Question

Using the Roback model, explain who bears a state or local property tax that confers no benefits. Explain what happens if the benefits of the goods supported by the tax accrue to labor; to capital investment.

Answer 1

Rosen-Roback model typically assumes (among other things) the
following:
• Labor is perfectly mobile
• Land is fixed
• Workers only care about nominal wages, cost of housing, and
local amenities
Shocks to the demand for and supply of labor translate into
housing prices.
We think that labor isn’t perfectly mobile and that land isn’t
completely fixed. So what happens then?

Imagine two cities (among many), A and B. Each produces a good
that is sold “internationally” and the price is the same everywhere.
In Rosen-Roback worker i in city c has the following utility:
Uic = wc − rc + Ac
where w is the nominal wage, r is the cost of housing, and A are
all the nice (or bad) things in city c.
We want to limit the elasticity of labor, i.e. the ability or
willingness of workers to move across cities. Any ideas?

Very simply, we can give workers preferences between cities:
Uic = wc − rc + Ac + eic
where e gives worker i’s preference for city c.
Then relative preferences between our cities A and B are given by
eia − eib.
These preferences are distributed across all workers, so we need to
specifiy this distribution:
eia − eib ∼ U[−s,s]
What does s control?
So the model tells us that the decision to move between cities is a
function of:
• Nominal wages
• Housing costs
• Amenities
• Individual preferences
Given this we know a worker will choose to live in city A iff
eia − eib > (wb − rb) − (wa − ra)
| {z }
Difference between real wage
+ (Ab − Aa)
| {z }
Difference in amenities
Now we can begin to discuss the labor market.

I want to determine the wages in both cities and the number of
workers in both cities where N = Na + Nb. The condition we want
is that the marginal worker is indifferent between cities. So some
workers will not be indifferent, i.e. they will have a preference for
their own city.
The local labor supply curve is then upward sloping
wb = wa + (rb − ra) + (Aa − Ab) + s
Nb − Na
N
You can think of s as regulating how much the number of workers
in city B changes in response to a change in the wage in city B.
We also need to specify labor demandAssume that firms in each city use a Cobb-Douglas production
function to generate output:
log ya = Xa + hNa + (1 − h)Ka
In a competitive economy factors are paid their marginal product
so that wages in city B are given by:
wc = Xc + (h − 1)Nc + (1 − h)Kc + log h
Now we have supply of labor and demand for labor, but we are
also interested in the cost of housing.Housing demand is just a rearrangement of the labor supply
equation:
rb = (wb − wa) + ra + (Ab − Aa) − s
Nb − Na
N
This equation gives us how the price of housing is related to the
number of workers, but we need to now how the supply of housing
varies with demand:
rb = z + kbNb
This is not a structural equation, it is a reduced form attempt to
acknowledge that the elasticity of housing supply k varies across
cities.

We solve for the equilibrium in the labor market by equating labor
supply and demand and in the housing market by equating housing
supply and demand.
Doing that would be annoying. Instead we will analyze several
extensions and experiments:
• Shock to labor demand
• Economies of agglomeration

Shock to labor demand
Assume that in the “next” period productivity in city b increases:
Xb2 = Xb1 + ∆
Let’s examine the effect under some special cases:
• s = 0: housing supply inelastic
• s = ∞: labor immobile
• s = 0: labor completely mobile, housing supply elastic
• kb = 0: housing supply completely elastic

Inelastic housing supply
All of the increase goes to land owners (becomes absorbed in
housing prices). So Rosen-Roback is a special case of this model.
Now let’s examine what happens if labor is immobile (s = ∞).

s = ∞: Labor immobile
• Workers in B benefit.
• Housing prices unchanged (no one moves), so landowners see
no difference.
Now let’s see what happens when labor is perfectly mobile ( s = 0
) and housing supply is elastic.

Evidence seems to indicate that firms and workers in cities are more
productive and that there are benefits to being close to each other.
Why might this be true?
• Thick labor markets
• Thick markets for intermediate goods
• Knowledge spillovers
• other things?

Why might productivity increase if there are lots of firms offering
lots of jobs and lots of workers looking for jobs?

• Firms can invest in special technologies
• Workers can invest in human capital
This has some empirical implications:
• Should see more turnover for young workers in dense areas
• Should see less turnover for older workers in dense areas
We tend to see these results, but evidence is not definitive.

Thick markets for intermediate goods work in a similar way:
• Special repair services (airplanes, trains, sophisticated
scientific equipment)
• Financial services like capital financing, accounting
• Legal services
• Software engineering
Evidence indicates that firms located near similar firms make more
intensive use of specialized inputs than similar firms located far
away from other firms.

An annual Local Property Tax (LPT) charged on all residential properties in the State came into effect in 2013. The LPT is collected by the Revenue Commissioners.

If you own a residential property in the State, you are liable for payment of the tax. (This includes local authorities and social housing organisations.) See ‘Who is liable to pay LPT’ below.

Residential property is any building or structure (or part of a building) which is used as, or is suitable for use as, a dwelling and includes grounds of up to one acre. The LPT does not apply to development sites or farmland.

The tax payable is based on the market value of relevant properties. The LPT is a self-assessment tax so you calculate the tax due based on your own assessment of the market value of the property. Revenue does not value properties for LPT purposes but provides guidance on how to value your property – see ‘Valuing your property’ below.

Revenue offers a range of methods for paying the tax.

An annual Local Property Tax (LPT) charged on all residential properties in the State came into effect in 2013. The LPT is collected by the Revenue Commissioners.

If you own a residential property in the State, you are liable for payment of the tax. (This includes local authorities and social housing organisations.) See ‘Who is liable to pay LPT’ below.

Residential property is any building or structure (or part of a building) which is used as, or is suitable for use as, a dwelling and includes grounds of up to one acre. The LPT does not apply to development sites or farmland.

The tax payable is based on the market value of relevant properties. The LPT is a self-assessment tax so you calculate the tax due based on your own assessment of the market value of the property. Revenue does not value properties for LPT purposes but provides guidance on how to value your property – see ‘Valuing your property’ below.

Revenue offers a range of methods for paying the tax.