Question

A brewery produces regular beer and a lower-carbohydrate "light" beer. Steady customers of the brewery buy 11 units of regular beer and 15 units of light beer monthly. While setting up the brewed produce the beers, the management decides to produce extra beer, beyond that needed to satisfy customers. The cost per unit of regular beer is $38,000 and the cost per unit of light beer is $60,000. Every unit of regular beer brings in $100,000 in revenue, while every unit of light beer brings in $300,000 in revenue. The brewery needs at least $9,000,000 in revenue, and has determined that the total demand will be at least 50 units of beer. Complete parts (a) and (b). (a) How much of each type of beer should be made so as to minimize total production costs? ___ units of regular beer and ____ units of light beer should be made, for a minimum total cost of $___. (Type integers or decimals.) (b) Suppose the minimum revenue is increased to $9,600,000. Use shadow costs to calculate the total production cost. The shadow cost of production is $_____. The increased minimum revenue corresponds to a total production cost of $____ (Type integers or decimals.)

Answer 1

To minimize total production costs, the number of units of regular beer and light beer that should be made needs to be determined. The shadow cost of production can be calculated using the shadow price of revenue.

Step-by-step explanation:

To minimize total production costs, we need to determine the number of units of regular beer and light beer that should be made.

Let's assume we produce x units of regular beer and y units of light beer.

The cost of regular beer is $38,000 per unit, so the total cost of regular beer is 38,000x.

The cost of light beer is $60,000 per unit, so the total cost of light beer is 60,000y.

The minimum revenue required is $9,000,000, which is the revenue generated by regular beer (100,000x) plus the revenue generated by light beer (300,000y).

Therefore, we have the following equations:

38,000x + 60,000y = Total cost

100,000x + 300,000y = Revenue

To find the minimum total cost, we need to solve these equations.

(b) To calculate the shadow cost of production, we can use the shadow price of revenue. Shadow price is the change in revenue for a small change in production quantity. The shadow cost is the negative of the shadow price. So if the shadow price of revenue is $10,000, the shadow cost of production would be -$10,000.

Let's assume the shadow price of revenue is p. In this case, the shadow cost of regular beer production is -p and the shadow cost of light beer production is -3p.