There are two opposing effects underlying the paradox. The engineering effect is where improved efficiency means each unit of the service requires less of the resource. But through the demand effect, improved efficiency lowers the price of the service, which increases service demand, thereby increasing demand for the resource. When the demand effect is large enough, it outweighs the engineering effect, and Jevons’ paradox occurs.

To model Jevons’ paradox, we need two markets: one for the resource, and one for the service. Since the resource is the input used to produce the output, the markets are connected. Hence, any supply and demand explanation of Jevons’ paradox requires two supply and demand graphs.

Let’s use the example of fuel efficiency. The service is vehicle travel, which is produced using the resource of gasoline. Vehicle travel is priced in dollars per mile, and gasoline is priced in dollars per gallon; fuel efficiency is miles per gallon. So the Jevons question is: does improved fuel efficiency lead to increased usage of gasoline?

Graphically, an increase in fuel efficiency shows up as a rightward-shift in the travel supply curve, as each unit of travel is now cheaper to produce, and either a left or right shift in gasoline demand, depending on the elasticity of travel demand.

Elastic demand: Jevons’ paradox, gasoline use increases

When demand for vehicle travel is elastic, the shift in the travel supply curve increases the quantity of travel demanded by enough that gasoline usage increases. Here, efficiency increases by 50%, but travel quantity increases by more than 50%, so the gasoline demand curve shifts right, and the quantity of gasoline rises.

Inelastic demand: no paradox, gasoline use falls

But when demand for travel is inelastic, the travel supply curve shifts right, but the increase in travel quantity is smaller, so gasoline usage decreases. Here, efficiency increases by 50%, but the quantity of travel increases by less than 50%, so the gasoline demand curve shifts left, resulting in lower gasoline usage.

See here for a formal model. Note the similarity to my model of housing produced using land.

Appendix: Wikipedia doesn’t get it

The graph on the current Wikipedia page is incoherent. It combines the travel and gasoline markets, with the fuel price on the y-axis and the quantity of travel on the x-axis. But there should be two graphs: (fuel price, fuel quantity) and (travel price, travel quantity). They never show the quantity of fuel, and hence cannot actually demonstrate the core result of fuel quantity increasing.

More generally, zoned capacity can be stranded and infeasible due to redevelopment costs. It’s cheap to tear down a bungalow, so zoning for a 5-storey apartment building is feasible capacity. But it’s very expensive to tear down a low-rise apartment building; capacity for one extra floor is infeasible. Each scenario adds four floors of housing, but only one will actually get built. Not all zoned capacity is the same.

We can think of zoned capacity as the housing supply curve Q_S(P) evaluated at infinite prices: Q_S(∞). If rents reached $10,000/month, it could be worth it to tear down those 4-storey buildings and redevelop at 5-storeys. When NIMBYs say “we have enough zoned capacity”, they are asking us to wait until rents rise to unbearable levels. This just shows that they don’t care about housing affordability.

This graph shows how two cities with the same theoretical zoned capacity can have very different affordability outcomes. City A is mostly bungalows and has liberal zoning, so 80% of the capacity can be built with rents below $3,000. In contrast, City B is mostly apartment buildings with restrictive zoning, so rents need to rise above $8,000 to produce the same number of homes.

In practice, we want to know how much of the nominal capacity can be produced at reasonable rents. We always need to ask “capacity at what price level?”. And the point of upzoning is to shift the supply curve and add capacity that is profitable to build at lower prices. The holy grail is zoned capacity that brings down rents.

So how do we measure feasible zoned capacity? One approach is a parcel-level ‘pro forma’ model, where for each parcel we calculate the optimal project and the probability of development, given current prices and construction costs; see the Terner housing policy simulator and this Grattan report. A more technical approach is to estimate a structural model and construct the aggregate supply curve. Rollet’s model could be used to quantify how much of NYC’s zoned capacity is feasible at specific rent levels.

Read more in my deep dive on upzoning:

Building Abundance

Upzoning

In 2015, the city of Auckland had roughly 350,000 units of ‘zoned capacity’: homes that could theoretically be built under the existing zoning. So you would expect that adding more zoned capacity would have no effect, like pushing on a string. And yet, after Auckland upzoned in 2016 to allow townhomes and apartments, housing construction surged, leading…

Read more

a month ago · 20 likes · Michael Wiebe

Consider a city of detached houses, like Vancouver.1 If you’re a developer trying to build apartments, you need to buy an existing house and redevelop it. But the average price of a house has risen from $400,000 in 2000 to over $3,000,000 today, reflecting a massive increase in demand from wealthy buyers to live in Vancouver. This translates to a large increase in assessed land values. So demand for detached houses sets a floor on land costs, since you need to outbid a house builder to acquire land.

For example, suppose a house builder is willing to pay $2M for a teardown bungalow, on which they can build a house that sells for $3M.2 Then as an apartment developer, you need to offer at least $2M to buy land. And when demand for houses goes up, house prices increase, so house builders will bid even more for teardowns. If house prices hit $3.5M, you now have to pay $2.5M for land, so your apartment project won’t pencil until rents go up.

In this scenario, higher land values represent higher input costs that drive up the price of new apartments. Land behaves like any other construction material, where higher input costs mean higher output prices.

But there’s an important case where this doesn’t hold: when the city upzones to allow higher-density housing. If we change the zoning from detached housing to allow apartment buildings, we create valuable development rights that developers will pay a premium for.

For example, take the bungalow above. When zoned for a single house, a builder was willing to offer $2M. Since apartments are not allowed, the apartment developer bids nothing. But if we upzone for 12-unit apartments, the developer is now an active bidder. Say the apartment building is worth $12M, and demolition and construction costs are $9M. Then the developer can pay $3M for land; assuming developers are competitive, this is also the parcel’s market price.

So upzoning increases the land value from $2M to $3M. But this is not an increase in input costs for building apartments. Initially, the apartment building wasn’t even allowed, meaning the land cost was infinite. Bringing that cost down to $3M is a definite improvement. Rather, the increased land value reflects the increase in total housing value, since a six-storey apartment building has more homes than a detached house.3

From this angle, high land values—when caused by upzoning—are a sign of success, because land is being employed for its most valuable use.4 When land is expensive, you should economize on it, by densifying and using less land per home. Restrictive zoning lowers land values by preventing developers from buying land and using it to build apartments. Low land values driven by constrained housing supply are a policy failure.

It is true that allowing apartment developers to bid on the land increases land costs for house builders. But this is a good thing. When zoning blocks apartments, it makes land artificially cheap by taking developers out of the market. This is a regressive subsidy from apartment renters to house buyers. If people want to live in detached houses, they should pay the market price.

Ultimately, high land values are caused by high housing demand. Demand for detached houses pushes up the floor on land prices. And demand for apartments leads to higher land values when we upzone to allow them. But if demand is the root cause, then “lowering land values” is not a solution.5 We care about lowering housing prices, and the way to do that is by allowing more density to reduce land cost per home and increase housing supply; land prices are just a symptom. Manhattan would have much lower land values if it was zoned for detached houses, but also much higher housing prices.6

1

It is still mostly houses:

To capture this core mechanism, I use the simplest possible model: a one-time build-out of single-family houses on greenfield land. Developers buy land from farmers and build houses. An urban growth boundary (UGB) is the zoning constraint on the supply of land, and we upzone by expanding the UGB and reallocating land from agricultural to residential. In general, upzoning reallocates land from low- to high-density, so this is analogous to upzoning from detached houses to apartments. (See here for a formal model.)

Setup

We’re building out a new city of single-family houses on undeveloped agricultural land. Each house requires one parcel of land, and an urban growth boundary limits the size of the city by restricting the number of parcels available to build on. We don’t model location, which means assuming all land is in the same location. We also assume all behavior takes place during a single time period. There are three actors in the model: landowners, developers, and homebuyers.

Landowners can farm their land or sell to a developer. All land is the same, so landowners get the same farming payoff. This means that, without an UGB, landowners offer their land to developers at a price equal to this payoff. When developers want more land than the UGB allows, land becomes artificially scarce and the price is bid up above the farming payoff. Now, landowners can sell at an inflated price that includes a scarcity premium.

Developers buy parcels of land and pay fixed construction costs to build houses. We assume developers are competitive, so profits are zero: the revenue from selling a house is equal to construction and land costs. Finally, homebuyers buy houses, demanding a larger total quantity when prices are lower. All houses are the same, and sell for the same price.

From this setup, we get a general equilibrium model of land and housing. In the land market, landowner supply and developer demand determine the price of land; and in the housing market, developer supply and homebuyer demand determine the price of housing.1 Since land is the input used to build housing, developers derive their demand for land from homebuyers’ demand for housing. Similarly, the supply of housing is derived from landowners’ supply of land. Developers are the link between the two markets.

The graphs below show an equilibrium where the UGB is binding.2 In the land market, the supply curve is flat until we reach the UGB, where it becomes vertical. We say that landowners are ‘perfectly elastic’, meaning they will offer any quantity at the same price (here, the farming payoff of 40). The UGB makes supply ‘perfectly inelastic:’ no matter how much prices rise, landowners cannot sell a larger quantity of land. The demand curve intersects the supply curve where it is vertical, which means the UGB is a constraint on residential land; if we could expand the UGB, developers would buy more land.

Because the UGB is binding, residential land is a fixed factor, so its price is determined as a residual, as the housing price minus construction costs. Land is artificially scarce and includes a scarcity premium, making it more expensive than agricultural land. Here, the residential land price is bid up to 100 and the farming payoff is 40, so the scarcity premium is 100 - 40 = 60. When zoning is not a constraint, residential and agricultural land sells for the same price, and the scarcity premium is zero.

In the housing market, we assume that homebuyers have downward-sloping demand: at lower prices, they are willing to buy a larger quantity of houses. Below the UGB, the housing supply curve is horizontal, and we calculate it by adding construction costs (80) to the land supply curve; developers build houses by paying land and construction costs (40 + 80 = 120). Because landowners are perfectly elastic suppliers, so too are developers. But once the UGB binds, the supply curve contains the scarcity premium and becomes vertical. (Similarly, the land demand curve is calculated as the housing demand curve minus construction costs.)

With a binding UGB, housing supply is constrained, pushing up house prices. If we could expand the UGB, we would move down the demand curve, increasing quantity and reducing housing prices.

Upzoning: expanding the UGB

So let’s expand the growth boundary and see what happens. Upzoning means shifting the land supply curve to the right, because we’re allowing more land to be used for housing. Each house uses one parcel of land, so more land means that the housing supply curve also shifts right, leading to lower housing prices. In the land market, increased supply leads to lower land prices, decreasing the scarcity premium on residential land.

In the graphs below, we expand the UGB from 120 parcels to 155. This reduces housing prices from 180 to 145, land prices from 100 to 65, and the scarcity premium from 60 to 25. So the constraint is still binding, but not as stringently.

Here upzoning reduces the land price of existing residential land. This is the cross-parcel effect, where increasing the supply of land (by 35 parcels) reduces the price of the original 120 residential parcels. When land is constrained and priced as a residual, lower housing prices translate to lower residual land values. (Generally, upzoning from low- to high-density reduces the price of existing high-density land.)

In contrast, the own-parcel effect is the increase in value for the 35 newly upzoned parcels. Initially, these were worth the farming payoff of 40. Upzoning them to residential increases their value to the new land price of 65, for a gain of 25. This is a windfall gain for those landowners, but the increase in price is not relevant to developers, since the land wasn’t even available to them when it was zoned for farming. All that matters for the housing market is that the stock of residential land is larger.

Also note that if we upzoned even more, by increasing the UGB to 180 parcels (or more), then the scarcity premium would be reduced to 0. Residential land would be so abundant that it would sell for the same price as agricultural land. In this case, the own-parcel effect is also 0: with no scarcity premium, agricultural landowners do not benefit from acquiring development rights. And when land is abundant, any additional upzonings have no effect on land prices.

This is the YIMBY endgame: upzoning so much that zoning is no longer a constraint, which achieves the maximum reduction in housing prices. In general, the policy goal is to have equal land prices for otherwise similar low- and high-density land.

Do higher construction costs increase housing prices?

One counterintuitive argument is that higher construction costs do not lead to higher housing prices, because market prices are fixed and higher costs just reduce land values. With our general equilibrium model, we can sort this out. When the UGB is not binding, prices are set by marginal costs, so higher costs do lead to higher housing prices. Demand for land falls, but with no effect, since land is priced at the farming payoff.

But when the UGB is a binding constraint, prices are set by scarcity, not marginal costs. In this case, a small increase in costs reduces the scarcity premium on land without affecting housing prices, as demonstrated below. Costs increase by 40, and land prices fall by 40. The housing supply curve shifts up with no effect, since we’re eating into the scarcity premium.3

So whether costs affect housing prices depends on whether the zoning constraint is binding. If costs increase enough, developers don’t want to build as many houses and the UGB stops binding. Then we’re back in the scenario where construction costs are passed through to housing prices.

Do development taxes increase housing prices?

If we tax developers for building houses, do we get higher housing prices or lower land prices? As with construction costs, it depends on whether the UGB is a binding constraint. First, a primer on tax incidence. Even though developers are charged the tax when they buy land (they bear the ‘statutory incidence’), they can pass on the tax to landowners or homebuyers, due to our assumption of making zero profit. So the tax is actually paid by landowners or homebuyers (they bear the ‘economic incidence’), and this is determined by who is more responsive to price changes (i.e., more price-elastic).

When the UGB is binding, landowners cannot supply more land, no matter how high prices are. Here landowners are unresponsive to prices (perfectly inelastic), and developers can pass on the tax to landowners by paying lower land prices. And when the UGB is slack, landowners are perfectly elastic, since they offer any quantity of parcels at the farming payoff. Here, landowners are more price responsive than homebuyers, so developers can pass on the tax to homebuyers by increasing housing prices.

The graphs below show a non-binding UGB. When zoning is not a constraint, the tax reduces developers’ demand for land, shifting the demand curve down. The tax acts like a cost, shifting the housing supply curve up, which reduces the quantity and increases the price of housing.

Next let’s see the binding UGB case. Here the tax also reduces demand for land, but because the zoning constraint makes landowners inelastic, developers can reduce their land payment without affecting the quantity of parcels. The housing supply curve shifts up, but this doesn’t change anything. Essentially, the tax comes out of the scarcity premium with no effect on housing prices, exactly the same as the cost increase above. If the tax is large enough, then the housing supply curve shifts up and the UGB stops binding, and we switch back to the case above where the tax increases house prices.

The political incentives here can be quite perverse. If the government wants to raise revenue while appearing to tax landowner windfalls, they are incentivized to impose an UGB, thereby creating a scarcity premium to tax. But as we saw when studying upzoning, the zoning constraint creates a scarcity premium only by raising housing prices! So even though the tax itself doesn’t affect the housing market, the combination of restrictive zoning and development taxes does result in higher prices. Even if the tax revenue is used to fund subsidized housing, the best case scenario is that housing prices fall back to their original level. We cannot improve affordability by first creating scarcity.

We can also interpret a development tax as value capture from upzoning, taxing the ‘land lift’ on upzoned land that increases in value. Initially, the UGB is set at zero parcels and no development is allowed; land is priced at the farming payoff of 40. Then we upzone by expanding the UGB to 120 parcels, and the upzoned land increases in price to 100. We’ve created a scarcity premium of 100 - 40 = 60 that the government can tax.

But the scarcity premium exists only because we did a limited upzoning. If we had upzoned to 180 parcels, the scarcity premium would be zero and the government would have nothing to tax. Since the UGB isn’t binding, any tax would fall on homebuyers, raising prices. So a government that depends on revenue from development taxes is incentivized to do limited upzonings to create a taxable scarcity premium. A government that cared about housing affordability would upzone as much as possible to minimize housing prices.

Land value taxes don’t affect housing prices

In contrast to development taxes, a land value tax does not affect housing prices. While a development tax is imposed only when a development occurs, a land value tax is charged to the landowner unconditionally, even when no housing is built. This is the key difference that makes land value taxes non-distortionary and hence a favorite of economists. Landowners are taxed on their farming payoff, so even though a developer has to pay a tax to own land, they also pay lower land prices, so the effects cancel out.

The graphs below show a 50% proportional tax on the value of the land. The demand curve for land shifts down, same as the development tax. But now the land supply curve also shifts down, reflecting that landowners are taxed on their outside option. Since both curves shift, there’s no effect on the quantity of land, and the housing market is unchanged. Land prices fall and housing prices are the same. When the UGB is binding, an LVT reduces the scarcity premium on land, so it can be used as a value capture mechanism.

An LVT also has no effect on housing prices when the UGB is slack. Again, since both curves shift down in the land market, there’s no effect on quantity, and hence no change in the housing market.

Land prices are driven by housing demand

Finally, let’s consider the argument that upzoning is the main factor driving up land prices. While it is true that upzoning can have a positive own-parcel effect (because residential land is priced above agricultural land), rising land prices are generally driven by increased housing demand.

When the UGB is binding, land prices are directly determined by the level of housing demand (via the demand curve intercept y). Shifting up the housing demand curve raises house prices, which also increases developer demand for land, thereby increasing land prices. So it makes sense that a city with growing demand will have higher housing and land prices.

We can confirm this by looking at a natural experiment where upzoning isn’t allowed. First Shaughnessy, a heritage-protected mansion neighborhood in Vancouver, has seen a 5x increase in land values even without any upzonings. This suggests that housing demand has increased substantially in Vancouver.

Conclusion

A simple supply and demand model of land and housing can explain how upzoning works, whether increased construction costs and taxes lead to higher housing prices, and how housing demand drives land prices. This model assumes one housing good, one zone, and one location, as the simplest model that captures how upzoning works. In future posts, I will extend it to have two zones (house- and apartment-zoning), two housing goods (houses and apartments), and two locations (city and suburb).

See here for the formal model.

So to evaluate whether upzoning works, we first check whether increased housing supply results in lower home prices. As a second step, we can check whether existing high-density land becomes cheaper. This gives us a simple empirical test using land prices: take the set of high-density land that existed before upzoning, and track its price over time. However, no one has done this (yet).

For example, Auckland had 389 parcels zoned for terraced housing and apartments (THA) whose zoning status was unchanged by the 2016 upzoning.1 The THA zoning category increased in size to 13,341 parcels, as lower-density zones were upzoned to THA. Intuitively, the land price of the initial 389 THA parcels will fall, at least relative to single-family land, as competition from the thousands of new THA parcels leads to expectations of lower apartment prices and hence lower residual land values. So we can see whether upzoning worked by testing if increasing the supply of high-density land made it cheaper.2

São Paulo also upzoned in 2016, changing the allowed density on a block-by-block basis. On average, the built area ratio (BAR)3 increased by 35%, with about half of blocks being upzoned. We can use this to test whether upzoning reduced the price of existing higher-density land at a fine-grained level. For example, if upzoning increases the size of the BAR=4.0 category, then we expect apartment prices to fall and the price of the initial BAR=4.0 land to decrease. For categories that decreased in size, we might expect land prices to increase, for example, as detached houses become scarcer and more expensive, leading to higher residual land values.4 Here’s a hypothetical graph:

If you have access to data on land prices or assessed land values in Auckland or São Paulo, please reach out! Existing research has not looked at this question, and it seems straightforward to answer. Showing that upzoning reduces high-density land prices would give us empirical evidence for upzoning in addition to the main evidence on housing supply and prices.

The setting is a 'mover' event study, where individuals move across cities; the treatment is the change in city environment. In M21, this is the change in cluster size, defined as the number of inventors in the same research field and city. The treatment variable is Size_diff = Size_post - Size_pre. The event study interacts Size_diff with event time indicators.

The sample is movers: inventors who change cities exactly once. Stayers are excluded from the sample (i.e., there are no never-treated observations). It's a staggered rollout design, with inventors moving in different calendar years. The graph above includes individual, year, and city fixed effects. Seems fine, right?

When I simulate data for N=1000 individuals moving between C=20 cities, with true effect B=1, we get the expected event study graph (using the same fixed effects):

Here's the graph for N=1000 individuals and C=1000 cities. This looks very similar to the original graph! So the city fixed effects are the key.

The core issue is cities being observed only as origins or only as destinations. By definition, origin = (Post==0), so origin-only cities have Post=0 for all observations. Similarly, destination-only always have Post=1. For these observations, the city fixed effect is collinear with Post.

In the event study, with t=-1 omitted, this means the post-move indicators are collinear with the city FE. In the extreme case, where origins and destinations are disjoint, no cities are both origin and destination, so the regression is unidentified.

With C=1000 cities, only 5% of city-year pairs are observed as both origin and destination. So when including city fixed effects, the effective sample is very small, and we get a big jump in confidence interval width. With C=20 cities, this is 78%, so the confidence intervals are fine.

Note that the jump in confidence interval width is just a function of the omitted period. When I omit t=0 instead of t=-1, then the pre-move indicators are nearly collinear with the city fixed effect, and the confidence intervals are now wider pre-move.

Final note: the pattern of large standard errors could also be explained by treatment effect heterogeneity. In this case, however, it is due to the city fixed effects driving collinearity. When I redo the original graph using real data and remove the city fixed effects, the jump in width disappears.

Link to simulation code.

But why are there two panels? An event study gives us only one set of coefficients, i.e., the treatment effect in each event year. What does the spike in Figure 6a mean? Why isn’t t=-1 omitted as the reference year? What’s going on with the x-axis labels?

To explain this, let’s go back to Figure 5 (which happens to look very similar to Figure 6). M21 is running a regression of log patents (y) for inventor i on current log cluster size (S) as well as five leads and five lags of cluster size. Here’s M21 Equation 2:

The lead terms (B_1 to B_5) capture how current patenting is affected by future cluster size, while the lag terms (B_{-5} to B_{-1}) capture lagged effects of past cluster size. M21 says this is “studying the dynamic response of productivity following a change in cluster size”, but also that “this is not a standard event study”. (p.3351)

Thanks to Schmidheiny and Siegloch (2023), we know that this is a distributed lag model. They prove that event study (ES) and distributed lag (DL) models are equivalent.1 But ES models give you estimates of the cumulative effect of a treatment, and DL models give you the incremental effect, i.e., changes in the cumulative effect. So the DL estimates need to be aggregated to get the total treatment effect. In M21 Fig 5, the beta terms are the incremental effects and the mu terms are the aggregated cumulative effect.2

Moreover, this is the most extreme case of a DL model. Cluster size is a continuous variable and changes every year, so this is equivalent to an event study where a different event occurs every year (i.e., a change in cluster size). The simplest case is a binary treatment that turns on once; here we have a time-varying continuous treatment dose. M21 does not discuss the identifying assumptions required here.

Now let’s revisit the event study in Figure 6. M21 seems to think that this is also a DL model, with incremental beta estimates in the top panel and aggregate mu estimates in the bottom panel. The spike in Figure 6a is exactly what a proper treatment effect looks like in a DL model, where there is an incremental effect in the treatment year, and nothing in other years. But this is wrong: an event study already gives us aggregate treatment effects, so aggregating again means calculating the cumulative response function of a cumulative effect. So Figure 6b is meaningless and should be removed.

And the text clearly states that Figure 6 is attempting an event study, based on interacting a treatment variable with event time indicators:

In Figure 6 I replicate [Figure 5] using only variation from inventors who move across cities. [...] I use the subset of inventors who change city once to estimate a variant of equation (2), where t = 0 marks the time of the move, and the timing relative to an inventor’s move is interacted with the corresponding cluster size before the move and after the move. [...] Thus, this specification is an event study, based on a “pure” movers design solely exploiting variation in cluster size induced by moves.” (p.3352)

But as I discuss in my comment, this is not a proper event study specification. Instead of using the change in cluster size associated with moving across cities, M21 interacts pre-move average cluster size with the pre-move event time indicators, and post-move average cluster size with the post-move event time indicators. M21 doesn’t show a regression equation; my reconstruction is:

But this doesn’t use variation in cluster size from moving; it is identified even if there is no change in size (Size_pre = Size_post). In my comment, I show that a correct mover event study using ∆Size = Size_post - Size_pre as the treatment variable gives a null result.

So why does Figure 6a have that spike in t=0? Equation 1 is not using variation from moving, so there shouldn’t be a spike. This specification is actually testing for heterogeneity in the cross-sectional relationship between (averaged) cluster size and patenting by event time. So the beta coefficients should all be equal, if they estimate the same constant elasticity.

The problem is that the M21 code does not follow my Equation 1 above, due to a coding error. It doesn’t generate an interaction of Size_post with the event time indicator for t=0, but instead uses the regular time-varying cluster size variable (ln S_it). So B_0 is estimated using data from all event years, instead of capturing the effect in t=0. In Figure A1 of my comment, I show the results using the original and corrected code. The large spike disappears:

To sum up: M21 Figure 6 shows an attempted event study that is confused with a distributed lag model. Figure 6b calculates the cumulative response function of a cumulative effect, and is unnecessary. Figure 6a uses an incorrect event study specification, and the large effect is due to a coding error anyway.

M21 studies agglomeration effects for innovation: do bigger technology clusters have more patenting? The main results are based on inventor-level regressions of patents on cluster size, which give a positive correlation. To claim a causal effect, M21 uses an event study and IV strategy to rule out selection and omitted variable bias.

To test for selection bias, M21 uses an event study based on inventors moving across cities. Since moving changes cluster size, we should see a clear effect on patenting; if inventors select into bigger cities, we’ll see a trend before the move. I show that the original event study uses an incorrect model: it doesn’t actually use variation from moving. When I run a proper mover event study, I get a null result.

M21 uses an instrumental variable to address omitted variable bias, such as local subsidies that drive both cluster size and patenting. The instrument is based on the number of inventors in the same field who work for firms in other cities. Due to a coding error, the data is not sorted by city, so the instrument is constructed incorrectly. I correct the code and find a null result.

The event study and IV are necessary to establish that the baseline regressions show causation, nor correlation. Since they both fail, the main takeaway of my comment is that the positive correlation between cluster size and patenting may not be causal.

I was hired by Coefficient Giving to do a small extension of this paper, since it’s the main evidence on technology agglomeration effects that informs their housing policy funding. But the more I looked at the paper, the more problems I found. Let’s go through the other eight problems:

Issue 3: Early on, I would sometimes get different results when running the baseline regression to get the main patenting-size elasticity. It turns out this is due to M21 using many-to-many merges in the data cleaning, so the results are actually unreproducible. Each time you run the code, you get a slightly different sample.

Issue 4: Table 6 studies the effect of cluster size on patent quality, measured using citations. M21 claims to use log citations, but the code actually does log(y+0.00001). When I use log(y+1) or Poisson, the effect switches from positive to negative.

As in Chen and Roth, this happens because the extensive margin effect is positive, and a smaller constant puts less weight on the intensive margin effect. Here, c=0.00001 means valuing the extensive margin as a 40,000% change in the intensive margin.

M21 also calculates citation- per-patent using citations-per-coauthor and patents-per-coauthor; but this nullifies the per-coauthor adjustment. The correct approach is to use citations per coauthor / whole patents.

So the original M21 finding is reversed: patent quality is in fact decreasing in cluster size. The actual story may be: larger clusters produce more patents, but they are lower quality, e.g., strategic patents used in IP lawsuits. (For example, bigger firms produce strategic patents and locate in big cities.)

Issue 5: there’s another substantive coding error when M21 estimates heterogeneous elasticities by cluster size in Table 8. Here the goal is to test whether agglomeration effects are bigger in bigger clusters. M21 reports limited heterogeneity.

M21 interacts log cluster size with size quartile indicators, but incorrectly omits the quartile indicators themselves. This forces the baseline level of patents to be equal across quartiles, biasing the estimates. When I do the correct interaction, I find substantial heterogeneity across size quartiles. The patenting-cluster size elasticity is actually much bigger in large clusters.

M21 does an exercise to quantify the aggregate benefits of agglomeration. The idea: equalize cluster size and calculate the decrease in the total number of patents. Since inventors are reallocated to smaller clusters, total output falls. But since M21 understates the heterogeneity in elasticities by size quartiles, M21 also understates the aggregate effects. The aggregate losses appear to be 3x larger!

But this isn’t the final word. First, the elasticity may not be causal. Second, we need to account for the negative effect on patent quality. When I redo the exercise using quality-adjusted patents, I find a similar aggregate loss as the original. (See appendix fn.32)

Issue 6: M21 uses a distributed lag model in Figure 5, regressing log patents on five leads and five lags of log cluster size. This is a demanding model, and M21 doesn’t discuss identifying assumptions. (More discussion here.)

There’s another coding error: M21 defines leads and lags using Stata’s _n operator, which refers to the observation number. But with an unbalanced panel, the previous observation is not the previous year. (You should use the panel operators L. and F.) When I fix the code, the estimate loses significance.

This also applies to Fig A1, which restricts to the subset of pharmaceutical patents. When I correct the code, the sample size drops to N=58, so the regression fails entirely. Moreover, the M21 code doesn’t even match the published figure: there’s no B_0 term in Fig A1, but it’s there when I run the original code. But Fig A1 is based on Equation 2, which clearly has B_0.

Issue 7: M21 Table A.8 tests whether the elasticity is larger in higher-quality clusters. There are two issues here: the code is supposed to use modal cluster size, but M21 now takes average size. And M21 doesn’t adjust for team size, when the main results use fractional attribution for coauthored patents. After correcting, the estimates are half as large.

Issue 8: M21 varies the time unit of the data, from 1-month to 3-year periods. The idea, according to M21, is to address missing observations; when inventors do not patent, they don’t have a row in the data. M21 claims this creates a downward bias at shorter time units, because more of the extensive margin effect is missing.

This interpretation is incorrect. Regardless of the time unit, there are no observations with patent=0, so no extensive margin effect can be estimated. Moreover: at shorter time units, coauthors make up a bigger share of cluster size, and patents-per-coauthor is mechanically linked to the number of coauthors. This creates a mechanical negative bias.

Issue 9: M21 claims that controlling for team size (number of coauthors on a patent) increases the elasticity. This is incorrect: patents are already measured on a per-coauthor basis, so adjusting for team size again means controlling for the same variable twice.

This is especially confused. Team size is not a confounder, but a mediator: cluster size affects patenting directly, and indirectly by enabling larger teams, where team size independently affects patenting. Controlling for a mediator means shutting down the causal pathway we’re trying to estimate.

Issue 10: When inventors do not patent in a year, they don’t have a row in the data; we know they have zero patents, but their city is unobserved. M21 interpolates these observations for gaps of 1 or 2 years, to capture part of the extensive margin effect.

There’s a coding error: for two-year gaps, only the second year of the gap is filled. Fixing this increases the elasticity, since more observations are interpolated, leading to a bigger extensive margin.

So there are a lot of issues in M21. The major ones are the event study and IV, the citation results, and the heterogeneity by cluster size. Should you believe in agglomeration effects for innovation? My advice: don’t update too much on this paper. (Although I do have ideas on how to fix it.)

I started working on this 3 years ago. I emailed Moretti about the event study in July 2023, and never got a response. But he was a referee at AER and offered some useful feedback. It would have been helpful for Coefficient Giving to get this feedback sooner!

This has happened to me several times now, where the author never responds to questions. If you publish research, and it’s actually being used to inform decision-making, please respond to replicators! (Another case was my replication of Atwood 2022 on the long-run economic effects of the measles vaccine.)

Again, read my full comment here. Please support my work by becoming a paid subscriber.

I. An incoherent theory of land values

Condon’s thesis is that rising land values are responsible for increasing housing costs:

The thesis of this volume […] is that the housing crisis is not caused by an impeded supply of homes but by the asset value of the land below them. This Rent value is inflated when higher-density homes are authorized, much to the disappointment of those who argue that the housing-affordability problem is caused by constrained supply. (p.177)

In other words, since land is an input into the production of housing, higher land values will drive up housing costs. And upzoning in particular is a primary cause of higher land values. This view is superficially plausible. As a naive empiricist, Condon has observed rising housing prices and land values coincide with upzonings. But what is the actual causal relationship underlying these correlations?

Condon thinks that upzoning drives land values, which in turn causes higher housing prices. Right away we can find a fatal flaw in this view: in First Shaughnessy, a heritage-protected mansion neighborhood in Vancouver, land values quintupled between 2006 and 2017 (before decreasing after a property tax was implemented). Over the same period, building values doubled.1

The entire neighborhood of First Shaughnessy is a heritage conservation area, so we have a clean natural experiment: very few upzonings are happening here (and any rezoned parcels are removed from the data), and yet land values are soaring. Moreover, the pattern in the rest of Vancouver is almost identical; so land values are increasing whether or not upzonings occur.2

But then what is causing the increase in land values? Condon has no answer. He could appeal to speculators bidding up parcels in anticipation of future upzonings, but when the entire neighborhood is heritage-protected, this is obviously strained. Hence, we can discard his thesis.

So what is the correct model of housing and land values?

Consider a simplified setting where land is zoned for either detached houses or six-storey apartments; developers can build houses on house-zoned or apartment-zoned land, and can build apartments only on apartment-zoned land. Upzoning means reallocating a parcel from house-zoning to apartment-zoning. This makes apartment-zoned land more abundant and hence cheaper (Figure 1b), while house-zoned land becomes scarcer and more expensive (Figure 1a). Since land is an input into housing, upzoning makes apartments cheaper and houses more expensive.3 (See here for a full writeup.)

A supply and demand model of upzoning and land prices

Condon does get one thing right: the upzoned parcel itself increases in value as it switches markets (from P1H to P2A). But this ‘own-parcel effect’ is irrelevant for evaluating upzoning. Developers make decisions based on the market-wide price, not the price of a single parcel. Upzoning reduces costs for apartment-developers, because it reduces the price of apartment-zoned land. The upzoned parcel itself was previously legally unavailable to apartment-developers, so making it available at some price is an unambiguous improvement.

In this model, housing demand is the main driver of land values. When demand to live in the city increases while supply is constrained, housing prices go up, which raises developers’ willingness-to-pay for land. This is why land values in First Shaughnessy quintupled: Vancouver became more attractive to global-elite CEO types who want to live in a mansion. Hence, the cause of the housing crisis is housing demand growing faster than supply, with rising land values being a symptom. So Condon has misdiagnosed the problem, confusing cause with effect.

We can also see the problem with Condon’s thesis by applying it more generally. His view implies that it’s never worth it to upgrade something, because upgrading just increases the cost. For example, processing iron ore into steel just means you have to pay more for the finished product. Let’s apply Condon’s thesis from above to steel:

the [steel] crisis is not caused by an impeded supply of [steel] but by the asset value of [iron ore]. This Rent value is inflated when [ore-processing is] authorized, much to the disappointment of those who argue that the [steel]-affordability problem is caused by constrained supply.

But of course, manufacturers can’t use iron ore, just as apartment-developers can’t use house-zoned land. And the increased availability of steel makes it cheaper, so manufacturers are actually better off, just as apartment-developers are better off when apartment-zoned land is cheaper.

II. A dogmatic rejection of supply and demand

In Condon’s worldview, supply and demand cannot explain the housing market:

The theory of supply and demand has been undercut by the observed economic reality […] that no matter how many new housing units a metropolitan area adds, housing prices continue to rise. (p.56)

Condon attributes the problem to land values:

[The] increase in urban land price seems to be caused by factors other than the supply and demand for new homes. The increase has to do with the innate limits on the availability of urban land and with how these limits make land perform uniquely in the global marketplace for “real” assets. (p.56)

For Condon, housing demand doesn’t affect land prices, so he needs to find some other factor to explain why land values have increased. As we’ve seen, his attempt to pin the problem on upzoning fails, and with it, his attempt to disprove supply and demand.

In fact, a supply and demand model can neatly explain the data. Demand to live in cities like Vancouver has increased, and restrictive zoning has prevented supply from keeping up. This supply and demand mismatch causes both higher housing prices and higher land values.

Condon is fond of claiming that Vancouver’s city center has tripled the stock of homes, yet prices remain very high. He is simply unable to conceive of demand being a cause of housing prices.

III. A reliance on theoretical edge cases

To see how Condon uses extreme theoretical edge cases, let’s consider his numerical example showing that upzoning merely increases land values:

Imagine a parcel of 4,000 square feet with an allowable floor surface ratio of one (FSR 1) that sells for $2 million dollars prior to rezoning. If the allowable density is doubled to a floor surface ratio of two (FSR 2), the redevelopment value increases in kind, forcing a near doubling in the value of the land to $4 million. Why?

When the city authorities a doubling of market density without requiring affordability, the residual land price goes up in response. The city gets more efficient use of the land with the new density – all good – but gets no substantial decrease in the [price] per square foot of new housing. […]

In this example, with a market sale price of $1,000 per square foot of usable interior space, the developer would calculate a land-price residual of $500 per buildable square foot because that is what is left after paying for construction (at $250 per square foot) and for fees, constants, and profits (at a total of $250 per square foot). The developer can afford only up to $500 per buildable square foot, or roughly $4 million for land (i.e., 8,000 interior square feet at $500 per buildable square foot). […] But note the land-price residual has doubled from $2 million to $4 million. And if the developer refuses to pay the $4 million, some other developer certainly will. (p.195-96)4

Note the last sentence: this entire argument depends on this tacked-on clause, though Condon does not realize it. When would a developer be willing to pay $4 million? Only when demand for housing is perfectly elastic, so the demand curve is horizontal (see Fig 2a below). In this case, shifting the supply of housing doesn’t reduce the price, because supply always intersects demand at the same price. So Condon is simply assuming that upzoning does not reduce housing prices.

If we relax this assumption, then the demand curve slopes downward (as usual), and increasing supply reduces the price (Fig 2b).5 For example, if a mass upzoning increases supply and causes housing prices to fall to $900psf, then the land price residual is now $400psf instead of $500psf. Hence, developers would only be willing to pay $400*8000 = $3.2M, instead of $4M.6 Notice what happened: land values went up and housing prices went down; but according to Condon, this should be impossible.

The effect of increased housing supply for different demand curves

Another problem is that Condon is focused on the land price of a single parcel, instead of the market-wide price of each type of land. When the parcel is upzoned, it switches from 1-FSR to 2-FSR. This makes the set of all 1-FSR parcels smaller by one, and the set of all 2-FSR parcels bigger by one. So 1-FSR land becomes more expensive and 2-FSR land becomes cheaper. But Condon completely ignores this effect, because he does not track land prices by type before and after the upzoning.

Finally, this argument doesn’t even support Condon’s thesis. His claim is that higher land values are the cause of higher housing prices, and that upzoning is the main cause of rising land values. But in this example, the goalposts have shifted: now the goal is to show that upzoning does not reduce housing prices. That’s because Condon doesn’t have a model of prices; by assuming perfectly elastic demand, he effectively takes prices as fixed, and determined outside of the model. Hence, even if we set aside the reliance on a theoretical edge case, Condon is unable to support his thesis that upzoning increases housing prices.

IV. A backwards analysis of inclusionary zoning

Condon favors inclusionary zoning (IZ) as a solution to the housing crisis, because it reduces land values. Under the ‘density bonus’ version of IZ discussed by Condon (from Cambridge, MA), developers are granted permission to build extra density in exchange for making all units in the project Affordable (that is, available at subsidized prices).

Continuing the numerical example from above, if the city imposes a housing price of $750psf on the new Affordable units, then the developer’s willingness-to-pay for land is: $750psf - $500psf = $250psf. For 8,000sf (at FSR=2), this means the developer will pay up to $2M for the parcel, compared to $4M in the market-rate upzoning scenario.

For Condon, IZ reduces housing prices because it reduces land values:

In this example, if a nonprofit housing corporation or co-op is assumed, the selling price per square foot drops from $1,000 to $750, with all of this price reduction attributable to lower land cost (p.197).

But this gets the causality backwards. The lower housing price is not ‘attributable’ to the lower land price; rather, the regulated housing price directly causes the lower land price. By artificially controlling the housing price, we reduce developers’ willingness-to-pay for land, which in turn reduces the land price. So the land price is a symptom, not a cause.7

Condon mistakenly implies that IZ works like a cost-reduction policy, by reducing the price of housing inputs (namely, land). But IZ is a non-market policy. It has no direct effect on the market forces that led to a housing price of $1,000psf.8 Using IZ to reduce land costs is like dipping a thermometer in cold water to reduce the air temperature.

This is an important point. IZ doesn’t address the question of how we make market-rate housing more affordable. Even though the vast majority of people live in market housing, Condon has no answers for them. Because he rejects supply and demand, Condon has given up on fixing the housing market, and instead wants to switch to a non-market approach.

While IZ is better than doing nothing, it is still worse than simple upzoning. With Condon’s version of IZ, the supply of market-rate housing cannot respond to increases in high-end demand, because only new subsidized units are allowed. This causes a demand cascade, where unabsorbed demand at the top of the market cascades down and increases competition for low-end homes, which in turn forces poorer residents to look for non-market housing. Hence, while IZ does increase the supply of subsidized homes, it also increases the need for subsidized housing. IZ is self-defeating.9

Original meme for context

V. A failure of value capture

And we can do land value capture in a market system. Using my upzoning example above, when we relax the assumption of perfectly elastic demand, housing prices fall from $1000psf to $900psf and the land value increases from $2M to $3.2M. Since the minimum amount the landowner is willing to accept (their reservation price) is $2M, the city could impose a windfall tax of $1.2M on the landowner. In this case, the surplus created by upzoning is split between lower market-rate housing prices and higher land values, which are taxed for public benefit. Again, higher land values are consistent with lower housing prices.10

Because Condon insists that only upzoning can cause higher land values, his approach to value capture is woefully incomplete. As we’ve seen, land values in First Shaughnessy have multiplied, despite zero upzonings. This land lift is driven by increased demand, and cannot be captured by regulations that apply only conditional on redevelopment. Condon’s fixation on upzoning has blinded him to billions of dollars in land lift that should have been captured for public benefit. The solution is to tax land values unconditionally, as with a land value tax.11

Moreover, the same focus on upzoning means that Condon prescribes taxing only new housing, while leaving old housing untouched; this makes him a favorite among NIMBY activists. In fact, we can see in the graph above that the 2019 property tax coincided with a decrease in land values; but Condon appears uninterested in raising taxes that affect all homeowners.

VI. A partial grasp of development taxes

Condon does not have a model of the land market as a whole. Instead he analyzes a single parcel, which leads to a misleading analysis of development charges:

Development taxes […] will affect the residual value of each buildable square foot of a subject parcel. When development taxes are added, the land-price residual goes down, and when development costs are removed, the land-price residual goes up. In this way, development taxes do not add to housing prices for new owners but reduce the residual value per buildable square foot of the development site, thus lowering the money that a developer is able to offer the landowner (p.197-198).12

This is true for a limited context. Specifically, when the tax amount is low enough that the project remains feasible, then the developer does pay less for land. Housing prices are set by the total quantity of homes supplied and demanded; since these haven’t changed, housing prices are unaffected by the tax. So development taxes are another way to implement land value capture.

But more generally, excessive taxes can kill projects, which reduces supply and raises prices. Obviously a tax of $100M would not merely reduce the land price residual. Instead, it would lower the developer’s willingness-to-pay below the landowner’s reservation price, so the transaction would fail and the project would be cancelled. Since the tax reduces the quantity of homes supplied, it raises housing prices. In general, to evaluate the effect of a tax on housing prices, we need to look at the change in the total number of homes.13

Condon doesn’t seem to recognize the risk of killing projects:

However, a development tax imposed late in the approval process may make the business case for a project unworkable. At that point, either the project will be cancelled or the price for the land will be renegotiated down. Municipalities are therefore in a position to moderate or eliminate land-price inflation by signalling their intention, years ahead of time, to impose a development tax. One might call this process disciplining the land market. (p.163)

Again, if projects are cancelled because a tax, then supply is reduced and housing prices are higher compared to the no-tax scenario. The same is true if the tax prevents projects from being started in the first place. Condon does not show an awareness of how taxes affect housing prices, since he is narrowly focused on minimizing land values.

It’s worth spelling out what Condon’s strategy of “disciplining the land market” means. For Condon, housing does not follow supply and demand, so reducing market-rate housing prices is off the table. Instead, our only hope is to make land cheap enough for non-market projects to pencil. This requires imposing enough costs on developers that they stop building new projects, and hence stop offering to buy land at high prices. In turn, landowners would eventually realize that their land won’t sell for a gain, and that they might as well sell it to a non-market developer with a lowball offer. Hence, we get cheap land for non-market housing.14

But as I noted above, this strategy would be a disaster. It does nothing to absorb demand increases, so demand cascades would occur continuously, raising prices and pushing more and more people into needing subsidized housing. Fortunately, Condon’s views are incorrect, and we can improve affordability by upzoning to increase housing supply.

library(VancouvR)
library(tidyverse)

# time series plot of average land and building values in First Shaughnessy

# remove upzoned parcels and character infill
  # if zoning changes, then no longer FSD
  # hence, require that zoning is FSD in all years
# use assessment date: July 1 of previous year

fsd_data <- search_cov_datasets("tax")$dataset_id |>
  map_df(\(ds)get_cov_data(ds, where = "zoning_district= 'FSD'",
         select = "pid,land_coordinate,zoning_district,neighbourhood_code,report_year,current_land_value,current_improvement_value")) |>
  mutate(n=n(),.by=land_coordinate) # count number of times a parcel appears

est_data <- tibble(Date=as.Date(c("2018-02-01")), label=c("Announced"))

png(file=paste0("shaughnessy.png"), width=750, height=400, res=90)
fsd_data |>
  # keep parcels with 18 observations as FSD
  # if >18, then multiple conversion dwelling: multiple dwellings per parcel
  filter(n==length(unique(fsd_data$report_year))) |>
  summarize(`Land value`=mean(current_land_value),
            `Building value`= mean(current_improvement_value),
            .by=report_year) |>
  mutate(Date=as.Date(paste0(as.integer(report_year)-1,"-07-01"))) |>
  arrange(Date) |>
  pivot_longer(cols=c(`Land value`, `Building value`), names_to="type", values_to="value") |>
  ggplot(aes(x=Date, y=value, colour=type)) +
  geom_line() +
  geom_point(shape=21) +
  geom_vline(data=est_data, aes(xintercept=Date), colour="black", linetype="dashed") +
  scale_y_continuous(labels=\(x) scales::dollar(x, scale=10^-6, suffix="M")) +
  scale_color_manual(values=c("Land value"="#1F77B4", "Building value"="#FF7F0E")) +
  labs(
    title="First Shaughnessy land and building values",
    x="Assessment year",
    y="Average value",
    colour=NULL,
    caption="Note: extra school tax announced in 2018, implemented in 2019."
  ) +
  guides(linetype="none") +   # removes legend entry for vline
  theme(plot.caption = element_text(hjust = 0) # left-align caption
        , legend.position = "right",
        legend.justification = "center"
        ) 
dev.off()

# same graph for rest of Vancouver
nonfsd_data <- search_cov_datasets("tax")$dataset_id |>
  map_df(\(ds)get_cov_data(ds, where = "zoning_district != 'FSD'",
                           select = "pid,land_coordinate,zoning_district,neighbourhood_code,report_year,current_land_value,current_improvement_value")) |>
  mutate(n=n(),.by=land_coordinate) # count number of times a parcel appears

png(file=paste0("non_shaughnessy.png"), width=750, height=400, res=90)
nonfsd_data |>
  filter(is.na(current_land_value)==0 & is.na(current_improvement_value)==0) |>
  summarize(`Land value`=mean(current_land_value),
            `Building value`= mean(current_improvement_value),
            .by=report_year) |>
  mutate(Date=as.Date(paste0(as.integer(report_year)-1,"-07-01"))) |>
  arrange(Date) |>
  pivot_longer(cols=c(`Land value`, `Building value`), names_to="type", values_to="value") |>
  ggplot(aes(x=Date, y=value, colour=type)) +
  geom_line() +
  geom_point(shape=21) +
  geom_vline(data=est_data, aes(xintercept=Date), colour="black", linetype="dashed") +
  scale_y_continuous(labels=\(x) scales::dollar(x, scale=10^-6, suffix="M")) +
  scale_color_manual(values=c("Land value"="#1F77B4", "Building value"="#FF7F0E")) +
  labs(
    title="Vancouver land and building values (excluding First Shaughnessy)",
    x="Assessment year",
    y="Average value",
    colour=NULL,
    caption="Note: additional school tax announced in 2018, implemented in 2019."
  ) +
  guides(linetype="none") +   # removes legend entry for vline
  theme(plot.caption = element_text(hjust = 0) # left-align caption
        , legend.position = "right",
        legend.justification = "center"
  ) 
dev.off()

2

We see a similar increase in land values in the rest of Vancouver, where upzoning happens much more frequently compared to First Shaughnessy. So upzoning cannot be main explanatory factor.

4

Condon has a meme version of this example, where the key point is that housing prices are held fixed by assumption.

By the same logic, upzoning agricultural land to single-family residential doesn’t reduce housing prices, because it just increases land values.

12

Condon also has a meme version of this argument.

The key assumption is that fees are less than the land price residual; if fees are larger, then the residual is negative and the project fails.

When we upzone land from single-family zoning to apartment zoning, we change the allocation of a city’s fixed stock of land. Upzoning makes apartment-zoned land more abundant and hence cheaper, while single-family-zoned land becomes scarcer and more expensive. Because land is an input into the production of housing, upzoning reduces the cost of building apartments, and hence is a key policy option for reducing housing costs.

In this post I’ll set up a two-sector model of land and show the effects of upzoning. (If you get stuck, try reading this post first.) House developers can use either single-family-zoned land or apartment-zoned land to produce houses, while apartment developers can use only apartment-zoned land. Since it’s more lucrative to build an apartment building than a house on a given parcel, apartment developers are willing to pay more for land; as a result, the price of apartment-zoned land is higher than the price of single-family-zoned land.

Upzoning moves a parcel from the single-family- to the apartment-zoned market. This makes apartment-zoned land more abundant, reducing its price; because land is an input into the production of apartments, upzoning reduces the price of apartments. In the extreme case, if we upzone all land, then house developers are indifferent between the two types of land, and we get an equilibrium where land prices are equal across submarkets.

A two-sector model of land

Consider a representative house developer with perfect substitutes preferences over house-zoned and apartment-zoned land. For the house developer, a parcel is a parcel, since they can build a house on it regardless of the zoning. (Parcels of different zoning types are identical in size.) Hence, they treat both types of land as interchangeable, and will choose the lower-price option. To simplify, I assume they have a fixed budget to spend; in a complete model, developer demand for the input (land) is derived from the price of the output (housing). Also note that we are considering vacant parcels with no teardown cost.

Click the footnote to see the underlying demand functions.1

The graph below shows the demand curve for house-zoned land, with the price of of house-zoned land (P_H) on the y-axis. Recall that this is an inverse demand function, so we read it as: for a given price, what quantity of land is demanded? For example, when the price is 2, quantity demanded is 12.

Here, I’m showing an equilibrium where the price of the substitute good (apartment-zoned land) is P_A=12, so when P_H<12, the house developer buys only house-zoned land and spend their entire budget on it (bottom curved segment). The horizontal segment is where P_H=P_A, so the house developer is indifferent; any quantity along this segment is equally good (and the remaining budget is spent on apartment-zoned land). When P_H>12, apartment-zoned land is cheaper, so demand for house-zoned land is 0 and the demand curve overlaps the y-axis.

For the apartment-zoned land market, we have to aggregate the demand of both house and apartment developers (see graph below). Apartment developers spend their whole budget on apartment-zoned land, regardless of how cheap house-zoned land gets. Because they have higher willingness to pay, their quantity demanded is larger at higher prices. (Note that I’m using continuous quantities, which allows buying an infinitesimally-small quantity of land; in practice, the demand curve would intersect the y-axis at some price.)

As before, house developers will spend everything on apartment-zoned land if it is cheap enough (bottom curved segment in blue). If apartment-zoned land is more expensive than house-zoned land, the demand curve is 0 and overlaps the y-axis (and they go all-in on house-zoned land). Here, the equilibrium has the substitute price at P_H=2, so when P_A=P_H=2, they are indifferent between the two, corresponding to the horizontal segment.

Adding up the sector demand curves horizontally, we get aggregate demand for apartment-zoned land (see below). In the upper-left curved segment we will get a corner solution where apartment developers buy only apartment-zoned land, and house developers buy only house-zoned land. (When P_A>2, the aggregate demand curve is equal to the apartment developers’ demand curve.) The horizontal segment is the interior solution where house developers buy some apartment-zoned land and prices are equal. (The bottom-right curved segment is the other corner solution where both developers buy only apartment zoned land and P_A<P_H; I’ll discuss this case later.)

Here’s an equilibrium for the corner solution. To maintain incentive compatibility, we require that P_A>P_H, so that the house developer chooses the cheaper option (house-zoned land). This holds when the (fixed) supply of house-zoned land is large and the supply of apartment-zoned land is small.

At these prices, all supply is demanded, so this is an equilibrium. Note that we also have a market clearing condition for supply across sectors: S_A+S_H=S=13, where S is the total stock of land. (I use S=13 to make the math work.)

Small upzoning

Let’s upzone a small amount by moving one parcel from the house market to the apartment market (so S_A increases from 1 to 2 and S_H falls from 13 to 12). This shifts S_A to the right, which reduces P_A from 12 to 6. At the same time, S_H shifts left, increasing P_H from 2 to 2.2.

Because the house developer treats the two land types as substitutes, these price changes also ‘shift’ the demand curves. (Recall from last time that this is actually a movement along the 3D demand surface.) House-zoned land becomes more expensive, so demand for its substitute (apartment-zoned land) increases; this is the shift up/right in demand for apartment-zoned land (from D1 (blue) to D2 (red)).

Similarly, apartment-zoned land becomes cheaper, so demand for its substitute (house-zoned land) falls, shown by the shift down/left in demand for house-zoned land. These changes in the demand curves occur away from the intersections with the supply curves, so the equilibria are unaffected.

So upzoning makes apartment-zoned land more abundant, and the price has to fall to clear the market. This reduces the price of land, making apartments cheaper to produce. Hence, upzoning reduces the price of apartments. (The model here shows only the input market (for land), but intuitively, decreased input prices translate to decreased output prices.)

Conversely, houses become a bit more expensive, since house developers have to compete over a more limited stock of land. This is progressive, because houses are inherently more of a luxury good than apartments (ask yourself: which has more square footage, a private yard, private parking?). (Note that upzoning does not literally remove land from house developers; instead, upzoning moves land to a submarket with higher prices, where house developers are unwilling to pay the market price.)

With this model, we can see where the “but upzoning raises land costs” argument goes wrong. By changing the allocation of land across zoning types, upzoning makes apartment-zoned land cheaper and house-zoned land more expensive. While the upzoned parcel itself increases in price (from 2 to 6), that parcel was previously unavailable to apartment developers at any price. Adding it to the apartment-zoned market makes apartment developers better off, because P_A falls from 12 to 6. The fallacy in “but upzoning raises land costs” lies in looking at a single parcel instead of the market-wide price.

Large upzoning

Next let’s explore the interior solution where P_A=P_H. Here we shift supply enough that we reach the horizontal segment of the perfect substitutes demand function. In this case, P_A=P_H=P≈2.8; this is the maximum possible reduction in the price of apartment-zoned land. We can choose S_A≥4.33 and S_H≤8.66, subject to S_A+S_H=13; here I’ve used S_A=S_H=6.5.

Since house developers are indifferent between the types of land, we can be anywhere on the horizontal segment as long as market clearing holds. Because the demand curve is horizontal, these movements don’t affect prices, and any allocation of supply is an equilibrium.

Maximum upzoning

Finally, let’s show the case where the entire stock of land is upzoned for apartments: S_H=0 and S_A=S=13. This is really the same case as the large upzoning above, because we again have P_A=P_H=P≈2.8, and the only difference is that we move all of the way to the edge of the interior solution.

Note that this is the minimum P_A achievable, since all supply is allocated to apartment-zoned land. So when S is fixed, we can’t drive P_A down to the initial P_H=2, because we have to clear the market with demand from both types of developer, whereas P_H=2 comes from a market with only house developers (and hence less total demand).

When S_H=0 we can also have equilibria with P_H>P_A, but these have the same equilibrium price P_A and quantity S_A. (With larger values of P_H, the demand curve for apartment-zoned land ‘shifts’ up, but the intersection of supply and demand is unchanged.)

Discussion

A full general equilibrium model would include the output markets for houses and apartments, and show explicitly how output prices for housing are linked to input prices for land. One interesting case to look at would be where mass upzoning reduces PH and hence further reduces house developers’ demand for house-zoned land. For example, a huge increase in the supply of apartments downtown induces people to move out the exurbs.

Another extension is the N-sector model with multiple types of apartments. For example, the N=3 case could have houses, 3-storey apartments, and 6-storey apartments. This would also shed light on how development feasibility changes over time. For example, as demand for houses increases, so does the price; this raises house developers’ willingness-to-pay for land. If demand from house developers is sufficiently higher than demand from 3-storey apartment developers, then no 3-storey apartments will be built.

Notice that in the small upzoning, the upzoned parcel increases in price from 2 to 6. This ‘land lift’ is the motivation for value capture policies that tax windfall gains from upzoning. But in the large upzoning, the land lift per parcel is much smaller: from 2 to 2.8. This shows the inherent tension in trying to both (a) improve housing affordability, and (b) collect taxes from new developments. Local governments are incentivized to do small spot-upzonings to maximize land lift2, but because this limits the stock of apartment-zoned land, it comes at the expense of making apartments more affordable. Instead of taxing land only when it is developed, a better approach is to tax land unconditionally, like with a land value tax.

Another extension is to model anticipation effects, where landowners expect upzoning to occur in the future, which raises the price of house-zoned land today.

Code for graphs here.

A supply and demand equilibrium

Consumers start with a willingness to pay for Old and New apartments. They buy the housing type with the largest difference between valuation and price: v-p. To satisfy the unit demand constraint, their demand only counts in the market of the type they choose. So one rich person could buy up all of the homes, but the unit demand constraint prevents this, by requiring them to buy their highest-valued type, and removing their demand for the other types. Here, there are five consumers, and everyone value New apartments more than Old.

I’ll show an initial equilibrium where Person 1 buys a New apartment, and Persons 2 and 3 buy Old. Because Person 1 buys New, their demand is removed from the Old market. Hence, the aggregate demand curve for Old apartments looks like this:

Since each person demands one unit, aggregate demand tops out at four. Next, here’s the supply and demand equilibrium for Old apartments. The supply curve is vertical, representing a fixed stock of Old apartments. Since the supply curve overlaps the demand curve for some segment, technically the equilibrium is not well-defined. However, this is not a big deal, because we can pretend the supply curve is slightly sloping upward, so the intersection is unique (though this brings in continuous quantities). Alternatively, if we had a large number of consumers, the demand curve would be a straight line that intersects supply uniquely; working with discrete quantities is just a pain.

So Persons 2 and 3 each get an Old apartment, and the price is 2. Note that Persons 4 and 5 are initially priced-out, and don’t get a home. We can interpret this as having to move away, or living in their car (or being homeless).

Next we have the aggregate demand curve for New apartments. Here we show Persons 1, 4, and 5, because Persons 2 and 3 buy Old apartments, so their demand for New is removed (by the unit demand constraint).

In the initial equilibrium, only Person 1 gets a new apartment, and the price is 4. The supply curve is vertical, representing zoning constraints: developers cannot build any more, even if prices went up. Note that Persons 4 and 5 are priced out here as well.

Vacancy chains

To show a vacancy chain, we increase the supply of New apartments. This will allow Person 2 to upgrade, vacating their Old apartment, which in turn allows Person 4 to get a home (instead of being homeless).

The higher supply of New apartments reduces the price, as we move down the demand curve. This lower price induces Person 2 to re-evaluate their housing choice. Their willingness to pay is 3.7 for New apartments, and 2.2 for Old. The price for New apartments was 4, so they chose Old. But now the price is 2, so their payoff for a New apartment is 3.7 - 2 = 1.7, and the payoff for an Old apartment is 2.2 - 2 = 0.2 Hence, the New apartment is a better deal, so they switch markets from Old to New.

Person 2 upgrading means that we also change which market their demand contributes to. So we subtract their demand from the Old demand curve, and add it to the New demand curve, shifting it to the right. For New apartments, this pushes up the price. Note that we can end up out of equilibrium if this increase in price is not consistent with Person 2’s willingness to pay. Here, I’ll use the modelling flexibility we have from the supply and demand curve overlapping to choose a new equilibrium price of 3.

Since Person 2 upgrades, we remove their demand for Old apartments, shifting it left. Now Persons 3 and 4 are able to get an Old apartment, and the price falls to 1.6. (That is, supply intersects demand on the section of the demand curve where Persons 3 and 4 buy.) This is the vacancy chain in action: Person 4 was initially priced out and didn’t have a home, but by adding a New apartment, we have freed up an Old apartment for Person 4 to live in!

To make sure this is an equilibrium, let’s check the payoffs. Person 1 is happy to buy New, because they get 4.7 - 3 = 1.7 from New and 2.6 = 1.6 = 1 from Old. Person 2’s payoffs are 3.7 - 3 = 0.7 (New) and 2.2 - 1.6 = 0.6, so they buy New. Person 3 gets 2.2 - 3 < 0 for New and 2 - 1.6 = 0.4 from Old, so they buy Old. So each consumer’s choice is consistent, hence, these prices define an equilibrium.

Note that with unit demand, we are able to have consumers being priced out (like Person 4 initially). With continuous quantities, this was not possible, since they could always buy some infinitesimal quantity, and so can’t be priced out completely. But with discrete units, Person 4 initially doesn’t get anything, then thanks to the vacancy chain, they get an Old apartment.

Demand cascade and yuppie fishtank

Now let’s see how demand cascades and yuppie fishtanks work in the unit demand case. Since people can be priced out entirely, demand cascades will push people out of old apartments, and yuppie fishtanks will allow them to avoid being pushed out.

We’ll use the same willingness to pay as before, but this time we’ll start with Person 1 living outside of the city. Person 2 starts in a New apartment, so they don’t contribute to demand for Old apartments. That leaves Persons 3, 4, and 5 making up the aggregate demand curve for Old apartments.

There are two Old units, so the equilibrium has Person 3 and 4 each getting an Old unit at a price of 1.6.

For New apartments, Persons 2 and 5 contribute to the demand curve.

But there’s only one New unit supplied, so only Person 2 gets a New unit in the equilibrium. The price is 3, which ensures that everyone is choosing their best option. Person 5 is priced out of both housing types, and has to live in their car.

The demand cascade starts with Person 1 moving in and bidding for a New apartment. This shifts the demand curve to the right (from D1 to D2a), increasing the price from 3 to 4.

Since Person 1 is richer, they outbid Person 2 for the New apartment, so now Person 2 downgrades to an Old apartment. Hence, we remove their demand from the New demand curve; this does not affect the new equilibrium. Now only Persons 1 and 5 contribute to the aggregate demand curve.

Turning to Old market, we add in Person 2’s demand, shifting the demand curve to the right. This pushes up the price, from 1.6 to 2. Now Persons 2 and 3 get an Old apartment, and Person 4 is priced out and doesn’t have a home. This is the demand cascade: when high-income Person 1 moves in, they bid up the price of new apartments, leading Person 2 to downgrade to an old apartment, which leaves Person 4 being priced out. In short, when supply is restricted, rich people moving in forces poor people to move out.

But what if supply is not restricted, and grows to match the increase in demand? This is the yuppie fishtank: by building a new glass tower to contain Person 1, we absorb their demand and protect the old apartments from competition. So we shift supply from 1 to 2, reducing the price.

This lower price induces Person 2 to upgrade back to a New apartment. So the demand curve shifts right, and the price is 3. This is the same as the initial equilibrium: the yuppie fishtank has absorbed Person 1’s demand, keeping prices flat.

And since Person 2 is not competing for an Old apartment, the demand curve shifts left. Thanks to the yuppie fishtank, Person 4 gets to stay in their Old unit, and isn’t priced out.

Notice that a yuppie fishtank can cancel out a demand cascade caused by a rich newcomer, while a vacancy chain triggered by a rich local upgrading is an inverse demand cascade.

See here and here for code to produce the graphs.