So the Millbrook off-campus housing market — under our stylized assumptions — creates about $4.8 million in total monthly value. About one-third of that goes to renters as consumer surplus, and about two-thirds goes to landlords as producer surplus. (The asymmetry reflects the different slopes of the demand and supply curves: supply is more inelastic, so producers capture more of the surplus.)

Per-renter consumer surplus: $\$1,600,000 \div 8000 = \$200$/month per renter. The average renter in this market gets about $200 of monthly value beyond what they pay in rent. That number gives a sense of how much renters are "ahead" by being able to participate in the market at all.

Scenario 2 — The rent control proposal

Now imagine Walden County imposes a price ceiling at $1,250 (about $150 below equilibrium). What happens?

At a rent of $1,250: - Quantity demanded: $1800 - 0.05Q = 1250$, so $Q_d = (1800-1250)/0.05 = 11000$ units - Quantity supplied: $600 + 0.10Q = 1250$, so $Q_s = (1250-600)/0.10 = 6500$ units

Quantity supplied at the ceiling = 6,500 units. (Quantity demanded at the ceiling is 11,000, but only 6,500 can be rented because that's all that's offered.)

Shortage: $11,000 - 6,500 = 4,500$ units

Now let's compute surpluses:

Consumer surplus (for the 6,500 lucky renters who get units at the controlled price): the area under the demand curve from 0 to 6,500 minus the rent paid.

The demand curve at Q = 6500 gives a willingness to pay of $1800 - 0.05(6500) = 1800 - 325 = \$1475$.

Consumer surplus = (1/2) × 6500 × ($1800 - $1475) + 6500 × ($1475 - $1250) = (1/2) × 6500 × $325 + 6500 × $225 = $1,056,250 + $1,462,500 = $2,518,750

(The first term is the triangle above the price line; the second is the rectangle of "savings" the renters who would have paid the higher market price get from the lower controlled price.)

Hmm — that's actually higher than before! Let me recompute the unregulated case to compare correctly. In the unregulated case with the same demand curve: - Consumer surplus = (1/2) × 8000 × ($1800 - $1400) = $1,600,000

So consumer surplus rose from $1,600,000 (without rent control) to about $2,518,750 (with rent control)? That doesn't seem right — let me reconsider.

Actually, the issue is that when there's a shortage, the renters who can't get apartments at all lose surplus. Let's account for that.

With rent control, only 6,500 out of 11,000 demanders get units. The 4,500 demanders who don't get units lose all of their consumer surplus. Without rent control, those same 4,500 demanders... well, they wouldn't all be in the market, because at the higher equilibrium price they'd have dropped out.

Let me re-think. In the unregulated case, only 8,000 units are demanded at $1,400. The 8,000 demanders all get units. Consumer surplus is the triangle area = $1,600,000.

In the rent control case, 11,000 demanders show up at $1,250 (because the price is lower), but only 6,500 get units. Of those 6,500, who gets them? The standard assumption is "the highest-willingness-to-pay buyers" (who would have been willing to pay the most). With this assumption, the 6,500 who get units are those at the top of the demand curve.

If the 6,500 highest-willingness-to-pay buyers get the units at $1,250, then: - Consumer surplus = area below the demand curve from 0 to 6500 minus 6500 × $1250 - = (1/2) × 6500 × ($1800 - $1475) [the triangle above $1475] + 6500 × ($1475 - $1250) [the rectangle from $1475 to $1250] - = $1,056,250 + $1,462,500 - = $2,518,750

But that's still higher than the unregulated case. This is because in our model, the 6,500 "lucky" renters are getting the most they would pay (their willingness is $1,475 at Q=6500) for a price of $1,250 — a $225 discount per unit, on top of their original consumer surplus.

The issue is that with rent control, the lucky renters do get a windfall. The total consumer surplus can rise. What you've also lost is the consumer surplus of the 1,500 renters between Q = 6500 and Q = 8000 who would have bought at the unregulated price — they're now shut out and get nothing.

OK so the right comparison is more nuanced. Let me redo this carefully.

Unregulated equilibrium (Q = 8000, P = $1400): - CS = $1,600,000 - PS = $3,200,000 - TS = $4,800,000

With rent control (Q = 6500, P = $1250): - CS for the 6500 who get units = $2,518,750 (as computed above) - PS = (1/2) × 6500 × ($1250 - $600) = (1/2) × 6500 × 650 = $2,112,500 - TS = $2,518,750 + $2,112,500 = $4,631,250 - Deadweight loss = $4,800,000 - $4,631,250 = $168,750

OK so the deadweight loss is real. And the CS distribution shifted dramatically: - CS rose by $918,750 (some renters got a windfall) - PS fell by $1,087,500 (landlords lost a lot) - Net loss to society (deadweight loss) = $168,750

But this hides something important. The 1,500 "shut out" renters who would have rented at $1,400 but now can't get any unit at $1,250 — what happens to them? Some of them found alternatives (commuting from further away, doubling up, dropping out of MSU, sleeping in cars). The welfare cost of being shut out is not zero — it's roughly the consumer surplus they would have had at the unregulated price.

So let me add a more honest calculation: of the 1,500 shut-out renters, their average willingness to pay was somewhere between $1,400 and $1,475. They were going to get some surplus at $1,400. They get nothing now. That's an additional welfare loss above the simple deadweight loss calculation.

The bottom line for the welfare comparison: - Total surplus is lower with rent control than without - The distribution of surplus shifted in favor of the 6,500 lucky renters - The 1,500 shut-out renters bear the largest loss - Landlords also lose substantially

This is the formal version of the argument from Chapter 7: rent control redistributes surplus, but on net it destroys some.

Scenario 3 — A housing voucher alternative

Now consider a different policy: instead of capping rent, the county provides housing vouchers worth $150/month to 1,500 low-income tenants. What happens?

Vouchers shift the demand curve. If the demand curve was $P = 1800 - 0.05Q$ before, with a voucher to 1,500 tenants the demand effectively becomes "higher" — for those 1,500 tenants, their willingness to pay rises by $150 because the government pays the first $150.

A simplified analysis: the 1,500 tenants now demand the same units they would have demanded before but at a price $150 lower (out of pocket). The market clears at a higher equilibrium quantity and a slightly higher equilibrium price.

For simplicity, suppose the new equilibrium is at Q ≈ 8200 (a small increase due to the new effective demand) and P* ≈ $1410 (slightly higher because of demand shift). The numbers are stylized, but the comparison is the point.

Consumer surplus with vouchers: - For the 6,700 unsubsidized renters: pay $1,410, surplus is the triangle as before - For the 1,500 voucher recipients: pay $1,260 ($1,410 - $150 voucher), so surplus is much higher per renter

Producer surplus: slightly higher because price is slightly higher and quantity is higher

Government cost: 1,500 × $150 = $225,000/month in voucher payments

Total welfare effect: similar in magnitude to the rent control case but distributed very differently. The voucher recipients gain. The non-recipients are slightly worse off (slightly higher market rent). Landlords gain. The government pays.

The key difference from rent control: the voucher policy adds value to the market (the government's $225,000 becomes part of the surplus distribution) rather than destroying it through quantity reduction. There is no deadweight loss in the simple model. The government's spending becomes consumer surplus for the voucher recipients.

Of course, the government has to raise that $225,000 somehow — and raising it through taxation creates its own deadweight loss. The honest comparison would account for this.

But broadly: vouchers redistribute surplus toward the targeted population without reducing total surplus very much, while rent control redistributes among renters (winners and losers) and destroys some surplus on the way. Most welfare economists prefer vouchers for this reason.

What this case study illustrates

Lesson 1 — Surplus calculations are concrete. "Rent control reduces welfare" can sound abstract. "Rent control destroys $168,750 of monthly value in this stylized Millbrook market" is concrete enough to compare against the value of helping the lucky renters.

Lesson 2 — Distribution matters as much as totals. The total surplus comparison shows rent control is worse than the unregulated market. But the distributional comparison shows that some renters are much better off under rent control. Whether this trade-off is worth it depends on values, not on the model.

Lesson 3 — Vouchers usually beat price controls. When the goal is helping a specific group, direct subsidies generally outperform price controls in welfare terms. This is one of the most consistent findings in the welfare-economics literature.

Lesson 4 — The model leaves things out. The deadweight loss calculation doesn't capture the welfare cost of stress, displacement, or instability that rent control might mitigate. It also doesn't capture the welfare cost of having to move because of a sudden rent increase. Real welfare evaluation needs to add these things in qualitatively, since the surplus framework can't quantify them.

Lesson 5 — The numbers are stylized. Real welfare analyses use much more sophisticated models, more carefully estimated demand and supply curves, and explicit treatment of distribution. But the structure is the same: compute surpluses, compute changes, compute deadweight loss, and compare to alternatives.

Discussion questions

1. The case study found that rent control destroys about $168,750/month of surplus in Millbrook (in the stylized model). Per person, that's about $20/month for 8,000 affected households. Is that a big number or a small number? Compared to what?

2. The voucher alternative was found to be more efficient than rent control. But it requires the government to spend money (about $225,000/month). Where does the money come from? What's the welfare cost of raising it?

3. The case study's calculations assume that the 6,500 "lucky" renters are the highest-willingness-to-pay tenants. In reality, who actually gets rent-controlled apartments? Use the assumption to think about who really benefits.

4. The framework leaves out the welfare cost of "instability" — the value renters place on knowing their rent won't suddenly jump. Can you think of a way to incorporate this into the surplus calculation? What number would you put on it?

5. Imagine you are a policy advisor in Walden County. Based on this case study, what would you recommend? Be honest about what the framework tells you and what it leaves out.