0

I have :

  • a set of N locations which can be workplace or residence
  • a vector of observed workers L_i, with i in N
  • a vector of observed residents R_n, with n in N
  • a matrix of distance observed between all pair residence n and workplace i
  • a shape parameter epsilon

Setting N=3, epsilon=5, and

d = [1 1.5 3 ; 1.5 1 1.5 ; 3 1.5 1] #distance matrix
L_i = [13  69  18] #vector of workers in each workplace
R_n = [27; 63; 10]

I want to find the vector of wages (size N) that solve this system of N equations, enter image description here

with l all the workplaces.

Do I need to implement an iterative algorithm on the vectors of workers and wages? Or is it possible to directly solve this system ?

I tried this,

w_i = [1 ; 1 ; 1]
er=1
n =1
while  er>1e-3
    L_i =   ( (w_i ./ d).^ϵ ) ./ sum( ( (w_i ./ d).^ϵ), dims=1) * R 
    er = maximum(abs.(L .- L_i))
    w_i = 0.7.*w_i + 0.3.*w_i.*((L .- L_i) ./ L_i)
    n = n+1
end
Pol Cosentino
  • 61
  • 5

1 Answers1

1

If L and R are given (i.e., do not depend on w_i), you should set up a non-linear search to get (a vector of) wages from that gravity equation (subject to normalising one w_i, of course).

Here's a minimal example. I hope it helps.

# Call Packages
using Random, NLsolve, LinearAlgebra

# Set seeds
Random.seed!(1704)

# Variables and parameters
N    = 10
R    = rand(N)
L    = rand(N) * 0.5
d    = ones(N, N) .+ Symmetric(rand(N, N)) / 10.0 
d[diagind(d)] .= 1.0
ε    = -3.0

# Define objective function
function obj_fun(x, d, R, L, ε) 
    
    # Find shares
    S_mat  = (x ./ d).^ε
    den    = sum(S_mat, dims = 1)
    s      = S_mat ./ den

    # Normalize last wage
    x[end] = 1.0

    # Define loss function
    loss   = L .- s * R

    # Return 
    return loss
    
end

# Run optimization
x₀   = ones(N) 
res  = nlsolve(x -> obj_fun(x, d, R, L, ε), x₀, show_trace = true)

# Equilibrium vector of wages
w    = res.zero
Fabrizio Leone
  • 132
  • 5
  • What is the goal of x[end] = 1.0 ? Because if I remove it, I obtain different result and I retrieve the results from my iterative algorithm. – Pol Cosentino Feb 03 '23 at 13:12
  • By Walras' law, one the N equations is redundant. Therefore, you can only retrieve your w_i up to a normalisation. I normalised the last one, but it doesn't really matter in that example which one you choose. See, e.g., the discussion below equation (6) on page 201 of this paper: shorturl.at/yEFR5 – Fabrizio Leone Feb 03 '23 at 13:28
  • @FabrizioLeone: il link che hai messo non funziona - the link to the paper you provided is broken – Antonello Feb 03 '23 at 14:31
  • Thanks. Sorry about that, try [this one](https://uc269c590b8876a5b07159588eb5.dl.dropboxusercontent.com/cd/0/inline2/B1zYzHX6rNIQ5um74h__p_DRH_0e5_ImkvJiLLJ1Qs2LCO1t0GvP8tZX1bxlZg7lYwVAnvXQ94LXkNM9TksBDJJop272FrbHrLeL4_Dfn7j5csGB8C3imWJilLNSfrFH__iXLgbm0ETBSZAMYxTvWjcaE2oGDKQ-aylcZA7QD9_-Oz5G0k_FjzOHGakHHz1P6t4iJJHz9ZVfR9uIswMzqq-ErawPU_zroeuFvu6Dv58AWISfmff5Ih23zNKhKv6eOJlfrFz5I4ajwaEIl9Ll2-Gh5h9dcDqefz-bH1KiCSYuYYD1FjhmFzdWN2F6Rq2hevFOr6YgpGAmfOyzhEjlELwnYqyiERzoDQo8C-KrxoEGcqfK9Wk88IQIxDYo5NYo_VFheMeZ_MJFsXRz9H7ir5XRRGcTiaDGC59bvSmNfBx-Ow/file#) – Fabrizio Leone Feb 03 '23 at 14:55
  • @FabrizioLeone: Grazie, but still, Error 404 ("We can't find the page you're looking for.")... – Antonello Feb 03 '23 at 15:20
  • 1
    Ok, one more trial [here](https://www.sciencedirect.com/science/article/abs/pii/B9780444543141000045). See the authors' webpages for the undated version – Fabrizio Leone Feb 03 '23 at 15:43