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.