Cobb-Douglas Utility Function with R Optimization

In this post, let’s see how we can use R optimization function optim to calculate the Utility maximization from a Cobb-Douglas utility function.

The Cobb-Douglas utility function takes the following general form

Our budget constraint (\(m\)) for two goods (\(x\) and \(y\)) is defined by the price of \(x\) (\(p_{x}\)) multiplied by the quantity of \(x\), plus the price of \(y\) (\(p_{y}\)) multiplied by the quantity of \(y\)

We solve for \(y\) with

We can define the objective function we want to maximize in R with

# objective function
# par = x

f = function(par = 1, m = 100, a = 0.5, b = 0.5, px = 1, py = 1){
  y = (m - (px * par))/py # solve for y
  U = (par^a) * (y^b) # cobb-douglas utility function
  return(U)
}

We simply have to input the quantity of \(x\) and set the \(a\) and \(b\) values of the Cobb-Douglas function.

For example, if we feed \(x = 1\), we get a Utility of \(39\)

# x = 1
# U is 39 #
f(par = 1, m = 100, px = 1, py = 1, a = 0.2, b = 0.8)

Now we can use the optimizer optim with this objective function.

The optim is searching for the \(x\) value (par) that maximizes Utility. (\(y\) is calculated automatically given our budget constraint).

optim(par = c(1), # starting value #
 fn = f, # the objective function we created above
  a = 0.2, # Cobb-Douglas alpha value
  b = 0.8, # Cobb-Douglas beta value
  lower = 1, # lower bound for search
  upper = 100, # upper bound for search #
  method = "L-BFGS-B",
  control = list(fnscale = -1)) # maximize instead of minimize
  
# optim$par

The result from optim$par tell us that \(20\) is the optimal value for \(x\).

Let’s input our function the optim$par result.

out = optim(par = c(1), # starting value #
  fn = f,
  a = 0.2,
  b = 0.8,
  lower = 1, # lower bound for search
  upper = 100, # upper bound for search #
  method = "L-BFGS-B",
  control = list(fnscale = -1)) # maximize instead of minimize

# optimal x value
out$par # 20

# input of optim into our objective function f()
f(par = out$par, m = 100, px = 1, py = 1, a = 0.2, b = 0.8)

The maximum Utility for this budget constraint is \(60\).

We can verify with the package microecon the result, see the link for the package https://github.com/giacomovagni/microecon.

library(tidyverse)
library(ggthemes)
library(microecon)


library(microecon)
cobbs_douglas_utility(I = 100, a = 0.2, b = 0.8, px = 1, py = 1)