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

U (x, y) = x^{a} \cdot y^{b}

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 $y$

m = p_{x} \cdot x + p_{y} \cdot y

We solve for $y$ with

y = (m - p_{x} \cdot x) \frac{1}{p_{y}}

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)