# Consumer’s Choosing an Optimal Bundle – Utility Maximization

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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^.3 cobb.douglas3 <- function(x,y,z) x^.3*y^.3*z^.4 lientief3 <- function(x,y,z) min(x,y,z) linear.concave <- function(x,y,z) x^.3 + y^.2 + z^.5 mixed <- function(x,y,z) min(x, y)^.5*z^.5 addative <- function(x,y,z) 2*x+3*y+z # Choose the utility function to maximize utility <- cobb.douglas3 # Choose the prices px <- 1 py <- 1 pz <- 1 # Choose the total budget b <- 100 # Let's see how much utility we get out of setting # x=1 and y=1 Uof(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 function utility <- linear.concave xv <- NULL yv <- 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

plot(yv, pxv, type="l", main="Demand for Y(px)", xlab="Quantity of Y", ylab="Price of X")

# 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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