Optimal Pricing for a Simple Monopolist

# A single price monopolist is a monopolist because it is the only supplier of a particular product.  The monopolist therefore has the power to choose a price to sell the product at.
 
# Those who have a willingness to pay which is greater than the price will buy the good while those who have a willingness to pay for the good which is less than the chosen price will not but it.
 
# Our monopolist is a broadband internet supplier within a city.
 
# For now let's say they only offer one bundle.
 
# Let's generate our consumers
 
npeep <- 2000 # Number of potential consumers
 
wtp <- 45 + rnorm(npeep)*15 # Each person has a different willingness to pay which
 
# To figure out the demand curve we count the number of people willing to pay at least as much as the offering price.
 
maxop <- 90 # Max offering price
op <- 0:maxop # Offering price ranges from 0 to maxop
qd <- rep(NA,length(op)) # Quantity demanded
for (i in 1:length(op)) qd[i] <- sum(wtp>=op[i])
 
mc <- qd*.01 # Marginal cost is increasing though this is not a neccessity
# For something like broadband services we might think that up to a point marginal costs might be decreasing since the cost of adding one more customer might be less than the cost of adding the previous customer.
 
plot(qd, op, type="l", xlab="Quantity", ylab="Price, Marginal Cost - Red", 
      main="Demand for Broadband Internet", lwd=2)
abline(h=0, lwd=2)
lines(qd, mc, col="red", lwd=2)

 
# The monopolist must choose a price in which to sell services at.
 
# If the monopolist chooses mc=p then the monopolist will not make any money but the consumers will be very happy.
 
# We know that the optimal point for the monopolist is at the point where marginal revenue curve intersects the marginal cost curve.
 
# Let's see if we can find it.
 
tr <- tp <- tc <- rep(NA,length(op)) # Total revenue, total profit, total cost vectors
 
# Calculate total cost
 
qd.gain <- qd[-length(qd)]-qd[-1]
qd.gain[length(qd.gain)+1] <- qd.gain[length(qd.gain)]
for (i in 1:length(op)) tc[i] <- sum((mc*qd.gain)[length(qd):i])
 
tr <- qd*op
tp <- tr-tc
 
minmax <- function(...) c(min(...),max(...))
plot(minmax(op),minmax(tr,tp), type="n", ylab="Total Revenue - Blue, Total Profit - Red", 
     xlab="Price", main="We can see optimal pricing\nfor the monopolist is around 39 dollars")
grid()
abline(h=0, lwd=2)
abline(v=39, col="red", lwd=2)
lines(op,tr, col="blue", lwd=3)
lines(op,tp, col="red", lwd=2)

# We can see at the price around 18 which would be the optimal price for the consumer, the supplier is making almost no profits.
 
# The last thing we might wish to consider to Total Surplus or total system efficiency which is defined as that which the consumer benefits by purchasing a good below the consumers willingness to pay plus that of the suppliers profit at that price.
 
cs <- tr
for (i in 1:length(op)) cs[i] <- sum((wtp[wtp>=op[i]]-op[i]))
 
tts <- cs+tp
 
op[tts==max(tts)] # Check the optimatal societal price
 
plot(c(min(op),max(op)),c(min(cs,tp),max(cs,tp)), type="n",
     main="Optimal societal pricing is at\n mc=wtp which is $19",
     xlab="Price",
     ylab="purple=CS, blue=PS, black=TS")
lines(op, cs, col="purple", lwd=2)
lines(op, tp, col="blue", lwd=2)
lines(op, tts, lwd=2)
abline(h=0,col="red", lwd=2)