Week 11 · What wages and housing costs reveal about amenities
Read: Brueckner, Chapter 11; Albouy (2016).
How do you put a dollar value on sunshine, a coastline, or low crime? You can’t buy them directly, but you can see what people give up to have them. That is the idea behind quality-of-life rankings in urban economics.
Return to spatial equilibrium across cities (Why Cities Exist). If workers can move freely, then in every city
\[ w_i - r_i + a_i = \bar{V}, \]
where \(w_i\) is the wage, \(r_i\) the cost of housing, and \(a_i\) the dollar value of the city’s amenities. Rearranging gives the value of amenities relative to other cities:
\[ a_i - a_j = (r_i - r_j) - (w_i - w_j). \]
A city has valuable amenities if people pay more for housing than its wages justify, or accept lower wages than its housing costs require. This framework comes from Roback (1982) and is the basis of modern quality-of-life indexes.
The same data reveal two different things. High wages and high housing costs mean firms value the place: it is productive. High housing costs with low wages mean households value the place: it is pleasant. Real indexes refine this with the share of income spent on housing, taxes, and the cost of other goods, but the logic is the same.
Enter yearly wages and housing costs for two cities, and see what the model implies about how residents value their amenities.
dA = (rA_ - rB) - (wA - wB) // amenity value of A relative to B
Plot.plot({
height: 220, marginLeft: 210,
x: {label: "Dollars per year: City A minus City B →", grid: true},
y: {label: null},
marks: [
Plot.ruleX([0]),
Plot.barX([
{k: "Higher housing cost in A", v: rA_ - rB},
{k: "Higher wage in A", v: wA - wB},
{k: "Implied amenity value of A", v: dA}
], {x: "v", y: "k", fill: d => d.k.startsWith("Implied") ? "#881124" : "#999"})
]
})Albouy (2016) uses wages, housing costs, and land values across U.S. metro areas to measure, in dollars, both how productive each place is and how much its residents value its quality of life, and then combines them into the total value of a city’s amenities. The paper turns the simple equation above into a full accounting of what places are worth. As you read, notice how it handles land values, the cost of goods other than housing, and taxes, and think about how each choice could change a city’s ranking.
The model treats every move as a free choice among cities. In practice, people move where they know people. Bailey et al. (2018) show that places linked by past migration remain socially connected for generations, and that connectedness predicts future migration. A city’s “outside option” therefore depends on who you know elsewhere, which helps explain why some places with similar wages and amenities attract very different numbers of newcomers.
Jennifer Roback’s 1982 framework built on work by Sherwin Rosen, who showed in 1979 that wage differences across cities could be used to value climate and other local conditions. Their combined approach, often called the Rosen–Roback model, underlies many of the “best places to live” rankings you see today, even when the rankings do not say so.