# High order approximation of a multi-country trade model
**URL:** https://discourse.julialang.org/t/high-order-approximation-of-a-multi-country-trade-model/83225
**Category:** Finance and Economics
**Tags:** performance, differentiation, forwarddiff, finitediff
**Created:** [June 23, 2022, 7:55am UTC](https://discourse.julialang.org/t/high-order-approximation-of-a-multi-country-trade-model/83225 "2022-06-23T07:55:27Z")
**Posts on this page:** 3
**Page:** 2
### Author:  [@daisukeadachi](https://discourse.julialang.org/u/daisukeadachi)
#### Post date: [July 6, 2022, 1:13pm UTC](https://discourse.julialang.org/t/high-order-approximation-of-a-multi-country-trade-model/83225/22 "2022-07-06T13:13:42Z")
Thanks so much. This helps to improve the performance. But I have a related question. Why does the `@btime` result change in the following code?
```julia
#---------------------------------------
# housekeeping
#---------------------------------------
using FiniteDifferences, Profile, LinearAlgebra, BenchmarkTools
#---------------------------------------
# internal functions
#---------------------------------------
# function about equilibrium condition (excess labor demand)
@views function es_loop(wzet,y,x,kappa,sig,N,tau_mat,cma,cld)
# define the matrix dimension
broadcast!(.^, tau_mat, wzet[N+1], kappa)
# armington condition
# 1. compute the change in the market access (cma)
cma_mat = x .* (tau_mat .* wzet[1:N]) .^ (1-sig)
for i = 1:N
cma .+= cma_mat[i:i,:]
end
# 2. sum over the change in labor demand across destination (cld)
cld_mat = y .* (tau_mat .* wzet[1:N]) .^ (1-sig) .* wzet[1:N]' ./ cma
for j = 1:N
cld .+= cld_mat[:,j:j]
end
excess_loop = wzet[1:N] - cld
return excess_loop
end
#---------------------------------------
# parameters and data
#---------------------------------------
N = 3; # number of countries
sig = 4; # elasticity of substitution
homebias = 0.9; # the size of the domestic consumption tendency
X = homebias * Matrix{Float64}(I,N,N) + (1-homebias) * 1/N * ones(N,N);
y = X./sum(X,dims=2);
x = X./sum(X,dims=1);
# trade cost setting
kappa = vcat(zeros(N-1,N), [ones(N-1); 0]'); # the kernel of the trade cost change
tau_mat = ones(N,N); # pre-allocate the tau matrix
wzet = ones(N+1); # initial point
# pre-allocate
cma = zeros(1,N); # change in the market access
cld = zeros(N,1); # change in labor demand
cma_mat = zeros(N,N);
cld_mat = zeros(N,N);
#---------------------------------------
# check performance
#---------------------------------------
es_loop(wzet,y,x,kappa,sig,N,tau_mat,cma,cld)
```
Specifically, the last line `es_loop(wzet,y,x,kappa,sig,N,tau_mat,cma,cld)` returns a correct answer as follows:
```julia
3×1 Matrix{Float64}:
0.0
0.0
0.0
```
However, when I change the last line into `@btime es_loop(wzet,y,x,kappa,sig,N,tau_mat,cma,cld)`, it returns a wrong answer as follows:
```julia
240.423 ns (3 allocations: 336 bytes)
3×1 Matrix{Float64}:
-12.700585535625594
-12.700585535625592
-12.700585535625592
```
---
### Author:  [@baggepinnen](https://discourse.julialang.org/u/baggepinnen)
#### Post date: [July 6, 2022, 1:52pm UTC](https://discourse.julialang.org/t/high-order-approximation-of-a-multi-country-trade-model/83225/23 "2022-07-06T13:52:29Z")
`@btime` runs the same code several times, could it be because you expect some array (`cma` for instance) to be filled with zeros (or some other value) when the algorithm starts, and then you overwrite it so that it does not have the same initial value the next time `@btime` runs the expression?
---
### Author:  [@daisukeadachi](https://discourse.julialang.org/u/daisukeadachi)
#### Post date: [July 6, 2022, 2:38pm UTC](https://discourse.julialang.org/t/high-order-approximation-of-a-multi-country-trade-model/83225/24 "2022-07-06T14:38:32Z")
Yes, of course. This makes perfect sense. Thanks so much, and I will work on a different approach for a while if I can compute the fourth order derivatives in a reasonable time 🙂
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