new-function-interaction.Rmd

---
title: "new_* function interactions"
author: "Matthew Kuperus Heun"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{new_* function interactions}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
library(magrittr)
library(matsbyname)
library(Recca)
```

<!-- Establish some helpful LaTeX shortcuts for equations -->
\newcommand{\transpose}[1]{#1^\mathrm{T}}
\newcommand{\inverse}[1]{#1^{\mathrm{-}1}}
\newcommand{\mat}[1]{\mathbf{#1}}
\newcommand{\colvec}[1]{\mathbf{#1}}
\newcommand{\rowvec}[1]{\transpose{\colvec{#1}}}
\newcommand{\inversetranspose}[1]{\transpose{\left( \inverse{\mat{#1}} \right)}}
\newcommand{\transposeinverse}[1]{\inverse{\left( \transpose{\mat{#1}} \right)}}
\newcommand{\hatinv}[1]{\inverse{\widehat{#1}}}

\newcommand{\matwithdims}[3]{\underset{\mathrm{#2} \times \mathrm{#3}}{\mat{#1}}}
\newcommand{\colvecwithdims}[2]{\underset{\mathrm{#2} \times 1}{\colvec{#1}}}

\newcommand{\Lpp}{\matwithdims{L}{p}{p}}
\newcommand{\Lip}{\matwithdims{L}{i}{p}}


## Introduction

The `Recca` package provides several `new_*` functions that allow the user to estimate a new version
of an energy conversion chain (ECC)
based on localized changed to one portion of the ECC.
The functions can have some interesting interactions, 
particularly if they are called sequentially.
This vignette show how to understand and deal with those interactions.
The functions `new_k_ps()` and `new_Y()` are used for this purpose.


## Setup

To investigate the interactions, 
we first make the simplest possible ECC that retains the features to be tested,
a minimum working example (MWE).
This example has two resource industries (R1 and R2),
one intermediate industry (I), and
two final demand sectors (Y1 and Y2).
The R1 industry makes product R1p.  
The R2 industry makes product R2p. 
Industry I makes product Ip.
Product Ip is consumed by final demand sectors Y1 and Y2.

The resources ($\mat{R}$), 
use ($\mat{U}$), 
make ($\mat{V}$), 
final demand ($\mat{Y}$), and 
unit summation ($\mat{S}_{units}$) matrices are given below.

```{r}
library(magrittr)
library(matsbyname)
library(Recca)
U <- matrix(c(0, 0, 10,
              0, 0, 10,
              0, 0,  0),
            byrow = TRUE, nrow = 3, ncol = 3,
            dimnames = list(c("R1p", "R2p", "Ip"), c("R1", "R2", "I"))) %>% 
  setrowtype("Products") %>% setcoltype("Industries")

# There is no EIOU, so U_feed and U are the same.
U_feed <- U

R_plus_V <- matrix(c(10,  0, 0,
                     0, 10, 0, 
                     0,  0, 4), 
                   byrow = TRUE, nrow = 3, ncol = 3, 
                   dimnames = list(c("R1", "R2", "I"), 
                                   c("R1p", "R2p", "Ip"))) %>% 
  setrowtype("Industries") %>% setcoltype("Products")

RV_mats <- separate_RV(U = U, R_plus_V = R_plus_V)
R <- RV_mats$R
V <- RV_mats$V

Y <- matrix(c(0, 0,
              0, 0, 
              2, 2),
            byrow = TRUE, nrow = 3, ncol = 2,
            dimnames = list(c("R1p", "R2p", "Ip"), c("Y1", "Y2"))) %>% 
  setrowtype("Products") %>% setcoltype("Industries")

S_units <- matrix(c(1, 
                    1, 
                    1), 
                  byrow = TRUE, nrow = 3, ncol = 1,
                  dimnames = list(c("R1p", "R2p", "Ip"), c("quad"))) %>% 
  setrowtype("Products") %>% setcoltype("Units")
```

We'll use Sankey diagrams to visualize several ECCs. 
The `make_sankey()` function creates the diagrams.
A Sankey diagram that shows the base energy conversion chain is shown below.
`make_sankey()` returns a list with a named item "Sankey" that we must extract
to insert the diagram into this vignette.

```{r}
make_sankey(R = R, U = U, V = V, Y = Y) %>% 
  extract2("Sankey")
```


## Adjust inputs to industry I

We can change the proportion of inputs to industry I using `new_k_ps()`.
To do so, we must first calculate the input-output structure of the ECC 
using the `calc_io_mats()` function.
Then, we call the `new_k_ps()` function with a new k vector
that indicates industry I now uses 20 quads of `R1` instead of a 50/50 split between `R1` and `R2`.

```{r}
library(Recca)
iomats <- calc_io_mats(R = R, U = U, U_feed = U_feed, V = V, Y = Y, S_units = S_units)
k_prime <- matrix(c(1,
                    0), 
                  byrow = TRUE, nrow = 2, ncol = 1,
                  dimnames = list(c("R1p", "R2p"), c("I"))) %>% 
  setrowtype("Products") %>% setcoltype("Industries")
change_input_proportions <- new_k_ps(c(iomats, 
                                       list(R = R, U = U, V = V, Y = Y,
                                            S_units = S_units, 
                                            k_prime = k_prime)))
# There is no EIOU, so U and U_feed are same.
change_input_proportions[["U_feed_prime"]] <- change_input_proportions[["U_prime"]]
make_sankey(R = change_input_proportions$R_prime, 
            U = change_input_proportions$U_prime, 
            V = change_input_proportions$V_prime, 
            Y = Y) %>% 
  extract2("Sankey")
```


## Change final demand

We can calculate the ECC that would be required to meet a new level of final demand
with the `new_Y()` function. 
The `new_k_ps()` and `new_Y()` functions interact in ways that might be surprising to the user, 
depending on whether the input-output structure of the ECC is re-calculated 
between sequential function calls. 


### Without recalculating the structure of the ECC

Now let's change final demand, saying that consumption by Y1 doubles.
But we'll call `new_Y()` *without* recalculating the structure of the ECC with the `calc_io_mats()` function.

```{r}
Y_prime <- matrix(c(0, 0,
                    0, 0, 
                    4, 2),
                  byrow = TRUE, nrow = 3, ncol = 2,
                  dimnames = list(c("R1p", "R2p", "Ip"), c("Y1", "Y2"))) %>% 
  setrowtype("Products") %>% setcoltype("Industries")
change_final_demand <- new_Y(c(iomats, list(Y_prime = Y_prime)))
make_sankey(R = change_final_demand$R_prime, 
            U = change_final_demand$U_prime, 
            V = change_final_demand$V_prime, 
            Y = change_final_demand$Y_prime) %>% 
  extract2("Sankey")
```

Industry I is back to using both R1p and R2p in a 50/50 split to make its output.
Both R1p and R2p now supply 15 quads to I to account for increased final demand.

The `new_Y()` function is performing exactly as expected, 
given that `iomats` was supplied to `new_Y()`.
`iomats` contains information about an ECC in which industry I's inputs are split 50/50 between R1p and R2p.
So when `new_Y()` calculates a new ECC, 
it assumes that industry I's needs are a 50/50 split between R1p and R2p.

In fact, it would be a mistake to think that `new_Y()` would give anything but a 50/50 split between R1p and R2p
when using the `iomats` description of the ECC.


### After recalculating the structure of the ECC

If you want `new_Y()` to start from the Sankey in which all energy supplied to industry I
comes from R1p, you need to first recalculate the input-output structure of the ECC
based on the ECC in which industry I uses only R1p.
I.e., you need to re-calculate an `iomats` object after calling `new_k_ps()`
and before calling `new_Y()`.
Then, you must submit that updated description of the input-output structure of the ECC to `new_Y()`.
The following code demonstrates the approach.

```{r}
iomats2 <- calc_io_mats(R = change_input_proportions$R_prime, 
                        U = change_input_proportions$U_prime, 
                        U_feed = change_input_proportions$U_feed_prime,
                        V = change_input_proportions$V_prime, 
                        Y = Y, S_units = S_units)
change_final_demand2 <- new_Y(c(iomats2, list(Y_prime = Y_prime)))
make_sankey(R = change_final_demand2$R_prime, 
            U = change_final_demand2$U_prime, 
            V = change_final_demand2$V_prime, 
            Y = change_final_demand2$Y_prime) %>% 
  extract2("Sankey")
```

The Sankey diagram above shows that we have successfully increased the demand from `Y1` while 
ensuring that all inputs to industry I come from R1p.
The "trick" is to recalculate the input-output structure of the ECC between the two steps.


# Conclusion

The `new_*` function of the `Recca` package can be called sequentially.
But the interactions among the `new_*` functions can be surprising
if their behavior is not fully understood. 
This vignette provides a detailed example using the
`new_k_ps()` and `new_Y()` functions.
If they are called sequentially *without* first recalculating the input-output structure of the ECC,
the call to `new_Y()` will be based on the original input-output structure of the ECC
(from before `new_k_ps()` was called).
If `calc_io_mats()` is called between the sequential calls to `new_k_ps()` and `new_Y()`
and the results are fed to `new_Y()`, 
the input-output structure of the ECC *after* calling `new_k_ps()` will be used for 
the calculations by `new_Y()`.