About Me Blog
Analyzing NetHack data, part 1: What kills the players Analyzing NetHack data, part 2: What players kill the most Dealing with heteroskedasticity; regression with robust standard errors using R Easy time-series prediction with R: a tutorial with air traffic data from Lux Airport Exporting editable plots from R to Powerpoint: making ggplot2 purrr with officer Forecasting my weight with R From webscraping data to releasing it as an R package to share with the world: a full tutorial with data from NetHack Getting data from pdfs using the pdftools package Getting the data from the Luxembourguish elections out of Excel Going from a human readable Excel file to a machine-readable csv with {tidyxl} How Luxembourguish residents spend their time: a small {flexdashboard} demo using the Time use survey data Imputing missing values in parallel using {furrr} Maps with pie charts on top of each administrative division: an example with Luxembourg's elections data Missing data imputation and instrumental variables regression: the tidy approach The year of the GNU+Linux desktop is upon us: using user ratings of Steam Play compatibility to play around with regex and the tidyverse {pmice}, an experimental package for missing data imputation in parallel using {mice} and {furrr} Building formulae Functional peace of mind Get basic summary statistics for all the variables in a data frame Getting {sparklyr}, {h2o}, {rsparkling} to work together and some fun with bash Importing 30GB of data into R with sparklyr Introducing brotools It's lists all the way down It's lists all the way down, part 2: We need to go deeper Keep trying that api call with purrr::possibly() Lesser known dplyr 0.7* tricks Lesser known dplyr tricks Lesser known purrr tricks Make ggplot2 purrr Mapping a list of functions to a list of datasets with a list of columns as arguments Predicting job search by training a random forest on an unbalanced dataset Teaching the tidyverse to beginners Why I find tidyeval useful tidyr::spread() and dplyr::rename_at() in action Easy peasy STATA-like marginal effects with R Functional programming and unit testing for data munging with R available on Leanpub How to use jailbreakr My free book has a cover! Work on lists of datasets instead of individual datasets by using functional programming Method of Simulated Moments with R New website! Nonlinear Gmm with R - Example with a logistic regression Simulated Maximum Likelihood with R Bootstrapping standard errors for difference-in-differences estimation with R Careful with tryCatch Data frame columns as arguments to dplyr functions Export R output to a file I've started writing a 'book': Functional programming and unit testing for data munging with R Introduction to programming econometrics with R Merge a list of datasets together Object Oriented Programming with R: An example with a Cournot duopoly R, R with Atlas, R with OpenBLAS and Revolution R Open: which is fastest? Read a lot of datasets at once with R Unit testing with R Update to Introduction to programming econometrics with R Using R as a Computer Algebra System with Ryacas

Object Oriented Programming with R: An example with a Cournot duopoly

I started reading Applied Computational Economics & Finance by Mario J. Miranda and Paul L. Fackler. It is a very interesting book that I recommend to every one of my colleagues. The only issue I have with this book, is that the programming language they use is Matlab, which is proprietary. While there is a free as in freedom implementation of the Matlab language, namely Octave, I still prefer using R. In this post, I will illustrate one example the authors present in the book with R, using the package rootSolve. rootSolve implements Newtonian algorithms to find roots of functions; to specify the functions for which I want the roots, I use R's Object-Oriented Programming (OOP) capabilities to build a model that returns two functions. This is optional, but I found that it was a good example to illustrate OOP, even though simpler solutions exist, one of which was proposed by reddit user TheDrownedKraken (whom I thank) and will be presented at the end of the article.

Theoretical background

The example is taken from Miranda's and Fackler's book, on page 35. The authors present a Cournot duopoly model. In a Cournot duopoly model, two firms compete against each other by quantities. Both produce a certain quantity of an homogenous good, and take the quantity produce by their rival as given.

The inverse demand of the good is :

$$P(q) = q^{-\dfrac{1}{\eta}}$$

the cost function for firm i is:

$$C_i(q_i) = P(q_1+q_2)*q_i - C_i(q_i)$$

and the profit for firm i:

$$\pi_i(q1,q2) = P(q_1+q_2)q_i - C_i(q_i)$$

The optimality condition for firm i is thus:

$$\dfrac{\partial \pi_i}{\partial q_i} = (q1+q2)^{-\dfrac{1}{\eta}} - \dfrac{1}{\eta} (q_1+q_2)^{\dfrac{-1}{\eta-1}}q_i - c_iq_i=0.$$

Implementation in R

If we want to find the optimal quantities \(q_1\) and \(q_2\) we need to program the optimality condition and we could also use the jacobian of the optimality condition. The jacobian is generally useful to speed up the root finding routines. This is were OOP is useful. First let's create a new class, called Model:

setClass(Class = "Model", slots = list(OptimCond = "function", JacobiOptimCond = "function"))

This new class has two slots, which here are functions (in general slots are properties of your class); we need the model to return the optimality condition and the jacobian of the optimality condition.

Now we can create a function which will return these two functions for certain values of the parameters, c and \inline \eta of the model:

my_mod <- function(eta, c) {
    e = -1/eta

    OptimCond <- function(q) {
        return(sum(q)^e + e * sum(q)^(e - 1) * q - diag(c) %*% q)
    }

    JacobiOptimCond <- function(q) {
        return((e * sum(q)^e) * array(1, c(2, 2)) + (e * sum(q)^(e - 1)) * diag(1, 
            2) + (e - 1) * e * sum(q)^(e - 2) * q * c(1, 1) - diag(c))
    }

    return(new("Model", OptimCond = OptimCond, JacobiOptimCond = JacobiOptimCond))

}

The function my_mod takes two parameters, eta and c and returns two functions, the optimality condition and the jacobian of the optimality condition. Both are now accessible via my_mod(eta=1.6,c = c(0.6,0.8))@OptimCond and my_mod(eta=1.6,c = c(0.6,0.8))@JacobiOptimCond respectively (and by specifying values for eta and c).

Now, we can use the rootSolve package to get the optimal values \(q_1\) and \(q_2\)

library("rootSolve")

multiroot(f = my_mod(eta = 1.6, c = c(0.6, 0.8))@OptimCond, start = c(1, 1), 
    maxiter = 100, jacfunc = my_mod(eta = 1.6, c = c(0.6, 0.8))@JacobiOptimCond)
## $root
## [1] 0.8396 0.6888
## 
## $f.root
##            [,1]
## [1,] -2.220e-09
## [2,]  9.928e-09
## 
## $iter
## [1] 4
## 
## $estim.precis
## [1] 6.074e-09

After 4 iterations, we get that \inline q_1 and \inline q_2 are equal to 0.84 and 0.69 respectively, which are the same values as in the book!

Suggestion by Reddit user, TheDrownedKraken

I posted this blog post on the rstats subbreddit on www.reddit.com. I got a very useful comment by reddit member TheDrownedKraken which suggested the following approach, which doesn't need a new class to be build. I thank him for this. Here is his suggestion:

generator <- function(eta, a) {
    e = -1/eta

    OptimCond <- function(q) {
        return(sum(q)^e + e * sum(q)^(e - 1) * q - diag(a) %*% q)
    }

    JacobiOptimCond <- function(q) {
        return((e * sum(q)^e) * array(1, c(2, 2)) + (e * sum(q)^(e - 1)) * diag(1, 
            2) + (e - 1) * e * sum(q)^(e - 2) * q * c(1, 1) - diag(a))
    }

    return(list(OptimCond = OptimCond, JacobiOptimCond = JacobiOptimCond))

}

f.s <- generator(eta = 1.6, a = c(0.6, 0.8))

multiroot(f = f.s$OptimCond, start = c(1, 1), maxiter = 100, jacfunc = f.s$JacobiOptimCond)
## $root
## [1] 0.8396 0.6888
## 
## $f.root
##            [,1]
## [1,] -2.220e-09
## [2,]  9.928e-09
## 
## $iter
## [1] 4
## 
## $estim.precis
## [1] 6.074e-09