Consider our model of the forestry auctions with loggers bidding in a sealed bid auction. Let v_i be logger i's value for the lot of trees, b_i(v_i) is the bidder's bid as a function of the bidder's value and p is the price paid by the winner. Write down the game.
Consider an imaginary data set with 1000 bids from a sealed bid auction with 2 bidders, where the bid distribution is a normal distribution with mean of 5 and standard deviation of 1. What is the mean of the value distribution? Hints: pnorm(q, m, s) calculates the cumulative probability up to q of a normal distribution with mean of m and standard deviation of s. While dnorm() calculates the density, you should use the numerical derivative h(q) = (H(q + epsilon) - H(q - epsilon)) / (2 epsilon) and set epsilon = 0.5.
Using the same set up as Question (2) but now there are really 6 bidders, what is the mean of the value distribution?
Consider a sealed bid auction where valuations are drawn from a uniform distribution from 0 to 1. In this case the equilibrium bidding strategy is b_i(v_i) = ((N-1) v_i)/N, where N is the number of bidders in the auction and v_i is the bidder i’s valuation.
Copy and paste the code below into RStudio. Run the code and determine the average value of bids, the estimated average of valuations and the actual average of values. Note use round(STUFF, 3) to round the answer to 3 digits.
set.seed(123456789)
M <- 2000 # number of simulated auctions.
data1 <- matrix(NA,M,12)
for (i in 1:M) {
N <- round(runif(1, min=2,max=10)) # number of bidders.
v <- runif(N) # valuations, uniform distribution.
b <- (N - 1)*v/N # bid function
p <- max(b) # auction price
x <- rep(NA,10)
x[1:N] <- b # bid data
data1[i,1] <- N
data1[i,2] <- p
data1[i,3:12] <- x
}
colnames(data1) <- c("Num","Price","Bid1",
"Bid2","Bid3","Bid4",
"Bid5","Bid6","Bid7",
"Bid8","Bid9","Bid10")
data1 <- as.data.frame(data1)
Copy and paste the code below into RStudio. Note that for this code to work you need to have the file “auction.csv” downloaded into the same folder as the working directory. One way to do this is to keep all files in the same folder and use the menu in RStudio to set working directory to where the current script is saved.
What are the mean and standard deviation of the normalized bids? Note report to 3 decimal places or less.
<<>>=
x <- read.csv("auctions.csv", as.is = TRUE)
lm1 <- lm(log_amount ~ as.factor(Salvage) + Acres +
Sale.Size + log_value + Haul +
Road.Construction + as.factor(Species) +
as.factor(Region) + as.factor(Forest) +
as.factor(District), data=x)
# as.factor creates a dummy variable for each entry under the
# variable name. For example, it will have a dummy for each
# species in the data.
y <- x[-lm1$na.action,]
y$norm_bid <- lm1$residuals
# lm object includes "residuals" term which is the difference
# between the model estimate and the observed outcome.
# na.action accounts for the fact that lm drops
# missing variables (NAs)
a. Just using the sealed bid auctions with 2 bidders, what is the mean of valuations for the loggers?
b. Compare it to the 2 bidder English auctions.
c. Consider the results above, is this evidence that the loggers are behaving rationally? Explain