# Consumer’s Choosing an Optimal Bundle – Utility Maximization

November 2, 2013
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`# The theoretical basis of classical consumer theory lies# in utility maximization. The idea is that consumers# make consumption decisions based on choosing a bundle # of goods that will maximize individual utility.# Despite this hypothesis being largely unsupported# by reproducible results indicating the superiority# any utility function over all other functions this# theory persists. # In this simulation I set up an easy framework for the # user to simulate the decision of the consumer as a# function of the utility function, price of goods,# and consumer budget.  # Uof is the function that takes a choice of# x and y and calculates the corresponding z as# well as expected utility. Uof <- function(XY) {  # x cannot be less than 0 or more than all of the  # budget expended on x  x <- min(max(XY[1],0), b/px)  # y cannot be less than 0 or more than the remaining  # budget left after x expenditures   y <- min(max(XY[2],0), (b - x*px)/py)   # z is purchased with whatever portion of the budget   # remains  z <- (b - x*px - y*py)/pz   # Display the quantity of x,y, and z chosen.  print(paste0("x:", x, " y:", y, " z:", z))   # Since optim minimizes a function I am making  # the returned value equal to negative utility.    -utility(x,y,z)} # I have defined a few different potential utility# functions. cobb.douglas2  <- function(x,y,z) x^.3*y^.3cobb.douglas3  <- function(x,y,z) x^.3*y^.3*z^.4lientief3      <- function(x,y,z) min(x,y,z)linear.concave <- function(x,y,z) x^.3 + y^.2 + z^.5mixed          <- function(x,y,z) min(x, y)^.5*z^.5addative       <- function(x,y,z) 2*x+3*y+z # Choose the utility function to maximizeutility <- cobb.douglas3 # Choose the pricespx <- 1py <- 1pz <- 1 # Choose the total budgetb <- 100 # Let's see how much utility we get out of setting# x=1 and y=1Uof(c(1,1)) # The following command will maximize the utility# subject the choice of the utility fuction, prices,# and budget.optim(c(1,1), Uof, method="BFGS") # In general it is a good idea not to use such # computational methods as this since by instead# solving mathematically in closed form for solutions# to utility maximization functions, you can discover# how exactly a change in one parameter in the model# may lead to a change in quantity demanded of a# type of good. # Of course you could do something fairly simple along# these lines in R as well. # For instance, define vectors: # Once again choose what utility functionutility <- linear.concave xv  <- NULLyv  <- NULL pxv <- seq(.25,10,.25) for (px in pxv) {  res <- optim(c(1,1), Uof, method="BFGS")  xv <- c(xv, max(res\$par[1],0))  yv <- c(yv, max(res\$par[2],0))} plot(xv, pxv, type="l", main="Demand for X(px)",     xlab="Quantity of X", ylab="Price of X") # The map of the demand function for X `
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`   plot(yv, pxv, type="l", main="Demand for Y(px)",     xlab="Quantity of Y", ylab="Price of X") `
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`  # The map of the demand function for y as a function# of the price of x. It is a little erratic probably# because of lack of precision in the optimization# algorithm.`

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