In teaching Behavioral Economics, optimization problems require some intuition. This intuition can be opaque without calculus literacy. Below is a simulation to demonstrate that the process for constrained optimization *works*. It has the added benefit of showing isoquants (by colored stripes in the image below), and the strict boundary condition of the efficiency frontier.

Basic constrained optimization problems are as follows:

I have made code in R to simulate the results for a two-part optimization process. The example uses as the functional form.

library(“plot3D”)

p1<-1

p2<-2

y<-10

x1<-runif(25000, min=0, max=y/p1)

#Checking any options inside the constraint or on the constraint

x2<-runif(25000,min=0, max=(y-p1*x1)/p2)

U<-sqrt(x1)+sqrt(x2)

out<-mesh(x1, x2, U)

points3D(x1, x2, U, xlab=”x1″, ylab=”x2″, zlab=”Utility”, phi=-90, theta=90)

plot(x1, U)

abline(v=x1[which(U==max(U))], col=”red”)

x1_star<-x1[which(U==max(U))]

x2_star<-x2[which(U==max(U))]

y-x1_star*p1-x2_star*p2

And it outputs the following plots (with minor variation). Note that the colored bands represent utility curves, *isoquants*. The end of the colored points represents the *efficiency frontier*.

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The actual solution is found by:

subject to:

The Lagrangian is then:

Leading to the first order conditions (derivatives of L):

Using these 3 conditions, we can find the equations:

Where, if then we can solve for the theoretical solutions:

These indeed match very closely with the real solutions.

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