# Tutorial 3: Create a Model

This tutorial walks through the steps to create a new model, first a one-region model and then a more complex two-region model.

While we will walk through the code step by step below, the full code for implementation is also available in the `examples/tutorial`

folder in the Mimi github repository.

Working through the following tutorial will require:

- Julia v1.1.0 or higher
- Mimi v0.8.0

If you have not yet prepared these, go back to the main tutorial page and follow the instructions for their download.

## Constructing A One-Region Model

In this example, we construct a stylized model of the global economy and its changing greenhouse gas emission levels through time. The overall strategy involves creating components for the economy and emissions separately, and then defining a model where the two components are coupled together.

There are two main steps to creating a component, both within the `@defcomp`

macro which defines a component:

- List the parameters and variables.
- Use the
`run_timestep`

function`run_timestep(p, v, d, t)`

to set the equations of that component.

Starting with the economy component, each variable and parameter is listed. If either variables or parameters have a time-dimension, that must be set with `(index=[time])`

.

```
using Mimi
@defcomp grosseconomy begin
YGROSS = Variable(index=[time]) # Gross output
K = Variable(index=[time]) # Capital
l = Parameter(index=[time]) # Labor
tfp = Parameter(index=[time]) # Total factor productivity
s = Parameter(index=[time]) # Savings rate
depk = Parameter() # Depreciation rate on capital - Note that it has no time index
k0 = Parameter() # Initial level of capital
share = Parameter() # Capital share
```

Next, the `run_timestep`

function must be defined along with the various equations of the `grosseconomy`

component. In this step, the variables and parameters are linked to this component and must be identified as either a variable or a parameter in each equation. For this example, `v`

will refer to variables while `p`

refers to parameters.

It is important to note that `t`

below is an `AbstractTimestep`

, and the specific API for using this argument are described in detail in the **userguide** in **Advanced Topics: Timesteps and available functions**.

```
function run_timestep(p, v, d, t)
# Define an equation for K
if is_first(t)
# Note the use of v. and p. to distinguish between variables and parameters
v.K[t] = p.k0
else
v.K[t] = (1 - p.depk)^5 * v.K[t-1] + v.YGROSS[t-1] * p.s[t-1] * 5
end
# Define an equation for YGROSS
v.YGROSS[t] = p.tfp[t] * v.K[t]^p.share * p.l[t]^(1-p.share)
end
end
```

Next, the component for greenhouse gas emissions must be created. Although the steps are the same as for the `grosseconomy`

component, there is one minor difference. While `YGROSS`

was a variable in the `grosseconomy`

component, it now enters the `emissions`

component as a parameter. This will be true for any variable that becomes a parameter for another component in the model.

```
@defcomp emissions begin
E = Variable(index=[time]) # Total greenhouse gas emissions
sigma = Parameter(index=[time]) # Emissions output ratio
YGROSS = Parameter(index=[time]) # Gross output - Note that YGROSS is now a parameter
function run_timestep(p, v, d, t)
# Define an equation for E
v.E[t] = p.YGROSS[t] * p.sigma[t] # Note the p. in front of YGROSS
end
end
```

We can now use Mimi to construct a model that binds the `grosseconomy`

and `emissions`

components together in order to solve for the emissions level of the global economy over time. In this example, we will run the model for twenty periods with a timestep of five years between each period.

- Once the model is defined,
`set_dimension!`

is used to set the length and interval of the time step. - We then use
`add_comp!`

to incorporate each component that we previously created into the model. It is important to note that the order in which the components are listed here matters. The model will run through each equation of the first component before moving onto the second component. - Next,
`set_param!`

is used to assign values to each parameter in the model, with parameters being uniquely tied to each component. If*population*was a parameter for two different components, it must be assigned to each one using`set_param!`

two different times. The syntax is`set_param!(model_name, :component_name, :parameter_name, value)`

- If any variables of one component are parameters for another,
`connect_param!`

is used to couple the two components together. In this example,*YGROSS*is a variable in the`grosseconomy`

component and a parameter in the`emissions`

component. The syntax is`connect_param!(model_name, :component_name_parameter, :parameter_name, :component_name_variable, :variable_name)`

, where`:component_name_variable`

refers to the component where your parameter was initially calculated as a variable. - Finally, the model can be run using the command
`run(model_name)`

. - To access model results, use
`model_name[:component, :variable_name]`

. - To observe model results in a graphical form ,
`explore`

as either`explore(model_name)`

to open the UI window, or use`Mimi.plot(model_name, :component_name, :variable_name)`

or`Mimi.plot(model_name, :component_name, :parameter_name)`

to plot a specific parameter or variable.

```
using Mimi
function construct_model()
m = Model()
set_dimension!(m, :time, collect(2015:5:2110))
# Order matters here. If the emissions component were defined first, the model would not run.
add_comp!(m, grosseconomy)
add_comp!(m, emissions)
# Set parameters for the grosseconomy component
set_param!(m, :grosseconomy, :l, [(1. + 0.015)^t *6404 for t in 1:20])
set_param!(m, :grosseconomy, :tfp, [(1 + 0.065)^t * 3.57 for t in 1:20])
set_param!(m, :grosseconomy, :s, ones(20).* 0.22)
set_param!(m, :grosseconomy, :depk, 0.1)
set_param!(m, :grosseconomy, :k0, 130.)
set_param!(m, :grosseconomy, :share, 0.3)
# Set parameters for the emissions component
set_param!(m, :emissions, :sigma, [(1. - 0.05)^t *0.58 for t in 1:20])
connect_param!(m, :emissions, :YGROSS, :grosseconomy, :YGROSS)
# Note that connect_param! was used here.
return m
end #end function
```

Note that as an alternative to using many of the `set_param!`

calls above, one may use the `default`

keyword argument in `@defcomp`

when first defining a `Variable`

or `Parameter`

, as shown in `examples/tutorial/01-one-region-model/one-region-model-defaults.jl`

.

Now we can run the model and examine the results:

```
# Run model
m = construct_model()
run(m)
# Check model results
m[:emissions, :E]
# Plot model results
Mimi.plot(m, :emissions, :E)
# Observe all model result graphs in UI
explore(m)
```

## Constructing A Multi-Region Model

We can now modify our two-component model of the globe to include multiple regional economies. Global greenhouse gas emissions will now be the sum of regional emissions. The modeling approach is the same, with a few minor adjustments:

- When using
`@defcomp`

, a regions index must be specified. In addition, for variables that have a regional index it is necessary to include`(index=[regions])`

. This can be combined with the time index as well,`(index=[time, regions])`

. - In the
`run_timestep`

function, unlike the time dimension, regions must be specified and looped through in any equations that contain a regional variable or parameter. `set_dimension!`

must be used to specify your regions in the same way that it is used to specify your timestep.- When using
`set_param!`

for values with a time and regional dimension, an array is used. Each row corresponds to a time step, while each column corresponds to a separate region. For regional values with no timestep, a vector can be used. It is often easier to create an array of parameter values before model construction. This way, the parameter name can be entered into`set_param!`

rather than an entire equation. - When constructing regionalized models with multiple components, it is often easier to save each component as a separate file and to then write a function that constructs the model. When this is done,
`using Mimi`

must be speficied for each component. This approach will be used here.

To create a three-regional model, we will again start by constructing the grosseconomy and emissions components, making adjustments for the regional index as needed. Each component should be saved as a separate file.

As this model is also more complex and spread across several files, we will also take this as a chance to introduce the custom of using Modules to package Mimi models, as shown below.

```
using Mimi
@defcomp grosseconomy begin
regions = Index() #Note that a regional index is defined here
YGROSS = Variable(index=[time, regions]) # Gross output
K = Variable(index=[time, regions]) # Capital
l = Parameter(index=[time, regions]) # Labor
tfp = Parameter(index=[time, regions]) # Total factor productivity
s = Parameter(index=[time, regions]) # Savings rate
depk = Parameter(index=[regions]) # Depreciation rate on capital - Note that it only has a region index
k0 = Parameter(index=[regions]) # Initial level of capital
share = Parameter() # Capital share
function run_timestep(p, v, d, t)
# Note that the regional dimension is used below and parameters and
variables are indexed by 'r'
# Define an equation for K
for r in d.regions
if is_first(t)
v.K[t,r] = p.k0[r]
else
v.K[t,r] = (1 - p.depk[r])^5 * v.K[t-1,r] + v.YGROSS[t-1,r] * p.s[t-1,r] * 5
end
end
# Define an equation for YGROSS
for r in d.regions
v.YGROSS[t,r] = p.tfp[t,r] * v.K[t,r]^p.share * p.l[t,r]^(1-p.share)
end
end
end
```

Save this component as `gross_economy.jl`

```
using Mimi #Make sure to call Mimi again
@defcomp emissions begin
regions = Index() # The regions index must be specified for each component
E = Variable(index=[time, regions]) # Total greenhouse gas emissions
E_Global = Variable(index=[time]) # Global emissions (sum of regional emissions)
sigma = Parameter(index=[time, regions]) # Emissions output ratio
YGROSS = Parameter(index=[time, regions]) # Gross output - Note that YGROSS is now a parameter
function run_timestep(p, v, d, t)
# Define an eqation for E
for r in d.regions
v.E[t,r] = p.YGROSS[t,r] * p.sigma[t,r]
end
# Define an equation for E_Global
for r in d.regions
v.E_Global[t] = sum(v.E[t,:])
end
end
end
```

Save this component as `emissions.jl`

Let's create a file with all of our parameters that we can call into our model. This will help keep things organized as the number of components and regions increases. Each column refers to parameter values for a region, reflecting differences in initial parameter values and growth rates between the three regions.

```
l = Array(Float64,20,3)
for t in 1:20
l[t,1] = (1. + 0.015)^t *2000
l[t,2] = (1. + 0.02)^t * 1250
l[t,3] = (1. + 0.03)^t * 1700
end
tfp = Array(Float64,20,3)
for t in 1:20
tfp[t,1] = (1 + 0.06)^t * 3.2
tfp[t,2] = (1 + 0.03)^t * 1.8
tfp[t,3] = (1 + 0.05)^t * 2.5
end
s = Array(Float64,20,3)
for t in 1:20
s[t,1] = 0.21
s[t,2] = 0.15
s[t,3] = 0.28
end
depk = [0.11, 0.135 ,0.15]
k0 = [50.5, 22., 33.5]
sigma = Array(Float64,20,3)
for t in 1:20
sigma[t,1] = (1. - 0.05)^t * 0.58
sigma[t,2] = (1. - 0.04)^t * 0.5
sigma[t,3] = (1. - 0.045)^t * 0.6
end
```

Save this file as `region_parameters.jl`

The final step is to create a module.

```
module MyModel
using Mimi
include("region_parameters.jl")
include("gross_economy.jl")
include("emissions.jl")
export construct_MyModel
function construct_MyModel()
m = Model()
set_dimension!(m, :time, collect(2015:5:2110))
set_dimension!(m, :regions, [:Region1, :Region2, :Region3]) # Note that the regions of your model must be specified here
add_comp!(m, grosseconomy)
add_comp!(m, emissions)
set_param!(m, :grosseconomy, :l, l)
set_param!(m, :grosseconomy, :tfp, tfp)
set_param!(m, :grosseconomy, :s, s)
set_param!(m, :grosseconomy, :depk,depk)
set_param!(m, :grosseconomy, :k0, k0)
set_param!(m, :grosseconomy, :share, 0.3)
# set parameters for emissions component
set_param!(m, :emissions, :sigma, sigma)
connect_param!(m, :emissions, :YGROSS, :grosseconomy, :YGROSS)
return m
end #function
end #end module
```

Save this file as `MyModel.jl`

We can now run the model and evaluate the results.

```
using Mimi
include("MyModel.jl")
using .MyModel
m = construct_MyModel()
run(m)
# Check results
m[:emissions, :E_Global]
# Observe model result graphs
explore(m)
```