Let's just say that economists don't really agree with this thesis, and most city-level empirical evidence points in the opposite direction. E.g.
We ultimately conclude, from both theory and empirical evidence, that adding new homes moderates price increases and therefore makes housing more affordable to low- and moderate-income families.
[...]
A few studies use panel data and find that the imposition of more stringent land-use controls
leads to lower supply and higher prices. Jackson (2016) uses longitudinal data from California’s
cities to assess the effect a city’s adoption of additional land-use regulations has on the number of
new construction permits issued, and finds that each additional land-use regulation adopted
reduced multifamily and single-family permits by an average of more than 6% and 3%, respectively,
and that regulations reducing allowable density had even larger effects. Zabel and Dalton
(2011) use longitudinal data from localities in Massachusetts and find that increases in minimum lot
sizes are followed by significant increases in prices. Looking at longitudinal data on municipalities
in the Boston (Massachusetts) metropolitan area, Glaeser and Ward (2009) find that the adoption of
stricter local regulations leads to higher house prices, but the coefficient falls in magnitude and
loses significance once they control for population demographics. They point out that this is
expected, if homes in other jurisdictions are seen as perfect substitutes. Thus, whereas supply
restrictions may increase prices in a market as a whole, they may not increase them disproportionately
in the particular locality where they are imposed due to spillover effects across jurisdictions.
Several other researchers use instrumental variables to try to more clearly assess the causal
effects regulatory restrictions have on housing supply and prices. Ihlanfeldt (2007) uses such an
approach to study regulation in localities in Florida and finds that predicted regulations significantly
increase the price of single-family homes. Saks (2008) uses instrumental variables and shows
that increases in labor demand lead to less residential construction and larger increases in housing
prices in metropolitan areas with more restrictive housing supply. Hilber and Vermeulen (2016)
show that changes in demand lead to increases in local house prices rather than increases in
supply in municipalities in England with greater regulatory restrictions, measured by the refusal
rate of proposed residential projects and the number of project approvals delayed more than
13 weeks.
In sum, the preponderance of the evidence shows that restricting supply increases housing
prices and that adding supply would help to make housing more affordable. [...] although it is surely true that land is constrained, especially in certain
markets (Saiz, 2010), land can be used more intensively to allow for more housing. The limits on the
land with which housing is bundled make housing different from many goods, but the difference is
one of degree: the supply of housing can and does increase even in constrained markets, and
prices should generally fall in response (see the review by Dipasquale, 1999; Mayer & Somerville,
2000).
[...] most housing filters down, or loses
value as it ages, representing new supply in submarkets at lower price points. [...] Indeed, recent research shows that filtering
was the primary source for additions to the affordable rental stock between 2003 and 2013,
whereas new construction was the largest contributor for the higher priced rentals, and tenure
conversion was the largest source for moderately priced rentals (Joint Center for Housing, 2015, fig.
14). Further, Weicher, Eggers, and Moumen (2016) report that 23.4% of the rental units that were
affordable to very low-income renters in the United States in 2013 had filtered down from higher
rent categories in 1985. Another 21.8% were conversions from formerly owner-occupied homes or
seasonal rentals.9 Most of the higher priced rental units that filtered down to become affordable in
2013 were moderate-rent units in 1985, but 15% of those that filtered down were high-rent units in
1985. Note that filtering occurs over a shorter time frame too; among affordable units in 2013,
19% had been higher rent units as recently as 2005.
Recent research analyzing the incomes of successive occupants of homes also suggests substantial
downward filtering, particularly of the rental stock due to tenure conversion; as the owner-occupied
stock ages, a portion converts to rental (Rosenthal, 2014). Rosenthal also finds, however,
that filtering rates are considerably lower in areas with high house price inflation, although
downward filtering still occurs.
In short, new construction is crucial for keeping housing affordable, even in markets where
much of the new construction is itself high-end housing that most people can’t afford. A lack of
supply to meet demand at the high end affects prices across submarkets and makes housing less
affordable to residents in lower-cost submarkets.
Finally getting to the discussion of the precise thesis you ask about:
Some skeptics argue that even if additional supply could help make housing more affordable in the
short run, it won’t in the long run because the additional supply will induce more demand, especially
among buyers or renters wealthier than the existing residents in the neighborhood (Redmond, 2015).
The claim is analogous to the argument that building more highways will not reduce congestion
because the lower cost of travel will simply cause more people to drive or to take that particular route
(Gorham, 2009). In this case, the argument is that by making the jurisdiction more affordable, adding
housing supply will attract new demand—both from current residents who would otherwise leave,
and from people living elsewhere who will now choose to move to the jurisdiction. Further, the argument goes, lower rents and prices may also induce latent demand—people who are living with
roommates or family members may choose to form their own households (Ellen & O’Flaherty, 2007) or
people may choose to invest in pied-à-terres in a city. That additional demand will drive prices back up
until supply can again respond, causing housing to be more affordable, at best, only cyclically,
according to the argument, and increasing the density of the jurisdiction, with the attendant costs
of congestion.
Although building additional highways does appear to induce more demand (Duranton &
Turner, 2011), in the case of housing, additional demand is unlikely to completely offset the new
supply. Such an offset requires demand curves to be perfectly elastic—or, in other words, it
assumes that neighborhoods and jurisdictions are perfect substitutes and that there are no
constraints on the ability and willingness of households to move. That is unrealistic. Moving
homes is not like driving a few extra miles (Lewyn, 2016), and costs associated with moving may be
high. Any additional demand induced by new housing is limited by personal and economic
constraints on the ability and willingness of households to move, restrictions on immigration, and
uncertainty and other factors that might inhibit renters and buyers from renting or buying in the
market in which housing supply increases. Indeed, mobility rates have fallen sharply over the past
several decades, and although the reasons for the decline are being debated, the decline reveals
significant constraints on the ability and willingness to move.
Thus, in the long run, whereas some additional households may be drawn from outside (or from
within the city) to buy or rent homes as supply increases, it is highly unlikely that prices will end up
at the same level that they would have reached absent any new supply. Finally, as noted above, the
empirical evidence shows that allowing more supply leads to lower housing prices; if adding
supply induced sufficient additional demand to offset the increased supply, the studies would
not find an association between supply and prices.
It can happen indeed in some very limited circumstances, and with localized [e.g. neighborhood] effect, which is probably on what this faulty generalization is based on:
Many renters in neighborhoods where market-rate housing is proposed express concern that the
construction of new housing will actually make their affordability problems worse by raising rents or
house prices, fueling gentrification, and potentially displacing existing residents (Atta-Mensah, 2017;
Savitch-Lew, 2017). Hankinson (2017) theorizes that renters’ opposition to local additions to supply is
driven by such worries; he argues that it is plausible that the construction of an attractive new building
will increase prices locally (by improving the physical landscape, bringing new amenities to the
neighborhood, and signaling that the neighborhood is improving), even as it reduces them citywide.
Testing this proposition empirically is quite challenging, given that developers will naturally be
attracted to areas where prices and rents are rising. There is evidence that improvements to
blighted housing can, in some circumstances, increase surrounding property values, even when
the new or improved housing is subsidized, low-income housing (Diamond & McQuade, 2016;
Schwartz, Ellen, Voicu, & Schill, 2006). The new housing studied, however, typically replaced
vacant, abandoned buildings and littered vacant lots, in essence removing a disamenity. [...]
There is little empirical evidence about the net effect new market-rate housing has on the prices or
rents of nearby homes, and what exists may not be causal. One recent study examines the effect of
market-rate single-family homes newly constructed on infill sites, and finds that newly constructed
single-family homes can have positive impacts on the sales price of other single-family homes nearby,
but the effect varies with context (Zahirovich-Herbert & Gibler, 2014). A study of multifamily high-rise
infill developments in Singapore found positive price effects on nearby houses (Ooi & Le, 2013), as did
a study of single multistory apartment buildings constructed in Helsinki (Kurvinen & Vihola, 2016).
These studies all consider property values and not rents, and none is able to prove a causal relationship
given that market-rate developments aim to target neighborhoods where they expect property
values to improve. Unfortunately, we found no study examining impacts on rents, although one study
by the California Legislative Analyst Office concluded that additional market-rate construction is linked
to lower displacement rates (Taylor, 2016). Examining low-income neighborhoods in the Bay Area
between 2000 and 2013, these researchers found that the production of market-rate housing was
associated with a lower probability that low-income residents in the neighborhood would experience
displacement. Although a singular study, the findings suggest that for neighborhoods in high-demand
cities, blocking market-rate construction may place greater pressures on the existing stock.
In short, although it is clear that the construction of new homes will moderate price and rent
increases citywide, neither theory nor empirical evidence provides clear guidance about when
localized spillover effects might occur and when they might actually cause an increase in the prices
and rents of immediately surrounding homes.
There's a more recent study on the core issue of the question, not covered in the above review:
Preliminary results using a spatial difference-in-differences approach suggest that any induced demand effects are overwhelmed by the effect of increased supply. In neighborhoods where new apartment complexes were completed between 2014-2016, rents in existing units near the new apartments declined relative to neighborhoods that did not see new construction until 2018. Changes in in-migration appear to drive this result. Although the total number of migrants from high-income neighborhoods to the new construction neighborhoods increases after the new units are completed, the number of high-income arrivals to previously existing units actually decreases, as the new units absorb a substantial portion of these households. On the whole, our results suggest that—on average and in the short-run—new construction lowers rents in gentrifying neighborhoods.
See also a blog discussing it (which also does a good job at debunking the roads analogy); in summary, the gist of the latter study is in this chart:
This chart shows that rental prices for apartments close to the new building fell relative to the prices for apartments located further away. The dashed “before” line has a negative slope, suggesting that prices declined the further you got from the site of the new building. The solid “after” line has a positive slope (prices increase the further you get from the new building). Overall, prices are higher (the solid line is above the dashed line), but prices actually went down next to the new building, and increased far less than in the area further away from the new building.
These data are a strong challenge to the induced demand theory. If a new building made an area more attractive, one would expect the largest effect in the area very near the building. But, consistent with the traditional “more supply reduces rents” view, the addition of more units in an area seems to have depressed rents (or at least rent increases) compared to buildings in the surrounding area.
In other words, even the theory that prices might increase locally in the neighborhood of a new building doesn't hold up, on average, to a difference-in-differences test. Without a difference-in-differences approach one can of course draw the wrong conclusion since overall demand in a city may be increasing faster than the overall supply.
Even more recently, another paper on NYC (not yet published in a peer-reviewed venue, but was mentioned in a recent NYT article):
I find that for every 10% increase in the housing stock within a 500-foot buffer,
residential rents decrease by 1%. The rent reduction is caused by the completion of new
high-rises rather than their approval. Across neighborhoods, the impact is smaller in more
central areas, presumably due to more elastic demand. Within neighborhoods, the impact
is smaller for lower-rent buildings. Finally, the negative impact appears to be driven by
supply effects rather than dis-amenity effects, like changes in neighborhood physical features,
blocked views, or shadows. New housing units alleviate the growing demand for existing
housing units and moderate the rapid growth of residential rents.
Residential property sales prices also decrease when new high-rises within 500 feet
are completed. [...] Because 99% of new high-rises are condos and rental
buildings, they do not significantly affect sales prices of co-ops and 1-5 family homes. The negative impact is bigger for closer substitutes, further confirming the negative impact is due to supply effects.
To address the hypothesis about amenity effects, I find new high-rises and their
high-income tenants attract new full-service restaurants, cafes, and coffee shops. These consumption amenities likely make neighborhoods more attractive and potentially increase rents
and sales prices (Couture and Handbury, 2017). However, the amenity effect is dominated by the supply effect, given that rents and sales prices still fall on net.
As was mentioned in a OP's comment below, living in a big city vs a small town (and vs the prairie) is an amenity effect. However, there seems to be a limit to this effect after which more housing supply lowers the (rental) price.
The amenity effect has been studied for decades but alas often in isolation, most prevalently perhaps with the the Roback model, which basically adapted Rosen's more general spatial equilibrium model to "nontraded" goods like housing. ("Nontraded" because Roback only considered effects on rents.) Using this model, one essentially gets (rental) price-amenity gradients. Roback assumed that "the cost of changing residences is zero" (i.e. costless migration). Modelling "partially open" cities, i.e. with non-zero (and non-infinite) relocation costs across its boundaries is a relatively more recent endeavour. Likewise for considering the combined effect of housing supply and amenities on rents. As one such theoretical work notes
as housing is supplied more elastically [...] the rent effect of amenities approaches zero.
