Question

A certain company produces three products. The cost for producing the first product is p1 dollars per unit, the cost of producing the second product is p2 dollars per unit, and the cost of producing the third is p3 dollars per unit. Assume the value derived from producing x1 units of the first product, x2 units of the second, and x3 units of the third is given by the utility function

U (x₁, x₂, x₃) = x₁x₂²x₃³

The larger the value of the utility function, the more worthwhile producing that many products is to the company. You can take it for granted that given some budget constraint as in (a) and (b) below, there are values of x1, x2, x3 that maximize utility subject to that constraint.
(a) Assume the company has a budget of 100, 000 dollars to spend on producing its products.
If p₁ = 1, p₂ = 3, and p₃ = 5, find the values of x₁, x₂, x₃ needed to maximize utility.

Answer 1

To maximize utility, we need to find the values of x1, x2, and x3 that satisfy the given budget constraint. By solving an optimization problem, we can find the optimal values of x1, x2, and x3 that maximize utility under the given budget constraint.

Step-by-step explanation:

To maximize utility, we need to find the values of x1, x2, and x3 that satisfy the given budget constraint. In this case, the company has a budget of $100,000 to spend on producing its products, and the costs per unit are p1 = $1, p2 = $3, and p3 = $5. The utility function U(x1, x2, x3) = x1x2²x3³ represents the value derived from producing x1 units of the first product, x2 units of the second product, and x3 units of the third product.

To find the values of x1, x2, and x3 that maximize utility, we can set up the following optimization problem:

Maximize U(x1, x2, x3) = x1x2²x3³

Subject to the constraint: p1x1 + p2x2 + p3x3 ≤ 100,000

By solving this optimization problem, we can find the optimal values of x1, x2, and x3 that maximize utility under the given budget constraint.