Yet more Armington.

Author: SebKrantz

src/main/optimal_network.jl

@@ -100,8 +100,13 @@ function optimal_network(param, graph; I0=nothing, Il=nothing, Iu=nothing, verbo
             recover_allocation = recover_allocation_mobility
         end
     elseif param.mobility == 0.5 && !param.cong
-        model = model_partial_mobility(optimizer, auxdata)
-        recover_allocation = recover_allocation_partial_mobility    
+        if all(sum(param.Zjn .> 0, dims = 2) .<= 1) # Armington case
+            model = model_partial_mobility_armington(optimizer, auxdata)
+            recover_allocation = recover_allocation_partial_mobility_armington    
+        else
+            model = model_partial_mobility(optimizer, auxdata)
+            recover_allocation = recover_allocation_partial_mobility    
+        end
     elseif param.mobility == 0 && !param.cong
         if all(sum(param.Zjn .> 0, dims = 2) .<= 1) # Armington case
             model = model_fixed_armington(optimizer, auxdata)

src/models/model_partial_mobility_armington.jl

@@ -0,0 +1,95 @@
+# The primal case, with labor mobility within regions, no cross-good congestion, and an Armington (1969) world where each location produces only one good
+
+function model_partial_mobility_armington(optimizer, auxdata)
+    
+    # Parameters and data
+    param = auxdata.param
+    graph = auxdata.graph
+    region = graph.region
+    if length(region) != graph.J
+        error("length(region) = $(length(region)) does not match number of nodes = $(graph.J)")
+    end
+    kappa_ex_init = auxdata.kappa_ex
+    A = auxdata.edges.A
+    Apos = auxdata.edges.Apos
+    Aneg = auxdata.edges.Aneg
+    psigma = (param.sigma - 1) / param.sigma
+    Hj = param.Hj
+    Lr = param.Lr
+    if length(param.omegar) != param.nregions
+        error("length(param.omegar) = $(length(param.omegar)) does not match number of regions = $(param.nregions)")
+    end
+    if length(Lr) != param.nregions
+        error("Populations Lr = $(length(Lr)) does not match number of regions = $(param.nregions)")
+    end
+
+    # Model
+    model = Model(optimizer)
+    set_string_names_on_creation(model, false)
+
+    # Variables + bounds
+    @variable(model, ur[1:param.nregions], container=Array, start = 0.0)               # Utility per capita in each region
+    @variable(model, Cjn[1:graph.J, 1:param.N] >= 1e-8, container=Array, start = 1e-6) # Good specific consumption
+    @variable(model, Qin[1:graph.ndeg, 1:param.N], container=Array, start = 0.0)       # Good specific flow
+    # NOTE: Fajgelbaum et al (2019) only optimize Lj and distribute it equally for goods with positive productivity
+    @variable(model, Lj[1:graph.J] >= 1e-8, container=Array)                           # Total labour
+    # Calculate start values for Lj
+    pop_start = (Lr ./ gsum(ones(graph.J), param.nregions, region))[region]
+    set_start_value.(Lj, pop_start)
+
+    # Parameters: to be updated between solves
+    @variable(model, kappa_ex[i = 1:graph.ndeg] in Parameter(i), container=Array)
+    set_parameter_value.(kappa_ex, kappa_ex_init)
+
+    # Objective
+    @expression(model, U, sum(param.omegar .* Lr .* ur))   # Overall utility
+    @objective(model, Max, U)
+
+    # Utility constraint (Lj * ur <= ... )
+    for j in 1:graph.J
+        Cj = sum(Cjn[j, n]^psigma for n in 1:param.N)^(1 / psigma)
+        @constraint(model, Lj[j] * ur[region[j]] - (Cj / param.alpha)^param.alpha * (Hj[j] / (1 - param.alpha))^(1 - param.alpha) <= -1e-8)
+    end
+
+    # Balanced flow constraints: same as with unrestricted mobility (no restrictions on goods)
+    @expression(model, Yjn[j=1:graph.J, n=1:param.N], param.Zjn[j, n] * Lj[j]^param.a)
+    @constraint(model, Pjn[j in 1:param.J, n in 1:param.N],
+        Cjn[j, n] + sum(A[j, i] * Qin[i, n] for i in 1:graph.ndeg) -
+        Yjn[j, n] + sum(
+            ifelse(Qin[i, n] > 0, Apos[j, i], Aneg[j, i]) *
+            abs(Qin[i, n])^(1 + param.beta) / kappa_ex[i]
+            for i in 1:graph.ndeg
+        ) <= -1e-8
+    )
+
+    # Labor resource constraints (within each region)
+    @constraint(model, -1e-8 .<= gsum(Lj, param.nregions, region) .- Lr .<= 1e-8)
+
+    return model
+end
+
+function recover_allocation_partial_mobility_armington(model, auxdata)
+    param = auxdata.param
+    graph = auxdata.graph
+    model_dict = model.obj_dict
+    results = Dict()
+
+    results[:welfare] = value(model_dict[:U])
+    results[:reg_pc_welfare] = value.(model_dict[:ur])
+    results[:Yjn] = value.(model_dict[:Yjn])
+    results[:Yj] = dropdims(sum(results[:Yjn], dims=2), dims = 2) 
+    results[:Cjn] = value.(model_dict[:Cjn])
+    results[:Cj] = dropdims(sum(results[:Cjn] .^ ((param.sigma-1)/param.sigma), dims=2), dims = 2) .^ (param.sigma/(param.sigma-1))
+    results[:Lj] = value.(model_dict[:Lj])
+    results[:Ljn] = (param.Zjn .> 0).* results[:Lj]
+    results[:cj] = ifelse.(results[:Lj] .== 0, 0.0, results[:Cj] ./ results[:Lj])
+    results[:hj] = ifelse.(results[:Lj] .== 0, 0.0, param.Hj ./ results[:Lj])
+    results[:uj] = param.u.(results[:cj], results[:hj])
+    # Prices
+    results[:Pjn] = shadow_price.(model_dict[:Pjn])
+    results[:PCj] = dropdims(sum(results[:Pjn] .^ (1-param.sigma), dims=2), dims=2) .^ (1/(1-param.sigma))    
+    # Network flows
+    results[:Qin] = value.(model_dict[:Qin])
+    results[:Qjkn] = gen_network_flows(results[:Qin], graph, param.N)
+    return results
+end