Employee productivity as function of number of workers revisited

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We have a mild obsession with employee productivity and how that declines as companies get bigger. We have previously found that when you treble the number of workers, you halve their individual productivity which is mildly scary.

Let’s try the FTSE-100 index of leading UK companies to see if they are significantly different from the S&P 500 leading American companies that we analyzed four years ago.

We will of course use the R statistical computing and analysis platform for our analysis, and once again we are grateful to Yahoo Finance for providing the data.

The analysis script is available as ftse100.R and is really simple:

## ftse100.R - Display employee productivity for FTSE-100 consitituents
## Copyright © 2010 Allan Engelhardt <http://www.cybaea.net/>
## All Rights Reserved.

## Get the index constituents.
ftse.100 <- read.csv(file = "http://uk.old.finance.yahoo.com/d/quotes.csv?s=@%5EFTSE&f=s&e=.csv", header = FALSE)
names(ftse.100) <- c("symbol")
data <- data.frame(symbol=NULL, employees=NULL, profit=NULL, sector=NULL)

## For each stock symbol, get employees, profit, and sector
for (symbol in ftse.100$symbol) {
    profile.url <- paste("http://uk.finance.yahoo.com/q/pr?s=", symbol, sep="")
    con <- url(profile.url, open = "r")
    text <- readChar(con, 2^24)     # enough bytes
    x <- sub('.*Number of employees:</td><td.*?>[[:space:]]*([[:digit:],]+).*', "\\1", text, ignore.case = TRUE)
    x <- gsub(',', '', x)
    empl <- tryCatch(as.integer(x), warning = function(x) NA)
    x <- sub('.*Net Profit.*?</td><td.*?>[[:space:]]*([+-]?[[:digit:],]+).*', '\\1', text)
    x <- gsub(',', '', x)
    profit <- tryCatch(as.integer(x)*1e6, warning = function(x) NA)
    sector <- sub('.*Sector:</td><td.*?>(.*?)</td>.*', '\\1', text)
    if (any(c(empl, profit) <= 0, is.na(c(empl, profit)))) {
        cat("Error parsing symbol", symbol, "see", profile.url, "\n")
    } else {
        data <- rbind(data, data.frame(symbol=symbol, employees=empl, profit=profit, sector=sector))

## Save the data so we don't have to hit Yahoo all the time.
save(data, file = "data.RData")

## Save plot to file:
#png(filename="ftse100.png", width=800, height=800, pointsize=14, bg="white", res=100)

opar <- par(cex.sub = sqrt(sqrt(2)), font.sub = 3, font.lab = 2)

## x and y coordinates of plot and plot limits
x <- with(data, employees)
y <- with(data, profit/employees)
xlim <- c(10^floor(log10(min(x))), 10^ceiling(log10(max(x))))
ylim <- c(10^floor(log10(min(y))), 10^ceiling(log10(max(y))))

## Set up to display different color and symbols
plot_col <- 1
plot_pch <- 1
markers <- 21:25
pchs <- rep(markers, ceiling(length(levels(data$sector))/length(markers)))
palette(rainbow(length(levels(data$sector)), start=3/6, end=6/6))

# Make empty plot:
plot(profit/employees ~ employees, data = data[FALSE, ], 
     type = "p", pch = pchs[plot_pch], col = plot_col,
     log="xy", xaxp = c(xlim, 1), yaxp = c(ylim, 1), xlim = xlim, ylim = ylim,
     main = "Profit per employee (FTSE 100)", xlab = "Employees", ylab = "Profit per employees (GBP)")

## Plot each sector
for (sector in levels(data$sector)) {
    plot.xy(xy.coords(with(data[data$sector == sector,], employees),
                      with(data[data$sector == sector,], profit/employees),
                      log = "xy", xlab = "", ylab = ""),
            type = "p", pch = pchs[plot_pch], col = plot_col, bg = plot_col)
    plot_pch <- plot_pch + 1
    plot_col <- plot_col + 1
legend(x = "bottomleft", legend = levels(data$sector), title = "Industry Sectors", 
       col = palette(), pt.bg = palette(), pch = pchs, cex = 2/3, pt.cex = 1, ncol = 2)

## Fit a linear model to the log-log data:
m <- lm(log10(y) ~ log10(x))
xl <- c(xlim[1]*5, xlim[2]/5)
yl <- 10^predict(m, data.frame(x = xl))
lines(xl, yl, col = "darkred", lty = "dashed", lwd = 2)
t <- sprintf("Power = %0.3g", m$coefficients[2])
text(xl[2], yl[2], t, adj = c(0.25, -1.5), col = "darkred", font = 2)

## All done.

Leave it to run and this is what you get:

The power law still broadly holds. In a large company, the productivity of the individual employee is only ¼ of the productivity in a company with one-tenth of the number of workers.

The analysis for the FTSE All-Share index is easy (ftse-all.R) and gives a slope of -0.7605541 for the 301 companies with the required information, which is much worse. More convincingly, fitting the companies with more than 1,000 employees (to avoid some bias of smaller companies needing to have large profits per employee in order to be big enough to afford a stock market listing) gives a slope of -0.2838.

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