Question 1 A boutique fruit juice manufacturer produces 2 types of juices, Apple and Fig daily with a total cost function: TC = 6A + A x F +9F where: A is the quantity of the Apple juice (in kegs) and F is the quantity of the Fig juice (in kegs). The prices that can be charged are determined by supply and demand forces and are influenced by the quantities of each type of juice according to the following equations: P₁ = 17-A + F for the price (in dollars per keg) of the Apple juice and PF = 23 + 2A-F for the price (in dollars per keg) of the Fig juice. The total revenue is given by the equation: TR = PAX A+ PF XF and the profit given by the equation Profit= TR-TC First, use a substitution of the price variables to express the profit in terms of A and F only. Using the method of Lagrange Multipliers find the maximum profit when total production (quantity) is restricted to 2023 kegs. Note A or F need not be whole numbers. Be sure to show that your solution is a maximum point.
Question 1 A boutique fruit juice manufacturer produces 2 types of juices, Apple and Fig daily with a total cost function: TC = 6A + A x F +9F where: A is the quantity of the Apple juice (in kegs) and F is the quantity of the Fig juice (in kegs). The prices that can be charged are determined by supply and demand forces and are influenced by the quantities of each type of juice according to the following equations: P₁ = 17-A + F for the price (in dollars per keg) of the Apple juice and PF = 23 + 2A-F for the price (in dollars per keg) of the Fig juice. The total revenue is given by the equation: TR = PAX A+ PF XF and the profit given by the equation Profit= TR-TC First, use a substitution of the price variables to express the profit in terms of A and F only. Using the method of Lagrange Multipliers find the maximum profit when total production (quantity) is restricted to 2023 kegs. Note A or F need not be whole numbers. Be sure to show that your solution is a maximum point.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Transcribed Image Text:Question 1
A boutique fruit juice manufacturer produces 2 types of juices, Apple and Fig daily with a total cost
function: TC = 6A + A x F +9F
where:
A is the quantity of the Apple juice (in kegs) and
F is the quantity of the Fig juice (in kegs).
The prices that can be charged are determined by supply and demand forces and are influenced by
the quantities of each type of juice according to the following equations:
P₁ = 17-A + F for the price (in dollars per keg) of the Apple juice and
P = 23+2A-F for the price (in dollars per keg) of the Fig juice.
The total revenue is given by the equation:
TR = PAXA + PF XF
and the profit given by the equation
Profit= TR-TC
First, use a substitution of the price variables to express the profit in terms of A and F only.
Using the method of Lagrange Multipliers find the maximum profit when total production (quantity)
is restricted to 2023 kegs. Note A or F need not be whole numbers.
Be sure to show that your solution is a maximum point.
Transcribed Image Text:Question 1: Answer
Question 1: Answer
The firm profit is calculated as:
Profit: PA x A + PF x F-6A - 9F
= (17 - A+F)A + (23 + 2A – F) F – (6A – AF – 9F)
= 11A - A^2 + AF + 14F + 2AF – F^2 – AF
= 11A - A^2 + 14F + 2AF – F^2
Thus, this shows the predicament of the firm to maximize
Profit: 11A + 14F + 2AF — A^2 – F^2
The subject of constraint: 2,023 = A + F
The Lagrange Function is:
L=11A + 14F + 2AF — A^2 − F^2 + λ(2,023 – A − F )
ƏL
(1)
JA
ƏL
ƏF
ƏL
əλ
= 11+2F-2A - λ = 0
= 14 + 2A - 2F - λ = 0
= 2,023-A-F = 0
(2)
(3)
From (1) and (2)
11 + 2F-2A = 14 +2A - 2F or 4F-3=4A
F-A=2=0.75 (4)
From (3) and (4)
A+ F = 2,023 - λ (Equation 1)
F-A=0.75 (Equation 2)
Thus, F = 1,011.125 and A = 1,011,125–0.75=1010.375
To demonstrate that it is a maximum point, we must compare the profit at
this point to the profit at the ends of the feasible region. (2023, 0) and (0, 2023)
are the endpoints.
Profit at (2023, 0) is: Profit = 11A - A^2 = 11(2023) - (2023)^2 = 112,306
Profit at (0, 2023) is: Profit = 14F - F^2 = 14(2023) - (2023)^2 = 112,306
Profit at (1010.375, 1011.125) = 11A - A^2 + 14F - F^2 + 0.75^2 =
11(1010.375) - (1010.375)^2 + 14(1011.125) - (1011.125)^2 +0.75^2 =
114,064.27
The highest profit, which happens when 1010.375 kegs of apple juice and
1011.125 kegs of fig juice are generated, is $114,064.27 since it exceeds the
profit at the feasible region's ends (1010.375, 1011.125).
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