Function Reference

A Country-wide Parameter struct

Parameter Struct

This struct defines the parameter type for the model. It reads default values from the file src/params.json. Those values can be overridden by supplying a Dict to the constructor with key => value pairs:

p = LandUse.Param(par = Dict(:η => 0.1))

creates a Param instance with agglomeration forces set to 0.1

Region type

Urban expansion model with flexible commuting cost and differential supply elasticity. This represents a single region. A region contains one potential urban area, and a rural area. A Region can be part of a wider Country.

Urban Model

This model has a fixed price of land ρr in the rural sector. Differently to our baseline model, where ρr is a GE object.

convert a country param to one param for each region

Population Density at location $l$

Population Density at location $l$. second version, independent of Lu

Computes the entries of the residual vector $u$

Eqsys!(F::Vector{Float64},m::Region,p::Param)

compute system of equations for the general (with flexible ϵ).

Eqsys!(F::Vector{Float64},m::Urban,p::Param)

compute system of equations for the general (with flexible ϵ).

GDP

housing supply at location $l$

Country Rural Market Clearing

Rural Market Clearing

Residential Land in Rural sector. equation (20)

Production of Rural Good

Production of Urban Good

cost of construction at location $l$

compute_ξw()

Computes input parameter ξw directly from data. Notice that $1 - \xi_w$ is the elasticity of speed to wage income. We measure from individual commuting data in ENL (Enquete National de Logement) an increase in commuting speed of about 11% from 1984 to 2013. This function returns

$\xi_w = 1 - \frac{11}{\%\Delta \theta_u}$

where $\%\Delta \theta_u$ stands for the percentage increase in urban wage from 1984 to 2013.

copy dropbox data to in/ folder

optimal rural good consumption at location $l$. Equation (7) divided by p

single spatial cross sections region

optimal urban good consumption at location $l$. Equation (8)

Amount of Urban Good (numeraire) required to build housing at $l$

urban good consumption in rural sector. Equation (8)

commuting distance

dash(it;par = Dict())

Helper function for quick Region dashboard.

dashboard(C::Vector{Country},it::Int)

Dashboard for a Country in period it

dashboard(M::Vector{Region},p::Param,i::Int; objvalue = nothing)

Model dashboard for a single Region.

dashk20(;save = false)

Produce and optionally save model output for 20 city version.

Obtain a Time Series from an array of Country as a DataFrame

Obtain a Time Series for a single region as a DataFrame

Estimate exponential model on spatial density.

Compute exponential decay model of density for each region in a country

normalize a vector wrt it's first element

FOC of rural firm wrt labor Lr

FOC of rural firm wrt land Sr

Get Fringe from indifference condition

At the fringe $\phi$ we have the condition

\[w(0) - \tau(\phi) = w_r\]

which can be rearranged to obtain a map from $w(0) - w_r = \tau(\phi)$.

The function takes $w(0) - w_r$ as argument x. then we give $\tau(\phi)$ to its inverse function to get back $\phi$

i0()

interact with the baseline single city version of the model. This basically calls dashboard(M::Vector{Region},p::Param,i::Int) after creating p from a set of user inputs.

i0k()

interact with the 2 city model.

integrate!(m::Region,p::Param)

perform numerical integration using gauss-laguerre.

integrate!(m::Urban,p::Param)

could be made much faster by filling a matrix col-wise with integration nodes and doing a matmul on it? have to allocate memory though.

inverse commuting cost. cost x → location. Notice we don't consider that cost is zero beyond ϕ: we want to find ϕ here to start with.

iobj()

Similar to i0 but prints the value of the objective function into one of the panels.

checks whether a certain row of the grid together with a starting value x0 produces a valid solution in all periods.

plot $\phi_k$ vs $L_{u,k}$ for all regions $k$ and different eps slopes. relates to https://github.com/floswald/LandUse.jl/issues/10

measure increase in land value at fringe. by default starts in 1960 and measures increase up to 2020.

https://github.com/floswald/LandUse.jl/issues/11

https://github.com/floswald/LandUse.jl/issues/15

https://github.com/floswald/LandUse.jl/issues/36

1. same growth in sectors i. high cbar vs low sbar: show implied city density time series to see that only that config works ii. show falling housing spending share as well
2. implications of growth in either sector only
3. identify commuting cost params by matching time series data

run model with flat epsilon

https://github.com/floswald/LandUse.jl/issues/59

run model with agglomeration forces

https://github.com/floswald/LandUse.jl/issues/60

run model with congestion forces

https://github.com/floswald/LandUse.jl/issues/64

run multi city with congestion

https://github.com/floswald/LandUse.jl/issues/66

five city cross section

https://github.com/floswald/LandUse.jl/issues/67

plot $\phi_k$ vs $\L{u,k}$ for all regions $k$. relates to https://github.com/floswald/LandUse.jl/issues/9

https://github.com/floswald/LandUse.jl/issues/21

https://github.com/floswald/LandUse.jl/issues/22

solve a Country at point x0

keywords:

• estimateθ: whether the sequence of $\theta_u$ should be a choice variable
• solve : solve the model (true) or return the model object for debugging?
• fit_allyears : try to match the population distribution in all available years or just a single year.

solve model at current Param p and starting at point x0

Keyword:

• estimateθ: whether the sequence of $\theta_u$ should be a choice variable or not. default false.

Produces output from 20-city model. In particular comparison with single city case, and extension with d1, d2.

5 country case

print default param to latex table

normalize a vector wrt it's maximal element

Runs K regions as a country and produces 2 plots

you just had a failed start at p

objective(x; moments = false, plot = false, save = false, fname = "moments")

moment objective function for an optimizer

objective1(;save = false)

run objective function at default parameter

produces all output from the model needed to compile the paper.

This uses current baseline parameters defined in params.json

p2Flux(p::Param)

map param to Flux Chain to get starting value

p2x(p::Param)

map param to x for objective function quick eval this is the inverse of x2dict.

aggregate per capita income

plot densities for all years

plotk20(;save = false)

Produce model output for 20 city version.

load population and area data

find closest year in population data to model years

prepare_data(p::Param; digits = 9)

Takes raw csv data and prepares to be used in model. writes to csv input files.

Smoothing of Productivity Series

We take the following steps to obtain a smoothed series for $\theta_r$ and $\theta_u$:

1. We obtain the estimated series at annual frequency.
2. We subset both series to start in 1840 and end in 2015 (rural productivity ends in that year)
3. We linearly interpolate the missing interwar years.
4. Smoothing is done via the [smooth] function from the QuantEcon package: we use the default Hann window and a 15-year window size. We experimented with the window size until high-frequency oscillations disappear.
5. Our rural productivity series gets very volatile from 2000 onwards, and in fact one would find a decreasing rural productivity series if we applied our smoother to post 2000 data. Therefore from the year 2000 onwards, we grow the smoothed series forward with 1% annual growth. Given that 2000 is very close to the final new steady state of the model, the actual choice of growth rate has only a small impact on our results.

house price function at $l$. equation (12)

rural house price from land price

rel_cityarea(m::Model)

Computes city area relative to total rural land use. This is the model moment related to the data moment artificialized land relative to agricultural land in 2015. We measure this to be 18.3% in the data.

return model relative population in each year to largest city largest city is city 1

run Single region model for all time periods

run Single region model for all time periods

runestim(;steps = 1000,fname = "moments")

Run the default differential evolution optimizer from BlackBoxOptim.jl

run Multi-region model for all time periods starting from the single city starting value.

collect valid starting values for the k country case by reusing the first period solution of each preceding step in θu along the sequence

startvals_k()

Find feasible starting values for multi city case (2 to 5 cities) and write to disk. By default return the saved dict with start param and initial guess vector x0.

helper function to prepare country param

returns a dict of empirical targets for the model

ts_plots(M,p::Param;fixy = false)

Time series plots for a single Region M.

update!(c::Country,p::Vector{Param},x::Vector{Float64})

Update a Country object with a current vector of choice variables x supplied by the solver. The ordering in x is:

1. b: ratio of labor to land in region 1
2. r: land rent
3. pr: relative price rural good
4. 4 : (K+3), Sr: amount of land use in rural production (complement of Srh)
5. (K+4) : (2K+3), Lu: urban pop in each k
6. (2K+4) : end, θu: urban prod in each k

update!(m::Region,p::Param,x::Vector{Float64})

update a single Region with a Param at choices x.

update!(m::Urban,p::Param,x::Vector{Float64})

update a single region a parameter vector at choices x.

Utility function equation (5)

wage at location $l$

rural wage from indifference condition at ϕ. Eq (11)

urban wage at location $l$

urban wage at center

x2dict(x)

map vector x to dict for Param

excess subsistence rural worker

excess subsistence urban worker

disposable income: GDP net of commuting costs

land price function at $l$. equation (15)

commuting cost: location x → cost

housing supply shifter at $l$

housing supply elasticity at $l$