The Armington case with cross-good congestion.

Author: SebKrantz

src/main/optimal_network.jl

@@ -79,8 +79,13 @@ function optimal_network(param, graph; I0=nothing, Il=nothing, Iu=nothing, verbo
         model = model_partial_mobility_cgc(optimizer, auxdata)
         recover_allocation = recover_allocation_partial_mobility_cgc
     elseif param.mobility == 0 && param.cong
-        model = model_fixed_cgc(optimizer, auxdata)
-        recover_allocation = recover_allocation_fixed_cgc
+        if all(sum(param.Zjn .> 0, dims = 2) .<= 1) # Armington case
+            model = model_fixed_cgc_armington(optimizer, auxdata)
+            recover_allocation = recover_allocation_fixed_cgc_armington
+        else
+            model = model_fixed_cgc(optimizer, auxdata)
+            recover_allocation = recover_allocation_fixed_cgc
+        end
     elseif param.mobility == 1 && !param.cong
         model = model_mobility(optimizer, auxdata)
         recover_allocation = recover_allocation_mobility

src/models/model_fixed_cgc_armington.jl

@@ -0,0 +1,79 @@
+# The primal case, no mobility, with cross-good congestion, and an Armington (1969) world where each location produces only one good
+
+function model_fixed_cgc_armington(optimizer, auxdata)
+
+    # Extract parameters
+    param = auxdata.param
+    graph = auxdata.graph
+    kappa_ex_init = auxdata.kappa_ex
+    A = auxdata.edges.A
+    Apos = auxdata.edges.Apos
+    Aneg = auxdata.edges.Aneg
+    Lj = param.Lj
+    m = param.m # 1:param.N: Vector of weights on each goods flow for aggregate congestion term
+    psigma = (param.sigma - 1) / param.sigma
+    beta_nu = (param.beta + 1) / param.nu
+
+    # Model
+    model = Model(optimizer)
+    set_string_names_on_creation(model, false)
+
+    # Variables
+    @variable(model, Djn[1:graph.J, 1:param.N] >= 1e-8, container=Array, start = 1e-6)             # Consumption per good pre-transport cost (Dj)
+    @variable(model, Qin_direct[1:graph.ndeg, 1:param.N] >= 1e-8, container=Array, start = 0.0)    # Direct aggregate flow
+    @variable(model, Qin_indirect[1:graph.ndeg, 1:param.N] >= 1e-8, container=Array, start = 0.0)  # Indirect aggregate flow
+    @variable(model, cj[1:graph.J] >= 1e-8, container=Array, start = 1e-6)                         # Overall consumption bundle, including transport costs
+
+    # 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)
+
+    # Defining Utility Funcion: from cj + parameters (by operator overloading)
+    @expression(model, uj, ((cj / param.alpha) .^ param.alpha .* (param.hj / (1-param.alpha)) .^ (1-param.alpha)) .^ (1-param.rho) / (1-param.rho))
+    @expression(model, U, sum(param.omegaj .* Lj .* uj))      # Overall Utility
+    @objective(model, Max, U)
+
+    # Create the matrix B_direct (resp. B_indirect) of transport cost along the direction of the edge (resp. in edge opposite direction)
+    B_direct = @expression(model, ((Qin_direct .^ param.nu) * m) .^ beta_nu ./ kappa_ex)
+    B_indirect = @expression(model, ((Qin_indirect .^ param.nu) * m) .^ beta_nu ./ kappa_ex)
+    # Final good constraints
+    @expression(model, Dj, sum(Djn .^ psigma, dims=2) .^ (1 / psigma))
+    @constraint(model, cj .* Lj + Apos * B_direct + Aneg * B_indirect - Dj .<= -1e-8)
+
+    # Balanced flow constraints
+    Yjn = param.Zjn .* Lj .^ param.a
+    @constraint(model, Pjn, Djn + A * Qin_direct - A * Qin_indirect - Yjn .<= -1e-8)
+
+    return model
+end
+
+function recover_allocation_fixed_cgc_armington(model, auxdata)
+    param = auxdata.param
+    graph = auxdata.graph
+    model_dict = model.obj_dict
+    results = Dict()
+    # Production: Fixed because just one good is produced by each location
+    Yj = dropdims(sum(param.Zjn, dims = 2) .* param.Lj .^ param.a, dims = 2)
+    WY = param.Zjn .> 0
+
+    results[:welfare] = value(model_dict[:U])
+    results[:Yjn] = WY .* Yj
+    results[:Yj] = Yj
+    results[:Ljn] = WY .* param.Lj
+    results[:Lj] = param.Lj
+    results[:Djn] = value.(model_dict[:Djn]) # Consumption per good pre-transport cost
+    results[:Dj] = dropdims(value.(model_dict[:Dj]), dims = 2)
+    results[:cj] = value.(model_dict[:cj])
+    results[:Cj] = results[:cj] .* param.Lj
+    results[:hj] = ifelse.(results[:Lj] .== 0, 0.0, param.Hj ./ results[:Lj])
+    results[:uj] = value.(model_dict[:uj])
+    # 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
+    Qin_direct = value.(model_dict[:Qin_direct])
+    Qin_indirect = value.(model_dict[:Qin_indirect])
+    results[:Qin] = max.(Qin_direct, Qin_indirect) - min.(Qin_direct, Qin_indirect)
+    results[:Qjkn] = gen_network_flows(results[:Qin], graph, param.N)
+    return results
+end