What are the optimal (utility maximizing) values of pens and notebooks?

To maximize utility with an income of $80 and prices of $30 per pen and $10 per notebook, the optimal values are approximately x = 2.67 pens and y = 8 notebooks. These values will yield the highest utility for the given budget and utility function.

## Calculating the Optimal Values of Pens and Notebooks

To find the optimal values of pens and notebooks, we need to maximize the utility function U=xy+10x, subject to the given constraints.
## Budget Constraint

Let's start by calculating the budget constraint. With an income of $80, and the price of a pen being $30 and the price of a notebook being $10, we can write the budget constraint as 30x + 10y = 80.
Now, we can rewrite the budget constraint in terms of one variable to solve for the other. Let's solve for x:
30x + 10y = 80
30x = 80 - 10y
x = (80 - 10y) / 30
Substituting this value of x in the utility function, we have:
U = ((80 - 10y) / 30)y + 10((80 - 10y) / 30)
Simplifying this expression, we have:
U = (8/3)y + (80/3) - (10/3)y
## Finding the Optimal Values

To find the optimal values of pens and notebooks, we need to maximize U. This can be done by finding the critical points of U with respect to y, and then checking the endpoints of the feasible region.
Taking the derivative of U with respect to y and setting it equal to zero:
dU/dy = (8/3) - (10/3) = 0
-2/3 = 0 (no solution)
Since there is no solution, we need to consider the endpoints of the feasible region.
When y = 0, we find x:
x = (80 - 10(0)) / 30
x = 8/3
When x = 0, we find y:
30(0) + 10y = 80
10y = 80
y = 8
So, the optimal values for pens and notebooks are x = 8/3 and y = 8, respectively.